Existence of moduli spaces for algebraic stacks
aa r X i v : . [ m a t h . AG ] M a y EXISTENCE OF MODULI SPACES FOR ALGEBRAIC STACKS
JAROD ALPER, DANIEL HALPERN-LEISTNER, AND JOCHEN HEINLOTH
Abstract.
We provide necessary and sufficient conditions for when an alge-braic stack admits a good moduli space. This theorem provides a generaliza-tion of the Keel–Mori theorem to moduli problems whose objects have pos-itive dimensional automorphism groups. We also prove a semistable reduc-tion theorem for points of algebraic stacks equipped with a Θ-stratification.Using these results we find conditions for the good moduli space to be sep-arated or proper. To illustrate our method, we apply these results to con-struct proper moduli spaces parameterizing semistable G -bundles on curvesand moduli spaces for objects in abelian categories. Contents
1. Introduction 22. Preliminaries 52.1. Reminder on local structure theorems for algebraic stacks 52.2. Reminder on mapping stacks and filtrations 62.3. The example of quotient stacks 73. Valuative criteria for stacks 83.1. Morphisms of stacks of filtrations 83.2. Property Θ- P S -complete morphisms 183.6. Unpunctured inertia 234. Existence of good moduli spaces 244.1. Reminder on maps inducing ´etale maps on good moduli spaces 244.2. Proof of the existence result 255. Criteria for unpunctured inertia 285.1. A variant of the valuative criteria 295.2. The proof of Theorem 5.2 325.3. The proof of Theorem 5.4 346. Semistable reduction and Θ-stability 356.1. The semistable reduction theorem 366.2. Comparison between a stack and its semistable locus 426.3. Application: Properness of the Hitchin fibration 457. Good moduli spaces for objects in abelian categories 487.1. Formulation of the moduli problem 497.2. Verification of the valuative criteria for the stack M A G -torsors 63Appendix A. Strange gluing lemma 67A.1. Gluing results 67 .2. Semistable reduction in GIT 69References 701. Introduction
In the study of moduli problems in algebraic geometry the construction ofmoduli spaces is a recurring problem. Given a moduli problem, described byan algebraic stack X , the ideal solution would be for X to be representable bya scheme or an algebraic space. This is never the case when objects parame-terized by X have non-trivial automorphism groups. In this case one hopes forthe existence of a universal map to an algebraic space q : X → X with usefulproperties.For algebraic stacks with finite automorphism groups the Keel–Mori theorem[KM97] gave a satisfactory existence result from the intrinsic perspective. It statesthat if X is an algebraic stack of finite type over a noetherian base whose inertiastack is finite over X , then there is a coarse moduli space q : X → X , which inaddition to being a universal map to an algebraic space is bijective on geometricpoints.The restriction to the case of finite automorphism groups is not necessary forthe construction of moduli spaces using GIT. Furthermore in many examples, suchas the moduli of vector bundles or coherent sheaves on a projective variety, onemust consider objects with positive dimensional automorphism groups in orderto construct moduli spaces which are proper.In [Alp13], the first author introduced the notion of a good moduli space foran algebraic stack X as an intrinsic formulation of many of the useful propertiesof the notion of a good quotient [Ses72], a specific type of GIT quotient includingall GIT quotients in characteristic 0. By definition, a good moduli space is amap q : X → X to an algebraic space such that the pushforward q ∗ of quasi-coherent sheaves is exact, and such that the canonical map O X → q ∗ ( O X ) is anisomorphism. This simple definition leads to many useful properties, includingthat q is universal for maps to an algebraic space, and that the fibers of q classifyorbit-closure equivalence classes of points in X .Our main result gives necessary and sufficient conditions under which an al-gebraic stack admits a good moduli space, and can be seen as uniting the maintheorem of geometric invariant theory with the intrinsic perspective of the Keel–Mori theorem. Theorem A (Theorem 4.1, Proposition 3.47, Proposition 3.45, Theorem 5.4) . Let X be an algebraic stack locally of finite type with affine diagonal over a quasi-separated and locally noetherian algebraic space S . Then X admits a good modulispace if and only if(1) X is locally linearly reductive (Definition 2.1);(2) X is Θ -reductive (Definition 3.10); and(3) X has unpunctured inertia (Definition 3.53).The good moduli space X is separated if and only if X is S -complete (Definition 3.37),and proper if and only if X is S -complete and satisfies the existence part of thevaluative criterion for properness. ssume in addition that S is of characteristic and X is quasi-compact. If X is S-complete, then (1) and (3) hold automatically. In particular, X admits aseparated good moduli space if and only if X is Θ -reductive and S-complete. Let us give an informal explanation of the above conditions. The first condi-tion is that closed points of X have linearly reductive stabilizers. In the languageof geometric invariant theory this would amount to the condition that the au-tomorphism groups of polystable objects are (linearly) reductive. The secondcondition is the geometric analog of the statement that filtrations by semistableobjects extend under specialization. This is formulated in terms of maps fromthe stack Θ := [ A / G m ] into X . The third condition is an analog of the conditionin the Keel–Mori theorem, it roughly states that the connected components ofstabilizer groups extend to closed points. In particular this condition is automaticif all stabilizer groups are connected (which happens for example for moduli ofcoherent sheaves). In Section 5 we provide several “valuative” criteria, in thesense that they involve only conditions on maps Spec( R ) → X where R is a dis-crete valuation ring (DVR), which are equivalent to unpunctured inertia underthe hypotheses (1) and (2) (see Theorem 5.2).Finally, S -completeness, where the S stands for “Seshadri,” is a geometricproperty that is reminiscent of classical methods of establishing separatednessof moduli spaces. More precisely we introduce a geometric notion of an ele-mentary modification (Definition 3.35) which relates two families over a DVRwhich are isomorphic at the generic point, and S -completeness states that anytwo families over a DVR which are isomorphic at the generic point differ by anelementary modification. It turns out that S-completeness has many desirableconsequences: namely, in characteristic 0, S-completeness implies both condi-tions (1) and (3) in Theorem A. This fact follows from the more general resultsof Proposition 3.45 and Theorem 5.4, which are characteristic independent. Ulti-mately, both S -completeness and Θ-reductivity are local criteria in the sense thateach is equivalent to a filling condition for G m -equivariant families over a suitablepunctured regular 2-dimensional scheme.The condition of linear reductivity is very strong in positive characteristic andit arises here through the recent local structure theorems on algebraic stacks from[AHR15, AHR]. In positive characteristics we would expect that an analogueof Theorem A holds with “good moduli space” replaced with “adequate modulispace” and “locally linearly reductive” replaced with “locally geometrically re-ductive.” The main obstacle to prove such a result is the lack of an analogue ofthe local structure theorem for such stacks. However, we are careful to proveintermediate results that do not require linear reductivity.Our second main theorem is an analog of Langton’s semistable reduction theo-rem [Lan75] for moduli of bundles, that works for a large class of algebraic stacksequipped with a notion of stability that induces a Θ-stratification, a geometricanalog of the notion of Harder–Narasimhan–Shatz stratifications. As in Lang-ton’s theorem, the statement is that if a family of objects parametrized by aDVR specializes to a point that is more unstable than the generic fiber of thefamily, then one can modify the family along the closed point to get a family thathas the same stability properties as the generic fiber. Surprisingly the existence ofmodifications can be obtained from the local geometry of Θ-stratifications. Theformal statement is the following. heorem B (Theorem 6.3) . Let X be an algebraic stack locally of finite type withaffine diagonal over a noetherian algebraic space S , and let S ֒ → X be a Θ -stratum(Definition 6.1). Let R be a DVR with fraction field K and residue field κ . Let ξ : Spec( R ) → X be an R -point such that the generic point ξ K is not mapped to S , but the special point ξ κ is mapped to S : Spec( K ) (cid:31) (cid:127) / / ξ K (cid:15) (cid:15) Spec( R ) ξ R (cid:15) (cid:15) Spec( κ ) ? _ o o ξ κ (cid:15) (cid:15) X − S (cid:31) (cid:127) j / / X S . ? _ ι o o Then there exists an extension R → R ′ of DVRs with K → K ′ = Frac( R ′ ) fi-nite and an elementary modification (Definition 3.35) ξ ′ of ξ | R ′ such that ξ ′ : Spec( R ′ ) → X lands in X − S . We may apply the above results to the semistable locus X ss ⊂ X defined bya class ℓ ∈ H ( X ; R ) via the Hilbert–Mumford criterion (see Definition 6.13).As many properties of X are inherited by the semistable locus, we can provideconditions on X ensuring that the semistable locus X ss admits a separated goodmoduli space and a further condition ensuring that the good moduli space isproper. To summarize, we have: Theorem C (Corollary 6.12, Proposition 6.14, Corollary 6.18) . Let X be an al-gebraic stack locally of finite type with affine diagonal over a quasi-separated andlocally noetherian algebraic space S , and let ℓ ∈ H ( X ; R ) be a class defining asemistable locus X ss ⊂ X which is part of a well-ordered Θ -stratification of X compatible with l . Then if X is either Θ -reductive, S -complete, or satisfies theexistence part of the valuative criterion for properness, then the same is true for X ss .In particular, if in addition S has characteristic , X → S is S-complete and Θ -reductive, and X ss → S is quasi-compact, then there exists a good moduli space X ss → X such that X is separated over S (and proper over S if X → S satisfiesexistence part of the valuative criterion for properness). We expect that in the semistable reduction theorem (Theorem B), weak Θ-strata (that only require canonical filtrations to exist after a purely inseparableextension) should be sufficient and these are available in greater generality in posi-tive characteristic. Similarly, in positive characteristic, we expect that Theorem Cholds with “good moduli space” replaced with “adequate moduli space.” The mainobstruction for these generalizations is a version of the local structure theoremwhere the embedding of a stratum is replaced with a radicial map.
Applications.
To illustrate our results we give some applications that may be ofindependent interest. First, we use the semistable reduction theorem to give aproof that the Hitchin fibration for semistable G -Higgs bundles is proper if thecharacteristic of the ground field is not too small (Corollary 6.21). This result isof course expected, but it doesn’t seem to appear in the literature.Second we apply our existence theorem to construct some new good modulispaces. Namely we construct proper good moduli spaces for semistable G -bundlesfor a Bruhat–Tits groups scheme G over a smooth geometrically connected projec-tive curve over a field of characteristic 0, generalizing work of Balaji and Seshadri See Proposition 6.14 for the precise compatability condition between l and the Θ-stratification. BS15] (Theorem 8.1). We also construct proper good moduli spaces for objects inabelian categories in Theorem 7.21 and Theorem 7.25. As a special case, we con-struct proper moduli spaces of semistable complexes with respect to a Bridgelandstability condition on a smooth projective variety X over a field of characteristic0. Whereas in these examples the lack of a convenient global quotient descriptionof the corresponding moduli problems seems to pose a serious obstruction to aconstruction using GIT, the verification of the conditions of our main theoremsturns out to be surprisingly simple. Acknowledgements.
We would like to thank Arend Bayer, Brian Conrad, Johan deJong, Maksym Fedorchuk, Jack Hall, Alexander Polishchuk, David Rydh, MichaelThaddeus, Yukinobu Toda and Xiaowei Wang for helpful conversations relatedto this project. We also thank Victoria Hoskins for organizing the workshop
NewTechniques in Geometric Invariant Theory where some of these ideas were firstdiscussed. The first author was partially supported by NSF grant DMS-1801976.The second author was partially supported by NSF grant DMS-1762669. Thethird author was partially supported by Sonderforschungsbereich/Transregio 45of the DFG. 2.
Preliminaries
Throughout we will fix a base S that will be a quasi-separated algebraic space,but of course the most interesting case for most readers will be when S = Spec( k )is the spectrum of a field.As our arguments build on the one hand on local structure theorems and onthe other hand on notions that came up in the study of notions of stability onalgebraic stacks, we briefly recall these results in this section.2.1. Reminder on local structure theorems for algebraic stacks.
For easeof notation let us introduce the following terminology.
Definition 2.1.
An algebraic stack X with affine stabilizers is locally linearlyreductive if every point specializes to a closed point and every closed point of X has a linearly reductive automorphism group.Note that in the case of a quasi-compact quotient stack X = [ X/G ] the closedpoints correspond to closed orbits of G on X , so in this case the above conditiononly requires that points contained in closed orbits have a linearly reductivestabilizer. In particular, a locally linearly reductive stack will often have geometricpoints with non-reductive stabilizers. Definition 2.2. If X is an algebraic stack and x ∈ | X | is a point with residualgerbe G x , we call an ´etale and affine pointed morphism f : ( W , w ) → ( X , x ) ofalgebraic stacks a local quotient presentation around x if (1) W ∼ = [Spec( A ) / GL N ]for some N and (2) f | f − ( G x ) is an isomorphism.The following is the key result on the local structure of locally linearly reductivestacks. Theorem 2.3. [AHR, Thm. 1.1]
Let S be a quasi-separated algebraic space. Let X be an algebraic stack locally of finite presentation with affine diagonal over S .If x ∈ | X | is a point with image s ∈ | S | such that the residue field extension κ ( x ) /κ ( s ) is finite and the stabilizer of x is linearly reductive, then there exists alocal quotient presentation f : ( W , w ) → ( X , x ) around x . n particular, if in addition X is locally linearly reductive, then there exist localquotient presentations around any closed point. Remark 2.4. If S is the spectrum of an algebraically closed field, the abovetheorem follows from [AHR15, Thm. 1.2]. In this case, one can arrange thatthere is a local quotient presentation ( W , w ) → ( X , x ) with W ∼ = [Spec( A ) /G x ],the quotient of an affine scheme by the stabilizer G x = Aut X ( x ) of x . Remark 2.5.
While GL N is linearly reductive in characteristic 0, it is not linearlyreductive in positive or mixed characteristic. For the same reason, the morphism[Spec( A ) / GL n ] → Spec( A GL N ) will only be an adequate moduli space (and nota good moduli space) in general.To prove the semistable reduction theorem, we will need a relative versionof the above local structure theorem where we fix a subgroup isomorphic to themultiplicative group G m of the stabilizer G x = Aut X ( x ), but do not assume G x tobe linearly reductive. A very general result of this form is the following theorem. Theorem 2.6. [AHHR]
Let X be an algebraic stack locally of finite presentationwith affine diagonal over a quasi-separated algebraic space S , and let G ⊂ GL N,S be a closed subgroup which is linearly reductive over S . If Y ⊂ X is a closedsubstack, then any representable and smooth (resp. ´etale) morphism [ Y /G ] → Y ,with Y → S affine, extends to a representable and smooth (resp. ´etale) morphism [ X/G ] → X with X → S affine, i.e. we have [ X/G ] × X Y ∼ = [ Y /G ] . Remark 2.7.
As the proof of the result has not yet appeared let us recall aspecial case, which will be sufficient for us if S = Spec( k ) is the spectrum ofa field and all stabilizer groups of X are smooth (a condition that is automaticin characteristic 0). Namely, if S = Spec( k ) is the spectrum of an algebraicallyclosed field, x ∈ X ( k ) with smooth automorphism group G x , Y = B k G x ⊂ X is the canonical inclusion, G rm ⊂ G x is a subgroup and Y = [Spec( k ) / G rm ], theabove result is a special case of [AHR15, Thm. 1.2].2.2. Reminder on mapping stacks and filtrations.
As in [Hal14] we willdenote by Θ := [ A / G m ] the quotient stack defined by the standard contractingaction of the multiplicative group on the affine line and by B G m = [pt / G m ],the classifying stack of the group G m . Both stacks are defined over Spec( Z ) andtherefore pull back to any base S . Note that since G m is a linearly reductivegroup, the structure morphisms Θ → Spec( Z ) and B G m → Spec( Z ) are goodmoduli spaces.Maps from Θ into a stack are the key ingredient to define stability notions onalgebraic stacks [Hal14, Hei17] and we need to recall some of their properties.By definition for any stack X and point Spec( k ) → S a map B G m,k → X isa point x ∈ X ( k ) together with a cocharacter G m,k → Aut X ( x ). As the actionof G m on a vector space is the same as a grading on the vector space, we oftenthink of a morphism B G m → X as a point of X equipped with a grading.Similarly, a vector bundle on Θ = [ A / G m ] is the same as a G m equivariantbundle on A and these are the same as vector spaces equipped with a filtration.So we think of morphisms f : Θ k → X as an object of x ∈ X ( k ) (the object f (1))together with a filtration of x and as f (0) = x as the associated graded object.In examples it is often easy to see that once one has found that some moduliproblem is described by an algebraic stack, the stacks of filtered or graded objectsare again algebraic. This turns out to be a general phenomenon, which we recall ext. For algebraic stacks X and Y over S , we denote byMap S ( Y , X )the stack over S parameterizing S -morphisms Y → X . If Y is defined over Spec( Z ),we will use the convention that Map S ( Y , X ) denotes the mapping stack Map S ( Y × S, X ).That these mapping stacks are again algebraic if Y = Θ or Y = B G m forquite general X follows from a general result established in [AHR] and [HLP14,Thm. 1.6]: if X is locally of finite presentation and quasi-separated over an al-gebraic space S with affine stabilizers, and Y is of finite presentation and withaffine diagonal over S such that Y → S is flat and a good moduli space, thenMap S ( Y , X ) is an algebraic stack locally of finite presentation over S , with affinestabilizers and quasi-separated diagonal. Moreover, if X → S has affine (resp.quasi-affine, resp. separated) diagonal, then so does Map S ( Y , X ). The stack of fil-trations Map S (Θ , X ) is denoted Filt( X ) in [Hal14], and the stack of graded objects Map S ( B G m , X ) is denoted Grad( X ).2.3. The example of quotient stacks.
To compute examples we recall thatstacks of filtrations and graded objects have a concrete description for quotientstacks. If X = [ X/G ] is a quotient stack locally of finite type over a field k , where G is a smooth algebraic group acting on a quasi-separated algebraic space X ,these mapping stacks have a classical interpretation [Hal14, Thm. 1.37]. To statethis recall that given λ : G m → G , one defines L λ = { l ∈ G | l = λ ( t ) lλ ( t ) − ∀ t } and P + λ = { p ∈ G | lim t → λ ( t ) pλ ( t ) − exists } . If G is geometrically reductive, then P + λ ⊂ G is a parabolic subgroup. There is asurjective homomorphism P + λ → L λ , defined by p lim t → λ ( t ) pλ ( t ) − .Similarly, one defines the functors: X λ := Map G m k (Spec( k ) , X ) (the fixed locus) X + λ := Map G m k ( A , X ) (the attractor)By [Dri13, Thm. 1.4.2], these functors are representable by algebraic spaces. More-over, there are the following natural morphisms: a closed immersion X λ ֒ → X , anunramified morphism X + λ → X (given by evaluation at 1) and an affine [AHR15,Thm. 2.22] morphism X + λ → X λ (given by evaluation at 0). If X is separated,then X + λ → X is a monomorphism.The k -points of X λ are simply the λ -fixed points, and if X is separated, the k -points of X + λ are the points x ∈ X ( k ) such that lim t → λ ( t ) · x exists. Thealgebraic space X λ inherits an action of L λ and X + λ inherits an action of P + λ such that the evaluation map ev : X + λ → X λ is equivariant with respect to thesurjection P + λ → L λ .We can now recall the description of our mapping stacks for quotient stacks: Proposition 2.8. [Hal14, Thm. 1.37]
Let X be a quasi-separated algebraic spacelocally of finite type over a field k equipped with an action of a smooth algebraicgroup G over k with a split maximal torus. Let Λ be a complete set of conjugacy lasses of cocharacters G m → G . Then there are isomorphisms Map k ( B G m , [ X/G ]) ∼ = G λ ∈ Λ [ X λ /L λ ];Map k (Θ , [ X/G ]) ∼ = G λ ∈ Λ [ X + λ /P + λ ] . Moreover, the morphism ev : Map k (Θ , [ X/G ]) → [ X/G ] is induced by the ( P + λ → G ) -equivariant morphism X + λ → X . The morphism ev : Map k (Θ , [ X/G ]) → Map( B G m , [ X/G ]) is induced by the ( P + λ → L λ ) -equivariant morphism X + λ → X λ . Valuative criteria for stacks
In this section, we introduce and study three valuative criteria for algebraicstacks—Θ-reductive morphisms (Definition 3.10), S-complete morphisms (Definition 3.37),and unpunctured inertia (Definition 3.53)—which appear in the formulation ofTheorem 4.1. Additionally, we introduce the notion of Θ-surjective morphisms in § Morphisms of stacks of filtrations.
It will be important to understandthe behavior of the stacks Map(Θ , X ) under morphisms X → Y , i.e., study thebehavior of filtrations on objects under morphisms. Lemma 3.1.
Let S be a quasi-separated algebraic space. Let f : X → Y be amorphism of algebraic stacks, locally of finite presentation and quasi-separatedover S , with affine stabilizers. Suppose f satisfies one of the following properties(a) representable;(b) monomorphism;(c) separated;(d) unramifed; or(e) ´etale,(f ) ´etale, surjective and representable.then Map S (Θ , X ) → Map S (Θ , Y ) has the same property.Proof. Properties (a) and (b) are clear. Property (c) follows from the valua-tive criterion and descent. Properties (d) and (e) follow from the formal lift-ing criterion and descent. For (f), it remains to show that Map S (Θ , X ) → Map S (Θ , Y ) is surjective. Let h : Θ k → Y s be a morphism over a geometricpoint s : Spec( k ) → S . We will use Tannaka duality to construct a lift to X .As any ´etale representable cover of B G m,k admits a section, we may choose alift B G m,k → X s of B G m,k ֒ → Θ k h −→ Y s . Let Θ [ n ] k = [Spec( k [ x ] /x n +1 ) / G m ]be the n th nilpotent thickening of B G m ֒ → Θ. Since f is ´etale, there existcompatible lifts Θ [ n ] k → X s of Θ [ n ] k ֒ → Θ k h −→ Y s . Since Θ k is coherently com-plete along B G m,k , by [AHR15, Cor. 3.6], there is an equivalence of categoriesMap k (Θ k , X s ) = lim ←− n Map k (Θ [ n ] k , X s ). This constructs the desired lift Θ k → X s of h . See also [Hal14, Lem. 4.33]. (cid:3) Property (f) is not preserved if the representability hypothesis is dropped. Forinstance, if X = B G m → B G m = Y is induced by G m → G m , t → t d for d > hen Map S (Θ , X ) → Map S (Θ , Y ) is not surjective. However, let us recall thefollowing useful lemma, whose proof relies on Theorem 2.6: Lemma 3.2 ([Hal14, Lem. 4.34]) . Let X be an algebraic stack of finite type withaffine diagonal over a noetherian algebraic space S . Then there is an algebraicspace X over S with X → S affine, a G nm action on X for some n ≥ , and asmooth, surjective and representable morphism [ X/ G nm ] → X such that the mor-phism Map S (Θ , [ X/ G nm ]) → Map S (Θ , X ) is smooth, surjective and representative. Property Θ - P . If f : X → Y is a morphism of algebraic stacks over analgebraic space S , we denote by ev( f ) the induced morphism of stacksev( f ) : Map S (Θ , X ) → X × Y , ev Map S (Θ , Y ) , λ (ev ( λ ) , f ◦ λ ) , i.e., this morphism takes an object together with a filtration in X and remembersthe object together with the induced filtration of the image in Y . It sits in acommutative diagram:Map S (Θ , X ) f ◦− ( ( ev ) ) ev( f ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ X × Y Map S (Θ , Y ) p / / p (cid:15) (cid:15) Map S (Θ , Y ) ev (cid:15) (cid:15) X f / / Y (3.1) Definition 3.3.
Let P be a property of morphisms of algebraic stacks. We saythat a morphism f : X → Y of algebraic stacks, locally of finite presentation andquasi-separated over an algebraic space S , with affine stabilizers, has property Θ - P if ev( f ) : Map S (Θ , X ) → X × Y , ev Map S (Θ , Y ) has property P . We say that X has property Θ - P if X → Spec( Z ) does.For example a morphisms f : X → Y is Θ -surjective if one can lift filtrationson any point f ( x ) to filtrations on x .The assignment f ev( f ) behaves well with respect to compositions andbase change. Namely, given a composition g ◦ f : X f −→ Y g −→ Z of morphisms ofalgebraic stacks over S , then ev( g ◦ f ) is naturally isomorphic to the compositionMap S (Θ , X ) ev( f ) −−−−→ X × Y Map S (Θ , Y ) id × ev( g ) −−−−−−−→ X × Y ( Y × Z Map S (Θ , Z )) ∼ = X × Z Map S (Θ , Z ) , and if X ′ f ′ / / (cid:15) (cid:15) Y ′ (cid:15) (cid:15) X f / / Y is a Cartesian diagram of algebraic stacks over S , thenMap S (Θ , X ′ ) ev( f ′ ) / / (cid:15) (cid:15) X ′ × Y ′ Map S (Θ , Y ′ ) (cid:15) (cid:15) / / Map S (Θ , Y ′ ) (cid:15) (cid:15) Map S (Θ , X ) ev( f ) / / X × Y Map S (Θ , Y ) / / Map S (Θ , Y )(3.2) s Cartesian. We conclude: Proposition 3.4.
Let P be a property of morphisms of algebraic stacks. If P is stable under composition and base change, then so is the property Θ - P . If P is stable under fppf (resp. smooth, resp. ´etale) descent, then Θ - P is stable underdescent by morphisms Y ′ → Y such that Map S (Θ , Y ′ ) → Map S (Θ , Y ) is fppf (resp.smooth and surjective, resp. ´etale and surjective). (cid:3) Lemma 3.5.
Let P be a property of representable morphisms of algebraic stacks.If P is stable under ´etale descent, then Θ - P is stable under descent by representable,´etale and surjective morphisms.Proof. This follows immediately from Proposition 3.4 and Lemma 3.1(f). (cid:3)
Lemma 3.6. If X is an algebraic stack with quasi-finite and separated inertia and T is a locally noetherian algebraic space, any morphism Θ T → X factors uniquelythrough Θ T → T .Proof. This follows from [Hal14, Lem. 1.29]. (cid:3)
Lemma 3.7.
Let S be a quasi-separated algebraic space. Let f : X → Y be amorphism of algebraic stacks, locally of finite presentation and quasi-separatedover S , with affine stabilizers. Assume that X and Y have separated diagonals.(1) The morphism ev( f ) is representable.(2) If f is separated, then so is ev( f ) .(3) If f is representable and separated, then ev( f ) is a monomorphism.(4) If X and Y have quasi-finite inertia, then ev( f ) is an isomorphism.(5) If f is ´etale, then so is ev( f ) .(6) If f is representable, ´etale, and separated, then ev( f ) is an open immer-sion.Proof. For (1), by diagram (3.1), it suffices to show that ev : Map S (Θ , X ) → X is representable, which is [Hal14, Lem. 1.10, Rem. 1.11].Part (2) follows from Lemma 3.1(c).For (3), to show that ev( f ) is a monomorphism, we need to show that forevery affine scheme Spec( R ), any commutative diagram of solid arrowsSpec( R ) / / (cid:15) (cid:15) X (cid:15) (cid:15) Θ R / / ssssss X × Y X can be filled in with a dotted arrow. As f is representable and separated, thebase change X × X × Y X Θ R → Θ R is a closed immersion containing the dense setSpec( R ); it is therefore an isomorphism.Part (5) follows directly from Lemma 3.1(e) using diagram (3.1). Part (6)follows directly from Parts (3) and (5) as ´etale monomorphisms are open immer-sions.For (4), it suffices by diagram (3.1) to show that ev : Map S (Θ , X ) → X is an isomorphism if X has quasi-finite inertia which follows immediately fromLemma 3.6. (cid:3) emark 3.8. The morphism ev( f ) is not in general quasi-compact. For an ex-ample, if f : B G m,k → Spec( k ), the morphism ev( f ) is the evaluation morphismis ev : Map S (Θ , B G m,k ) = F n ∈ Z B G m,k → B G m,k . Remark 3.9. If f is representable but not separated, then ev( f ) is not neces-sarily a monomorphism.3.3. Θ -reductive morphisms. In this section, we study the class of Θ-reductivemorphisms as introduced in [Hal14]. As before, we set Θ := [ A / G m ] defined overSpec( Z ). If R is a DVR with fraction field K , we set 0 ∈ Θ R := Θ × Spec( R ) tobe the unique closed point. Observe that a morphism Θ R \ → X is the dataof morphisms Spec( R ) → X and Θ K → X together with an isomorphism of theirrestrictions to Spec( K ). Definition 3.10.
