aa r X i v : . [ m a t h . AG ] N ov INVARIANT RINGS THROUGH CATEGORIES
JAROD ALPER AND A. J. DE JONG
Abstract.
We formulate a notion of “geometric reductivity” in an abstractcategorical setting which we refer to as adequacy. The main theorem statesthat the adequacy condition implies that the ring of invariants is finitely gen-erated. This result applies to the category of modules over a bialgebra, thecategory of comodules over a bialgebra, and the category of quasi-coherentsheaves on a finite type algebraic stack over an affine base.
Contents
1. Introduction 12. Setup 23. Axioms 34. Direct summands 45. Commutativity 56. Direct products 57. Symmetric products 68. Ring objects 69. Commutative ring objects and modules 810. Finiteness conditions 911. Adequacy 1012. Preliminary results 1213. The main result 1514. Quasi-coherent sheaves on algebraic stacks 1815. Bialgebras, modules and comodules 1916. Adequacy for a bialgebra 2017. Coadequacy for a bialgebra 21References 221.
Introduction
A fundamental theorem in invariant theory states that if a reductive group G overa field k acts on a finitely generated k -algebra A , then the ring of invariants A G is finitely generated over k (see [MFK94, Appendix 1.C]). Mumford’s conjecture,proven by Haboush in [Hab75], states that reductive groups are geometrically re-ductive; therefore this theorem is reduced to showing that the ring of invariantsunder an action by a geometrically reductive group is finitely generated, which wasoriginally proved by Nagata in [Nag64]. Nagata’s theorem has been generalized to various settings. Seshadri showed ananalogous result for an action of a “geometrically reductive” group scheme over auniversally Japanese base scheme (see [Ses77]). In [BFS92], the result is generalizedto an action of a “geometrically reductive” commutative Hopf algebra over a fieldon a coalgebra. In [KT08], an analogous result is proven for an action of a “geo-metrically reductive” (non-commutative) Hopf algebra over a field on an algebra.In [Alp08] and [Alp10], analogous results are shown for the invariants of certainpre-equivalence relations; moreover, [Alp10] systematically develops the theory ofadequacy for algebraic stacks.These settings share a central underlying “adequacy” property which we formulatein an abstract categorical setting. Namely, consider a homomorphism of commuta-tive rings R → A . Consider an R -linear ⊗ -category C with a faithful exact R -linear ⊗ -functor F : C −→
Mod A such that C is endowed with a ring object O ∈
Ob( C ) which is a unit for ⊗ . Forprecise definitions, please see Situation 2.1. One can then defineΓ : C −→
Mod R , F 7−→
Mor C ( O , F ) . Adequacy means (roughly) in this setting that Γ satisfies: if
A → B is a surjectionof commutative ring objects and if f ∈ Γ( B ), then there exists g ∈ Γ( A ) with g f n for some n >
0. The main theorem of this paper is Theorem 13.5 whichstates (roughly) that if Γ is adequate, then(1) Γ( A ) is of finite type over R if A is of finite type, and(2) Γ( F ) is a finite type Γ( A )-module if F is of finite type.Note that additional assumptions have to be imposed on the categorical setting inorder to even formulate the result.In the final sections of this paper, we show how the abstract categorical settingapplies to (a) the category of modules over a bialgebra, (b) the category of comod-ules over a bialgebra, and (c) the category of quasi-coherent sheaves on a finitetype algebraic stack over an affine base. Thus the main theorem above unifies andgeneralizes the results mentioned above, which was the original motivation for thisresearch.What is lacking in this theory is a practical criterion for adequacy. Thus we wouldlike to ask the following questions: Is there is notion of reductivity in the categoricalsetting? Is there an abstract analogue of Haboush’s theorem? We hope to returnto these question in future research. Conventions.
Rings are associative with 1. Abelian categories are additive cate-gories with kernels and cokernels such that Im ∼ = Coim for any morphism.2. Setup
In this section, we introduce the types of structure we are going to work with. Wekeep the list of basic properties to an absolute minimum, and later we introduceadditional axioms to impose.
Situation 2.1.