A morphism f : X → Y of locally noetherian algebraic stacksis Θ -reductive if for every DVR R , any commutative diagramΘ R \ / / (cid:15) (cid:15) X f (cid:15) (cid:15) Θ R / / < < ②②②②② Y (3.3)of solid arrows can be uniquely filled in. Remark 3.11.
Let S be a noetherian algebraic space and f : X → Y be a mor-phism of algebraic stacks, locally of finite type and quasi-separated over S , withaffine stabilizers. Then f is Θ-reductive if and only if ev( f ) : Map S (Θ , X ) → X × Y , ev Map S (Θ , Y ) satisfies the valuative criterion for properness with respectto DVR’s, that is, for every DVR R with fraction field K , any diagramSpec( K ) / / (cid:15) (cid:15) Map S (Θ , X ) ev( f ) (cid:15) (cid:15) Spec( R ) ♠♠♠♠♠♠ / / X × Y , ev Map S (Θ , Y )of solid arrows can be uniquely filled in. Note that the morphism ev( f ) is alwaysrepresentable (Lemma 3.7(1)) and locally of finite type. However, the morphismev( f ) is not in general quasi-compact (see Remark 3.8) and therefore ev( f ) isnot in general proper. Remark 3.12.
In the context of the previous remark, when Y has quasi-finiteinertia the morphism ev : Map S (Θ , Y ) → Y is an equivalence (Lemma 3.6), andev( f ) is isomorphic to ev : Map S (Θ , X ) → X . Therefore, f is Θ-reductive if andonly if X is Θ-reductive in the absolute sense (i.e. X → Spec( Z ) is Θ-reductive).In order to be consistent with the terminology of Θ-reductivity introduced in of[Hal14, Def. 4.16], we have deviated from the “property Θ- P ” naming convention.3.3.1. Examples illustrating Θ -reductivity. In the following examples, we workover a field k . The following proposition gives a criterion using the notation from § X/G ] is Θ-reductive.
Proposition 3.13.
Let X = [ X/G ] be a quotient stack, where X is a quasi-separated algebraic space locally of finite type over a field k and G is a (smooth ut not necessarily connected) split reductive algebraic group over k . Then X is Θ -reductive if and only if for every cocharacter λ : G m → G , the morphism X + λ → X is proper. Remark 3.14. If X is separated, then X + λ → X is proper if and only if it is aclosed immersion. Proof.
This follows easily from the explicit description of the mapping stackMap S (Θ , X ) in Proposition 2.8. Indeed, there is a factorizationev : [ X + λ /P + λ ] → [ X/P + λ ] → [ X/G ]and since G is reductive, each P + λ ⊂ G is a parabolic subgroup. Since the quotient G/P + λ is projective, the morphism [ X/P + λ ] → [ X/G ] is proper. Thus propernessof ev is equivalent to properness of X + λ → X . (cid:3) In order to develop some intuition for Θ-reductivity, we use this result toprovide some basic examples and counterexamples of Θ-reductivity. For an integer n , we denote by λ n : G m → G m the cocharacter defined by t t n ; in this way,the integers Z index the cocharacters of G m . Example 3.15 (Affine quotients) . Consider the action of G m on X = A via t · ( x, y ) = ( tx, t − y ). Then X + λ n = V ( y ) if n > A if n = 0 V ( x ) if n < λ n is [ X + λ n / G m ] → [ X/ G m ] which is induced by the inclusion X + λ n → X . We see directly that [ X/ G m ]is Θ-reductive.More generally, if X = Spec( A ) is an affine scheme of finite type over k withan action of a reductive algebraic group G , then [ X/G ] is Θ-reductive. Indeed, if λ : G m → G is a cocharacter, then A inherits a Z -grading A = L n ∈ Z A n . If I − λ denotes the ideal generated by homogeneous elements of strictly negative degree,then it is easy to see that X + λ = V ( I − λ ); see [Dri13, § X + λ → X is a closed immersion and the conclusion follows from the characterization inProposition 3.13. Example 3.16.
In contrast, quotients of schemes that are not affine are notalways Θ-reductive. Consider the action of G m on X = A \ t · ( x, y ) = ( tx, y ).Then X + λ n = { y = 0 } if n > X if n = 0 V ( x ) if n < X/ G m ] is not Θ-reductive as X + λ n → X is not proper for n > R , the algebraic stack Θ R \ G m on X = P via t · [ x, y ] = [ tx, y ] and on the nodal cubic C ⊂ P such hat the normalization P → X is G m -equivariant. Then X + λ n = P \ { } ⊔ { } if n > P if n = 0 P \ {∞} ⊔ {∞} if n < C + λ n = P \ { } if n > C if n = 0 P \ {∞} if n < C + λ n → C for n = 0 are induced by the normalization. We seethat [ P / G m ] and [ C/ G m ] are not Θ-reductive.3.3.2.
Properties of Θ -reductive morphisms. We now give a few properties of Θ-reductive morphisms. First observe that Θ-reductive morphisms are stable undercomposition and base change. We first show that one can check the lifting crite-rion of (3.3) after taking extensions of the DVR.
Proposition 3.17.
Let X → Y be a morphism of locally noetherian algebraicstacks, and consider a diagram of the form (3.3) . There exists a unique dottedarrow filling in the diagram if either(1) there exists a unique filling after passing to an unramifed extension R ⊂ R ′ of DVR’s which is an isomorphism on residue fields, such as the com-pletion of R , or(2) X → Y has affine diagonal, and there exists a filling after an arbitraryextension of DVR’s R ⊂ R ′ .In particular, to verify that a morphism of locally noetherian algebraic stacks is Θ -reductive, it suffices to check the lifting criterion (3.3) for complete DVRs.Proof. The first statement follows from an explicit descent argument similar to[Hei17, Rmk. 2.5]. Alternatively, if R ⊂ R ′ is an unramified extension of DVRswith isomorphic residue fields, thenΘ R ′ \ / / (cid:15) (cid:15) Θ R \ (cid:15) (cid:15) Θ R ′ / / Θ R is a flat Mayer–Vietoris square ([HR16, Defn. 1.2]) and thus by [HR16, Thm.A] is a pushout in the 2-category of algebraic stacks. This establishes the firststatement.For the second statement, we begin with the observation that if X → Y hasaffine diagonal and j : U → T is an open immersion of algebraic stacks over Y with j ∗ O U = O T , then any two extensions f , f : T → X of a Y -morphism U → X arecanonically 2-isomorphic. Indeed, since Isom T ( f , f ) → T is affine, the sectionover U induced by the 2-isomorphism f | U ∼ → f | U extends uniquely to a sectionof T .Consider a diagram (3.3), an extension of DVRs R ⊂ R ′ and a lifting Θ R ′ → X .The open immersion j : Θ R \ → Θ R satisfies j ∗ O Θ R \ = O Θ R and by flat basechange, the same property holds for the morphisms obtained by base changing j along Θ R ′ → Θ R , Θ R ′ × Θ R Θ R ′ → Θ R , and Θ R ′ × Θ R Θ R ′ × Θ R Θ R ′ → Θ R .By the above observation, there exists a canonical 2-isomorphism between thetwo extensions Θ R ′ × Θ R Θ R ′ ⇒ Θ R ′ → X which necessarily satsifies the cocyclecondition. By fpqc descent, the lifting Θ R ′ → X descends to a lifting Θ R → X . (cid:3) Θ-reductivity satisfies the following two descent properties. The second prop-erty is not used in this paper and is only included for thoroughness. roposition 3.18. Let X → Y be a morphism of locally noetherian algebraicstacks.(1) If Y ′ → Y is an ´etale, representable and surjective morphism, then X → Y is Θ -reductive if and only if X × Y Y ′ → Y ′ is Θ -reductive.(2) If X ′ → X is a finite, ´etale and surjective morphism, then X → Y is Θ -reductive if and only if X ′ → Y is Θ -reductive. Remark 3.19. If X → Y has affine diagonal, then (2) also holds, with a similarproof, if one replaces the words ‘finite, ´etale’ with ‘quasi-compact, universallyclosed.’ Proof.
For (1), to check Θ-reductivity of X → Y , by Proposition 3.17 we mayassume we have a diagram (3.3) where R is a complete DVR. As any ´etale, rep-resentable cover of Θ R has a section after a finite ´etale extension R ⊂ R ′ , wemay lift the composition Θ R ′ → Θ R → Y to Θ R ′ → Y ′ . The Θ-reductivity of X ′ := X × Y Y ′ → Y ′ shows that the lift Θ R ′ \ → X ′ extends uniquely to amorphism Θ R ′ → X ′ . This implies that the lift Θ R ′ \ → X extends uniquely toa morphism Θ R ′ → X as well, because both extension problems can be rephrasedin terms of sections of Θ R ′ × Y ′ X ′ ≃ Θ R ′ × Y X .Finally, the first few levels of the Cech nerve for the ´etale cover Θ R ′ → Θ R have the form . . . / / / / / / / / F j Θ R ′′′ j / / / / / / F i Θ R ′′ i / / / / Θ R ′ , for some complete DVR’s R ′′ i and R ′′′ j . The argument of the previous paragraphshows that for any of the DVR’s A = R ′ , R ′′ i , R ′′′ j the lift Θ A \ → X extendsuniquely to a lift Θ A → X of the map Θ A → Y . ´Etale descent now implies thatthe original lift Θ R \ → X extends uniquely to a morphism Θ R → X .For (2), the ‘only if’ direction follows since finite morphisms are Θ-reductive.Conversely, given a diagram (3.3), we may find a finite ´etale extension R ⊂ R ′ with fraction field K ⊂ K ′ such that the composition Spec( K ) → Θ R \ → X lifts to a map Spec( K ′ ) → X ′ . As X ′ → X is finite, we may extend this morphismuniquely to a map Θ R ′ \ → X ′ lifting Θ R ′ \ → Θ R \ → X . By Θ-reductivityof X ′ → Y , this map extends uniquely to a morphism Θ R ′ → X ′ . The compositionΘ R ′ → X ′ → X is an extension of Θ R ′ \ → Θ R \ → X , and is unique since X ′ → X is Θ-reductive. By an ´etale descent argument similar to the one given inPart (1), this descends uniquely to the desired lift Θ R → X . (cid:3) We now provide some important classes of Θ-reductive morphisms.
Proposition 3.20. (1) An affine morphism of locally noetherian algebraic stacks is Θ -reductive.(2) Let S be a locally noetherian scheme. Let G → S be a geometricallyreductive and ´etale-locally embeddable group scheme (e.g. reductive) act-ing on a locally noetherian scheme X affine over S . Then the morphism [ X/G ] → S is Θ -reductive.(3) A good moduli space X → X , where X is a locally noetherian algebraicstack with affine diagonal, is Θ -reductive. Remark 3.21.
In the case that S = Spec( k ) where k is an algebraically closedfield, Part (2) implies that [Spec( A ) /G ], where G is a geometrically reductivealgebraic group, is Θ-reductive. In the case that G is smooth, then this followsfrom the explicit calculation in Example 3.15. roof. For (1), since 0 ∈ Θ R has codimension 2 and Θ R is regular for a DVR R ,we have that (Θ R \ → Θ R ) ∗ O Θ R \ = O Θ R . Given an affine morphism f : X → Y ,we have canonical isomorphismsMap Y (Θ R \ , X ) ∼ = Map O Y − alg ( f ∗ O X , (Θ R \ → Y ) ∗ O Θ R \ ) ∼ = Map O Y − alg ( f ∗ O X , (Θ R → Y ) ∗ O Θ R ) ∼ = Map Y (Θ R , X ) . See also [Hal14, Prop. 1.19], which shows that ev( f ) is a closed immersion when f is affine.For (2), since Θ-reductive morphisms descend under representable, ´etale andsurjective morphisms (Proposition 3.18), we may assume that S is an affine noe-therian scheme and that G is a closed subgroup of GL N,S for some N . Wefirst show that B Z GL N = [Spec( Z ) / GL N ] is Θ-reductive, which implies that B S GL N = [ S/ GL N,S ] is also Θ-reductive. A morphism Θ R \ → X correspondsto a vector bundle E on Θ R \
0. If e E is any coherent sheaf on Θ R extending E , thenthe double dual e E ∨∨ is a vector bundle extending E . This provides the desired ex-tension Θ R → X . Since GL N,S /G is affine [Alp14, Thm. 9.4.1], B S G → B S GL N is affine. By Part (1), B S G is Θ-reductive. Since X is affine over S , [ X/G ] → B S G is affine which implies using again Part (1) that [ X/G ] is Θ-reductive.For (3), we may assume that X is quasi-compact. By [AHR], there exists an´etale cover Spec( B ) → X such that X × X Spec( B ) ∼ = [Spec( A ) / GL N ] for some N and B = A GL N . Since Θ-reductive morphisms descend under representable,´etale and surjective morphisms, this reduces the claim to the statement that[Spec( A ) / GL N ] → Spec( A GL N ) is Θ-reductive which follows from Part (2). (cid:3) Proposition 3.22.
A morphism f : X → Y of locally noetherian algebraic stacks,such that X and Y both have quasi-finite and separated inertia, is Θ -reductive.Proof. This follows from Lemma 3.6. (cid:3)
Specialization of k -points. Next we provide general criteria for when spe-cialization of k -points can be realized by a morphism from Θ k . Lemma 3.23.
Let X be an algebraic stack locally of finite type over a perfect field k such that either (1) X is locally linearly reductive or (2) X ∼ = [Spec( A ) / GL N ] for some N . Then any specialization x x of k -points where x is a closedpoint is realized by a morphism Θ k → X .Proof. The first case follows from the second by Theorem 2.3 while the secondcase follows from the Hilbert–Mumford criterion [Kem78, Thm. 4.2]. (cid:3)
The topology of k -points of Θ-reductive stacks is analogous to the topology ofquotient stacks arising from GIT. Lemma 3.24.
Let X be an algebraic stack locally of finite type over a field k suchthat either (1) X is locally linearly reductive or (2) X ∼ = [Spec( A ) / GL N ] for some N . If X is Θ -reductive, then the closure of any k -point p contains a unique closedpoint x .Proof. Assume that x and x ′ are two closed points in the closure of p . Afterreplacing k with an extension if necessary, we may assume that k is perfect, andthat x and x ′ are k -rational. It follows from Lemma 3.23 that these specializationscome from two filtrations f, f ′ : Θ k → X with f (1) ≃ f ′ (1) ≃ p , f (0) ≃ x and ′ (0) ≃ x ′ . The maps f and f ′ glue to define a map [ A k − { (0 , } / ( G m ) k ],and choosing one of the two G m factors we can apply Θ-reductivity to extendthis morphism to a map [ A / G m ] → X . Then γ (0 ,
0) is a specialization of both x ≃ γ (1 ,
0) and x ′ ≃ γ (0 , x and x ′ are closed implies that x ≃ γ (0 , ≃ x ′ . (cid:3) -surjective morphisms. In this section, we study the class of Θ-surjectivemorphisms. We will observe that Θ-surjective morphisms between locally lin-early reductive algebraic stacks necessarily map closed points to closed points(Lemma 3.27). This notion will play a fundamental role in our proof of Theorem 4.1;namely, we will use Θ-reductivity to ensure that we can find local quotient pre-sentations which are Θ-surjective (Proposition 4.4(1)).By Definition 3.3, a morphism f : X → Y (of algebraic stacks, locally of finitepresentation and quasi-separated over a quasi-separated algebraic space S , withaffine stabilizers) is Θ -surjective ifev( f ) : Map(Θ , X ) → X × Y , ev Map(Θ , Y )is surjective. From Proposition 3.4 and Lemma 3.5, Θ-surjective morphisms arestable under composition and base change, and they descend under representable,´etale and surjective morphisms. Remark 3.25.
The condition of Θ-surjectivity translates into the following liftingcriterion: For a field k , denote by i : Spec( k ) ֒ → Θ k the open immersion. Then f : X → Y is Θ-surjective if and only if for any algebraically closed field k , anycommutative diagram Spec( k ) i (cid:15) (cid:15) / / X f (cid:15) (cid:15) Θ k / / ; ; ①①①①① Y (3.4)of solid arrows can be filled in with a dotted arrow. Remark 3.26. If f is representable and separated, it follows from Lemma 3.7(3)that there is at most one lift in diagram (3.4), that is, f is Θ-universally injective(or equivalently Θ-radicial). This fails for non-separated morphisms.We also note that if f is proper, then the valuative criterion for propernessimplies that there exists a unique lift in the above diagram. Therefore properrepresentable morphisms are Θ-universally bijective.If X is an algebraic stack over a quasi-separated algebraic space S and s ∈ | S | ,let X s be the fiber product X × S Spec( κ ( s )), where κ ( s ) is the residue field of s . Lemma 3.27.
Let S be a quasi-separated algebraic space and f : X → Y be amorphism of algebraic stacks, locally of finite presentation over S with affinestabilizers. Suppose that Y is locally linearly reductive and f is Θ -surjective. If x ∈ | X | is a point with image s ∈ | S | such that x ∈ | X s | is closed, then f ( x ) ∈ | Y s | is closed.Proof. We immediately reduce to the case when S is the spectrum of an alge-braically closed field k and x ∈ | X | is a closed point. If f ( x ) is not closed,then there exists a specialization f ( x ) y of k -points to a closed point. By emma 3.23, there exists a morphism Θ k → Y realizing f ( x ) y . As thediagram Spec( k ) i (cid:15) (cid:15) x / / X f (cid:15) (cid:15) Θ k / / h ; ; ①①①①① Y can be filled in with a morphism h and x ∈ | X | is closed, h (0) = h (1). It followsthat f ( x ) = y is closed. (cid:3) Remark 3.28.
The converse of Lemma 3.27 is not true; see Example 3.34.For the construction of good moduli spaces we will need a variant of the aboveproperties. Let X and Y be algebraic stacks of finite type with affine diagonalover a noetherian algebraic space S , and let f : X → Y be a morphism. DefineΣ f ⊂ | X | be the set of points x ∈ | X | where f is not Θ-surjective at x , i.e., points x ∈ | X | where there exists a representative Spec( k ) → X of x with k algebraicallyclosed and a commutative diagram as in diagram (3.4) which cannot be filled in.By definition, Σ f is the image under p of the complement of the image of ev( f ) ,i.e.,(3.5) Σ f = p (cid:18)(cid:0) X × Y Map S (Θ , Y ) (cid:1) \ ev( f ) (Map S (Θ , X )) (cid:19) ⊂ | X | . Lemma 3.29.
Let X and Y be algebraic stacks of finite type with affine diagonalover a noetherian algebraic space S , and let f : X → Y be a representable, quasi-finite, and separated morphism. Suppose that either(1) Y admits a good moduli space; or(2) Y ∼ = [Spec( A ) / GL N ] for some N .Then the locus Σ f ⊂ | X | is closed.Proof. Zariski’s Main Theorem [LMB, Thm. 16.5] provides a factorization f : X i −→ e Y ν −→ Y where i is an open immersion and ν is a finite morphism. As ν is properand therefore Θ-surjective we have Σ i = Σ f . Thus, it suffices to assume that f isan open immersion. Let Z ⊂ Y be the reduced complement of X and let π : Y → Y denote the adequate moduli space. We claim that Σ f = π − ( π ( | Z | )) ∩ | X | .Indeed, the inclusion “ ⊂ ” is clear: the morphism Y \ π − ( π ( | Z | )) ֒ → Y is thebase change of the Θ-surjective morphism Y \ π ( | Z | ) ֒ → Y of algebraic spaces.For the inclusion “ ⊃ ,” let x ∈ π − ( π ( | Z | )) ∩ | X | and let x : Spec( k ) → X be arepresentative of x , where k is algebraically closed, with image s : Spec( k ) → S .Let x s ∈ | X s | be the image of Spec( k ) → X s and z ∈ | Z s | be the unique closedpoint in the closure of x s . If Y admits a good moduli space, it is in particularlocally linearly reductive. Therefore, in either case (1) or (2), we may applyLemma 3.23 to obtain a morphism Θ k → Y s realizing the specialization x s z .Since the commutative diagramSpec( k ) x / / (cid:15) (cid:15) X f (cid:15) (cid:15) Θ k / / ; ; ①①①①① Y does not admit a lift, x ∈ Σ f . As π − ( π ( | Z | )) ⊂ | Y | is closed, the conclusionfollows. (cid:3) roposition 3.30. Let X and Y be algebraic stacks, of finite type with affinediagonal over a noetherian algebraic space S , and let f : X → Y be a representable,quasi-finite and separated morphism. If Y is locally linearly reductive, then Σ f ⊂ X is constructible.Proof. By Theorem 2.3, the hypotheses imply that there exists a representable,´etale and surjective morphism g : Y ′ → Y , where Y ′ ∼ = [Spec( A ) / GL N ] for some N .Let X ′ = X × Y Y ′ with projections g ′ : X ′ → X and f ′ : X ′ → Y ′ . By Lemma 3.1(f),the morphism Map S (Θ , Y ′ ) → Map S (Θ , Y ) is surjective. Therefore by CartesianDiagram (3.2), the complement of Map S (Θ , X ′ ) in X ′ × Y ′ Map S (Θ , Y ′ ) surjectsonto the complement of Map S (Θ , X ) in X × Y Map S (Θ , Y ). It follows that Σ f = g ′ (Σ f ′ ). By Chevalley’s Theorem and Lemma 3.29, the locus Σ f is constructible. (cid:3) Let us give some simple examples and non-examples of Θ-surjectivity. In theseexamples, we work over a field k . Example 3.31. If φ : X → X is an adequate moduli space and U ⊂ X is anopen substack, then U is saturated (i.e. φ − ( φ ( U )) = U ) if and only if U ֒ → X isΘ-surjective. In this case, U ֒ → X is even a Θ-isomorphism. Example 3.32.
The open immersion Spec( k ) ֒ → [ A / G m ] is Θ-reductive but not Θ-surjective. Indeed, this is the prototypical example of a morphism that doesnot send closed points to closed points.
Example 3.33.
Consider the action of G m on X = A \ t · ( x, y ) = ( tx, y )(as in Example 3.16)) and the open immersion f : A ֒ → [ X/ G m ] of the locuswhere x is non-zero. Thenev( f ) : A = Map(Θ , A ) → Map(Θ , [ X/ G m ]) = A ⊔ (cid:0) G n< A \ (cid:1) which is the inclusion onto the first factor. Again, f is affine and hence Θ-reductive but not Θ-surjective. Example 3.34.
Let C ⊂ P be the nodal cubic with a G m -action and considerthe ´etale presentation f : [ W/ G m ] → [ C/ G m ] where W = Spec( k [ x, y ] /xy ) and G m acts with weights 1 and − x and y , respectively. Then f clearly mapsclosed points to closed points but we claim it is not Θ-surjective. Indeed, thereis no lift in the diagramSpec( k ) / / (cid:15) (cid:15) [Spec( k [ x, y ] /xy ) / G m ] f (cid:15) (cid:15) (cid:15) (cid:15) Θ / / ❧❧❧❧❧❧❧❧ [ C/ G m ]where Spec( k ) → [Spec( k [ x, y ] /xy ) / G m ] is defined by y = 0 and x = 0, andΘ → [ C/ G m ] is the composition of the morphism Θ → [Spec( k [ x, y ] /xy ) / G m ]defined by x = 0 and the morphism f .3.5. Elementary modifications and S -complete morphisms. Modifications and elementary modifications.
As in [Hei17, § R , plays an important role in ouranalysis of criteria for separatedness of good moduli spaces.(3.6) ST R := [Spec (cid:0) R [ s, t ] / ( st − π ) (cid:1) / G m ] , here s and t have G m -weights 1 and − π is a choice of uni-formizer for R . A different choice of π results in an isomorphic stack.Observe that ST R \ ∼ = Spec( R ) ∪ Spec( K ) Spec( R ), where K is the fraction fieldof R , because the locus where s = 0 in ST R is isomorphic to [Spec (cid:0) R [ s, t ] s / ( t − π/s ) (cid:1) / G m ] ∼ = [Spec( R [ s ] s ) / G m ] ∼ = Spec( R ) and the locus where t = 0 has asimilar description. A morphism h : ST R \ → X to an algebraic stack is thedata of two morphisms ξ, ξ ′ : Spec( R ) → X , where ξ := h | { s =0 } and ξ ′ := h | { t =0 } ,together with an isomorphism ξ K ≃ ξ ′ K . Definition 3.35.
Let X be an algebraic stack and let ξ : Spec( R ) → X be amorphism where R is a DVR with fraction field K .(1) A modification of ξ is the data of a morphism ξ ′ : Spec( R ) → X alongwith an isomorphism between the restrictions ξ | K ≃ ξ ′ | K .(2) An elementary modification of ξ is the data of a morphism h : ST R → X along with an isomorphism ξ ≃ h | { s =0 } .An elementary modification is clearly also a modification. Remark 3.36.
The terminology here is inspired by the terminology of [Lan75],but does not exactly coincide. Langton’s notion of “elementary modifications” offamilies of vector bundles over a DVR are examples of the notion of elementarymodification above which flip two-step filtrations. To see this, let X be a noether-ian scheme and Coh( X ) the stack of coherent sheaves on X . Let R be a DVRwith fraction field K and residue field κ . A quasi-coherent sheaf on X × ST R cor-responds to a Z -graded coherent sheaf L n ∈ Z F n on X R together with a diagramof maps(3.7) · · · s , , F n − s * * t j j F n s , , t l l F n +1 s * * t j j · · · t l l , such that st = ts = π . Moreover, F is coherent if each F n is coherent, s : F n − → F n is an isomorphism for n ≫ t : F n → F n − is an isomorphism for n ≪ F is a flat over ST R if and only if the maps s and t are injective, andthe induced map t : F n +1 /sF n → F n /sF n − is injective. (See Corollary 7.13 fora proof of that these properties characterize coherence and flatness of F .)Suppose that we have a coherent sheaf E on X R which is flat over R whoserestriction E κ to X fits into a short exact sequence(3.8) 0 → B → E κ → G → . (In Langton’s algorithm, one takes B ⊂ E κ to be the maximal destabilizingsubsheaf.) Let E ′ = ker( E → E κ → G ) . Then E ′ is flat over R and E K = E ′ K .Moreover, we have that πE ⊂ E ′ ⊂ E with E/E ′ = G and E ′ /πE = B ; thisimplies that E ′ κ fits into a short exact sequence(3.9) 0 → G → E ′ κ → B → . This data defines a coherent sheaf on X × ST R flat over ST R as follows: set F n to be E if n ≥ E ′ if n <
0. Let s and t act via · · · π * * E ′ π ) ) j j E ′ ) ) i i E ) ) π i i E * * π i i · · · π i i In general, the restriction of F to { s = 0 } = Spec( R ) is the colimit over the Z -sequence of maps s : F n → F n +1 . In our case, this restriction is E . Likewise, he restriction of F to { t = 0 } is the colimit over the t maps so its E ′ in our case.In general, the restriction of F to { s = 0 } = Θ κ is the (generalized) Z -filtration · · · ← F n /sF n − t ←− F n +1 /sF n +2 ← · · · which in our case corresponds to E ′ κ ⊇ G of (3.8). Similarly, the restriction of E to { t = 0 } corresponds to the filtration B ⊂ E κ of (3.9).An analogous construction shows that any finite sequence of steps in Langton’salgorithm can be realized by a single elementary modification.3.5.2. S-complete morphisms.
Definition 3.37.
We say that a morphism f : X → Y of locally noetherian al-gebraic stacks is S-complete if for any DVR R and any commutative diagramST R \ / / (cid:15) (cid:15) X f (cid:15) (cid:15) ST R / / < < ②②②②② Y (3.10)of solid arrows, there exists a unique dotted arrow filling in the diagram. Remark 3.38.
The motivation for the terminology “S-complete” comes fromSeshadri’s work on the S-equivalence of semistable vector bundles. Namely, if X is the moduli stack of semistable vector bundles over a smooth projective curve C over k , then X is S-complete (see e.g., Lemma 8.4). If R is a DVR with fractionfield K and residue field k , and E , F are two families of semistable vector bundleson C R which are isomorphic over C K , then S-completeness implies that the specialfibers E and F on C are S-equivalent. Remark 3.39.
S-complete morphisms are stable under composition and basechange. A morphism of quasi-separated and locally noetherian algebraic spacesis S-complete if and only if it is separated (Proposition 3.44). While affine mor-phisms are always S-complete (Proposition 3.42(1)), it is not true that sepa-rated, representable morphisms are S-complete. For instance, the open immer-sion ST R \ → ST R is not S-complete. This example also shows that S-completemorphisms do not satisfy smooth descent; however, S-completeness does descendalong representable, ´etale and surjective morphisms (Proposition 3.41).We now state properties of S-completeness analogous to Proposition 3.17, Proposition 3.18,and Proposition 3.20. In each case, the proof is identical. Proposition 3.40.
Let X → Y be a morphism of locally noetherian algebraicstacks, and consider a diagram of the form (3.10) , then there exists a uniquedotted arrow filling the diagram if either(1) there exists a unique filling after passing to an unramifed extension R ⊂ R ′ of DVR’s which is an isomorphism on residue fields, such as the com-pletion of R , or(2) X → Y has affine diagonal, and there exists a filling after an arbitraryextension of DVR’s R ⊂ R ′ .In particular, to verify that a morphism of locally noetherian algebraic stacks isS-complete, it suffices to check the lifting criterion (3.10) for complete DVRs. (cid:3) roposition 3.41. Let X → Y be a morphism of locally noetherian algebraicstacks.(1) If Y ′ → Y is an ´etale, representable and surjective morphism, then X → Y is S-complete if and only if X × Y Y ′ → Y ′ is S-complete.(2) If X ′ → X is a finite, ´etale and surjective morphism, then X → Y isS-complete if and only if X ′ → Y is S-complete. (cid:3) Proposition 3.42. (1) An affine morphism of locally noetherian algebraic stacks is S-complete.(2) Let S be a locally noetherian scheme. Let G → S be a geometricallyreductive and ´etale-locally embeddable group scheme (e.g. reductive) act-ing on a locally noetherian scheme X affine over S . Then the morphism [ X/G ] → S is S-complete(3) A good moduli space X → X , where X is a locally noetherian algebraicstack with affine diagonal, is S -complete. (cid:3) We now detail additional important properties of S-completeness.
Lemma 3.43. If X is an algebraic stack with quasi-finite and separated inertiaand T is a locally noetherian algebraic space, any morphism ST T → X factorsuniquely through ST T → T .Proof. This can be established with the same method as Lemma 3.6. (cid:3)
Proposition 3.44.
Let f : X → Y be a morphism of quasi-separated and locallynoetherian algebraic stacks such that X and Y both have quasi-finite and separatedinertia. Then f is S-complete if and only if f is separated.Proof. Let R be a DVR with fraction field K . By Lemma 3.43, any morphismfrom ST R to X or Y factors through ST R → Spec( R ). As ST R \ R ) S Spec( K ) Spec( R ),we see that the valuative criterion of Diagram 3.10 is equivalent to the valuativecriterion for separatedness. (cid:3) Proposition 3.45. If G is an algebraic group over a field k , then G is geomet-rically reductive if and only if B k G is S-complete. In particular, a closed pointof an S-complete locally noetherian algebraic stack with affine stabilizers has ageometrically reductive stabilizer.Proof. From Proposition 3.42(2), we know that if G is geometrically reductive,then B k G is S-complete. For the converse, we may assume that k is algebraicallyclosed. Suppose that G is not geometrically reductive. Then by considering theunipotent radical R u ( G ) of the reduced group scheme G red , the induced morphism B k R u ( G ) → B k G is affine. Similarly, by taking a normal subgroup G a ⊂ R u ( G ),there is an affine morphism B k G a → B k R u ( G ). The composition B k G a → B k R u ( G ) → B k G is affine. Since B k G is S-complete, by Proposition 3.42(1)so is B k G a , a contradiction. (cid:3) Remark 3.46.
The proof shows more generally that if X is a locally noetherianalgebraic stack that is S-complete with respect to DVRs essentially of finite typeover R , then any closed point of X has geometrically reductive stabilizer.Expanding on Proposition 3.42(3), we have the following criterion for when agood moduli space is separated. roposition 3.47. Let X be a locally noetherian algebraic stack with affine di-agonal over an algebraic space S , and let X → X be a good moduli space. Then(1) the morphism X → X is S-complete;(2) the morphism X → S is separated if and only if X → S is S-complete;and(3) the morphism X → S is proper if and only if X → S is of finite type,universally closed and S-complete. Remark 3.48.
If in addition X → S is of finite type, then in verifying that X is S-complete, it suffices to verify that for every DVR R essentially of finite typeover S , any commutative diagram (3.3) has a unique lift after an extension ofthe DVR R . Likewise, in verifying that X → S is universally closed, it suffices toverify the existence part of the valuative criterion for properness of X → S withrespect to DVR’s which are essentially of finite type over S . Proof.
Part (1) is Proposition 3.42(3). The implication ‘ ⇒ ’ in Part (2) followsfrom Part (1) and the fact that separated algebraic spaces are S-complete. Con-versely, suppose X is S-complete. Suppose f, g : Spec( R ) → X are two mapssuch that f | K = g | K . After possibly an extension of R , we may choose a liftSpec( K ) → X of f | K = g | K . Since X → X is universally closed, after possi-bly further extensions of R , we may choose lifts e f , e g : Spec( R ) → X of f, g suchthat e f | K ∼ = e g | K . By applying the S-completeness of X , we can extend e f , e g toa morphism ST R → X . As ST R → Spec( R ) is a good moduli space and henceuniversal for maps to algebraic spaces [Alp13, Thm. 6.6], the morphism ST R → X descends to a unique morphism Spec( R ) → X which necessarily must be equalto both f and g . We conclude that X is separated by the valuative criterion forseparatedness. Part (3) follows from Part (1) using the fact that X is universallyclosed if and only if X is. (cid:3) Remark 3.49.
Assume instead that X → X is an adequate moduli space (ratherthan good moduli space) while keeping the other hypotheses on X . The sameargument as above shows that if X is S-complete (resp. universally closed andS-complete), then X is separated (resp. proper). We suspect that the conclusionof all parts of Proposition 3.47 hold but at the moment we cannot show this aswe do not have a slice theorem to reduce to the case of [Spec( A ) /G ] with G geometrically reductive. Corollary 3.50.
Let X be a locally noetherian algebraic stack with affine diagonaland let X → X be a good moduli space. Let R be any DVR and consider twomorphisms ξ , ξ : Spec( R ) → X with ( ξ ) | K ∼ = ( ξ ) | K Then the following areequivalent:(1) ξ and ξ differ by an elementary modification,(2) ξ and ξ differ by a finite sequence of elementary modifications,(3) the compositions ξ i : Spec( R ) → X → X agree for i = 0 , .Proof. Clearly (1) ⇒ (2). The projection ST R → Spec( R ) is a good moduli spaceand hence universal for maps to algebraic spaces [Alp13, Thm. 6.6]. It followsthat any two maps which differ by an elementary modification induce the same R -point of X , and thus (2) ⇒ (3). The implication (3) ⇒ (1) follows from part(1) of Proposition 3.47. (cid:3) Remark 3.51.
The above conditions are not equivalent to saying that ξ and ξ are modifications such that the closures of ξ (0) and ξ (0) intersect. For instance, et X be the non-locally separated algebraic space obtained by taking the free Z / Z / x
7→ − x and swaps the origins. Then there are two distinct maps ξ , ξ : Spec( R ) → X with ξ | K = ξ | K and ξ (0) = ξ (0). Remark 3.52 (Hartog’s principle) . Both Θ-reductivity and S-completeness areconditions asserting the existence and uniqueness of extending morphisms alonga codimension two locus. One might be tempted to unify these two notions bydefining that a morphism f : X → Y of locally noetherian algebraic stacks satisfies Hartogs’s principle if for any regular local ring S of dimension 2 with closedpoint 0 ∈ Spec( S ), there exists a unique dotted arrow filling in any commutativediagram Spec( S ) \ / / (cid:15) (cid:15) X f (cid:15) (cid:15) Spec( S ) / / : : ✉✉✉✉✉ Y (3.11)of solid arrows. Any such morphism is necessarily both Θ-reductive and S-complete. Moreover, the analogues of Proposition 3.20 and Proposition 3.42 holdfor such morphisms. However, many algebraic stacks (e.g. the stack Coh( X ) ofcoherent sheaves on a proper scheme X over a field k ) are both Θ-reductive andS-complete but do not satisfy Hartog’s principle.3.6. Unpunctured inertia.
We now give the last of the properties that willturn out to be necessary for the existence of good moduli spaces.
Definition 3.53.
We say that a noetherian algebraic stack has unpuncturedinertia if for any closed point x ∈ | X | and versal deformation p : ( U, u ) → ( X , x ),where U is the spectrum of a local ring with closed point u , each connectedcomponent of the inertia group scheme Aut X ( p ) → U has non-empty intersectionwith the fiber over u . Remark 3.54.
The condition of unpuncturedness is related to the property ofpurity of the morphism Aut X ( p ) → U as defined in [RG71, § U is the spectrum of a strictly henselian localring, then purity requires that if s ∈ U is any point and γ is an associated point inthe fiber Aut X ( p ) s , then the closure of γ in Aut X ( p ) has non-empty intersectionwith the fiber over u .In Section 5, we will provide valuative criteria which can be used to verify thata stack has unpunctured inertia. In this section though, we provide only a fewsituations in which this condition is easy to check. Proposition 3.55. If X is a noetherian algebraic stack with quasi-finite inertia,then X has unpunctured inertia if and only if X has finite inertia.Proof. If X has finite inertia, then Aut X ( p ) → U is finite so clearly the imageof each connected component contains the unique closed point u ∈ U . For theconverse, we may assume that U is the spectrum of a Henselian local ring inwhich case Aut X ( p ) = G ⊔ H where G → U finite and the fiber of H → U over u is empty. If Aut X ( p ) is not finite, then H is non-empty and any connectedcomponent of H will have empty intersection with the fiber over u . (cid:3) roposition 3.56. Let X be a noetherian algebraic stack. If X has connectedstabilizer groups, then X has unpunctured inertia.Proof. This is clear, by definition all fibers of Aut X ( p ) → U are connected, so anyconnected component of Aut X intersects the component containing the identitysection. (cid:3) The following example shows that unpuncturedness need not be preservedwhen passing to open substacks.
Example 3.57.
Consider the action of G = G m ⋊ Z / X = A via t · ( a, b ) =( ta, t − b ) and − · ( a, b ) = ( b, a ). Note that every point ( a, b ) ∈ X with ab = 0 isfixed by the order 2 element ( a/b, − ∈ G . The algebraic stack [( X \ /G ] doesnot have unpunctured inertia by Proposition 3.55. However, it will follow fromProposition 5.7 that [ X/G ] has unpunctured inertia.4.
Existence of good moduli spaces
The goal of this section is to prove the following theorem providing necessaryand sufficient conditions for an algebraic stack to admit a good moduli space.
Theorem 4.1.
Let X be an algebraic stack, locally of finite type with affine di-agonal over a quasi-separated and locally noetherian algebraic space S . Then X admits a good moduli space if and only if(1) X is locally linearly reductive (Definition 2.1);(2) X is Θ -reductive (Definition 3.10); and(3) X has unpunctured inertia (Definition 3.53). Remark 4.2.
The theorem also holds if one replaces the condition (2) with(2 ′ ) for every DVR R essentially of finite type over S , any commutative dia-gram (3.3) has a unique lift.The idea of the proof is simple. We use the slice theorem (Theorem 2.3) toreduce to quotient stacks and glue the resulting moduli spaces. As this onlyworks ´etale locally we need to apply the slice theorem carefully in such a waythat preserves the stabilizer groups and the topology of finite type points in orderto ensure that the ´etale covering of the stack induces an ´etale covering on thelevel of good moduli spaces.4.1. Reminder on maps inducing ´etale maps on good moduli spaces. If f : X → Y is a morphism of algebraic stacks and x ∈ | X | , we say that f is stabilizerpreserving at x if there exists a representative e x : Spec( l ) → X of x (equivalently,for all representatives of x ), the natural map Aut X ( e x ) → Aut Y ( f ◦ e x ) is anisomorphism. Proposition 4.3.
Let X and Y be noetherian algebraic stacks with affine diagonal.Consider a commutative diagram X f / / π X (cid:15) (cid:15) Y π Y (cid:15) (cid:15) X g / / Y (4.1) here f is representable, ´etale and separated, and both π X and π Y are good mod-uli spaces (and in particular X and Y are locally linearly reductive). If f is Θ -surjective and f is stabilizer preserving at every closed point in X , then g is ´etaleand Diagram 4.1 is Cartesian.Proof. This result is essentially a stack-theoretic reformation of Luna’s funda-mental lemma [Alp10, Thm. 6.10]. To see why this version holds, we first reduceby ´etale descent to the case that Y is affine. If x ∈ | X | is a closed point, then y = π Y ( f ( x )) ∈ Y is necessarily locally closed so after replacing Y with anopen subspace, we may assume that y ∈ Y is closed. Since f is Θ-surjective, f ( x ) ∈ | Y | is a closed point by Lemma 3.27, and [Alp10, Thm. 6.10] implies thatthere is an open subspace U ⊂ X containing π X ( x ) such that g | U is ´etale and π − X ( U ) = U × Y Y . (cid:3) Proof of the existence result.
We first provide conditions on an algebraicstack ensuring that there are local quotient presentations which are Θ-surjectiveand stabilizer preserving. This is the key ingredient in the proof of Theorem 4.1.
Proposition 4.4.
Let Y be an algebraic stack, locally of finite type with affinediagonal over a quasi-separated and locally noetherian algebraic space S , and let y ∈ | Y | be a closed point. Let f : ( X , x ) → ( Y , y ) be a pointed ´etale and affinemorphism such that there exists an adequate moduli space π : X → X and f induces an isomorphism f | f − ( G y ) over the residual gerbe at y (e.g. f is a localquotient presentation).(1) If Y is Θ -reductive, then there exists an affine open subspace U ⊂ X of π ( x ) such that f | π − ( U ) is Θ -surjective.(2) If Y has unpunctured inertia, then there exists an affine open subspace U ⊂ X of π ( x ) such that f | π − ( U ) which induces an isomorphism I π − ( U ) → π − ( U ) × Y I Y .In particular, if Y is locally linearly reductive, is Θ -reductive and has unpuncturedinertia, then there exists a local quotient presentation g : W → Y around y whichis Θ -surjective and induces an isomorphism I W → W × Y I Y .Proof. For (1) let us first show that the morphism ev( f ) : Map S (Θ , X ) → X × Y Map S (Θ , Y ) is an open and closed embedding. As f is representable, ´etale andseparated, the map is an open embedding by Lemma 3.7. As X admits a goodmoduli space, it is Θ-reductive by Proposition 3.20, as is Y by assumption. Thusev( f ) is proper and in particular closed. Let Z ⊂ X × Y Map S (Θ , Y ) be the openand closed complement of Map S (Θ , X ). By Equation (3.5), the image p ( Z ) ⊂ | X | consists of the points where f is not Θ-surjective. By Proposition 3.30, the image p ( Z ) ⊂ | X | is constructible. On the other hand, since Y is Θ-reductive, theimage p ( Z ) is closed under specializations. Alternatively, one could invoke [Hal14, Lem. 4.36], which implies unconditionally that theimage in | X | of any open and closed substack of X × Y Map S (Θ , Y ) is constructible. It is here where the Θ-reductivity hypothesis on Y is used in an essential way. Note thatthe implication that p ( Z ) is closed would follow from the weaker condition of uniqueness oflifts in the valuative criterion (3.3) for DVR’s R essentially of finite type over S . This justifiesRemark 4.2. onsider an arbitrary diagram of solid arrows:Spec( k ) x / / i (cid:15) (cid:15) X f (cid:15) (cid:15) Θ k λ / / ; ; ①①①①① Y . As y = f ( x ) ∈ | Y | is a closed point, λ factors through the residual gerbe G y of y . The induced map G x → G y on residual gerbes is an isomorphism so λ liftsto a morphism Θ k → G x → X filling in the dotted arrow. It follows that f isΘ-surjective at x , i.e. x / ∈ p ( Z ).Let U ⊂ X \ π ( p ( Z )) be an open affine neighborhood of π ( x ), and let U = π − ( U ). We claim that f | U : U → Y is Θ-surjective. First, observe that theinclusion ι : U ֒ → X is a Θ-isomorphism (i.e. ev( ι ) is an isomorphism); seeExample 3.31. The composition U ֒ → X → Y induces a commutative diagramMap S (Θ , U ) ev( ι ) / / ev( f ◦ ι ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ U × X Map S (Θ , X ) / / (cid:15) (cid:15) Map S (Θ , X ) ev( f ) (cid:15) (cid:15) U × Y Map S (Θ , Y ) / / (cid:15) (cid:15) X × Y Map S (Θ , Y ) / / (cid:15) (cid:15) Map S (Θ , Y ) (cid:15) (cid:15) U / / X / / Y where all squares are Cartesian. The substack U was chosen precisely such that U × X Map S (Θ , X ) → U × Y Map S (Θ , Y ) is an isomorphism. It follows that ev( f ◦ ι ) is an isomorphism.For (2), it suffices to find an open neighborhood U ⊂ X of x such that f | U : U → Y induces an isomorphism I U → U × Y I Y . We have a Cartesian diagram I X / / (cid:15) (cid:15) X × Y I Y (cid:15) (cid:15) X / / X × Y X . Since f is ´etale and affine, the morphism I X → X × Y I Y is an open and closedimmersion; let Z ⊂ X × Y I Y be the open and closed complement, so the fiber overa point p : Spec( k ) → X consists of the complement of the subgroup Aut X ( p ) ⊂ Aut Y ( f ◦ p ) . Denote p : X × Y I Y → X . We know that x / ∈ p ( Z ) as f is stabilizerpreserving at x . Moreover, if we choose a versal deformation ( U, u ) → ( Y , y ) where U is the spectrum of a local ring, then using that Y has unpunctured inertia, weknow that the preimage of Z in X × Y I Y × Y U is empty; indeed, if there were anon-empty connected component of this preimage, it must intersect the fiber over u non-trivially contradicting that x / ∈ p ( Z ). This in turn implies that x / ∈ p ( Z ).Therefore, if we set U = X \ p ( Z ), the induced morphism I U → U × Y I Y is anisomorphism. (cid:3) Lemma 4.5.
Let X be a locally noetherian algebraic stack with affine diagonal.Suppose that { U i } i ∈ I is a Zariski-cover of X such that each U i admits a goodmoduli space and each inclusion U i ֒ → X is Θ -surjective. Then X admits a goodmoduli space. roof. Let π i : U i → U i denote the good moduli space. Since each inclusion U i ∩ U j ֒ → U i is Θ-surjective, there exist open subspaces U i,j ⊂ U i with π − i ( U i,j ) = U i ∩ U j (see Example 3.31). By universality of good moduli spaces [Alp13,Thm. 6.6], there are isomorphisms U i,j ∼ → U j,i providing gluing data for analgebraic space U . The morphisms π i glue to produce a good moduli space U → U . (cid:3) Using Proposition 4.4 and Lemma 4.5, we can now establish Theorem 4.1.
Proof of Theorem 4.1.
For the sufficiency of these three conditions, we follow theproof of [AFS17, Thm. 2.1]. First observe that by Lemma 4.5 and Proposition 4.4(1),it suffices to show that every closed point x ∈ | X | has an open neighborhood U admitting a good moduli space. By Proposition 4.4, there exists a local quo-tient presentation f : X = [Spec( A ) / GL N ] → X around x such that f is Θ-surjective and f induces an isomorphism I X → X × X I X . After replacing X with f ( X ), we may assume that f is surjective. Since I X → X × X I X is anisomorphism, every closed point of [Spec( A ) / GL n ] has linearly reductive stabi-lizer. It follows from [AHR] that [Spec( A ) / GL N ] is cohomologically affine. Welet π : X → X := Spec( A GL N ) be the induced good moduli space.Set X = X × X X . The projections p , p : X → X are also ´etale, affine,surjective, and Θ-surjective morphisms that induce isomorphisms I X → X × X I X . Since f is affine, X is cohomologically affine and admits a good moduli space π : X → X . By Proposition 4.3, both commutative squares in the diagram X p / / p / / π (cid:15) (cid:15) X f / / π (cid:15) (cid:15) X X q / / q / / X are Cartesian. Moreover, by the universality of good moduli spaces, the ´etalegroupoid structure on X ⇒ X induces a ´etale groupoid structure on X ⇒ X .The fact that f induces isomorphisms of stabilizer groups implies that ∆ : X → X × X is a monomorphism (see the argument of [AFS17, Prop. 3.1]). Thus, X ⇒ X is an ´etale equivalence relation and there exists an algebraic spacequotient X . It follows from descent that there is an induced morphism π : X → X which is a good moduli space.Conversely suppose that X admits a good moduli space π : X → X . Thenthe closed points of X have linearly reductive stabilizers. If x ∈ | X | is a pointand U ⊂ X is a quasi-compact open containing π ( x ), then x specializes to aclosed point x ∈ | π − ( U ) | which is necessarily also closed in X . As x has lin-early reductive stabilizer, we see that X is locally linearly reductive. Moreover,Proposition 3.20(3) implies that X is Θ-reductive. Establishing that X has un-punctured inertia will take more effort, and we will prove this in Theorem 5.2. (cid:3) We note another consequence of Proposition 4.4, which will be used in § Proposition 4.6.
Let X be an algebraic stack which is of finite type with affinediagonal over a field k . Suppose that X is Θ -reductive and there exists a singleclosed point x ∈ | X | which has a linearly reductive stabilizer. Then X admits a ood moduli space. If k is algebraically closed, then X ∼ = [Spec( A ) /G x ] , and if inaddition X is reduced, then X → Spec( k ) is the good moduli space.Proof. Choose a local quotient presentation f : ( X , x ) → ( X , x ) with X =[Spec( B ) / GL n ] such that x ∈ | X | is the unique point mapping to x . Since X is Θ-reductive, by Proposition 4.4(1), we can assume that f is Θ-surjective.This implies that f sends closed points to closed points and both projections X = X × X X ⇒ X send closed points to closed points. Since both X and X have a unique closed point and f induces an isomorphism of residual gerbes G x → G x , it follows that X has a unique closed point and that both projections X ⇒ X induce isomorphism of stabilizers at this point. Moreover, there aregood moduli spaces X → X and X → X . As in the proof of Theorem 4.1,Proposition 4.3 implies that the induced groupoid X ⇒ X is an ´etale equiva-lence relation, and the quotient X /X is a good moduli space for X . The finalstatement follows from [AHR, Thm. 2.9] coupled with the observation that if X is reduced, so is its good moduli space. (cid:3) Criteria for unpunctured inertia
In this section we establish criteria which imply that a stack has unpuncturedinertia. They are “valuative criteria” in the sense that they apply to families overdiscrete valuation rings. We will need the following notion:
Definition 5.1.
Let X be an algebraic stack over an algebraic space S , let R bea DVR over S with fraction field K and residue field κ , and let ξ : Spec( R ) → X be a morphism. A nearby modification of ξ is a morphism ξ ′ : Spec( R ) → X along with an isomorphism ξ ′ | K ≃ ξ | K such that the closures of ξ ′ (0) and ξ (0) in | X × S Spec( κ ) | have nonempty intersection.This notion is stronger than that of a modification, and weaker than thatof an elementary modification. Note however, that in an S -complete stack, thenotions of “modification,” “nearby modification,” and “elementary modification”coincide. We can now state the main results of this section: Theorem 5.2.
Let X be an algebraic stack locally of finite type and with affinediagonal over a quasi-separated and locally noetherian algebraic space S . If X hasa good moduli space, then X has unpunctured inertia. Moreover, if X is locallylinearly reductive and Θ -reductive, then the following conditions are equivalent:(1) For any essentially finite type DVR R , any morphism ξ : Spec( R ) → X ,and any g ∈ Aut X ( ξ K ) of finite order, there is an extension of DVR’s R ′ /R with fraction field K ′ and a nearby modification ξ ′ of ξ | R ′ such that g | K ′ extends to an automorphism of ξ ′ ;(2) For any essentially finite type DVR R , any morphism ξ : Spec( R ) → X ,and any geometrically connected component H ⊂ Aut X ( ξ K ) , there is anextension of DVR’s R ′ /R with fraction field K ′ and a nearby modification ξ ′ of ξ | R ′ and some g ∈ Aut X ( ξ ′ ) such that g | K ′ lies in H ;(3) X has unpunctured inertia; and(4) X has a good moduli space. This elaborates on Theorem 4.1, which stated the equivalence of (3) and (4).Recall that only the implication of (3) ⇒ (4) was shown in the proof of Theorem 4.1,which quoted Theorem 5.2 for the implication (4) ⇒ (3). xample 5.3. To illustrate the subtlety of the condition (1), let us exhibit in thecontext of Example 3.57 a map from a DVR to [ A /G ] (where G = G m ⋊ ( Z / R = k [[ z ]] and K = k (( z )). Consider ξ : Spec( R ) → A via z ( z , z ). Then g = ( z, − ∈ G ( K ) stabilizes ξ K but does not extendto G ( R ). Consider the degree 2 ramified extension R → R ′ with R ′ = k [[ √ z ]]and K ′ = k (( √ z )), and define ξ ′ : Spec( R ′ ) → X by √ z (( √ z ) , ( √ z ) ). Overthe generic point, ξ ′ is isomorphic as a point in [ A /G ] to the restriction ξ K ′ ,because ( √ z, − · ξ ′ K ′ = ξ K ′ . Under this isomorphism our generic automorphism g becomes g ′ = ( √ z, − − · g | K ′ · ( √ z, −
1) = (1 , −
1) which clearly extends to an R ′ -point.Our second main result states that when X is S -complete, which is the situationof most interest in applications, these conditions hold automatically. Theorem 5.4. If X is an algebraic stack which is locally linearly reductive and S -complete, then the conditions (1), (2), and (3) of Theorem 5.2 hold automatically. Recall from Proposition 3.45 that if a noetherian algebraic stack X with affinediagonal is S-complete and defined over Q , then X is locally linearly reductive. Itfollows that Theorem 5.4 and Theorem 4.1 imply the second part of Theorem A. Remark 5.5.