We consider the following systems of data:(1) R → A is a map of commutative rings, NVARIANT RINGS THROUGH CATEGORIES 3 (2) C is an R -linear abelian category,(3) ⊗ : C × C → C is an R -bilinear functor,(4) F : C →
Mod A is a faithful exact R -linear functor,(5) there is a given bifunctorial isomorphism γ F , G : F ( F ) ⊗ A F ( G ) −→ F ( F ⊗ G ) , (6) there exist functorial isomorphisms τ F , G , H : ( F ⊗ G ) ⊗ H −→ F ⊗ ( G ⊗ H )which are compatible with the usual associativity of tensor products of A -modules via γ , and(7) there is an object O of C endowed with functorial isomorphisms µ : O ⊗F → F , and µ : F ⊗ O → F such that F ( O ) = A and the isomorphismscorrespond to the usual isomorphisms A ⊗ A M = M ⊗ A A = M (via γ above).If an associativity constraint τ as above exists, then it is uniquely determined bythe condition that it agrees with the usual associativity constraint for A -modules(as F is faithful). Hence we often do not list it as part of the data, and we say “Let( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1”.Note that in particular O ⊗ O = O , and hence that O is a ring object of C (seeSection 8), and for this ring structure every object of C is in a canonical way an O -module. Definition 2.2.
In the situation above we define the global sections functor to bethe functor Γ :
C −→
Mod R , F 7−→ Γ( F ) = Mor C ( O , F ) . Note that Γ( F ) ⊂ F ( F ) since the functor F is faithful. There are canonical mapsΓ( F ) ⊗ R Γ( G ) −→ Γ( F ⊗ G ) defined by mapping the pure tensor f ⊗ g to the map O = O ⊗ O f ⊗ g −−−→ F ⊗ R G For any pair of objects F , G of C there is a commutative diagramΓ( F ) ⊗ R Γ( G ) (cid:15) (cid:15) / / Γ( F ⊗ G ) (cid:15) (cid:15) F ( F ) ⊗ A F ( G ) F ( F ⊗ G )In particular, there is a natural Γ( O )-module structure on Γ( F ) for every object F of C . 3. Axioms
The following axioms will be introduced throughout the text. For the convenienceof the reader, we list them here.
Definition 3.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axioms:(D) The category C has arbitrary direct summands, and ⊗ , F , and Γ commutewith these. ALPER AND DE JONG (C) There exist functorial isomorphisms σ F , G : F ⊗ G −→ G ⊗ F such that σ F , G is via F and γ compatible with the usual commutativity constraint M ⊗ A N ∼ = N ⊗ A M on A -modules.(I) The category C has arbitrary direct products, and F commutes with them.(S) For every object F of C and any n ≥ F ⊗ n −→ Sym n C ( F )such that the map of A -modules F ( F ⊗ n ) −→ F (Sym n C ( F )) factors throughthe natural surjection F ( F ) ⊗ n → Sym nA ( F ( F )), and such that Sym n C ( F ) isuniversal with this property.(L) Every object F of C is a filtered colimit F = colim F i of finite type objects F i such that F ( F ) = colim F ( F i ).(N) The ring A is Noetherian.(G) The functor Γ is exact.(A) For every surjection of weakly commutative ring objects A → B in C with A locally finite, and any f ∈ Γ( B ), there exists an n > g ∈ Γ( A ) such that g f n in Γ( B ).Terminology used above: An object F of C is finite type if F ( F ) is finite type, seeDefinition 10.1. A ring object A , see Definition 8.1, is weakly commutative if F ( A )is commutative, see Definition 9.1. An object F of C is locally finite if it is a filteredcolimit F = colim F i of finite type objects F i such that also F ( F ) = colim F ( F i ),see Definition 11.2. 4. Direct summands
We cannot prove much without the following axiom.
Definition 4.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axiom:(D) The category C has arbitrary direct summands, and ⊗ , F , and Γ commutewith these.This implies that C has colimits and that ⊗ , F and Γ commute with these. Lemma 4.2.