As the proof will show, Theorem 5.4 holds more generally if X isonly assumed to be S-complete with respect to DVRs essentially of finite typeover R . Combining this observation with Remark 3.46 and Remark 4.2, we infact obtain the following stronger version of the second part of Theorem A: if X isan algebraic stack of finite type with affine diagonal over a noetherian algebraicspace S of characteristic 0, then X admits a separated good moduli space if andonly if X is Θ-reductive and S-complete with respect to DVRs essentially of finitetype over R .We prove Theorem 5.4 and Theorem 5.2 below, after establishing some prelim-inary results.5.1. A variant of the valuative criteria.
It will be convenient to introduce avariant of the valuative criterion (1) in Theorem 5.2 for an algebraic stack X :(1 ′ ) For any DVR R with fraction field K , any morphism ξ : Spec( R ) → X ,and any generic automorphism g ∈ Aut X ( ξ K ) of finite order, there is anextension of DVR’s R ′ /R with fraction field K ′ and a modification ξ ′ of ξ | R ′ such that g | K ′ extends to an automorphism of ξ ′ .Unlike in the valuative criterion (1), criterion (1 ′ ) does not require that the mod-ification ξ ′ is a nearby modification. Criterion (1 ′ ) is not a sufficient conditionfor X to have unpunctured inertia even when X is locally linearly reductive andΘ-reductive, but it has useful formal properties. Note in particular that in an S -complete stack, any modification is an elementary modification and in par-ticular a nearby modification so condition (1 ′ ) is equivalent to condition (1) ofTheorem 5.2. Remark 5.6.
In the case that X = [ X/G ] is a noetherian quotient stack definedover a field k , X satisfies (1 ′ ) if and only if for every map ξ : Spec( R ) → X and g ∈ G ξ K ⊂ G ( K ) of finite order, there exists after an extension R ⊂ R ′ (with K ′ = Frac( R ′ )) an element h ∈ G ( K ′ ) such that h · ξ K ′ and h − g | K ′ h both extendto R ′ -points. Even in the case where X = V is a linear representation of a linearly eductive group G , we are not aware of a completely elementary proof of this fact,despite the purely representation-theoretic nature of this property. This is themost challenging part of the proof of Theorem 5.2. Proposition 5.7.
Let G be a geometrically reductive and ´etale-locally embeddablegroup scheme (e.g. reductive) over an algebraic space S and let W → S be anaffine morphism of finite type with an action of G . Then [ W/G ] satisfies thecriterion (1 ′ ), and in particular [ W/G ] satisfies condition (1) of Theorem 5.2 if S is separated. We will prove Proposition 5.7 at the end of this subsection, after establishingsome preliminary results.
Lemma 5.8.
The stack B Z GL N satisfies the condition (1 ′ ).Proof. In this case, because every vector bundle on Spec( R ) is trivializable, thecondition is equivalent to the claim that every element g ∈ GL N ( K ) is conjugateto an element of GL N ( R ) after passing to an extension of the DVR R . After anextension of R we may conjugate g to its Jordan canonical form. The fact that g has finite order implies that the diagonal entries of the resulting matrix are rootsof unity. Because the group of k th roots of unity is a finite group scheme, theentries of the Jordan canonical form must lie in R . (cid:3) Lemma 5.9.
Let p : X → Y be a proper representable morphism of noetherianstacks. If Y satisfies the valuative criterion (1 ′ ), then so does X .Proof. Since p is representable and separated, for any morphism ξ : Spec( R ) → X from a DVR, we have a closed immersion Aut X ( ξ ) ֒ → Aut Y ( p ◦ ξ ) of group schemesover Spec( R ). Furthermore, because p is proper, any modification of p ◦ ξ liftsuniquely to a modification of ξ . Therefore, given a generic automorphism of ξ , we may pass to an extension R ′ /R and modify p ◦ ξ | R ′ so that this genericautomorphism extends, and then this lifts uniquely to a modification of ξ | R ′ suchthat the given generic automorphism extends. (cid:3) Proof of Proposition 5.7.
It suffices to show that [Spec( A ) /G ] satisfies the crite-rion (1 ′ ), where G and Spec( A ) are defined over a DVR R and with A finitelygenerated over R . After passing to a finite extension of K = Frac( R ) we mayassume that G embeds as a closed subgroup G ֒ → GL N,R for some N . We maythen replace G with GL N,R and replace Spec( A ) with GL N,R × G Spec( A ), whichwill again be affine because G is geometrically reductive. Furthermore we canassume that A is reduced, because we are only considering maps from reducedschemes. So it suffices to prove the claim for [Spec( A ) / GL N,R ] for a reduced R -algebra A of finite type.Now consider a GL N -scheme X which is reduced and projective over Spec( A GL N )such that X contains Spec( A ) as a dense GL N -equivariant open subscheme andthe complement X \ Spec( A ) is the support of an ample GL N -invariant Cartierdivisor E . The construction in [Tel00, Lem. 6.1] for smooth schemes in character-istic 0 works here as well: We simply choose a closed G -equivariant embeddingSpec( A ) ֒ → A A GL N ( E ) for some locally free GL n -module over A GL N , and then let be the closure of Spec( A ) in P A GL N ( E ⊕ O ). Thus we have a diagram:Spec( A ) (cid:31) (cid:127) / / & & ◆◆◆◆◆◆◆◆◆◆ X (cid:15) (cid:15) Spec( A GL N )We claim that Spec( A ) is precisely the semistable locus of X with respect to O X ( E ) in the sense of [Ses77]. Indeed the tautological invariant section s : O X → O X ( E ) which restricts to an isomorphism over Spec( A ) shows that Spec( A ) ⊂ X ss .Conversely s n gives an isomorphism of A GL N -modules Γ(Spec( A ) , O X ( nE )) GL N ≃ A GL N for all n >
0. Under this isomorphism any invariant global section f ∈ Γ( X, O X ( nE )) GL N , after restriction to the dense open subset Spec( A ), agreeswith a section of the form gs n , where g is the pullback of a function under themap X → Spec( A GL N ). It follows that f = g · s n because X is reduced. Thisshows that X ss ⊂ Spec( A ).Now Lemma 5.8 implies that the criterion (1 ′ ) holds for Spec( A GL N ) × B GL N ,and hence Lemma 5.9 implies that the criterion holds for [ X/ GL N ]. So in or-der to establish the criterion for [Spec( A ) / GL N ], it suffices to show that givena point ξ : Spec( R ) → [ X/ GL N ] along with a finite order automorphism g of ξ ,if ξ K lies in the open substack [Spec( A ) / GL N ], then after passing to an exten-sion of R one can modify the pair ( ξ, g ) at the special point of Spec( R ) so thatthe image of ξ lies in [Spec( A ) / GL N ]. Note that the stabilizer group X -schemeStab GL N ( X ) ⊂ X × GL N is equivariant for the GL N action which acts by the givenaction on X and by conjugation on the GL N factor. It suffices to show that givenan R -point of Stab GL N ( X ) whose generic point lies in Spec( A ) × GL N , after pass-ing an extension of R there is a modification of the resulting map ξ : Spec( R ) → [Stab GL N ( X ) / GL N ] whose image lies in [Stab GL N (Spec( A )) / GL N ] = [Stab GL N ( X ) ∩ (Spec( A ) × GL N ) / GL N ].Note that Stab GL N ( X ) is projective over Spec( A GL N ) × GL N . We claim thatthe semistable locus of Stab GL N ( X ) for the action of G with respect to the pull-back of O X ( E ) is precisely Stab GL N (Spec( A )). Indeed, this follows from theHilbert–Mumford criterion [Ses77]. Any destabilizing one parameter subgroupfor ( x, g ) ∈ Stab GL N ( X ) is destabilizing for x ∈ X . Conversely, for every point( x, g ) ∈ Stab GL N ( X ) whose underlying point x ∈ X is unstable, Kempf’s theoremon the existence of canonical destabilizing flags [Kem78] implies that there is adestabilizing one parameter subgroup λ for x which commutes with Stab GL N ( x ),and this λ defines a destabilizing one parameter subgroup for the point ( x, g ). Sothe fact that any map Spec( R ) → [Stab GL N ( X ) /G ] whose generic point lies in[Stab GL N (Spec( A )) /G ] admits a modification which lies in [Stab GL N (Spec( A )) /G ]after passing to an extension of R follows from the classical semistable reductiontheorem in the setting of reductive group schemes [Ses77]. (cid:3) This is not stated in this way in Seshadri’s paper, but it follows from the results there: If X is projective over a finite type affine G -scheme and L is a G -ample bundle, then [ X ss /G ]admits an adequate moduli space Y which is projective over Spec(Γ( X, O X ) G ). So given a mapSpec( R ) → [ X/G ] whose generic point lands in [ X ss /G ], one can compose with the projectionto get a map Spec( R ) → [ X/G ] → Spec(Γ( X, O X ) G ). By construction one has a lift of thegeneric point along both maps [ X ss /G ] → Y → Spec(Γ( X, O X ) G ). So because both maps areuniversally closed, one can lift this to a map Spec( R ) → [ X ss /G ] after an extension of R . emark 5.10. By appealing to Theorem A.8, the proof in fact shows that aslightly stronger version of (1 ′ ) holds in which the extension K ′ /K of fractionfields is finite . It follows that the statements of Theorem 5.2 and Theorem 5.4remain true after replacing (1) and (2) with the stronger condition where theextension K ′ /K is required to be finite.5.2. The proof of Theorem 5.2.Lemma 5.11. If X is a noetherian algebraic stack with affine automorphismgroups, then the valuative criterion (1) implies the valuative criterion (2) inTheorem 5.2.Proof. It suffices to show that every connected component of the K -group Aut X ( ξ K )contains a finite type point of finite order. Let g ∈ Aut X ( ξ K ) be a finite typepoint. After a finite field extension we can decompose g = g s g u under the Jordandecomposition, where g s is semisimple and g u is unipotent. Now consider thereduced Zariski closed K -subgroup H ⊂ Aut X ( ξ K ) generated by g s . Because g s is semisimple, H is a diagonalizable K -group and hence every component of H contains an element of finite order. We may thus replace g s with a finite order el-ement in the same connected component of Aut X ( ξ K ) which still commutes with g u . If char( K ) >
0, then g u has finite order and we are finished. If char( K ) = 0,then g u lies in the identity component of G , so g lies on the same component asthe finite order element g s . (cid:3) Lemma 5.12.
Suppose X is a noetherian algebraic stack with affine stabilizergroups, and suppose that the criterion (2) of Theorem 5.2 holds in such a way thatone may always choose the nearby modification ξ ′ so that ξ ′ (0) is a specializationof ξ (0) in | X | . Then X has unpunctured inertia.Proof. Let x ∈ | X | be a closed point, and let p : ( U, u ) → ( X , x ) be a versaldeformation of x , and let H ⊂ Aut X ( p ) be a connected component. The image ofthe projection H → U is a constructible set whose closure contains u . It followsthat we can find an essentially finite type DVR R and a map Spec( R ) → U whosespecial point maps to u and whose generic point lies in the image of H → U . Afteran extension of the DVR R , we may assume that the generic point Spec( K ) → U lifts to H , and that the connected component H ′ ⊂ H | Spec( K ) containing thislift is geometrically connected. The hypotheses of the lemma imply that, afterpossibly further extending R , there exists a modification ξ ′ : Spec( R ) → X of ξ such that the closure of H ′ in Aut X ( ξ ) meets the fiber over 0 ∈ Spec( R ) and0 ∈ Spec( R ) still maps to u . By construction H ′ maps to H , which implies that H ⊂ Aut X ( p ) meets the fiber over u . (cid:3) Remark 5.13.
The valuative criterion (2) does not imply the valuative criterion(1) without additional hypotheses. Consider the group G m ⋉G a given coordinates( z, y ) and the product rule ( z , y ) · ( z , y ) = ( z z , z y + y ), and let G ⊂ ( G m ⋉ G a ) × A t be the hypersurface cut out by the equation ty = 1 − z . Then G is in fact a smooth subgroup scheme over A whose fiber over 0 is G a and whosefiber everywhere else is G m .Let X = B A G and consider the map ξ : Spec( k [[ t ]]) → X which is just thecompletion of the canonical map A t → X at the origin. Then all modificationsof ξ agree after composing with the projection X → A t , so after an extensionof DVR’s the automorphism group of ξ will be isomorphic to G k [[ t ]] . There isa generic automorphism of ξ given by the formula ( α, (1 − α ) /t ), where α is a on-identity n th root of unity. This automorphism does not extend to 0, andthe generic automorphism group is abelian and hence acts trivially on itself byconjugation. It follows that no extension and modification of ξ will allow thisgeneric automorphism to extend either. Lemma 5.14.
Let X be an algebraic stack locally of finite type with affine diagonalover a quasi-separated and locally noetherian algebraic space S . Suppose that X is Θ -reductive, and that either (1) X is locally linearly reductive or (2) X ∼ =[Spec( A ) / GL N ] for some N . Let R be a DVR and let ξ : Spec( R ) → X be amorphism. If ξ ′ is a nearby modification of ξ and g ∈ Aut X ( ξ ′ ) , then thereis a finite extension of DVR’s R ′ /R with fraction field K ′ and a modification ξ ′′ : Spec( R ′ ) → X of ξ such that g | K ′ extends to an automorphism of ξ ′′ and ξ ′′ (0) is a specialization of ξ (0) .Proof. Let us first reduce the case (1) to the case (2): Let κ be the residue field of R and let Z ⊂ X κ = X × S Spec( κ ) be the closure of the point p := ξ ′ (0) ∈ X κ . ByLemma 3.24 we know that Z has a unique closed point z ∈ | Z | , and in particular z is a specialization of both p and ξ (0) because ξ ′ is a nearby modification of ξ . Ifnecessary we pass to a finite extension of R so that we may assume that z ∈ Z ( κ )as well. Under the hypothesis (1), Proposition 4.6 implies that Z ≃ [Spec( A ) /G z ]for some affine G z -scheme Spec( A ). Embedding G z ⊂ GL N,κ for some N , we mayreplace G z with GL N and Spec( A ) with the affine scheme GL N × G z Spec( A ). Itsuffices to prove the claim for the stack Z , so for the remainder of the proof weassume we are in the case (2).Kempf’s theorem [Kem78] implies that after passing to a finite purely insepara-ble extension of κ , which can be induced by a suitable finite extension of DVR’s,there is a canonical filtration f : Θ κ → [Spec( A ) / GL N ] with an isomorphism f (1) ≃ p such that f (0) = z . The fact that f is canonical implies that any auto-morphism of p = f (1) extends to an automorphism of the map f . In particularthe restriction of g ∈ Aut X ( ξ ) to p = ξ ′ (0) extends uniquely to an automorphismof f which we also denote g .We now apply the strange gluing lemma (Corollary A.2), which states thatafter composing f with a suitable ramified cover ( − ) n : Θ κ → Θ κ , the data of themap ξ ′ : Spec( R ) → X and the filtration f : Θ κ → X , comes from a unique map γ : ST R → X , where f is the restriction of γ to the locus { s = 0 } and ξ ′ is therestriction of γ to the locus { t = 0 } . The uniqueness of this extension guaranteesthat the automorphism g of ξ ′ and f extends uniquely to an automorphism of γ , which we again denote g . Finally we construct our modification ξ ′′ as thecomposition ξ ′′ : Spec( R [ √ π ]) → ST R γ −→ X , where the first map is given in ( s, t, π ) coordinates by ( √ π, √ π, π ), which mapsthe special point of Spec( R [ √ π ]) to the point { s = t = π = 0 } of ST R . By con-struction the automorphism g of γ restricts to an automorphism of ξ ′′ extending g | K [ √ π ] , and the special point ξ ′′ (0) maps to the closed point z of Z , which is aspecialization of ξ (0). (cid:3) Proof of Theorem 5.2.
Lemma 5.11 shows that (1) ⇒ (2). Lemma 5.12 combinedwith Lemma 5.14 shows that under the locally linearly reductive and Θ-reductivehypotheses, (2) ⇒ (3). We have seen in Theorem 4.1 that under the locallylinearly reductive and Θ-reductive hypotheses (3) ⇒ (4), so what remains is toshow that (4) ⇒ (1). uppose that X → X is a good moduli space, and let ξ : Spec( R ) → X be amorphism, and let g be an automorphism of ξ K of finite order. Then we maychoose an ´etale map U → X whose image contains the image of Spec( R ) andsuch that U := X × X U ≃ [Spec( A ) /G ] for a reductive group G [AHR]. Afterreplacing R with an extension of DVR’s we may assume that ξ ′ lifts to a map ξ ′ : Spec( R ′ ) → U . Furthermore the map U → X is inertia preserving in thesense that I U ≃ I X × X U , which implies that g lifts to a finite order genericautomorphism g ′ of ξ ′ K ′ . By Proposition 5.7 the stack U satisfies condition (1)of Theorem 5.2. This provides a nearby modification of the map ξ ′ for which g ′ extends, and we can compose this with the map U → X to get a nearbymodification of the original map for which g | K ′ extends. (cid:3) The proof of Theorem 5.4.
Let X be a noetherian algebraic stack withaffine stabilizers, let p : Y → X be an ´etale map with Y ∼ = [Spec( A ) /G ], and let R be a complete DVR with fraction field K and residue field κ . Let x ∈ | X | be aclosed point such that p induces an isomorphism p − ( G x ) ≃ G x , where G x denotesthe residual gerbe of x . Lemma 5.15.
The functor
Map(Spec( R ) , Y ) → Map(Spec( R ) , X ) defined by com-position with p induces an equivalence between the full subgroupoids of maps takingthe special point of Spec( R ) to p − ( x ) and x respectively. The same is true forthe functor Map(ST R , Y ) → Map(ST R , X ) and the subgroupoid taking the point ∈ ST R ( κ ) to p − ( x ) and x respectively.Proof. The map p is ´etale and induces an equivalence between the residual gerbeof x ∈ | X | and p − ( x ) ∈ | Y | . It therefore induces an equivalence between the n th order neighborhoods of these residual gerbes, so any map Spec( R ) → X mapping0 to x lifts uniquely along p over any nilpotent thickening of 0 ∈ Spec( R ). Theresult then follows from Tannaka duality and the fact that Spec( R ) is coherentlycomplete along its special point, so a compatible family of lifts over nilpotentthickenings of 0 corresponds to a unique lift of the map Spec( R ) → X along p . The same argument applies to ST R , which is coherently complete along theinclusion ( B G m ) κ ֒ → ST R at the point 0. (cid:3) Now let ξ ′ : Spec( R ) → Y be an R -point mapping the special point to p − ( x ),and let ξ = p ◦ ξ ′ . Lemma 5.16. If X is S -complete, then p induces an isomorphism Aut Y ( ξ ′ K ) → Aut X ( ξ K ) .Proof. For any map f : ST R \ → X , let f and f denote the two R -pointsresulting from f . For any stack X and ξ ∈ X ( R ), we have an equivalence ofgroupoids: { automorphisms of ξ K } ≃ { maps ST R \ → X + equivalences f ≃ ξ ≃ f } Because both Y and X are S -complete, restriction gives an equivalence ofgroupoids Map(ST R , − ) → Map(ST R \ , − ) for both stacks. It follows that(5.1) Aut X ( ξ K ) ≃ { maps ST R → X + equivalences f ≃ ξ ≃ f } and likewise for Y . If ξ maps the special point of Spec( R ) to x ∈ X , then anymap f : ST R → X which admits an isomorphism f ≃ ξ must also map (0 ,
0) to x ,because x is closed. Now the previous lemma implies that p induces a bijection f sets on the right hand side of (5.1) for X and Y , and thus also on the left handside. (cid:3) Remark 5.17.
Note that the conclusion of this lemma also applies withoutassuming that Y = [Spec( A ) /G ]—it suffices to assume Y is S -complete, or thatthe map p is affine (which implies that Y is S-complete). Proof of Theorem 5.4.
We first verify criterion (1) of Theorem 5.2. Consider amap from a DVR ξ : Spec( R ) → X . Let x ∈ | X | be a closed point in the closureof ξ (0), and let p : [Spec( A ) / GL N ] → X be a local quotient presentation around x (see Definition 2.2). Then after an extension of DVR’s R ′ /R we may lift ξ to amap ξ ′ : Spec( R ′ ) → [Spec( A ) / GL N ]. Now Lemma 5.14 allows one to constructa modification of ξ ′ , after replacing R ′ with a further finite extension, which mapsthe special point to the closed point p − ( x ) ∈ [Spec( A ) / GL N ]. It follows fromLemma 5.16 that the mapAut [Spec( A ) / GL N ] ( ξ ′ K ′ ) → Aut X ( p ◦ ξ ′ K ′ )is an isomorphism of K ′ -groups. In particular given a finite order element g ∈ Aut X ( ξ K ), one may lift this to ξ ′ after replacing R ′ with a further extension.We know that the criterion (1) of Theorem 5.2 holds for [Spec( A ) / GL N ] byProposition 5.7, and after replacing R ′ with a further extension this producesa nearby modification for which g | K ′ extends. Composing with p gives a nearbymodification of the original map ξ for which g extends. The same argument showsthat X satisfies the criterion (2).Finally, the previous paragraph shows that in verifying the criterion (2), wecould choose the modification ξ ′ of ξ in such a way that ξ ′ (0) is a specializationof ξ (0). It follows from Lemma 5.12 that X has unpunctured inertia. (cid:3) Semistable reduction and Θ -stability In this section we explain how completeness properties of stacks induce similarproperties of the substack of semistable objects, if these are defined using thetheory of Θ-stability. Our key result is Theorem 6.3 that is inspired by Langton’salgorithm for semistable reduction for families of torsion-free sheaves on a projec-tive variety. Recall from Remark 3.36 that this algorithm starts with a family ofbundles parametrized by a DVR R such that the generic fiber is semistable andthe special fiber is unstable, and then applies elementary modifications to arriveat a semistable family. Surprisingly, it turns out that his construction admits ananalog that relies only on the geometry of the algebraic stack representing themoduli problem, not on the particular type of objects classified by the moduliproblem. The structure we will need is that of a Θ-stratification from [Hal14,Def. 2.1] that formalizes the notion of canonical filtrations in geometric terms. Definition 6.1.
Let X be an algebraic stack locally of finite type over a noether-ian algebraic space S .(1) A Θ -stratum in X consists of a union of connected components S ⊂ Map S (Θ , X ) such that ev : S → X is a closed immersion.(2) A Θ -stratification of X indexed by a totally ordered set Γ is a cover of X by open substacks X ≤ c for c ∈ Γ such that X ≤ c ⊂ X ≤ c ′ for c < c ′ , alongwith a Θ-stratum S c ⊂ Map S (Θ , X ≤ c ) in each X ≤ c whose complement is S c ′ It will be convenient for us to identify a Θ-stratum S with the closedsubstack it defines on X , i.e., we will sometimes say that a closed substack S ⊂ X is a Θ-stratum, if there exist a union of connected components S ′ ⊂ Map S (Θ , X )such that ev : S ′ → S ⊂ X is an isomorphism.Our notation differs slightly from [Hal14], which denotes the stack Map S (Θ , X )by Filt( X ) to promote the analogy of maps Θ k → X as filtered objects in X . Inaddition Map S ( B G m , X ) is denoted Grad( X ) in order to promote the analogy ofmaps B k G m → X as graded objects in X . Given a Θ-stratification, we denotethe open substack X ss := X ≤ as the semistable locus. For any unstable point x ∈ X ( k ) \ X ss ( k ), the Θ-stratification determines a canonical filtration f : Θ k → X with f (1) ≃ x , which we refer to as the HN filtration .Restricting a map f : Θ → X to B G m ֒ → Θ defines a map ev : Map S (Θ , X ) → Map S ( B G m , X ) which corresponds to “passing to the associated graded object”of the filtration f . Composition with the projection Θ → B G m defines a sec-tion σ : Map S ( B G m , X ) → Map S (Θ , X ) of the map ev which corresponds to the“canonical filtration of a graded object.” These maps define a canonical A de-formation retract of Map S (Θ , X ) onto Map S ( B G m , X ), and in particular inducebijections on connected components [Hal14, Lem. 1.24]. We refer to the union ofconnected components Z ⊂ Map S ( B G m , X ) corresponding to S as the center ofthe Θ-stratum S . The result is a diagram Z (cid:31) (cid:127) σ / / S ev h h (cid:31) (cid:127) ev / / X . The semistable reduction theorem.Theorem 6.3 (Langton’s algorithm) . Let X be an algebraic stack locally of finitetype with affine diagonal over a noetherian algebraic space S , and let S ֒ → X bea Θ -stratum. Let R be a DVR with fraction field K and residue field κ . Let ξ R : Spec( R ) → X be an R -point such that the generic point ξ K is not mapped to S , but the special point ξ k is mapped to S : Spec( K ) (cid:31) (cid:127) / / ξ K (cid:15) (cid:15) Spec( R ) ξ R (cid:15) (cid:15) Spec( κ ) ? _ o o ξ κ (cid:15) (cid:15) X − S (cid:31) (cid:127) j / / X S . ? _ ι o o Then there exists an extension R → R ′ of DVRs with K → K ′ = Frac( R ′ ) finite and an elementary modification ξ ′ R ′ of ξ R ′ such that ξ ′ R ′ : Spec( R ′ ) → X lands in X − S . Remark 6.4. In the proof of the above result we will apply the non-local slicetheorem (Theorem 2.6) for algebraic stacks. As the proof of this result has notappeared, we give an alternative argument using [AHR15, Thm. 1.2], which re-quires the additional hypothesis that S is the spectrum of an algebraically closedfield and that for any x ∈ X ( k ), the automorphism group G x is smooth – thissuffices, in particular, for stacks over a field of characteristic 0. his theorem is stated for a single stratum, but it immediately implies a versionfor a stack with a Θ-stratification: Theorem 6.5 (Semistable reduction) . Let X be an algebraic stack, locally of finitetype with affine diagonal over a noetherian algebraic space S , with a well-ordered Θ -stratification. Then for any morphism Spec( R ) → X , after an extension R → R ′ of DVRs with K → K ′ = Frac( R ′ ) finite there is a modification Spec( R ′ ) → X ,obtained by a finite sequence of elementary modifications, whose image lies in asingle stratum of X .Proof. Beginning with a map ξ R : Spec( R ) → X such that ξ K ∈ S c and ξ κ ∈ S c for c > c , we may apply Theorem 6.3 iteratively to obtain a sequence of finiteextensions of R and elementary modifications of ξ with special point in S c i for c > c > · · · . Each S c i meets ξ K , so the well-orderness condition guaranteesthat this procedure terminates, and it can only terminate when c i = c . (cid:3) Remark 6.6. In the relative situation when X is defined over a base algebraicstack S , one can base change the structure of a Θ-stratification along a smoothmap S ′ → S , so both Theorem 6.3 and Theorem 6.5 extend immediately to thecase of a quasi-separated and locally noetherian base stack S .6.1.1. Langton’s algorithm in the basic situation. The main idea of the proof isto reduce to the situation where X = [Spec( A ) / G m ] is the quotient of an affinescheme by an action of G m , Z = [(Spec A ) G m / G m ] is the substack defined bythe fixed point locus of the action and S = [Spec( A/I + ) / G m ] is the attractingsubstack, where I + := ( M n> A n )is the graded ideal generated by elements of positive weight. In this basic situationthe theorem will then follow from an elementary calculation. We will first explainthe proof of this special case and then show how to reduce to the basic situation. Lemma 6.7. In the setting of Theorem 6.3 suppose in addition that X = [Spec( A ) / G m ] for a graded ring A = L n ∈ Z A n and that S = [Spec( A/I + ) / G m ] . Then the con-clusion of Theorem 6.3 holds.Proof. Let us denote X := Spec( A ) and S := Spec( A/I + ). As X → X is a G m -torsor, we can lift ξ to a map ξ ′ R : Spec( R ) → Spec( A ), obtaining a diagramSpec( K ) (cid:31) (cid:127) / / ξ ′ K (cid:15) (cid:15) Spec( R ) ξ ′ R (cid:15) (cid:15) Spec( κ ) ? _ o o ξ ′ κ (cid:15) (cid:15) X − S (cid:31) (cid:127) j / / Spec( A ) Spec( A/I + ) . ? _ ι o o As ξ ′ κ ∈ S = Spec( A/I + ) and A/I + is generated by elements of non-positiveweight, the G m -orbit of ξ ′ κ , corresponding to a map of graded algebras A/I + → k [ t ± ] where t has weight − 1, extends to an equivariant morphism A κ → S . Thusthe G m -orbits of the points ξ ′ K , ξ ′ R , ξ ′ κ define a diagram: G m,K (cid:31) (cid:127) / / f K (cid:15) (cid:15) G m,Rf R (cid:15) (cid:15) G m,κ ? _ o o _(cid:127) (cid:15) (cid:15) A κf κ (cid:15) (cid:15) X − S (cid:31) (cid:127) j / / Spec( A ) Spec( A/I + ) . ? _ ι o o e know that f R ( I + ) ∈ π ( R [ t ± ]) since f κ factors through A/I + , and we have K [ t ± ] · f R ( I + ) = K [ t ± ] since the image of f K does not intersect S .Let a i ∈ I d i be homogeneous generators of I + . Then for all i we have f R ( a i ) = ǫ i π n i t − d i for some n i > ǫ i ∈ R × ∪ { } . As f R ( I + ) is not 0 we can define md := min i (cid:26) n i d i (cid:12)(cid:12)(cid:12) f R ( a i ) = 0 (cid:27) and let R ′ := R [ π d ]. Since n i − d i md ≥ i , we can write f R ( a i ) = ǫ i ( π n i − dimd )( π md x − ) d i = ǫ i ( π n i − dimd ) s d i . Since f R maps elements of negative weight to R [ t ], we have a homomorphism ofgraded rings f ′ R ′ : A → R ′ [ s, t ] / ( st − π md ) = R ′ [ t, π md t − ] ⊂ R ′ [ t, t − ]Furthermore, composing with the map setting s = 1 at least one f ′ R ′ ( a i ) is notmapped to 0 mod π d , i.e. f ′ R ′ | { s =1 } : Spec( R ′ ) Spec( A a i ) ⊂ X − S . Thegraded homomorphism f ′ R ′ defines a morphism[Spec (cid:0) R ′ [ s, t ] / ( st − π md ) (cid:1) / G m ] → X = [ X/ G m ] . As π md is not a uniformizer for R ′ , this is not quite an elementary modification.However, we can embed R ′ [ s, t ] / ( st − π m/d ) ⊂ R ′ [ s /m , t /m ] / ( s /m t /m − π /d ).If we regard s /m and t /m as having weight 1 and − R ′ [ s /m , t /m ] / ( s /m t /m − π /d )) → Spec( R ′ [ s, t ] / ( st − π m/d )) is equivariantwith respect to the group homomorphism G m → G m given in coordinates by z z m . The resulting compositionST R ′ → [Spec (cid:0) R ′ [ s, t ] / ( st − π md ) (cid:1) / G m ] → X is the desired modification of ξ R . (cid:3) Reduction to quasi-compact stacks. We first show that by replacing X by asuitable open substack we may assume that X is quasi-compact. Lemma 6.8. In the setting of Theorem 6.3, let σ : Z → S be the center of the Θ -stratum ev : S ֒ → X . Then for any point x ∈ | Z | and any open substack U ⊂ X containing σ ( x ) , there is another open substack with σ ( x ) ∈ V ⊂ U such that S ∩ V is a Θ -stratum in V .Proof. We only need to find a substack V ⊂ X containing σ ( x ) such that for any f : Θ k → X , where k is a field, with f ∈ S and f (1) ∈ V , we have f (0) ∈ V aswell. Let U ′ = (ev ◦ σ ) − ( U ) ⊂ Z , and let Z ′ = Z \ U ′ be its complement. Thenthe open substack V := U \ ( U ∩ ev (ev − ( Z ′ ))) ⊂ X satisfies the condition. (cid:3) .1.3. Reminder on the normal cone to a Θ -stratum. The main problem in find-ing a presentation of the form [Spec( A/I + ) / G m ] ⊂ [Spec( A ) / G m ] is that for anarbitrary morphism [Spec( A ) / G m ] → X the preimage of the Θ-stratum need notbe defined by the ideal generated by the elements of positive weight. To findpresentations for which this happens, we need to recall that the weights of the G m -action of the restriction of the conormal bundle of a Θ-stratum to its center Z are automatically positive. This property was already important in the work ofAtiyah–Bott [AB83] and it appears in the language of spectral stacks in [Hal14, § Lemma 6.9. In the setting of Theorem 6.3, let σ : Z → S be the center of the Θ -stratum ev : S ֒ → X and x ∈ Z ( k ) be a k -point. By abuse of notation we willalso denote σ ( x ) ∈ X ( k ) by x .(1) Let T X ,x = L n ∈ Z T X ,x,n be the decomposition of the tangent space at x into weight spaces with respect to the G m -action induced form the canon-ical cocharacter λ x : G m → Aut X ( x ) . Then we have T S ,x = L n ≥ T X ,x,n .(2) G m acts with non-negative weights on Lie( Aut X ( x )) .Proof. Let us first show that L n ≥ T X ,x,n ⊆ T S ,x . Let t ∈ X ( k [ ǫ ] /ǫ ) be a tangentvector in T X ,x,n for some n ≤ 0, i.e. t comes equipped with an isomorphism t mod ǫ ∼ = x .This means that we have a commutative diagram[Spec( k [ ǫ ] /ǫ ) / G m ] t / / X , [Spec( k ) / G m ] ( x,λ x ) ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ ?(cid:31) O O where G m acts on Spec( k [ ǫ ] /ǫ ) via ( λ, ǫ ) λ n ǫ . In other words, we have acommutative diagram G m × Spec( k [ ǫ ] /ǫ ) ( λ,ǫ λ n ǫ ) / / * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Spec( k [ ǫ ] /ǫ ) t (cid:15) (cid:15) X . If n ≥ A , i.e., we get an extension A × Spec( k [ ǫ ] /ǫ ) ( λ,ǫ λ n ǫ ) / / * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ Spec( k [ ǫ ] /ǫ ) t (cid:15) (cid:15) X and this defines an extension of t to a k [ ǫ ] /ǫ -valued point of Map S (Θ , X ).Conversely, an extension of the constant map [ A / G m ] → [Spec( k ) / G m ] → X to [ A × Spec( k [ ǫ ] / ( ǫ )) / G m ] → X automatically factors through the first infinites-imal neighborhood of x ∈ X . On a versal first order deformation this correspondsto a homomorphism of graded algebras k [ ǫ , . . . , ǫ d ] / ( ǫ i ) i =1 ,...d → k [ λ, ǫ ] / ( ǫ ),where we can choose ǫ i to be homogeneous for the G m -action defined by λ x .This has to vanish on those tangent directions ǫ i on which λ x acts with negativeweights. This shows (1). imilarly for (2), when we regard x as a k point of Z ֒ → S ⊂ Map S (Θ , X ), itcorresponds to a map which factors as Θ k → B k G x ֒ → X , where we abbreviated G x = Aut X ( x ). We know that Aut S ( x ) → Aut X ( x ) is an equivalence, so bythe classification of G x -bundles on [ A / G m ] (see [Hei17, Lem. 1.7] or [Hal14,Prop. A.1]) this implies that for the canonical cocharacter λ x : G m → G x we have G x = P ( λ x ) as an algebraic group. In particular this means that G m acts withnon-negative weights on the Lie algebra of G x = P ( λ x ). (cid:3) Reduction to the basic situation - Case of smooth stabilizers over a field. Lemma 6.10. Let X be an algebraic stack of finite type with affine diagonal overan algebraically closed field k . Let S ⊂ X be a Θ -stratum with center σ : Z → S ,and let x ∈ Z ( k ) be a point such that x := σ ( x ) has a smooth automorphismgroup. Then there is a smooth representable morphism p : [Spec( A ) / G m ] → X whose image contains x and such that p − ( S ) = [Spec( A/I + ) / G m ] ֒ → [Spec( A ) / G m ] . Proof. The point x has a canonical non-constant homomorphism G m → Aut Z ( x ),which induces a canonical homomorphism λ : G m → G x := Aut X ( x ). We mayreplace G m with its image in G x and thus assume that λ is injective. As weassumed that G x is smooth the quotient G x /λ ( G m ) is smooth, so we may apply[AHR15, Thm. 1.2] to obtain a smooth representable morphism p : [Spec( A ) / G m ] → X together with a point w ∈ Spec( A )( k ) in p − ( x ) which is fixed by G m and suchthat p − ( B k G x ) ∼ = B k G m . The isomorphism p − ( B k G x ) ∼ = B k G m implies thatthe relative tangent space to ˜ p : Spec( A ) → X at w is naturally identified withLie( G x ) / Lie( G m ) on which G m acts with non-negative weights by part (2) ofLemma 6.9.Note that connected components of Spec( A ) G m can be separated by invariantfunctions, so we may replace Spec( A ) with a G m -equivariant affine open neigh-borhood of w so that Spec( A ) G m is connected. It follows that Spec( A/I + ) isconnected as well.This implies that S A := [Spec( A/I + ) / G m ] ⊆ [Spec( A ) / G m ] is isomorphic to aconnected component of Map(Θ , [Spec( A ) / G m ]) and Z A := [Spec( A G m ) / G m ] ⊂ S A is the center of S A . As p ( x ) ∈ Z connectedness now implies that p ( Z A ) ⊂ Z and therefore we also have [Spec( A/I + ) / G m ] ⊂ p − ( S ).To conclude that S A ∼ = p − ( S ) after possibly shrinking A , it suffices to checkthat the inclusion [Spec( A/I + ) / G m ] ⊆ p − (ev ( S )) of closed substacks of [Spec( A ) / G m ]is an isomorphism locally at w . Consider the pull-back: p − (ev ( S )) = Spec( B ) (cid:15) (cid:15) (cid:31) (cid:127) / / Spec( A ) p (cid:15) (cid:15) S (cid:31) (cid:127) / / X Then B is a graded ring and we still have an exact sequence T p,w → T Spec( B ) ,w → T S ,x . As G m acts with non-negative weight on the relative tangent bundle at w andalso on T S ,x by Lemma 6.9, this shows that G m acts with non-negative weights n T Spec( B ) ,w . In particular the maximal ideal m w ⊂ B of w is generated byelements of non-positive weight locally at w .Therefore, after possibly shrinking A we may assume that B = L n ≤ B n isnon-positively graded. As Spec( A/I + ) ⊂ Spec( A ) was the contracting subschemefor G m we find that locally around w we thus have p − ( S ) ⊂ Spec( A/I + ) locallyaround w . This proves our claim. (cid:3) Reduction to the basic situation - general case. Lemma 6.11. In the setting of Theorem 6.3 where S ֒ → X is a Θ -stratum and X is quasi-compact, there is a smooth representable morphism p : [Spec( A ) / G m ] → X such that p − ( S ) is the Θ -stratum p − ( S ) = [Spec( A/I + ) / G m ] ֒ → [Spec( A ) / G m ] , and S is contained in the image of p .Proof. Because X is finite type over the base space S , we may apply Lemma 3.2to obtain a smooth, surjective and representable map p : [Spec( A ) / G nm ] → X such that Map(Θ , [Spec( A ) / G nm ]) → Map(Θ , X ) is also smooth, surjective andrepresentable. From Proposition 2.8, we know that Map(Θ , [Spec( A ) / G nm ]) isthe disjoint union indexed by cocharacters G m → G nm of stacks of the form[Spec( A/I + ) / G nm ], where I + is the ideal generated by positive weight elementswith respect to a given cocharacter. Choosing different connected components ifnecessary and forgetting all but the relevant cocharacter in each component, wecan construct a non-positively graded algebra C = L n ≤ C n along with a smoothsurjective representable map [Spec( C ) / G m ] → S .We now discard the previously constructed Spec( A ) and apply the relative slicetheorem (Theorem 2.6) to the smooth surjective map [Spec( C ) / G m ] → S , wherewe regard S as a closed substack of X . This provides a map p : [Spec( A ′ ) / G m ] → X along with an isomorphism C ≃ A ′ /I S , where I S ⊂ A ′ is the ideal correspondingto p − ( S ). By construction C has no positive weight elements, so the ideal I + generated by positive weight elements of A ′ is contained in I S .Because p is smooth, the relative cotangent complex of Spec( C ) ֒ → Spec( A ′ ) is p ∗ ( L S / X ). In particular, the fiber of the conormal bundle of Spec( C ) ֒ → Spec( A ′ )has positive weights at every point of Spec( C ) G m by Lemma 6.9. One may there-fore find a collection of positive weight elements of I S which generate the fiber of I S at every closed point of Spec( C ) G m .Moreover, as C is non-positively graded, the orbit closure of every point inSpec( C ) meets the fixed locus Spec( C ) G m . So by Nakayama’s lemma we canactually find a collection of homogeneous elements of I + which generate the fiberof I S at every point of Spec( C ) and hence in a G m -equivariant open neighborhoodof Spec( C ) ֒ → Spec( A ′ ). We may thus invert a weight 0 element a ∈ A ′ sothat these elements of I + generate ( I S ) a ⊂ A ′ a and C = A ′ /I S = A ′ a / ( I S ) a isunaffected.In particular we have shown that after inverting a weight 0 element of A ′ , wehave a smooth map p : [Spec( A ′ ) / G m ] → X such that [Spec( A ′ /I + ) / G m ] = p − ( S )and the map Spec( A ′ /I + ) → S is surjective. (cid:3) We can now prove the semistable reduction theorem: Proof of Theorem 6.3. Consider a map ξ : Spec( R ) → X as in the statement ofthe theorem. Observe that for any smooth map p : Y → X such that S induces Θ-stratum p − ( S ) in Y and the image of p contains the image of ξ , if we knowthe conclusion of the theorem holds for Y then the conclusion holds for X as well:indeed after an extension of R we may lift ξ to a map ξ ′ : Spec( R ′ ) → Y , constructan elementary modification in Y such that the new map ξ ′′ : Spec( R ′ ) → Y lies in Y \ p − ( S ), and observe that the composition of this elementary modification with p gives an elementary modification of ξ such that the new map p ◦ ξ ′′ : Spec( R ′ ) → X lies in X \ S .Using this observation and the fact that ξ k lies in S , we may use Lemma 6.8 toreplace X with a quasi-compact open substack, then use Lemma 6.11 to constructa smooth map p : [Spec( A ) / G m ] → X whose image contains the image of ξ andfor which S induces a Θ-stratum. Then we are finished by Lemma 6.7. (cid:3) Comparison between a stack and its semistable locus. As an imme-diate consequence of the semistable reduction theorem, we have the following: Corollary 6.12. Let X be an algebraic stack locally of finite type with affine di-agonal over a noetherian algebraic space S . Let X = S c ∈ Γ X ≤ c be a well-ordered Θ -stratification of X . If X → S satisfies the existence part of the valuative crite-rion for properness, then so does X ≤ c → S for every c ∈ Γ . In particular, if thesemistable locus X ss := X ≤ is quasi-compact, then X ss → S is universally closed.Proof. Consider a DVR R and a map Spec( R ) → S along with a lift Spec( K ) → X ≤ c . If X → S satisfies the existence part of the valuative criterion, then afteran extension of R one can extend this lift to a lift Spec( R ′ ) → X of Spec( R ) → S .By hypothesis the generic point lies in X ≤ c , so by Theorem 6.5 after passing to afurther extension of R there is a sequence of elementary modifications resulting ina modification Spec( R ′ ) → X ≤ c . Note that because Spec( R ) is the good modulispace of ST R , and good moduli spaces are universal for maps to an algebraicspace [Alp13, Thm. 6.6], any elementary modification of a map Spec( R ) → S istrivial. It follows that our modified map Spec( R ′ ) → X ≤ c is a lift of the originalmap Spec( R ) → S . (cid:3) Next let us briefly recall the notion of Θ-stability from [Hal14, Def. 4.1 & 4.4]and [Hei17, Def. 1.2]. Definition 6.13. Given a cohomology class ℓ ∈ H ( X ; R ), we say that a point p ∈ | X | is unstable with respect to ℓ if there is a filtration f : Θ k → X with f (1) = p ∈ | X | and such that f ∗ ( ℓ ) ∈ H (Θ k ; R ) ≃ R is positive. The Θ -semistable locus X ss is the set of points which are not unstable.The above definition is simply an intrinsic formulation of the Hilbert–Mumfordcriterion for semistability in geometric invariant theory. We are somewhat flexiblewith what type of cohomology theory we use: if X is locally finite type over C wemay use the Betti cohomology of the analytification of X , if X is locally finite typeover another field k , we can use Chow cohomology, and in general one may usethe Neron–Severi group N S ( X ) R for H ( X ; R ). In [Hal14, § Proposition 6.14. Let X be an algebraic stack locally of finite type with affinediagonal over a noetherian algebraic space S , and let X ss be the Θ -semistablepoints with respect to a class ℓ ∈ H ( X ; R ) . Suppose that either(a) X ss is the open part of a Θ -stratification of X , i.e. X ss = X ≤ , such thatfor each HN filtration g : Θ k → X of an unstable point one has g ∗ ( ℓ ) > in H (Θ k ; R ) , or b) X ss ⊂ X is open and X → S is Θ -reductive.Then(1) if X → S is S -complete, then so is X ss → S , and(2) if X → S is Θ -reductive, then so is X ss → S . In the proof, we will need the following: Lemma 6.15. Under the hypotheses of Proposition 6.14, given a filtration f : Θ k → X such that f (1) is semistable with respect to ℓ , then f ∗ ( ℓ ) = 0 if and only if f (0) is semistable as well.Proof. The proof is a geometric reformulation of the corresponding argument forsemistability for vector bundles. One direction is easy: for any semistable point x ∈ X ( k ) and any cocharacter λ : G m → G x , the restriction of ℓ to H ([Spec( k ) / G m ]; R ) ≃ R along the resulting map f λ : Θ k → [Spec( k ) / G m ] → X must vanish, becausethe invariants for λ and λ − differ by sign and are both non-positive.For the converse suppose that f ∗ ( ℓ ) = 0 and f (0) / ∈ X ss . We claim that thereis a filtration g : Θ k → X of f (0) with g ∗ ( l ) > G m on f (0) induced from the filtration f : Θ k → X . This is automaticin case (a) as HN filtrations are canonical. For case (b), since X → S is Θ-reductive, the representable map Map(Θ , X ) → X satisfies the valuative criterionfor properness, so the fiber of this map over f (0) ∈ X ( k ), which is denotedFlag( f (0)) is an algebraic space of finite type over k which satisfies the valuativecriterion for properness. The action of G m by automorphisms of f (0) gives a G m -action on Flag( f (0)). Given some point g ∈ Flag( f (0))( k ) for which g ∗ ( ℓ ) > G m → Flag( f (0)) of g . Because Flag( f (0)) satisfiesthe valuative criterion for properness, this map extends to an equivariant map A k → Flag( f (0)). This map sends 0 ∈ A k to a fixed point for the action of G m on Flag( f (0)), which corresponds to a G m -invariant filtration g of f (0), and g ison the same connected component of Flag( f (0)) as g , so g ∗ ( ℓ ) = g ∗ ( ℓ ) > R = k [[ π ]] the completion of the local ring of the affine line withcoordinate π at 0. Then the map f R : Spec( R ) → [Spec( k [ π ]) / G m ] = Θ k f −→ X and g : Θ k → X define the datum needed to apply the gluing lemma Corollary A.2,which says that after restricting f R to R ′ = R [ π /n ] for n ≫ F R ′ : ST R ′ → X such that F | t =0 ∼ = f R ′ and F | s =0 ∼ = g . Let π ′ = π /n denote the uniformizer in R ′ .As f R ′ was the restriction of a map f A : A k → Θ k → X we find that thismorphism extends canonically to F : [Spec( k [ π ′ , s, t ] / ( st − π ′ )) / G m ] = [Spec( k [ s, t ]) / G m ] → X . By uniqueness of the extension F R ′ and the fact that g is fixed by the G m -actionon f (0) induced by f , this morphism comes equipped with a descent datum forthe standard G m -action on A = Spec( k [ π ′ ]). We therefore obtain F : [Spec( k [ π ′ , s, t ] / ( st − π ′ )) / G m ] = [ A k / G m ] → X . where the action of the second copy of G m is with weight − s and trivial on t . Choosing a different basis for the cocharacter lattice of G m , we see that thisis equivalent to the usual action of G m on A k . In particular, every cocharacter λ : G m → G m has the form λ ( t ) = ( t a , t b ) for some pair h a, b i . f a, b ≥ 0, then the point (1 , ∈ A k has a limit under λ ( t ) as t → 0, andrestricting F to the corresponding line we obtain a filtration f h a,b i : Θ k → X , with f (1) ≃ f h a,b i (1). By construction the original filtration f corresponds to f h , i .Note that F ∗ ( ℓ ) | [0 / G m ] defines a character χ of G m such that for a, b > f ∗h a,b i ( ℓ ) is given by the canonical pairing ( h a, b i , χ ) between charactersand cocharacters. As F | { s =0 } = g was assumed to be destabilizing and F wasdefined by the subgroup h− , i we have ( h− , i , χ ) = g ∗ ( l ) > 0. We also have( h , i , χ ) = f ∗h , i ( ℓ ) = 0 by hypothesis. Thus for a ≫ b > f ∗h a,b i ( ℓ ) > f (1) was semistable. (cid:3) Remark 6.16. In Theorem 7.25 below we will use a slightly more general notionof stability: we replace the weight of f ∗ ( ℓ ) with any function ℓ from the set offiltrations in X to a totally ordered real vector space V , and define x ∈ X to besemistable if ℓ ( f ) ≤ f with f (1) = x . Then the proof aboveapplies verbatim, provided that 1) ℓ ( f ) is locally constant in algebraic families offiltrations, and 2) for any map F : [ A k / G m ] → X the function on the cocharacterlattice λ ℓ ( f λ ) ∈ V is linear, where f λ denotes the filtration associated to thecocharacter λ as in the proof of Lemma 6.15. The second condition is equivalent torequiring the function λ ℓ ( g λ ) is linear, where g λ denotes the constant filtrationof F (0 , 0) induced from the Z -grading of F (0 , 0) associated to the cocharacter λ . Remark 6.17. The proof of Lemma 6.15 is a special case of the techniqueused to prove the perturbation theorem on filtrations [Hal14, Thm. 3.60]. Thistheorem constructs a bijection between filtrations of a point x ∈ X which are“close” to a given filtration f and filtrations of the associated graded object f (0) ∈ Map k ( B G m , X ) which are “close” to the canonical filtration defined bythe action of G m on f (0). In this language, the proof that f (0) is semistable if f (1) is semistable and f ∗ ( ℓ ) = 0 amounts to the observation that if f (0) had adestabilizing filtration as a graded object, then because the canonical filtrationof f (0) has weight 0, one can find destabilizing filtrations of f (0) which are ar-bitrarily close to the canonical filtration, and then one can identify these withdestabilizing filtrations of f (1) using the perturbation theorem. Proof of Proposition 6.14. Consider a DVR R and a diagramSpec( R ) ∪ Spec( K ) Spec( R ) / / (cid:15) (cid:15) X ss (cid:15) (cid:15) ST R ♠♠♠♠♠♠♠♠ / / S . By hypothesis we can fill the dotted arrow uniquely to a map ST R → X . Weclaim that in fact the map ST R → X factors through X ss . Because X ss is open,it suffices to check that the unique closed point maps to X ss . By hypothesis thepoint ( π, s, t ) = (0 , , 0) and the point ( π, s, t ) = (0 , , 1) map to X ss . Restrictingthe map ST R → X to the locus Θ k ≃ { s = 0 } and Θ k ≃ { t = 0 } give filtrations f and f in X of points in X