Assume that we are in Situation 2.1 and that the axiom (D) holds.Then Γ has a left adjoint O ⊗ R − : Mod R −→ C with O ⊗ R R ∼ = O , and F ( O ⊗ R M ) = A ⊗ R M . Moreover, for any object F of C there is a canonical isomorphism F ⊗ ( O ⊗ R M ) = ( O ⊗ R M ) ⊗ F which reducesto the obvious isomorphism on applying F .Proof. For any R -module M choose a presentation L j ∈ J R → L i ∈ I R → M → O ⊗ R M = Coker( M j ∈ J O −→ M i ∈ I O )where the arrow is given by the same matrix as the matrix used in the presentationfor M . With this definition it is clear that F ( O ⊗ R M ) = A ⊗ R M . Moreover, sincethere is an exact sequence M j ∈ J O −→ M i ∈ I O −→ O ⊗ R M −→ C ( O ⊗ R M, F ) = Mor R ( M, Γ( F )). We leavethe proof of the last statement to the reader. (cid:3) NVARIANT RINGS THROUGH CATEGORIES 5
In the situation of the lemma we will write M ⊗ R F instead of the more clumsynotation M ⊗ R O ⊗ F . Remark . Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1, and further assume(D) holds. By Lemma 4.2 above, we have a diagram of functorsMod R O⊗ R − + + A ⊗ R − ( ( QQQQQQQQQQQQ C F (cid:15) (cid:15) Γ l l Mod A where F ◦ ( O ⊗ R − ) = ( A ⊗ R − ), and O ⊗ R − is a left adjoint to Γ.5. Commutativity
Definition 5.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axiom:(C) There exist functorial isomorphisms σ F , G : F ⊗ G −→ G ⊗ F such that σ F , G is via F and γ compatible with the usual commutativity constraint M ⊗ A N ∼ = N ⊗ A M on A -modules.As in the case of the associativity constraint, if such maps σ F , G exist, then theyare unique. 6. Direct products
Definition 6.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axiom:(I) The category C has arbitrary direct products, and F commutes with them.If this is the case, then the category C has inverse limits and the functor F commuteswith them, which is why we use the letter (I) to indicate this axiom.In the following lemma and its proof we will use the following abuse of notation.Suppose that F , G are two objects of C , and that α : F ( F ) → F ( G ) is an A -modulemap. We say that α is a morphism of C if there exists a morphism a : F → G in C such that F ( a ) = α . Note that if a exists it is unique. Lemma 6.2.
Assume we are in Situation 2.1 and that (I) holds. Let F , G be twoobjects of C . Let α : F ( F ) → F ( G ) be an A -module map. The functor C −→
Sets , H 7−→ { ϕ ∈ Mor C ( G , H ) | F ( ϕ ) ◦ α is a morphism of C} is representable. The universal object G → G ′ is a surjection.Proof. Since C is abelian, any morphism π : G → H factors uniquely as
G →H ′ → H where the first map π ′ is a surjection and the second is an injection. If F ( π ) ◦ α = F ( a ) is a morphism of C , then a factors through H ′ and we see that F ( π ′ ) ◦ α is a morphism of C . Hence it suffices to consider surjections. Considerthe set T = { π : G → H π } of surjections π such that F ( π ) ◦ α is a morphism of C .Set G ′ = Im( G −→ Y π ∈ T H π ) . The rest is clear. (cid:3)
ALPER AND DE JONG Symmetric products
We introduce the axiom (S) and show that either axiom (I) or (C) implies (S).
Definition 7.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axiom:(S) For every object F of C and any n ≥ F ⊗ n −→ Sym n C ( F )such that the map of A -modules F ( F ⊗ n ) −→ F (Sym n C ( F )) factors throughthe natural surjection F ( F ) ⊗ n → Sym nA ( F ( F )), and such that Sym n C ( F ) isuniversal with this property.Note that if axiom (S) holds, then the universality implies the rule F Sym n C ( F )is a functor. Moreover, for every n, m ≥ n C ( F ) ⊗ Sym m C ( F ) −→ Sym n + m C ( F ) . If axiom (D) holds as well, then this will turn L n ≥ Sym n C ( F ) into a weakly com-mutative ring object of C (see Definitions 8.1 and 9.1 below). Lemma 7.2.
In Situation 2.1, if either axiom (C) or (I) holds, then axiom (S)holds.Proof.