ss , and if one has f ∗ ( ℓ ) < f ∗ ( ℓ ) > f (1) ∈ X ss . Therefore f ∗ ( ℓ ) = 0 for bothfiltrations, and it follows from Lemma 6.15 that f (0) ∈ X ss as well.For the corresponding claim for Θ-reductivity is proved similarly. For theanalogous filling diagram, we start with a map f : Θ R \ { (0 , } → X ss and fillit to a map ˜ f : Θ R → X . We claim that (0 , 0) maps to X ss as well, and hence ecause X ss ⊂ X is open it follows that ˜ f lands in X ss . Because the restriction f K of f to Θ K ⊂ Θ R \ { (0 , } maps to X ss , we know from Lemma 6.15 that f ∗ K ( ℓ ) = 0. The function f f ∗ ( ℓ ) ∈ R , regarded as a function on Map(Θ , X ),is locally constant. It therefore follows that the restriction ˜ f k : Θ k → X of ˜ f alsohas ˜ f ∗ k ( ℓ ) = 0. It follows that ˜ f k (0) ∈ X ss . (cid:3) Corollary 6.18. Let X be an algebraic stack locally of finite type with affinediagonal over a noetherian algebraic space S defined over Q . Assume that X → S is S -complete and Θ -reductive. Let X ss ⊂ X be the Θ -semistable locus with respectto some class ℓ ∈ H ( X ; R ) . If X ss ⊂ X is a quasi-compact open substack, then X ss admits a good moduli space which is separated over S . Furthermore if in addition X → S satisfies the existence part of the valuative criterion for properness and X ss is the open part of a well-ordered Θ -stratification of X , then the good modulispace for X is proper over S .Proof. The map X ss → S is S -complete and Θ-reductive by Proposition 6.14.By Theorem A, X ss admits a separated good moduli space X → X . Underthe additional hypotheses, X ss → S satisfies the existence part of the valua-tive criterion for properness by Corollary 6.12, and hence X → S is proper byProposition 3.47. (cid:3) Application: Properness of the Hitchin fibration. Let us illustratehow the semistable reduction theorem (Theorem 6.5) can be used to simplify andextend classical semistable reduction theorems for principal bundles and Higgsbundles on curves.The setup for these results is the following (see e.g., [Ngˆo06, § C be asmooth projective, geometrically connected curve over a field k and G a reductivealgebraic group. As the notions are slightly easier to formulate over algebraicallyclosed fields and the valuative criteria allow for extensions of the ground field, wewill assume that k is algebraically closed in this section.We denote by Bun G the stack of principal G -bundles on C , i.e., for a k -scheme S we have that Bun G ( S ) is the groupoid of principal G -bundles on C × S . Fix aline bundle L on C . A G -Higgs bundle with coefficients in L on C is a pair ( P , φ )where P is a G -bundle on C and φ ∈ H ( C, ( P × G Lie( G )) ⊗ L ). We denote byHiggs G the stack of G -Higgs bundles with coefficients in L .The stack Higgs G comes equipped with the forgetful morphism Higgs G → Bun G and the Hitchin morphism h : Higgs G → A G . Here A G ∼ = L ri =1 H ( C, L d i ),where d , . . . , d r are the degrees of homogeneous generators of k [Lie( G ) ∗ ] G and h is defined by mapping ( P , φ ) to the characteristic polynomial of φ .On both Bun G and Higgs G there is a classical notion of stability, which isdefined in terms of reductions to parabolic subgroups.Let us recall how this notion is related to Θ-stability. For vector bundles thereis an equivalence (Proposition 2.8, [Hei17, Lem. 1.10])Map(Θ , Bun GL n ) ∼ = (cid:28) ( E , E i ) i ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) E ∈ Bun GL n , E i ⊆ E i +1 ⊆ E subbundles E i = E for i ≫ , E i = 0 for i ≪ (cid:29) which is given by assigning to a weighted filtration of a vector bundle E thecanonical G m -equivariant degeneration of E to the associated graded bundle.This construction has an analog for principal bundles. To state this we fix(as in Proposition 2.8) a complete set of conjugacy classes of cocharacters Λ ⊂ Hom( G m , G ). As in § P + λ ⊆ G the parabolic subgroup defined y λ and by L λ ⊂ P + λ the Levi subgroup defined by λ which is isomorphic to thequotient of P + λ by its unipotent radical U + λ ⊂ P + λ . Then there is an equivalence(see e.g., [Hei17, Lem. 1.13])Map(Θ , Bun G ) ∼ = a λ ∈ Λ Bun P + λ . For Higgs bundles note that the forgetful map Higgs G → Bun G is representableand therefore Map(Θ , Higgs G ) ⊂ Map(Θ , Bun G ) × Bun G Higgs G , i.e., a filtrationof a Higgs bundle is the same as a filtration of the underlying principal bundlethat preserves the Higgs field φ .Recall that a G -bundle E is called semistable, if for all λ and all E λ ∈ Bun P + λ with E λ × P + λ G ∼ = E we have deg( P λ × P + λ Lie( P + λ )) ≤ 0. Similarly a Higgs bundleis called semistable if the same condition holds for all reductions that respectthe Higgs field φ . This stability notion can be viewed as Θ-stability inducedfrom the so called determinant line bundle L det on the stack Bun G whose fiberat a point P (resp. a point ( P , φ )) is given by the one dimensional vector spacedet( H ( C, P × G Lie( G ))) ⊗ det( H ( C, P × G Lie( G ))) − (see [Hei17, § ss G ⊆ Bun G the open substack of semistable bundlesand for by Bun ss P + λ ⊆ Bun P + λ the open substack of bundles such that the associated L λ -bundle is semistable.Finally let us recall how the notion of Harder–Narasimhan reduction can beused to equip the stacks Bun G and Higgs G with a (well-ordered) Θ-stratificationif the characteristic of k is not too small, i.e., such that Behrend’s conjectureholds for G (see [Hei08a, Thm. 1] for explicit bounds depending on G ; note thatchar 2 has to be excluded for groups of type B n , D n as well).For any unstable G -bundle P there exists a canonical Harder–Narasimhan re-duction P HN to a parabolic subgroup P + λ , where λ is uniquely determined up toa positive integral multiple. We denote by d := deg( P HN ) : Hom( P λ , G m ) → Z χ deg( P HNλ × χ G m )the degree of P HN and by Bun d, ss P + λ ⊂ Bun ss P + λ the connected component defined by d . The instability degree of P λ is defined asideg( P ) := deg( P HN × P + λ Lie( P )) . Behrend showed that the morphism Bun d,ssP + λ → Bun G defined by the inclusion P + λ ⊂ G is radicial if the degree d is the degree of a canonical reduction [Beh] andthe map is an embedding if Behrend’s conjecture holds for G [Hei08b, Lem. 2.3]this condition is satisfied if the characteristic of k is not too small with respectto G (e.g., > G if the char-acteristic of k is not too small. The same arguments apply for Higgs bundles andthis shows the following lemma. emma 6.19. If the characteristic of k is large enough so that Behrend’s conjec-ture holds for G , then the Harder–Narasimhan stratifications of Bun G and Higgs G form a well-ordered Θ -stratification. To apply the semistable reduction theorem to Higgs G → A G we need to showthat this morphism satisfies the existence part of the valuative criterion for proper-ness. The existence result is probably well known (see e.g. [CL10, § Lemma 6.20. Suppose that the characteristic of k is not a torsion prime for G and very good for G . Let R be a DVR with fraction field K , and let ( E K , φ K ) ∈ Higgs G ( K ) be a Higgs bundle such that h ( E K , φ K ) ∈ A G ( K ) extends to A G ( R ) .Then there exists an extension R → R ′ of DVRs with K → K ′ = Frac( R ′ ) finiteand a point ( E ′ R , φ ′ R ) ∈ Higgs G ( R ′ ) extending ( E K , φ K ) .Proof. First let us assume that the derived group of G is simply connected. Thegeneric point of C will be denoted by η , g = Lie( G ) and car := g //G is the spaceof characteristic polynomials of elements of g .Let ( E K , φ K ) ∈ Higgs G ( K ) be a Higgs bundle such that h ( E K , φ K ) ∈ A G ( R ) ⊂ A G ( K ).We argue as in [CL10, § K we may assumethat E K is trivial at the generic point K ( η ) of C K . Choosing trivializations of E | K ( η ) and L | η identifies φ K with an element in X K ∈ g ( K ( η )). To concludethe argument as in loc.cit., it is sufficient to show that after passing to a finiteextension of K we can conjugate X K to an element of g ( R ( η )), because this allowsone to extend φ K to the trivial bundle over R ( η ) and as sections of affine bundlesextend canonically in codimension 2, the Higgs field φ K will then define a Higgsfield for any extension E R of E K that is trivial over R ( η ).We denote by X K = X sK + X nK the Jordan decomposition of X K into thesemisimple and nilpotent part.As h ( E , φ ) extends to R we know that the image of X K in car = g //G definesan R ( η )-valued point. We can use the Kostant section car → g to obtain Y R ∈ g ( R ( η )) with h ( Y R ) = h ( X K ).We claim that we can modify Y R such that its generic fiber Y K is semisimple.To see this let us consider the Jordan decomposition Y K = Y sK + Y nK . By ourassumptions on the characteristic of K the main result of [McN05] shows thatthere exists a parabolic subgroup P + λ ⊂ G defined by a cocharacter λ : G m → G K such that Y nK is contained in the Lie algebra of the unipotent radical of P + λ and as Y sK is in the centralizer of Y nK the element Y K also lies in Lie( P + λ ). As parabolicsubgroups extend over valuation rings, we find that Y R is contained in a parabolicsubgroup P R ⊂ G R and we can choose λ to be a cocharacter defined over R aswell.As P ( λ ) is defined to be the set of points p ∈ G such that lim t → λ ( t ) pλ ( t ) − exists, the limit lim t → λ ( t ) · Y R will be an R -valued point Y ′ R such that Y ′ K = Y sK is semisimple.As the semi-simple part of X K is the unique closed orbit in the conjugacy classof X K we know that X sK and Y sK lie in the same closed orbit. As we assumed thatthe derived group of G is simply connected and that p is not a torsion prime for G the centralizer Z G ( X s ) is a connected reductive group [Ste75, Thm. 0.1]. BySteinberg’s theorem, any Z G ( X s ) torsor over K ( η ) splits after a finite extensionof K , so after possibly extending K the elements X sK and Y sK are conjugate. Thus fter conjugating X K we may assume that X sK = Y sK , i.e. we may assume thatthe semisimple part of X K extends to R .Now we can apply the previous argument to X K , namely the element X K iscontained in a parabolic subalgebra defined by a cocharacter λ , such that X nK iscontained in its unipotent radical, so that for some a ∈ K the element λ ( a ) .X nK will extend to R as well.Finally for any group G we can consider a z -extension0 → Z → e G → G → Z is a central torus and the derived group G ′ of e G is simply connected.Then the map Bun e G → Bun G is a smooth surjection. Moreover the covering G ′ → G is separable, because we assumed that the fundamental group of G has no p -torsion. Therefore Lie( e G ) ∼ = Lie( Z ) ⊕ Lie( G ) and therefore the mapHiggs e G → Higgs G also admits local sections. Thus it suffices to prove the resultfor e G . (cid:3) The semistable reduction theorem (Theorem 6.5) now allows us to deduce: Corollary 6.21. Suppose that the characteristic p of k is large enough such thatBehrend’s conjecture holds for G , such that p is not a torsion prime for G andsuch that p is very good for G , then the Hitchin morphism h : Higgs ss G → A G satisfies the existence part of the valuative criterion for properness, i.e. if R isa DVR with fraction field K and x K : Spec( K ) → Higgs ss G is a morphism suchthat h ( x K ) : Spec( K ) → A G extends to R , then there exists an extension R → R ′ of DVRs with K → K ′ = Frac( R ′ ) finite and a morphism x R ′ : Spec( R ′ ) → Higgs ss G ( R ′ ) extending x K . Note that in characteristic 0 this result is due to Faltings [Fal93, Thm. II.4] andfor the regular part of the Hitchin fibration this is due to Chaudouard–Laumon[CL10, Thm. 8.1.1]. Over the complex numbers, the result can also be deducedfrom results of Simpson as explained in [Cat18]. Remark 6.22. Since Higgs ss G is quasi-compact, the conclusion is equivalent tosaying that the Hitchin morphism is universally closed. In particular the inducedmorphism on an adequate moduli space will be proper. Proof. By Lemma 6.20 we can find an extension of x K to x R . As Higgs G admits awell ordered Θ-stratification by Harder–Narasimhan reductions we can thereforeapply the semistable reduction Theorem 6.5 to conclude. (cid:3) Good moduli spaces for objects in abelian categories In this section we study the moduli functor for objects in a k -linear abeliancategory A , following the foundational work of Artin and Zhang [AZ01], whoexplained that many of the results known for categories of quasi-coherent sheaveson a scheme can be carried out in an abstract setting. This construction has beenstudied more recently in [Gai05, AP06, CG13]. The general setup is very usefulas it for example gives an easy way to formulate moduli problems for objectsin the heart of different t-structures in the derived category of coherent sheaveson a scheme. It turns out that this setup leads to moduli problems in which he conditions of Θ-reductivity, S -completeness, and unpunctured inertia can bechecked rather easily. Following the convention of [AZ01], throughout this chapterwe exceptionally fix a base ring k , which is allowed to be any commutative ring.7.1. Formulation of the moduli problem. Let us start by recalling the setupof [AZ01]. Let A be a k -linear abelian category that is assumed to be cocomplete,i.e. arbitrary small colimits exist in A . To formulate a reasonable moduli problemwe first need to recall some finiteness conditions on objects. Recall that an object E ∈ A is • finitely presentable (also known as compact ) if the canonical map(7.1) colim α ∈ I Hom( E, F α ) → Hom( E, colim α ∈ I F α )is an isomorphism for any (small) filtered system { F α } α ∈ I in A ; • finitely generated if (7.1) is an isomorphism for any filtered system ofmonomorphisms in A , or equivalently, if E = S α E α for a filtered systemof subobjects, then E = E α for some α [Pop73, Prop. 3.5.6]; and • noetherian if every ascending chain of subobjects of E terminates, orequivalently, if every subobject of E is finitely generated.We denote by A fp the full subcategory of A consisting of finitely presentableobjects. Example 7.1. If A = Mod R for a (possibly non-commutative) ring R , an object E ∈ A is finitely presentable (resp. finitely generated, resp. noetherian) if andonly if the corresponding module is. The analogous statement holds if A =QCoh( X ) for a scheme X .We say that A is • locally of finite type if every object in A is the union of its finitely gener-ated subobjects; • locally finitely presented if every object in A can be written as the filteredcolimit of finitely presentable objects, and A fp is essentially small; • locally noetherian if it has a set of noetherian generators.If A is locally noetherian, then finitely generated, finitely presentable, and noe-therian objects coincide [AZ01, Prop. B1.3], and the category A fp is closed underkernels and hence abelian. Our main results will assume that A is locally noe-therian.The next ingredient to the formulation of moduli problems is the observationthat the existence of colimits allows one to define a tensor product, which in turnprovides a notion of base change.More precisely, there is a canonical k -bilinear functor(7.2) ( − ) ⊗ k ( − ) : Mod k × A → A which is characterized by the formulaHom A ( M ⊗ k E, F ) = Hom Mod k ( M, Hom A ( E, F ))for objects E, F ∈ A and a k -module M . Explicitly, if one presents M as thecokernel of a morphism k I → k J for index sets I and J , then M ⊗ k E can becomputed as coker( E I → E J ) where the morphism E I → E J is induced by thematrix defining k I → k J .The functor ( − ) ⊗ k ( − ) commutes with filtered colimits and is right exact ineach variable. If M ∈ Mod k is flat and A is locally noetherian then M ⊗ k ( − ) isexact [AZ01, Lem. C1.1]. efinition 7.2. [AZ01, § C1] We say that an object E ∈ A is flat if ( − ) ⊗ k E : Mod k → A is exact.This tensor product leads to a base change formalism as follows. Definition 7.3 (Base change categories) . [AZ01, § B2] For a commutative k -algebra R , let A R denote the category of R -module objects in A , i.e., pairs ( E, ξ E )where E ∈ A and ξ E : R → End A ( E ) is a morphism of k -algebras, and a morphism( E, ξ E ) → ( E ′ , ξ E ′ ) in A R is a morphism E → E ′ in A compatible with the actionsof ξ E and ξ E ′ .For a commutative k -algebra R , A R is an R -linear abelian category [AZ01,Prop. B2.2], and A k = A . Given a homomorphism of commutative rings φ : R → R , the forgetful functor φ ∗ : A R → A R is faithfully exact, commutes with filtered colimits and faithful, and φ ∗ is fullyfaithful if φ is surjective [AZ01, Prop. B2.3]. Moreover, φ ∗ admits a left adjoint φ ∗ = R ⊗ R ( − ) : A R → A R by [AZ01, Prop. B3.16]. Remark 7.4. The property of being locally noetherian is not stable under basechange, but if A is locally noetherian and R is an essentially finite type k algebra,then A R is locally noetherian [AZ01, Cor. B6.3].The above constructions allow one to prove descent if A is locally noetherian[AZ01, Thm. C8.6], i.e., if R → S is a faithfully flat map of commutative k -algebras then A R is equivalent to the category of objects in A S equipped with adescent datum. Remark 7.5 ( A X for stacks over k ) . As the assignment R A R satisfies descentif A is locally noetherian, it defines a stack A in the fppf topology on k -Alg. Asin [LMB] this extends the category A R naturally not only to schemes but also toalgebraic stacks, i.e. for any algebraic stack X over k we can define A X := Map Fibered Cat /k - Alg ( X , A ) . If X is the quotient stack for a groupoid of affine schemes X = [ X ⇒ X ] with X i = Spec( R i ), then descent implies that the category A X is naturally equivalentto the category of objects of A X equipped with a descent datum. We will usethis description for the stacks Θ and ST R .Faithfully flat descent also allows one to extend the functor R ⊗ R ( − ) : A R → A R above to a functor f ∗ : A Y → A X for any morphism of stacks f : X → Y . Toprove extension theorems, we will need the following construction. Lemma 7.6 (Push-forward in A ) . Suppose that A is locally noetherian. If f : X → Y is a quasi-compact morphism with affine diagonal of algebraic stacksthen the restriction functor f ∗ : A Y → A X admits a right adjoint f ∗ which com-mutes with filtered colimits and flat base change.Proof. Let us first prove the claim when Y = Spec( A ) is affine. In this case wecan choose a presentation of X by a groupoid in affine schemes X ≃ [Spec( R ) ⇒ Spec( R )], and an object E ∈ A X is described by an object E ∈ A R along witha descent datum, i.e. denoting by d i : R → R the structure maps of the presen-tation this is an isomorphism φ : R ⊗ d ,R E ≃ R ⊗ d ,R E , satisfying a cocycle ondition. Then using the fact that homomorphisms in A [Spec( R ) ⇒ Spec( R )] arehomomorphisms in A R which commute with the respective cocycles, one mayverify directly that(7.3) f ∗ ( E ) = ker (cid:16) E φ ◦ η − η −−−−−→ B ⊗ d ,B E (cid:17) ∈ A A , where η i : E → B ⊗ d i ,B E for i = 0 , B ⊗ d i ,B ( − ) and the forgetful functor ( d i ) ∗ : A B → A B . Both objects in (7.3)are regarded as objects of A A via the canonical forgetful functor. The fact that f ∗ commutes with filtered colimits can be deduced from the formula (7.3) and thefact that filtered colimits are exact in A A [AZ01, Prop. B2.2]. The fact that f ∗ commutes with flat base change can be deduced from the fact that if A → A ′ isa flat ring map, then A ′ ⊗ A ( − ) is exact and hence commutes with the formationof the kernel in (7.3).Now let Y be an algebraic stack with affine diagonal. For any E ∈ A X andany morphism Spec( R ) → Y we have shown that the object ( f R ) ∗ ( E | X R ) ∈ A R isdefined and its formation commutes with flat base change. Faithfully flat descentimplies that these objects descend to a unique object f ∗ ( E ) ∈ A Y . Faithfully flatdescent also shows that the resulting functor f ∗ : A X → A Y is right adjoint to f ∗ . (cid:3) Remark 7.7. If X is a separated scheme and f : X → Spec( R ) a morphism thenthe above construction reproduces the usual Cech description of the push forward,i.e. given E ∈ A X choose a covering X = F i U i → X by open affines then (7.3)reduces to f ∗ ( E ) = ker (cid:18) M i ( f i ) ∗ ( E | U i ) → M i The category fibered in groupoids M A is a stack in the big fppftopology on k - alg and extends naturally to a stack on the big fppf topology onschemes over k .Proof. As we already quoted the result [AZ01, Thm. C8.6] that the categories A R satisfy flat descent, we only need to check that the conditions of flatnessand finite presentation are preserved by descent and pull-back. Given a ringmap R → R , the pullback functor R ⊗ R ( − ) preserves flat objects by [AZ01,Lem. C1.2], and it preserves finitely presentable objects because its right adjointcommutes with filtered colimits. If R → R is a faithfully flat map of k -algebraswe have R ⊗ R ( M ⊗ R ( − )) ≃ ( R ⊗ R M ) ⊗ R ( R ⊗ R ( − )), and the exactnessof a sequence in A R can be checked after applying the functor R ⊗ R ( − ), so E ∈ A R is flat if R ⊗ R E is. Also, one can directly verify from the descriptionof Hom in the category of descent data for the map R → R that any descentdata for a finitely presentable object E ∈ A R is a finitely presentable object inthe category of descent data. (cid:3) arning 7.10 (Passing to subcategories changes the moduli functor) . For theabove definition to make sense, the assumption that the category A is cocompleteand thus rather big is essential, as we defined R -modules to be elements of A equipped with additional structure. One might be tempted to replace this largecategory by a smaller ind-category generated by some of some subclass of finitelypresented objects, but this will change the moduli problem M A .For example if we take A to be the category of representations of the funda-mental group of a projective variety X , vector bundles with flat connections andfinite dimensional representations of the fundamental group are equivalent cate-gories, but as this equivalence is not algebraic it does not extend to an algebraicequivalence that identifies families over X R for k -algebras R . Indeed families ofrepresentations of π ( X ) are defined using finitely presented modules over thegroup algebra R [ π ( X )], whereas families of sheaves with connection are definedin terms of quasi-coherent sheaves of R ⊗ k D X -modules, and these larger categoriesare not equivalent. If one instead considered M A where A is the ind-completionof the category of vector bundles with connection, then this moduli functor differsfrom both of these, because a finitely presented R [ π ( X )] module need not be afiltered colimit of finite ones.7.2. Verification of the valuative criteria for the stack M A . To apply ourexistence results we will check Θ-reductivity and S -completeness for the stack M A with respect to discrete valuation rings which are essentially of finite typeover k . The first step is to show that, as for module categories, G m -equivariantobjects can be interpreted as graded objects. Let us recall these notions.Recall from [AZ01, § B7] that the category of Z -graded objects A Z consists offunctors Z → A , where Z is regarded as a category with only identity morphisms.Concretely all objects in A Z are of the form L n ∈ Z E n . Given a Z -graded k -algebra A , a Z -graded A -module object is a Z -graded object E = L n ∈ Z E n ∈ A Z with thestructure of an A -object such that multiplication A ⊗ k E → E maps A n ⊗ k E m to E n + m . The category A Z A of Z -graded A -module objects is abelian and locallynoetherian if A A is [AZ01, Prop. B7.5].Now let A be a Z -graded k -algebra, then objects of A [Spec( A ) / G m ] are by def-inition objects E ∈ A A together with a descent datum, i.e., a coaction σ : E → A [ t ] t ⊗ A E compatible with the coaction σ A : A → k [ t ] t ⊗ k A ∼ = A [ t ] t (induced fromthe Z -grading of A ), i.e., σ is a morphism in A A , where A [ t ] t has the A -modulegiven by σ A , such that the diagrams of objects in A A E σ / / σ (cid:15) (cid:15) A [ t ] t ⊗ A E t tt ′ (cid:15) (cid:15) A [ t ] t ⊗ A E id ⊗ σ / / A [ t, t ′ ] tt ′ ⊗ A E E σ / / id $ $ ❏❏❏❏❏❏❏❏❏❏❏ A [ t ] t ⊗ A E t (cid:15) (cid:15) E commute.An object E = L n ∈ Z E n ∈ A Z A induces a coaction σ E on E where σ | E n is theinclusion of E n into the summand A h t n i ⊗ A E ⊂ A [ t ] t ⊗ A E . The assignment E ( E, σ E ) defines a functor Can : A Z A → A [Spec( A ) / G m ] . Proposition 7.11. Let k be a commutative ring and let A be a locally noether-ian k -linear abelian category. Let A be a Z -graded k -algebra. Then the functor Can : A Z A → A [Spec( A ) / G m ] is an equivalence of categories between A [Spec( A ) / G m ] and the category A Z A of Z -graded A -module objects in A which restricts to an quivalence between M A ([Spec( A ) / G m ]) and the groupoid of objects L n ∈ Z E n in A Z A such that each E n ∈ A A is flat and E n = 0 for n ≪ and n ≫ .Proof. As for graded modules, we can give an inverse to the functor Can. Let E ∈ A A and σ : E → A [ t ] t ⊗ A E be a coaction. For each integer d , we define E n := ker( E σ − t n ⊗ id −−−−−−→ A [ t ] t ⊗ A E ) , where t n ⊗ id : E ∼ = A ⊗ A E → A [ t ] t ⊗ A E is induced by the A -module homo-morphism A → A [ t ] t defined by 1 t n . It remains to show that the naturalmap L n ∈ Z E n → E is an isomorphism. Since A A is locally finitely presentable, itsuffices to show that Hom A A ( X, L n ∈ Z E n ) → Hom A A ( X, E ) is an isomorphismfor all X ∈ A fp A . As X is finitely presentable, the coaction σ induces a coactionof A -modules σ X : Hom A A ( X, E ) → Hom A A ( X, A [ t ] t ⊗ A E ) ∼ = A [ t ] t ⊗ A Hom A A ( X, E ) . compatible with σ A . As Hom A A ( X, − ) is left exact, we have an identification ofHom A A ( X, E n ) withHom A A ( X, E ) n := ker (cid:0) Hom A A ( X, E ) σ X − t n ⊗ id −−−−−−−→ A [ t ] t ⊗ A Hom A A ( X, E ) (cid:1) Hom A A ( X, L d ∈ Z E n ) = L d ∈ Z Hom A A ( X, E ) n → Hom A A ( X, E ) is an isomor-phism by the usual argument: clearly Hom A A ( X, E ) n ∩ Hom A A ( X, E ) n ′ = 0 if n = n ′ and if α ∈ Hom A A ( X, E ), we can write σ X ( α ) as a finite sum P n t n ⊗ α n where all but finitely many of the α n are zero and the coaction axioms of σ X imply that α n ∈ Hom A A ( X, E ) n and that α = P n α n . (cid:3) We can use the above result to describe objects of A over Θ R = [Spec( R [ x ]) / G m ]for any k -algebra R and over ST R := [Spec (cid:0) R [ s, t ] / ( st − π ) (cid:1) / G m ] for any DVR R over k with uniformizing parameter π . Both descriptions are in terms of filteredobjects. Corollary 7.12. Suppose that A is locally noetherian. Let R be a k -algebra thenthe category A Θ R is equivalent to the category of sequences of morphisms E : · · · → E n +1 x −→ E n → · · · in A R , such that the restriction of E • along Spec( R ) ֒ → Θ R is colim E i , and • along B G m,R ֒ → Θ R is L n ∈ Z E n /x ( E n +1 ) .This equivalence restricts to an equivalence between M A (Θ R ) and the groupoid of Z -weighted filtrations · · · ⊂ E n +1 ⊂ E n ⊂ · · · of an object E ∞ in A R such that E