Suppose (C) holds. If F is an object of C , using the maps σ F , F we get anaction of the symmetric group S n on n letters on F ⊗ n (to see that it is an actionof S n apply the faithful functor F ). Thus, Sym n C ( F ) can be defined as the cokernelof a map M τ ∈ S n F ⊗ n −→ F ⊗ n where in the summand corresponding to τ we use the difference of the identity andthe map corresponding to τ .Suppose (I) holds. Let F be an object of C . The quotient F ⊗ n → Sym n C ( F ) ischaracterized by the property that if a : F ⊗ n → G is a map such that F ( a ) factorsthrough F ( F ) ⊗ n → Sym nA ( F ( F )) then a factors in C through the map to Sym n C ( F ).To prove such a quotient exists apply Lemma 6.2 to the map M τ ∈ S n F ( F ) ⊗ n −→ F ( F ) ⊗ n mentioned above. (cid:3) Ring objects
Definition 8.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1.(1) A ring object A in C consists of an object A of C endowed with maps O → A and µ A : A ⊗ A → A which on applying F induce an A -algebra structureon F ( A ).(2) If A is a ring object of C , then a (left) module object over A is an object F endowed with a morphism µ F : A ⊗ F → F such that F ( A ) ⊗ A F ( F ) → F ( F ) induces an F ( A )-module structure on F ( F ). NVARIANT RINGS THROUGH CATEGORIES 7 If A is a ring object of C , then Γ( A ) inherits an R -algebra structure in a naturalmanner. In other words, we have the following diagram of rings R / / (cid:15) (cid:15) A (cid:15) (cid:15) Γ( O ) / / Γ( A ) / / F ( A )In the same vein, given a A -module F the global sections Γ( F ) are a Γ( A )-modulein a natural way. Let Mod A denote the category of A -modules. Lemma 8.2.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. If A is a ringobject in C , then the category Mod A is abelian.Proof. Let ϕ : F → G be a map of A -modules. Set K = Ker( ϕ ) and Q = Coker( ϕ )in C . We claim that both K and Q have natural A -module structure that turn theminto the kernel and cokernel of ϕ in Mod A . To see this for K consider the map A ⊗ K → A ⊗ F → F
Its composition with the map to G is zero as ϕ is a map of A -modules. Hence wesee that it factors into a map A ⊗ K → K . To get the module structure for Q , notethat the sequence A ⊗ F → A ⊗ G → A ⊗ Q → F . Hence the module structure on G inducesone on Q . We omit checking that these structures do indeed give the kernel andcokernel of ϕ in Mod A . (cid:3) Let us use Hom A ( − , − ) for the morphisms in the category Mod A . Note thatΓ( F ) = Mor C ( O , F ) = Hom A ( A , F )for F ∈
Mod A . The map from the left to the right associates to f : O → F themap A = A ⊗ O ⊗ f −−−→ A ⊗ F µ F −−→ F . Lemma 8.3.
In Situation 2.1 assume axiom (D) and let A be a ring object in C .Then the functor Γ :
Mod A −→ Mod Γ( A ) has a right adjoint A ⊗ Γ( A ) − : Mod Γ( A ) −→ Mod A . We have
A ⊗ Γ( A ) Γ( A ) = A and F ( A ⊗ Γ( A ) M ) = F ( A ) ⊗ Γ( A ) M .Proof. The proof is identical to the argument of Lemma 4.2 using that Γ( F ) =Hom A ( A , F ) for any A -module F . (cid:3) Remark . Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Assume axiom(D). Let A be a ring object, and let S be a set. We can define the polynomialalgebra over A as the ring object A [ x s ; s ∈ S ] = A ⊗ Γ( A ) (Γ( A )[ x s ; s ∈ S ])Explicitly A [ x s ; s ∈ S ] = L I A x I where I runs over all functions I : S → Z ≥ withfinite support. The symbol x I = Q x I ( s ) s indicates the corresponding monomial. ALPER AND DE JONG
The multiplication on A [ x s ; s ∈ S ] is defined by requiring the “elements” of A tocommute with the variables x s .A homomorphism A [ x s ; s ∈ S ] → B of ring objects is given by a homomorphism A → B of ring objects together with some elements y s ∈ Γ( B ) which commute withall elements in the image of F ( A ) → F ( B ).9. Commutative ring objects and modules
Definition 9.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. A ring object A is called weakly commutative if F ( A ) is commutative. Lemma 9.2.