n /E n +1 ∈ A R is flat and finitely presented, E n = E ∞ for n ≪ and E n = 0 for n ≫ .Proof. By Proposition 7.11, an object E in A Θ R is a Z -graded R [ x ]-module objectof A R . This corresponds to a Z -graded object E = L n ∈ Z E n in A R togetherwith a multiplication x : E → E mapping E n +1 to E n . The restriction of E along the open immersion Spec( R ) ֒ → Θ R is the G m -invariants (i.e. the degree0 component) of the Z -graded R [ x ] x -module object E ⊗ R [ x ] R [ x ] x ∈ A R [ x ] x . We ompute that E ⊗ R [ x ] R [ x ] x = E ⊗ R [ x ] (colim( · · · x −→ R x −→ R x −→ · · · )= colim( · · · x −→ E x −→ E x −→ · · · )= M n ∈ Z colim( · · · x −→ E n +1 x −→ E n x −→ · · · )whose G m -invariants is colim E n . The restriction of E along the closed immersion B G m,R ֒ → Θ R is the object E ⊗ R [ x ] ( R [ x ] /x ) ∼ = E/xE in A R with the Z -grading E/xE = L n ∈ Z E n /x ( E n +1 ). This proves the first claim.If E ∈ A Θ R is flat and finitely presented, then so is the corresponding objectin A R [ x ] (also denoted by E ). Since E ∈ A R [ x ] is flat, it is torsion free and thus x : E → E is injective. The base change E ⊗ R [ x ] ( R [ x ] /x ) = L E n /E n +1 is aflat and finitely presented object in A R which implies that the sum is finite andthe summands are finitely presentable and flat. This implies that E n stabilizesfor n ≪ 0. Also, if E ∈ A R [ x ] is finitely presentable, it must admit a surjection R [ x ] ⊗ R F → E for some F ∈ A fp R , corresponding to a map F → E in A R . Because F is finitely presentable it must factor through L n ≤ N E n ⊂ E for some N , andhence the image of R [ x ] ⊗ R F → E lies in L n ≤ N E n as well, which implies that E n = 0 for n ≫ · · · ⊂ E n +1 ⊂ E n ⊂ · · · satisfies the conditions above, then each E n is constructed as a finite sequence of extensions of flat and finitely presentableobjects in A R and is thus flat and finitely presentable. It follows that the graded R [ x ]-module objects R [ x ] ⊗ R E n are flat and finitely presentable as objects of A R [ x ] . Furthermore, the object E ∈ A Z R [ x ] corresponding to this Z -weightedfiltration can be constructed as a finite sequence of extensions of objects of theform R [ x ] ⊗ R E n h− n i , where the h− n i denotes a grading shift so that the resultingobject is homogeneous of degree n . Hence E ∈ A Θ R is finitely presentable andflat. (cid:3) Corollary 7.13. Suppose that A is locally noetherian. Let R be a DVR over k with uniformizing parameter π and residue field κ . The category A ST R is equiva-lent to the category of diagrams in A R (7.4) E : · · · s , , E n − s * * t j j E n s , , t l l E n +1 s * * t j j · · · t l l , satisfying st = ts = π , such that the restriction of E • along Spec( R ) s =0 ֒ −−→ ST R is colim( · · · s −→ E n − s −→ E n s −→ · · · ) , • along Spec( R ) t =0 ֒ −−→ ST R is colim( · · · t ←− E n − t ←− E n t ←− · · · ) , • along Θ κ s =0 ֒ −−→ ST R is the object corresponding to the sequence ( · · · t ←− E n /sE n − t ←− E n +1 /sE n t ←− · · · ) , and • along Θ κ t =0 ֒ −−→ ST R is ( · · · s −→ E n − /tE n s −→ E n /tE n +1 s −→ · · · ) .This equivalence restricts to an equivalence between M A (ST R ) and the groupoidconsisting of objects E such that: (a) s and t are injective, (b) s : E n − /tE n → E n /tE n +1 is injective for all n , (c) each E n ∈ A R is finitely presentable, (d) s : E n − → E n is an isomorphism for n ≫ , and (e) t : E n → E n − is anisomorphism for n ≪ . roof. The equivalence between A ST R and the category of diagrams as in (7.4) isargued as before (Corollary 7.12). Let us first show that flatness is characterizedby conditions (a) and (b). Suppose E ∈ A ST R is flat, then the pullbacks of E to R [ s, t ] / ( st − π ) and κ [ s ] (by setting t = 0) are both flat and in particular torsionfree which gives conditions (a) and (b). Conversely if E is given by the diagram(7.4) flatness is a local condition and (a) implies that the restriction of E to s = 0(or t = 0) is torsion free and thus flat. Condition (b) implies that the restrictionto κ [ s ] is s − torsion free and so E is also flat at the origin s = 0 = t by applying[AZ01, Lem. C1.12].To check that a finitely presentable, flat object E satisfies conditions (c)–(e)note that these are closed under cokernels, so we only need to check these for agenerating class of objects. Now if M ∈ A R is an R -module, then R [ s, t ] / ( st − π ) ⊗ R M ≃ M n< M · t − n ⊕ M ⊕ M n> M · s n , is a Z -graded R [ s, t ] / ( st − π )-module for which t is an isomorphism on negativelygraded pieces, and s is an isomorphism on positively graded pieces. Thereforefor any finitely presentable M ∈ A R , R [ s, t ] / ( st − π ) ⊗ R M ∈ A Z R satisfies theconditions (c)–(e) of the lemma, and the same holds if M ∈ A Z R is graded objectwith finitely many non-trivial graded pieces which are each finitely presentable.As any finitely generated object of A Z R [ s,t ] / ( st − π ) admits a surjection from anobject of this form, any E ∈ ( A Z R [ s,t ] / ( st − π ) ) fp admits a presentation of the form E ≃ coker( R [ s, t ] / ( st − π ) ⊗ M → R [ s, t ] / ( st − π ) ⊗ M ) for some M , M ∈ ( A Z R ) fp ,which proves (c)–(e) for E .Conversely, suppose that the diagram (7.4) satisfies the conditions (c)–(e) ofthe lemma. Then (d) and (e) imply that for N ≫ 0, the canonical homomorphismof graded R [ s, t ] / ( st − π )-modules R [ s, t ] / ( st − π ) ⊗ R L − N ≤ n ≤ N E n → E issurjective. Let K = L n K n be the kernel of this homomorphism. Because R [ s, t ] / ( st − π ) ⊗ R L − N ≤ n ≤ N E n satisfies conditions (d) and (e), it follows that K satisfies these conditions as well. Therefore R [ s, t ] / ( st − π ) ⊗ R L − M ≤ n ≤ M K n → K is surjective as well for M ≫ 0. By (c) each K n is the kernel of a surjectionof finitely presented objects, and it is thus finitely generated. We have thusexpressed E as the cokernel E = coker (cid:0) R [ s, t ] / ( st − π ) ⊗ R M | n |≤ M K n → R [ s, t ] / ( st − π ) ⊗ R M | n |≤ N E n (cid:1) of a homomorphism from a finitely generated object to a finitely presented object,so E is finitely presented. (cid:3) Now our extension results will follow from a basic result about extensions incodimension 2 that again carries over from quasi-coherent modules. Lemma 7.14. Let j : U → X be an open subscheme of a regular noetherianscheme of dimension whose complement is -dimensional. Then j ∗ : A U → A X maps flat objects to flat objects, and induces an equivalence between the fullsubcategory of flat objects over X and over U , with inverse given by j ∗ : A X → A U .Proof. It suffices to show that j ∗ preserves flat objects, and that both the unitand counit of the adjunction between j ∗ and j ∗ are equivalences on flat objects.The property of being an isomorphism is local by descent, so we may assume hat X = Spec( R ) is affine and U is the complement of a single closed point.Localizing further it suffices to consider the case of X = Spec( R ) for a regularring R of dimension 2 and U ⊂ X the complement of the closed point p whosemaximal ideal is generated by a regular sequence x, y ∈ R . In particular U =Spec( R x ) ∪ Spec( R y ).By construction (Lemma 7.6) we then have j ∗ ( E ) = ker( E | R x ⊕ E | R y → E | R xy ) . So the natural map j ∗ ( j ∗ ( E )) → E is an equivalence for any E as A U was definedby descent from affine schemes.Conversely if E ∈ A R is flat we can tensor E with the left exact sequence0 → R → R x ⊕ R y → R xy to find that the sequence0 → E → E | R x ⊕ E | R y → E | R xy is still exact, so E ∼ = j ∗ ( E | U ).Finally we must show that j ∗ preserves flat objects. By [AZ01, Lem. C1.12]we only need to show that Tor R ( R/ p , j ∗ E ) = 0for all prime ideals p of R . (Here Tor is defined as usual by choosing projectiveresolutions of R -modules.) For prime ideals in U this follows from flatness of E , so it suffices to show that Tor ( κ, j ∗ ( E )) = 0 for any flat E ∈ A U , where κ = R/ ( x, y ) is the residue field of the missing point, i.e. we need to show thattensoring j ∗ E with the Koszul complex0 → R → R ⊕ R → R → κ → → j ∗ ( E ) x ⊕ ( − y ) −−−−−→ j ∗ ( E ) ⊕ j ∗ ( E ) y ⊕ x −−−→ j ∗ ( E ) . This is the pushforward of the tensor product of E with the short exact sequenceof flat objects on U , 0 → O U → O U ⊕ O U → O U → j ∗ is left exact. (cid:3) This construction now allows us to check our conditions for the existence ofgood moduli spaces for M A . In the following we will assume that A is locallynoetherian and use the result [AZ01, Cor. B6.3] stating that then for any k -algebrathat is essentially of finite type (i.e. a localization of a finitely generated k -algebra)the category A R is again locally noetherian and thus finitely presentable andnoetherian are equivalent.We start with S -completeness. Lemma 7.15. If A is locally noetherian, then M A is S -complete with respect toany DVR R that is essentially of finite type over k .Proof. Let us denote by j : ST R \ ⊂ ST R the inclusion and take any E ∈ M A (ST R \ j ∗ ( E ) is flat, so we only need to show that it isfinitely presentable, i.e. we have to check conditions (c)–(e) of Corollary 7.13.We shall compute j ∗ ( E ) explicitly. Let us denote by K the fraction field of R .As E is flat, it is defined by an object F ∈ A K and two R -module subobjects E , E ⊂ F such that K ⊗ R E i → F is an isomorphism. et j i : Spec( R ) → ST R for i = 1 , j : Spec( K ) → ST R their intersection. Then by j ∗ ( E ) = ker (( j ) ∗ ( E ) ⊕ ( j ) ∗ ( E ) → ( j ) ∗ ( F ))To compute this, we describe it as graded R [ s, t ] / ( st − π )-module. Under thisdescription j is given by the graded inclusion R [ s, t ] / ( st − π ) ⊂ R [ t ± ] and j bythe graded inclusion R [ s, t ] / ( st − π ) ⊂ R [ s ± ]. ThusAs the maps E i → F are injective, we may thus identify j ∗ ( E ) ⊂ ( j ) ∗ ( F ) withthe intersection of two subobjects ( j ) ∗ ( E ) and ( j ) ∗ ( E ) under the equivalenceof Proposition 7.11, we compute( j ) ∗ ( F ) = R [ t ± ] ⊗ R F = M n F t n , ( j ) ∗ ( E ) = E ⊗ R R [ t ± ] = M n E t n , ( j ) ∗ ( E ) = E ⊗ R R [ s ± ] ≃ M n ∈ Z ( π − n · E ) t n ⊂ ( j ) ∗ ( F )where in the third line we have used the identification s = t − π . We computethat: j ∗ ( E ) ≃ M n ∈ Z (cid:0) E ∩ ( π − n · E ) (cid:1) t n ⊂ M n ∈ Z F t n . Now each graded piece of j ∗ ( E ) is finitely presentable because they are sub-objects of the noetherian object E (using that A R is locally noetherian). Theunion of the ascending sequence · · · ⊂ E ∩ ( π − n · E ) ⊂ E ∩ ( π − n − · E ) ⊂ · · · is E because K ⊗ R E ≃ F , and because E is finitely generated, this unionmust stabilize. By symmetry the same argument applies E thus j ∗ ( E ) is finitelypresentable. (cid:3) Next we show Θ-reductivity. Lemma 7.16. If A is locally noetherian, then M A is Θ -reductive with respect toany DVR R that is essentially of finite type over k .Proof. As in the proof of Lemma 7.15 let us denote by K the fraction field of R , j : U ֒ → Θ R the complement of the closed point and we take E ∈ M A ( U ). Thenby Lemma 7.14, j ∗ ( E ) is flat and so we need to show that it is finitely presentable.We pass to the presentation A R → Θ R , where the open subset U ⊂ A R cor-responding to U is covered by the two affine subschemes defined by R [ x ] ⊂ K [ x ]and R [ x ] ⊂ R [ x ± ]. Now E ∈ A U corresponds to an object F ∈ A ( K ), a R -submodule object E ⊂ F such that K ⊗ R E → F is an isomorphism, and aweighted descending filtration · · · F n +1 ⊂ F n ⊂ · · · ⊂ F satisfying the hypothesesof Corollary 7.12, then j ∗ ( E ) corresponds to the graded R [ x ]-module object j ∗ ( E ) = M n ∈ Z (cid:0) F n ∩ E (cid:1) x − n ⊂ M n ∈ Z F x − n = (Spec( K ) → Θ R ) ∗ ( F ) . Because A R is locally noetherian each graded piece G n := F n ∩ E of j ∗ ( E ) willbe finitely presentable, the maps G n +1 → G n are injective, G n = 0 for n ≫ G n = E for n ≪ 0. Thus j ∗ ( E ) is finitely presentable by Corollary 7.13. (cid:3) The valuative criteria for universal closedness turn out to be satisfied as well: emma 7.17. If A is a locally noetherian abelian category, the stack M A satis-fies the valuative criterion for universal closedness, i.e. the existence part of thevaluative criterion for properness, with respect to DVR’s which are essentially offinite type over k .Proof. If R is a DVR, the statement that an object E ∈ A R is flat if and only ifit is torsion free follows from [AZ01, Lem. C1.12] as the condition is equivalentto the vanishing of Tor . If j : Spec( K ) → Spec( R ) denotes the inclusion of thegeneric point, then for any E ∈ A K , we can write j ∗ ( E ) = S α F α as a directedunion of finitely generated (hence finitely presentable) subobjects which must betorsion free. If E is finitely generated then E = S α K ⊗ R F α must stabilize, sothere is some flat and finitely presentable object F α extending E . (cid:3) We will also use the following below: Lemma 7.18. Suppose that A is locally noetherian. If M A is an algebraic stackwith affine stabilizers, κ is a field of finite type over k , and E ∈ M A ( κ ) representsa closed point, then E is a semisimple object in A κ .Proof. Because E is finitely presented, it can not be expressed as an infinite sumof non-zero objects. Therefore, we only have to show that every finite filtrationof E splits. Now by Corollary 7.13 any finite filtration of E corresponds to a mapΘ κ → M A mapping 1 E . Because E is a closed point, the resulting map mustfactor through a map Θ κ → B κ Aut M A ( E ). We know from the classification oftorsors on Θ κ ([Hei17, Lem. 1.7] or [Hal14, Prop. A.1]) that any such map factorsthrough the projection Θ κ → B κ G m , and thus the corresponding filtration of E κ is split. (cid:3) Lemma 7.19. If M A is an algebraic stack locally of finite presentation over k ,then the diagonal of M A is affine.Proof. If R is a valuation ring over k with fraction field K and E, F ∈ M A ( R ),then F → K ⊗ R F is injective and hence so is the restriction map Hom R ( E, F ) → Hom K ( K ⊗ R E, K ⊗ R F ) ≃ Hom R ( E, K ⊗ R F ). This implies the valuative criterionfor separatedness of the diagonal of M A .For any ring R over k and E, F ∈ M A ( R ), we claim that the functor R ′ /R Hom R ′ ( R ′ ⊗ R E, R ′ ⊗ R F ) is represented by a separated algebraic space Hom R ( E, F )locally of finite presentation over R . First, observe that the subfunctor P ⊂ Aut R ( E ⊕ F ) classifying automorphisms of the form [ A C D ] is representable by aclosed subspace, because it is the preimage of the (closed) identity section underthe map of separated R -spaces Aut R ( E ⊕ F ) → Aut R ( E ⊕ F ) given by (cid:20) A BC D (cid:21) (cid:20) B (cid:21) . Next observe that we have a group homomorphism P → Aut R ( E ) × Aut R ( F )over R given by (cid:20) A C D (cid:21) ( A, D ) . The preimage of the (closed) identity section is the subgroup classifying auto-morphisms of the form [ C ], which is canonically identified with the functorHom( E, F ). Thus Hom( E, F ) is a closed subgroup of Aut R ( E ⊕ F ), and it isrepresentable, separated, and locally finitely presented over R . rom the functor of points definition of the algebraic space X := Hom R ( E, F ),there is a natural action of G m which scales the homomorphism. Furthermore,the resulting map G m × X → X extends (uniquely) to A × X , i.e. X = X + in theterminology of Section 2.3. Under these hypotheses, the morphism X + → X G m isaffine [AHR]. Also, the fixed locus X G m is the zero section X G m ≃ Spec( R ) ֒ → X ,and hence X = X + = Hom R ( E, F ) is affine as well.The algebraic R -space Isom R ( E, F ) is the closed subspace of Hom R ( E, F ) × Hom R ( F, E ) obtained as the preimage of the identity section under the map ofseparated R -schemes Hom R ( E, F ) × R Hom R ( F, E ) → Hom R ( E, E ) × Hom R ( F, F ).Hence Isom R ( E, F ) is affine, i.e. M A has affine diagonal. (cid:3) Construction of good moduli spaces. We now apply the previous dis-cussion to construct moduli good moduli spaces for objects in a k -linear abeliancategory A . As our results require linear reductivity we will now need to assumethat k is a noetherian commutative ring over a field of characteristic 0.Furthermore we assume that M A , the moduli functor of flat families of finitelypresentable objects of Definition 7.8, is an algebraic stack locally of finite typeand with affine diagonal over k . Example 7.20. Let X be a projective scheme over an algebraically closed field k , and consider the heart of a t -structure C ⊂ D b ( X ) which is noetherian andsatisfies the “generic flatness property” [AP06, Prop. 3.5.1]. Then if we considerthe ind-completion A := Ind( C ), M A is an open sub-functor of the moduli functor D bpug ( X ) of universally glueable relatively perfect complexes on X and is hencean algebraic stack locally of finite type with affine diagonal over k [Lie06], [Stacks,Tag 0DPW]. By [Pol07, Prop. 3.3.7], M A as defined above agrees with the mod-uli functor constructed in [AP06], which is the most commonly studied modulifunctor for flat families of objects in the heart of a t -structure.The first result concerns a situation in which no additional stability conditionis required. Theorem 7.21. Let k be a noetherian ring of characteristic and A be a locallynoetherian, cocomplete and k -linear abelian category. Assume that M A is an al-gebraic stack locally of finite type over k . Then any quasi-compact closed substack X ⊂ M A admits a proper good moduli space, and in this case points of X mustparameterize objects of A of finite length.Proof. The stack X is Θ-reductive (Lemma 7.16) and S -complete (Lemma 7.15)with respect to essentially finite type DVR’s because both properties pass to closedsubstacks, and it has affine diagonal by Lemma 7.19. Therefore Theorem A andRemark 5.5 imply the existence of a separated good moduli space X → X . Since X satisfies the existence part of the valuative criterion for properness with respectto essentially finite type DVR’s (Lemma 7.15), X is proper by Proposition 3.47and Remark 3.48. The fact that closed points of X are represented by semisimpleobjects in A κ for fields κ of finite type over k is Lemma 7.18. (cid:3) In general we will need a notion of “semistable” objects in A . As in applicationsdifferent notions of stability are used, we will use an abstract setup that includesmany of these. We will illustrate how classical notions of stability fit into thiscontext below.For any connected component ν ∈ π ( M A ), we let M ν A ⊂ M be the correspond-ing open and closed substack. Our notion of semistability on M ν A will be encoded y a locally constant function on | M A | p ν : π ( M A ) → V where V is a totally ordered abelian group, p ν ( E ) = 0 for any E ∈ M ν A , and p ν is additive in the sense that p ν ( E ⊕ F ) = p ν ( E ) + p ν ( F ). We will say thata point of M ν A represented by E ∈ A κ for some algebraically closed field κ over k , is semistable if for any subobject F ⊂ E , p ν ( F ) ≤ Note that this definition is unaffected by embedding V in a larger totally orderedgroup, so we may assume that V is a totally order vector space over R by theHahn embedding theorem.Using Corollary 7.12 to identify maps f : Θ κ → M A with Z -weighted descend-ing filtrations · · · ⊂ E w +1 ⊂ E w ⊂ · · · in A κ , we define a locally constant function ℓ : | Map k (Θ , M ν A ) | → V by the formula ℓ ( · · · ⊂ E w +1 ⊂ E w ⊂ · · · ) := X w wp ν ( E w /E w +1 ) . Lemma 7.22. A point x ∈ | M ν A | is unstable if and only if there is some f ∈| Map k (Θ , M ν A ) | with f (1) = x and ℓ ( f ) > .Proof. If F ⊂ E is a destabilizing subobject, then we can simply consider thefiltration where E = 0, E = F , and E w = E for w ≤ 0. This filtration has ℓ ( E • ) = deg ν ( F ) > Z -weighted filtration suchthat ℓ ( · · · ⊂ E w +1 ⊂ E w ⊂ · · · ) := P w wp ν ( E w /E w +1 ) > p v ( E ) = P w p ν ( E w /E w +1 ) = 0 it follows that for some index i we have p v ( E i ) = X w ≥ i p ν ( E w /E w +1 ) > (cid:3) Example 7.23 (Gieseker stability) . Let A = QCoh( X ) denote the categoryof quasi-coherent sheaves on a scheme X which is projective over a field k ofcharacteristic 0. Then M A is the usual stack of flat families of coherent sheaveson X . Fix a numerical K -theory class γ ∈ K num0 ( X ) corresponding to sheaveswhose support has dimension d , fix an ample line bundle O X (1) on X , and let M γ A denote the open and closed substack of objects of class γ . For any E ∈ Coh( X ),let P ( E ) = α n ( E ) t n + · · · + α ( E ) denote its Hilbert polynomial with respect to O X (1).We say that a coherent sheaf F of class γ is semistable with respect to O X (1)if for all nonzero subsheaves E ⊂ F , the asymptotic inequality p γ ( E ) := α d ( F ) P ( E ) − α d ( E ) P ( F ) ≤ t ≫ 0. If E ⊂ F is a nonzero subsheaf with dim(supp( E )) < d , then α d ( E ) = 0 but α d ( F ) > 0, so E destabilizes F . Therefore a semistable sheaf mustbe pure, and in this case P ( E ) /α d ( E ) and P ( F ) /α d ( F ) are the reduced Hilbertpolynomials, so our definition is equivalent to the classical definition of Giesekersemistability [HL10, Def. 1.2.4]. Note also that P ( F ) = P ( γ ) only depends on γ , so p γ defines a function p γ : M A → V d , where V d denotes the vector space of Note that because the flag space Map(Θ , M ν A ) × ev , M ν A , [ E ] Spec( κ ) of [ E ] : Spec( κ ) → M ν A is an algebraic space locally of finite type over κ , if there is a destabilizing subobject of E afterbase change to an arbitrary field extension κ ⊂ κ ′ , then there is a destabilizing subobject for E over κ , so this definition does not depend on the choice of representative. olynomials of degree ≤ d totally ordered by asymptotic inequality as t → ∞ .The function p γ is locally constant in flat families and additive in short exactsequences. Example 7.24 (Bridgeland stability) . Consider a projective scheme X over analgebraically closed field k of characteristic 0. A Bridgeland stability condition isdetermined by the heart of a t -structure C ⊂ D b ( X ), and a central charge homo-morphism Z : K (D b ( X )) → C , which we assume factors through the numerical k -theory K num0 (D b ( X )), i.e. the quotient K (D b ( X )) by the kernel of the Eulerpairing χ ( − , − ). The central charge Z is required to be a stability function on C with the Harder–Narasimhan property [Bri07, Prop. 5.3]. The simplest exampleis when X is a projective curve, C = Coh( X ) is the usual t -structure, and Z ( E ) = − deg( E ) + i rank( E ) . For a stability condition ( C , Z ) on a general X , we take the formula above asthe definition of rank and degree of objects in C . As before we let A := Ind( C ).Given a class γ ∈ K num0 (D b ( X )), there is an open and closed substack M γ A ⊂ M A whose k -points classify objects E ∈ A of numerical class γ . Then Bridgelandsemistability on the stack M γ A is determined by the degree function p γ ( E ) := deg( E ) rank( γ ) − deg( γ ) rank( E ) , so E ∈ M γ A is unstable if and only if there is a subobject F ⊂ E such that p γ ( F ) > p γ ( E ) = 0. Theorem 7.25. Let k be a noetherian ring of characteristic and A be a locallynoetherian, cocomplete and k -linear abelian category. Assume that M A is analgebraic stack locally of finite type over k . Let ν ∈ π ( M A ) be a connectedcomponent, and let p ν : π ( M A ) → V be an additive function defining a notion ofsemistability on M ν A , as above.If the substack of semistable points M ν, ss A ⊂ M ν A is open and quasi-compactthen M ν, ss A admits a separated good moduli space. If in addition M ν, ss A is the openpiece of a Θ -stratification of M ν A , then M ν, ss A admits a proper good moduli space.Proof. We have already seen that M ν A has affine diagonal (Lemma 7.19), andwith respect to essentially finite type DVR’s M ν A is Θ-reductive (Lemma 7.16), S -complete (Lemma 7.15), and satisfies the existence part of the valuative criterionfor properness (Lemma 7.17). It follows from Proposition 6.14 and Remark 6.16that M ν, ss A is Θ-reductive and S -complete with respect to essentially finite typeDVR’s as well , so it has a separated good moduli space M ν, ss A → M space byTheorem A and Remark 5.5. For the final statement, Lemma 7.17 and Corollary 6.12applied to the Θ-stratification of M ν A imply that M ν, ss A satisfies the existence partof the valuative criterion for properness with respect to essentially finite typeDVR’s and hence M is proper over Spec( k ) by Proposition 3.47 and Remark 3.48. (cid:3) Example 7.26 (Gieseker stability, continued) . The Harder–Narasimhan strat-ification with respect to Gieseker semistability defines a Θ-stratification of thestack Coh( X ) (this is essentially the content of [Nit11], using different language),and the semistable locus is open and bounded. Therefore Theorem 7.25 provides Technically it follows from the proof of Proposition 6.14 as we are only know here that M ν A is Θ-reductive and S-complete with respect to essentially of finite type DVR’s. n alternate construction of a proper good moduli space for Gieseker semistablesheaves. Example 7.27 (Bridgeland stability, continued) . We consider Bridgeland sta-bility conditions whose central charge factors through a fixed surjective homo-morphism cl : K (D b ( X )) → K num0 (D b ( X )) → Γ, where Γ is a finitely generatedfree abelian group, and we let Stab Γ ( X ) denote the space of Bridgeland stabilityconditions whose central charge factors through cl and which satisfy the supportproperty with respect to Γ. This is given the structure of a complex manifold insuch a way that the map Stab Γ ( X ) → Hom(Γ , C ) which forgets the t -structure isa local isomorphism [Bri07, Thm. 1.2].A stability condition ( C , Z ) is algebraic if Z (Γ) ⊂ Q + i Q . If ( C , Z ) is algebraic,then C is noetherian [AP06, Prop. 5.0.2]. Let Stab ∗ Γ ( X ) ⊂ Stab Γ ( X ) be a con-nected component which contains an algebraic stability condition σ = ( C , Z )for which: • the heart C satisfies the generic flatness condition, and • M γ, ssInd( C ) is bounded for every γ ∈ Γ.Then by [PT15, Prop. 4.12] the same is true for any algebraic stability conditionin the connected component Stab ∗ Γ ( X ). Furthermore, if σ = ( C , Z ) ∈ Stab Γ ( X )is algebraic and satisfies the generic flatness and boundedness conditions, thenthe Harder–Narasimhan filtration defines a Θ-stratification of M γ Ind( C ) whose openpiece is M γ, ssInd( C ) [Hal14, Prop. 5.40, § M γ, ssInd( C ) has a proper good moduli space for any algebraic stability condition inStab ∗ ( X ).Finally, we claim that for an arbitrary stability condition σ ∈ Stab ∗ ( X ), onecan find an algebraic stability condition σ ′ which defines the same moduli functor M γ, ssInd( C ) , so M γ, ssInd( C ) has a proper good moduli space for any stability conditionin Stab ∗ ( X ).To establish this claim, fix a class γ ∈ Γ. For any γ ′ ∈ Γ which is linearlyindependent of γ over Q , consider the real codimension 1 subset W γ ′ := { σ = ( C , Z ) ∈ Stab ∗ Γ ( X ) | Z ( γ ′ ) ∈ R > · Z ( γ ) } . If one restricts to a small compact neighborhood B ⊂ Stab ∗ Γ ( X ) containing σ ,then there is a finite subset S ⊂ Γ such that for any S ′ ⊂ S the moduli functor M γ, ssInd( C ) is constant for all σ ∈ C S ′ ∩ B [Tod08, Prop. 2.8], where C S ′ := (cid:18) [ γ ′ ∈ S ′ W γ ′ (cid:19) \ [ γ ′ / ∈ S ′ W γ ′ . M γ, ssInd( C ) is constant for σ ∈ B ∩ C S ′ because M γ, ssInd( C ) can only change if the set ofclasses γ ′ ∈ Γ with Z ( γ ′ ) ∈ R · Z ( γ ) changes, and the set of such γ ′ is constantfor σ ∈ C S ′ ∩ B .The condition that σ ∈ S γ ′ ∈ S ′ W γ ′ amounts to the claim that if W ⊂ Γ Q isthe span of γ and the γ ′ ∈ S ′ , then dim Q ( Z ( W )) = 1. We may write Z = Z ⊕ Z under a choice of splitting Γ Q ≃ W ⊕ U , and the condition now amounts torank( Z ) = 1. As rational points are dense in the space of rank 1 real matrices,we may find an arbitrarily small perturbation Z ′ of Z which is rational, and thisperturbed central charge defines our new algebraic stability condition σ ′ ∈ B ∩ C S ′ . . Good moduli spaces for moduli of G -torsors To illustrate our general theorems we now construct good moduli spaces forsemistable torsors under Bruhat–Tits group schemes. This generalizes the re-sults obtained by Balaji and Seshadri who constructed such moduli spaces forgenerically split groups over the complex numbers. In [Hei17], the third authoranalyzed the coarse moduli space for the moduli of stable bundles, whose exis-tence is guaranteed by the Keel–Mori theorem. Here we extend this analysis toinclude semistable bundles which are not stable. As in this article we are inter-ested in existence theorems for good moduli spaces (instead of adequate moduli)we will have to assume that we work over a base field k of characteristic 0 in thissection.Let us briefly introduce the setup from [Hei17]. Let C be a smooth geometri-cally connected, projective curve over a field k and G /C a smooth Bruhat–Titsgroup scheme over C , i.e., G is smooth a affine group scheme over C that hasgeometrically connected fibers, such that over some dense open subset U ⊂ C thegroup scheme is reductive and over all local rings at points p in Ram( G ) := C r U the group scheme G | Spec( O C,p ) is a connected parahoric Bruhat–Tits group. Thesimplest examples are of course reductive groups G × C .The stack of G -torsors is denoted by Bun G and this is a smooth algebraic stack.To define stability one usually chooses a line bundle on Bun G . As explainedin [Hei17, § L det given by the adjoint representation, i.e., the fiberat a bundle E ∈ Bun G p is L det , E = det( H ∗ ( C, ad( E ))) ∨ , where ad( E ) = E × G Lie( G p /C ) is the adjoint bundle of E .Next any collection of characters χ ∈ Q p ∈ Ram( G ) Hom( G p , G m ) defines linebundles on the classifying stacks B G p and one obtains a line bundle L χ on Bun G ,by pull back via the map Bun G → B G p defined by restriction of G torsors on C to the point p . We will denote by L det ,χ := L det ⊗ L χ , call the correspondingnotion of stability χ -stability and denote by Bun χ − ss G ⊂ Bun G the substack of χ -semistable torsors.Under explicit numerical conditions on χ this satisfies the positivity assumptionof loc.cit. [Hei17, Prop. 3.3], i.e., the restriction of L det to the affine GrassmannianGr G ,p classifying G bundles together with a trivialization on C r p is nef. Theparameter χ will be called positive if L det ,χ is ample on Gr G ,p for all p . It is calledadmissible if χ furthermore satisfies the numerical condition of [Hei17, Sec. 3.F]. Theorem 8.1 (Good moduli for semistable G -torsors) . Assume k is a field ofcharacteristic , C is a smooth, projective, geometrically connected curve over k , G is a parahoric Bruhat–Tits group scheme over k and χ is a admissible stabilityparameter. Then Bun χ − ss G admits a proper good moduli space M G . As remarked before, in the case that G is a generically split group scheme, thespace M G was constructed by Balaji and Seshadri [BS15].To prove the theorem we only need to check that Bun G satisfies the assumptionsof our main Theorems A, B and C, i.e., we need to show that the line bundle L det ,χ defines a well-ordered Θ-stratification of Bun G , that this stack satisfiesthe existence part of the valuative criterion for properness and that Bun χ − ss G isΘ-reductive and S -complete. This will be done in a series of Lemmas. emma 8.2. The canonical reduction of G -torsors defines a Θ -stratification on Bun G with semistable locus Bun χ − ss G . This stratification admits a well-ordering. Let us briefly recall the context of χ -stability. To simplify the presentationwill assume that our base field is algebraically closed. Then by [Hei17, Lem.3.9] any map f : Θ → Bun G arises as a Rees construction Rees( E P , λ ) from a G -bundle E , together with a reduction E P to a parabolic subgroup P ⊂ G and ageneric cocharacter λ : G m,k ( C ) → G k ( C ) that is dominant for P η . The argumentof the proof implies that all components of Map(Θ , Bun G ) can be identified withcomponents of Bun P for parabolics P equipped with a dominant λ . As reductionsto parabolics are determined by the induced filtration of the adjoint bundle, thisimplies in particular that the forgetful morphism from any component to Bun G is quasi-compact. We denote by wt E P ( λ ) the weight of f ∗ ( L χ ). Proof of Lemma 8.2. In [Hal14, Thm. 2.7, Simplif. 2.8, Simplif. 2.9] a list of nec-essary and sufficient conditions for a stratification to be a Θ-stratification is given.We will first verify these criteria (which we number as in loc.cit.), then brieflyrecall how the general proof works in the specific context of G -bundles in orderto illustrate the relation to classical arguments of Behrend. It suffices to assumethe ground field k is algebraically closed [Hal14, Lem. 2.14]. Existence and uniqueness of HN filtrations (1) : HN filtrations are maps f : Θ → Bun G corresponding to the canonical para-bolic reductions for unstable G -bundles constructed in [Hei17, Sec. 3.F]. To definecanonical reductions in this setup we fixed an invariant inner form ( , ) on thegeneric cocharacters of G defining the norm || · || we showed [Hei17, Prop. 3.6]that for any admissible χ and any G -torsor E there exists a canonical reduction E P to a parabolic subgroup P ⊂ G defined by a dominant cocharacter λ : G m,η → G η maximizing µ max ( E ) := max( wt EP ( λ ) || λ || ).The uniqueness of the reduction followed by checking that for any choice of ageneric maximal torus T η ⊂ G E η ∼ = G η that contains a maximal split torus the mapsending a Borel subgroup B η ⊂ G η containing T η to the cocharacter defined by − wt E B ( · ) defines a complementary polyhedron in the sense of Behrend [Beh95,Def. 2.1]. Consistency of HN filtrations (5) : Any canonical reduction f : Θ → Bun G of f (1) also defines the canonical re-duction of the associated graded bundle f (0). Indeed, the perspective of comple-mentary polyhedra shows that a canonical reduction E P can also be characterizedby the property that (1) wt E P ( λ ′ ) > λ ′ that is non-negative onany root of P , and (2) such that for any reductions E B of E P to a Borel subgroups B ⊂ P we have wt E B (ˇ α ) < α in the centralizer of λ [Beh95, Sec. 3] [HS10, Thm 4.3.2]. Specialization of HN filtrations (2 ′ ) : For any family E R ∈ Bun G ( R ) defined over a DVR R with fraction field K and residue field κ we have µ max ( E K ) ≤ µ max ( E κ ) and equality holds only if thecanonical filtration over K extends to the family, by [Hei17, Lem. 3.17]. Local finiteness (4) : or any family of G -bundles over a finite type scheme U , only finitely manyconnected components of Map(Θ , Bun G ) are necessary to realize the HN filtrationof every fiber E u for u ∈ U . This follows from the fact that there are only finitelymany λ such the canonical reduction of E u for some u ∈ U has type P λ , which isestablished in the proof of [Hei17, Prop. 3.18]. Completing the proof: The function wt EP ( λ ) || λ || defines a locally constant real valued function µ on thecomponents of Map(Θ , Bun G ) containing a HN filtration. We have verified that µ satisfies the conditions of [Hal14, Thm. 2.7, Simplif. 2.8, Simplif. 2.9] and thusdefines a weak Θ-stratification on Bun G in which the HN filtrations of unsta-ble points correspond to canonical reductions of G -bundles. In our context, theargument goes as follows:“Local finiteness” and quasi-compactness of Bun P λ → Bun G implies that thestratification of Bun G defined by µ max is constructible (See also the proof of[Hei17, Prop. 3.18]). Since the invariant µ max is semicontinuous, this implies thatfor any constant c the substacks Bun µ max ≤ c G defined by the condition µ max ( E ) ≤ c are open. To show that this defines a Θ-stratification we are therefore left toshow that the closed substacks Bun µ max ≤ c G \ Bun µ max Assume for simplicity that the ground field k is algebraically closed.The notion of χ -stability is controlled by the class ℓ = c ( L χ ) ∈ H (Bun G ; Q ). If p ∈ C is a regular point for G , then generic cocharacters of G induce cocharactersin G p , and under this map the norm || λ || on generic cocharacters can be inducedfrom a conjugation invariant norm on cocharacters of G p . It follows that || λ || isinduced by a class in H (Bun G ; Q ) defined via pullback(Sym ( N ∗ Q )) W ≃ H ( B G p ; Q ) → H (Bun G ; Q ) , along the restriction morphism Bun G → B G p , where N denotes the coweightlattice of G p and W denotes the Weyl group. herefore the function µ ( E P , λ ) = wt EP ( λ ) || λ || is a standard numerical invariant inthe sense of [Hal14, Def. 4.7] satisfying condition (R) by [Hal14, Lem. 4.10,Lem. 4.12].It is also not difficult to check this directly (See [Hal14, Rem. 4.11]). So in theproof of Lemma 8.2 one could also apply [Hal14, Thm. 4.38], which provides ashorter list of criteria for a numerical invariant to define a weak Θ-stratification.In particular this implies that condition (5) above is automatic here. Lemma 8.4. The stack Bun χ − ss G is S -complete and locally linearly reductive.Proof. S -completeness holds because of the existence of a blow up of ST R tolinking two specializations. L det ,χ is positive on the exceptional lines, so if theblow-up was necessary, one of the bundles was unstable.In particular every closed substack of Bun Bun χ − ss G G is again S -complete, so thatby Proposition 3.45 the automorphism groups of closed points are geometricallyreductive. As we assumed our base field to be of characteristic 0 in this section,these groups are linearly reductive. (cid:3) Lemma 8.5. The stack Bun χ − ss G is Θ -reductive.Proof. This again follows from semi-continuity of the numerical invariant as in[Hei17, Lem. 3.17]. We briefly recall the argument: Let R be a DVR withfraction field K and residue field κ . Let f : Θ R \ → Bun χ − ss G be a morphism.The restriction of f to Spec( R ) defines a family E R of G -torsors over C R .The restriction of f | Θ K defines a filtrations on E K and after possibly passing toa finite extension of R we may by [Hei17, Rem. 3.10] assume that this filtrationis given by a reduction E P K of E K to a parabolic subgroup P ⊂ G which is definedby a generic cocharacter λ : G m → G k ( η ) .As G k ( η ) / P k ( η ) is projective, any reduction of E K ( η ) to P extends to an opensubset U ⊂ C R whose the complement consists of finitely many closed points ofthe special fiber. Also the reduction over U κ extends canonically to a reduction E P ′ κ to a parabolic subgroup P ′ ⊂ G κ over C κ . Now, as in loc. cit. at any point p the adjoint bundle ad( E P ′ κ ) ⊂ ad( E κ ) is the saturation of the adjoint bundle atthe generic fiber. As χ is admissible this implies that either the reduction E P K extends over C R or the weight of the reduction E P ′ κ is strictly larger than theweight of f | Θ K , which is 0 as f is a reduction in Bun χ − ss G . As by assumption E κ is χ -semistable the weight cannot increase, so the reduction extends to C R . Nowwe can apply Lemma 6.15 to find that this filtration also lies in Bun χ,ss G and thusdefines an extension of f to Θ R . (cid:3) Proof of Theorem 8.1. We just proved that Bun χ − ss G is S -complete, locally lin-early reductive and Θ-reductive.By [Hei17, Prop. 3.3] the stack Bun G satisfies the existence criterion for proper-ness, i.e., if R is a DVR with fraction field K and E K ∈ Bun G ( K ) is a G -torsorover C K then there exists a finite extension R ′ of R such that E K extends to atorsor over C R . Therefore we can apply Theorem C to deduce the existence of aproper good moduli space. (cid:3) ppendix A. Strange gluing lemma In this section, we establish two gluing results: Theorem A.1 and Theorem A.5.Additionally, we apply both results to give a refinement of the classical semistablereduction theorem in GIT (Theorem A.8).A.1. Gluing results. Let R be a DVR with fraction field K and residue field κ , and let π ∈ R be a uniformizer parameter. For n > n, R = [Spec( R [ s, t ] / ( st n − π )) / G m ]where the G m -action is encoded by giving s weight n > t weight − 1. We have a closed immersion Θ κ ֒ → ST n, R defined by s = 0 and an openimmersion Spec( R ) ֒ → ST n, R defined by t = 0.We will denote 0 ∈ Spec( R ) as the closed point and 1 ∈ Θ k as the open point.Observe that any morphism ST n, R → X restricts to morphisms f : Θ κ → X and ξ : Spec( R ) → X along with an isomorphism φ : ξ (0) ≃ f (1) in X ( κ ). Theorem A.1. Let X be an algebraic stack with affine diagonal and locally offinite presentation over an algebraic space S . Let R be a DVR with residue field κ and consider morphisms f : Θ κ → X and ξ : Spec( R ) → X over S together withan isomorphism φ : ξ (0) ≃ f (1) . For all n ≫ , there is a morphism ST n, R → X unique up to unique isomorphism extending the triple ( f, ξ, φ ) . This theorem is inspired by the perturbation theorem [Hal14, Prop. 3.53],which is an analogous result for constructing map [ A κ / G m,κ ] → X from mapsfrom the loci { s = 0 } and { t = 0 } . Corollary A.2. In the context of Theorem A.1, for n ≫ the data of the mor-phisms f and ξ | Spec( R [ π /n ]) with isomorphism φ extends canonically to a mor-phism ST , R [ π /n ] → X .Proof. Compose the uniquely defined map ST n, R → X of Theorem A.1 with thecanonical map ST , R [ π /n ] → X induced by the map of graded algebras R [ s, t ] / ( st n − π ) → R [ π /n ][ s /n , t ] / ( s /n t − π ), where s /n has weight 1. (cid:3) In order to establish Theorem A.1, we will need to recall the following factconcerning pushouts. Lemma A.3. If Spec( A ) → Spec( B ) is a closed immersion and Spec( A ) → Spec( C ) is a morphism, then Spec( A ) (cid:15) (cid:15) / / Spec( C ) (cid:15) (cid:15) Spec( B ) / / Spec( B × A C ) is a pushout diagram in the category of algebraic stacks with affine diagonal.Proof. Ferrand established that the diagram is a pushout in the category of ringedspaces [Fer03, Thm. 5.1]. Temkin and Tyomkin establish that it is a pushoutin the category of algebraic spaces [TT16, Thm. 4.2.4]. The lemma follows byapplying [TT16, Thm 4.4.1] or [Hal17, Lem. A.4] to the pullback of the diagramunder an affine presentation of an algebraic stack X with affine diagonal. (cid:3) emma A.4. Let C = R [ t, π/t, π/t , . . . ] ⊂ R [ t ] t . The commutative diagram Spec( κ ) / / (cid:15) (cid:15) Θ κ (cid:15) (cid:15) Spec( R ) / / [Spec( C ) / G m ](A.1) is cartesian and a pushout diagram in the category of algebraic stacks with affinediagonal.Proof. Lemma A.3 implies that both diagramsSpec( κ [ t ] t ) / / (cid:15) (cid:15) Spec( κ [ t ]) (cid:15) (cid:15) Spec( R [ t ] t ) / / Spec( C ) G m × Spec( κ [ t ] t ) / / (cid:15) (cid:15) G m × Spec( κ [ t ]) (cid:15) (cid:15) G m × Spec( R [ t ] t ) / / G m × Spec( C )are pushout diagrams in the category of algebraic stacks with affine diagonal.Since (A.1) is the G m -quotient of the left diagram above, the statement followsfrom descent. (cid:3) Proof of Theorem A.1. By Lemma A.4, the triple ( f, ξ, φ ) glues to a morphism[Spec( C ) / G m ] → X unique up to unique isomorphism. Write C as a union C = S C n , where C n := R [ t, π/t n ] ⊂ R [ t ] t . Note that C n ∼ = R [ s, t ] / ( st n − π ) soin particular [Spec( C n ) / G m ] ∼ = ST n, R and that Spec( C ) → Spec( C n ) is G m -equivariant. As X → S is locally of finite presentation, for n ≫ C ) / G m ] → X factors uniquely as [Spec( C ) / G m ] → [Spec( C n ) / G m ] → X . (cid:3) To setup the second gluing result, for n > R [ t/π n ] ⊂ K [ t ] and the quotient stackΘ R,n = [Spec( R [ t/π n ]) / G m ]where t has weight − 1. We have a closed immersion B R G m ֒ → Θ R,n defined by t/π n = 0 and an open immersion Θ K ֒ → Θ R,n defined by π = 0. Observe thatany morphism Θ R,n → X restricts to morphisms g : B R G m → X and λ : Θ K → X along with an isomorphism φ : g | B K G m ≃ λ | B K G m . Theorem A.5. Let X be an algebraic stack with affine diagonal and locally offinite presentation over an algebraic space S . Let R be a DVR with fraction field K and consider morphisms g : B R G m → X and λ : Θ K → X over S togetherwith isomorphism φ : g | B K G m ≃ λ | B K G m . For all n ≫ , there is a morphism Θ R,n → X unique up to unique isomorphism extending the triple ( g, λ, φ ) . Remark A.6. Observe that Θ R,n ∼ = Θ R and that the above theorem states thatany triple ( g, λ, φ ) extends uniquely to a morphism Θ R → X after precomposing λ : Θ K → X with the isomorphism Θ K → Θ K , defined by t π n t , for n ≫ Lemma A.7. Let D = R [ t, t/π, t/π , . . . ] ⊂ K [ t ] . The commutative diagram B K G m / / (cid:15) (cid:15) B R G m (cid:15) (cid:15) Θ K / / [Spec( D ) / G m ](A.2) s cartesian and a pushout diagram in the category of algebraic stacks with affinediagonal.Proof. The proof is identical to the proof of Lemma A.4 using that D = R [ t ] t × K [ t ] t K [ t ]. (cid:3) Proof of Theorem A.5. By Lemma A.4, the triple ( g, λ, φ ) glues to a morphism[Spec( D ) / G m ] → X unique up to unique isomorphism. Writing D = S n R [ t/π n ]and using that X → S is locally of finitely presention, we have that for n ≫ D ) / G m ] → X factors uniquely through [Spec( R [ t/π n ]) / G m ] to yield amap [Spec( R [ t/π n ]) / G m ] → X . (cid:3) A.2. Semistable reduction in GIT.Theorem A.8. Let X be a noetherian algebraic stack with affine diagonal. As-sume that either (1) there is a good moduli space π : X → X or (2) X ∼ = [Spec( A ) / GL n ] and that the adequate moduli space π : X → X = Spec( A GL N ) is of finite type.Given a DVR R with fraction field K and a commutative diagram Spec( K ) / / (cid:15) (cid:15) X π (cid:15) (cid:15) Spec( R ) / / X there exists an extension of DVRs R → R ′ with K → K ′ := Frac( R ′ ) finitetogether with a morphism h : Spec( R ′ ) → X fitting into a commutative diagram Spec( K ′ ) / / (cid:15) (cid:15) Spec( K ) / / (cid:15) (cid:15) X π (cid:15) (cid:15) Spec( R ′ ) / / h ❥❥❥❥❥❥❥❥❥❥ Spec( R ) / / X such that h (0) ∈ | X × X Spec( κ ′ ) | is a closed point, where κ ′ is the residue field of R ′ . Remark A.9. If R is universally Japanese (e.g. excellent), then it can be ar-ranged that R → R ′ is finite. Remark A.10. It follows from the valuative criterion for universally closedness([LMB, Thm. 7.3]) that there exists a lift Spec( R ′ ) → X with K → K ′ a finitelygenerated field extension. Even in the case that X = [Spec( A ) /G ] → Spec( A G ) = X with G linearly reductive and A finitely generated over a field, the above resultdoes not seem to appear in the literature. Proof of Theorem A.8. Base changing by Spec( R ) → X , we reduce to the casethat X = Spec( R ). We are given a K -point x K ∈ X ( K ) and after possibly afinite extension of K (and a corresponding extension of R ), the unique closedpoint in π − ( π ( x K )) is represented by a K -point x ′ K . The closure { x ′ K } is flatover Spec( R ) and it follows from [EGA, IV.17.16.2] that after a finite extensionof K , the morphism { x ′ K } → Spec( R ) has a section z R . Note that K -points z K and x ′ K are isomorphic. By the Hilbert–Mumford criterion (Lemma 3.23), thespecialization x K z K can be realized by a morphism λ : Θ K → X after possiblya further finite extension of K .Restricting λ to 0 yields a map B K G m → X . Since Ψ : Map R ( B R G m , X ) → X ,induced by precomposing with Spec( R ) → B R G m , satisfies the valuative criteria or properness, B K G m → X extends to a map g : B R G m → X such that z R is iso-morphic to Ψ( g ). Since X → Spec( R ) is finitely presented ([AHR15, Thm. A.1]),Theorem A.5 implies that the maps λ : Θ K → X and g : B R G m → X glue to amap [Spec( R [ t/π n ]) / G m ] → X . Restricting this map to t/π n − π yields a lift h : Spec( R ) → X of x K .Finally, if h (0) ∈ | X κ | is not closed, then after possibly a finite extensionof the residue field κ , we may use the Hilbert–Mumford criterion to constructa map η : Θ κ → X κ which realizes the specialization of h (0) to a closed point.Corollary A.2 implies that after a finite extension of R , h and η extends to amap ST R → X . Restricting this map to s = t = √ π yields the desired extensionSpec( R [ √ π ]) → X . (cid:3) References [AB83] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces , Philos.Trans. Roy. Soc. London Ser. A (1983), no. 1505, 523–615.[AFS17] J. Alper, M. Fedorchuk, and D. I. Smyth, Second flip in the Hassett-Keel program:existence of good moduli spaces , Compos. Math. (2017), no. 8, 1584–1609.[AHHR] J. Alper, J. Hall, D. Halpern-Leistner, and D. Rydh, A non-local slice theorem forschemes and stacks , in preparation.[AHR] J. Alper, J. Hall, and D. Rydh, The ´etale local structure of algebraic stacks , in prepa-ration.[AHR15] , A Luna ´etale slice theorem for algebraic stacks , 2015, arXiv:1504.06467.[Alp10] J. Alper, On the local quotient structure of Artin stacks , Journal of Pure and AppliedAlgebra (2010), no. 9, 1576 – 1591.[Alp13] , Good moduli spaces for Artin stacks , Ann. Inst. Fourier (Grenoble) (2013),no. 6, 2349–2402.[Alp14] , Adequate moduli spaces and geometrically reductive group schemes , Algebr.Geom. (2014), no. 4, 489–531.[AP06] D. Abramovich and A. Polishchuk, Sheaves of t-structures and valuative criteria forstable complexes , Journal fur die reine und angewandte Mathematik (Crelles Journal) (2006), no. 590, 89–130.[AZ01] M. Artin and J. J. Zhang, Abstract hilbert schemes , Algebras and representation theory (2001), no. 4, 305–394.[Beh] K. Behrend, The lefschetz trace formula for the moduli stack of principal bundles , avail-able online, .[Beh95] , Semi-stability of reductive group schemes over curves , Math. Ann. (1995),no. 2, 281–305.[Bri07] T. Bridgeland, Stability conditions on triangulated categories , Ann. of Math. (2) (2007), no. 2, 317–345.[BS15] V. Balaji and C. S. Seshadri, Moduli of parahoric G -torsors on a compact riemannsurface , J. Algebraic Geom. (2015), no. 1, 1–49.[Cat18] M. de Cataldo, in preparation , 2018.[CG13] J. Calabrese and M. Groechenig, Moduli problems in abelian categories and the recon-struction theorem , arXiv preprint arXiv:1310.6600 (2013).[CL10] P.-H. Chaudouard and G. Laumon, Le lemme fondamental pond´er´e. I. Constructionsg´eom´etriques , Compos. Math. (2010), no. 6, 1416–1506.[Dri13] V. Drinfeld, On algebraic spaces with an action of G m , 2013, arXiv:1308.2604.[EGA] A. Grothendieck, ´El´ements de g´eom´etrie alg´ebrique , I.H.E.S. Publ. Math. 4, 8, 11, 17,20, 24, 28, 32 (1960, 1961, 1961, 1963, 1964, 1965, 1966, 1967).[Fal93] G. Faltings, Stable G -bundles and projective connections , J. Algebraic Geom. (1993),no. 3, 507–568.[Fer03] D. Ferrand, Conducteur, descente et pincement , Bull. Soc. Math. France (2003),no. 4, 553–585.[Gai05] D. Gaitsgory, The notion of category over an algebraic stack , arXiv preprintmath/0507192 (2005).[Hal14] D. Halpern-Leistner, On the structure of instability in moduli theory , 2014,arXiv:1411.0627. Hal17] J. Hall, Openness of versality via coherent functors , J. Reine Angew. Math. (2017),137–182.[Hei08a] J. Heinloth, Bounds for Behrend’s conjecture on the canonical reduction , Int. Math.Res. Not. IMRN (2008), no. 14.[Hei08b] , Semistable reduction for G -bundles on curves , J. Algebraic Geom. (2008),no. 1, 167–183.[Hei17] , Hilbert-Mumford stability on algebraic stacks and applications to G -bundles oncurves , ´Epijournal Geom. Alg´ebrique (2017), Art. 11, 37.[HL10] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves , second ed.,Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010.[HLP14] D. Halpern-Leistner and A. Preygel, Mapping stacks and categorical notions of proper-ness , 2014, arXiv:1402.3204.[HR16] J. Hall and D. Rydh, Mayer-Vietoris squares in algebraic geometry , arXiv e-prints(2016), arXiv:1606.08517, arXiv:1606.08517.[HS10] J. Heinloth and A. Schmitt, The cohomology rings of moduli stacks of principal bundlesover curves , Doc. Math. (2010), 423–488.[Kem78] G. R. Kempf, Instability in invariant theory , Annals of Mathematics (1978), no. 2,299–316.[KM97] S. Keel and S. Mori, Quotients by groupoids , Ann. of Math. (2) (1997), no. 1,193–213.[Lan75] S. G. Langton, Valuative criteria for families of vector bundles on algebraic varieties ,Ann. of Math. (2) (1975), 88–110.[Lie06] M. Lieblich, Moduli of complexes on a proper morphism , J. Algebraic Geom. (2006),no. 1, 175–206.[LMB] G. Laumon and L. Moret-Bailly, Champs alg´ebriques , Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge., vol. 39, Springer-Verlag, Berlin, 2000.[McN05] G. J. McNinch, Optimal SL(2) -homomorphisms , Comment. Math. Helv. (2005),no. 2, 391–426.[Ngˆo06] B. C. Ngˆo, Fibration de Hitchin et endoscopie , Invent. Math. (2006), no. 2, 399–453.[Nit11] N. Nitsure, Schematic Harder-Narasimhan stratification , Internat. J. Math. (2011),no. 10, 1365–1373.[Pol07] A. Polishchuk, Constant families of t -structures on derived categories of coherentsheaves , Mosc. Math. J. (2007), no. 1, 109–134, 167.[Pop73] N. Popescu, Abelian categories with applications to rings and modules , Academic Press,London-New York, 1973, London Mathematical Society Monographs, No. 3.[PT15] D. Piyaratne and Y. Toda, Moduli of Bridgeland semistable objects on 3-folds andDonaldson-Thomas invariants , ArXiv e-prints (2015), arXiv:1504.01177.[RG71] Michel Raynaud and Laurent Gruson, Crit`eres de platitude et de ivit´e. Techniques de“platification” d’un module , Invent. Math. (1971), 1–89.[Ses72] C. S. Seshadri, Quotient spaces modulo reductive algebraic groups , Ann. of Math. (2) (1972), 511–556; errata, ibid. (2) 96 (1972), 599.[Ses77] , Geometric reductivity over arbitrary base , Advances in Math. (1977), no. 3,225–274.[Stacks] The Stacks Project Authors, Stacks Project , http://stacks.math.columbia.edu , 2018.[Ste75] R. Steinberg, Torsion in reductive groups , Advances in Math. (1975), 63–92.[Tel00] C. Teleman, The quantization conjecture revisited , Ann. of Math. (2) (2000), no. 1,1–43.[Tod08] Y. Toda, Moduli stacks and invariants of semistable objects on K surfaces , Adv. Math. (2008), no. 6, 2736–2781.[TT16] M. Temkin and I. Tyomkin, Ferrand pushouts for algebraic spaces , Eur. J. Math. (2016), no. 4, 960–983.(2016), no. 4, 960–983.