In Situation 2.1. If A is a weakly commutative ring and I ⊂ A is aleft ideal, then I is a two-sided ideal and A / I is a weakly commutative ring.Proof. Consider the image I ′ of the multiplication A ⊗ I → A . By assumption F ( I ′ ) = F ( I ), hence we have equality. The final assertion is clear. (cid:3) In order to define the tensor product of two modules over a ring object we use thenotion of commutative modules.
Definition 9.3.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1.(1) A ring object A is called commutative if there exists an isomorphism σ : A ⊗ A → A ⊗ A which under F gives the usual flip isomorphism andwhich is compatible with the multiplication (so in particular A is weaklycommutative).(2) A module object F over a ring object A is said to be commutative if thereexists an isomorphism σ : F ⊗ A → A ⊗ F which on applying F gives theusual flip isomorphism.It is clear that if axiom (C) holds, then any weakly commutative ring object iscommutative and all module objects are automatically commutative. Let us de-note Mod c A the category of all commutative A -modules. This category always hascokernels, but not necessarily kernels. Lemma 9.4.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Let A be acommutative ring object of C . The category Mod c A is abelian in each of the followingcases: (1) axiom (C) holds, or (2) the ring map F ( A ) → F ( A ) ⊗ A F ( A ) is flat.The second condition holds for example if A → F ( A ) is either flat or surjective.Proof. In case (1) we have Mod A = Mod c A so the statement follows from Lemma8.2. For case (2), let ϕ : F → G be a map of commutative A -modules. We set K = Ker( ϕ ) and Q = Coker( ϕ ) in C , and we know that these are kernels andcokernels in Mod A . The diagram with exact rows F ⊗ A / / σ (cid:15) (cid:15) G ⊗ A / / σ (cid:15) (cid:15) Q ⊗ A / / (cid:15) (cid:15) A ⊗ F / / A ⊗ G / / A ⊗ Q / / NVARIANT RINGS THROUGH CATEGORIES 9 defines the commutativity map σ for Q . But in general we do not know that themap K ⊗ A → F ⊗ A is injective. After applying F this becomes the map F ( K ) ⊗ A F ( A ) → F ( F ) ⊗ A F ( A )By our discussion in Section 8 we know that B = F ( A ) is a commutative A -algebra,and F ( K ) ⊂ F ( F ) is an inclusion of B -modules. Note that for a B -module M wehave M ⊗ A B = M ⊗ B ( B ⊗ A B ). Hence the injectivity of the last displayed mapis clear if property (2) holds, and in this case we get the commutativity restraintfor K also. (cid:3) If A is a commutative ring object of C and F , G are module objects over A , and F is commutative then we define F ⊗ A G := Coequalizer ofgoing aroundboth ways A ⊗ F ⊗ G σ ⊗ / / µ ⊗ (cid:15) (cid:15) F ⊗ A ⊗ G ⊗ µ (cid:15) (cid:15) F ⊗ G / / F ⊗ G
Then it is clear that there is a canonical isomorphism γ A : F ( F ) ⊗ F ( A ) F ( G ) −→ F ( F ⊗ A G )which is functorial in the pair ( F , G ). In particular, it is clear that there arefunctorial isomorphisms µ A : A ⊗ A F −→ F , µ A : F ⊗ A A −→ F for any commutative A -module F (via σ and the multiplication map for F ). Lemma 9.5.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Let A be acommutative ring object of C . Assume the category Mod c A is abelian. Then ( R → F ( A ) , Mod c A , ⊗ A , F, γ A , A , µ A ) is another set of data as in Situation 2.1. Furthermore, if axiom (D) is satisfied for ( R → A, C , ⊗ , F, γ, O , µ ) , then it is also satisfied for ( R → F ( A ) , Mod c A , ⊗ A , F, γ A , A , µ A ) .Proof. This is clear from the discussion above. (cid:3)
In the situation of the lemma we have the global sections functorΓ A : Mod A −→ Mod R , F 7−→
Hom A ( A , F ) . We have seen in Section 8 that for an object
F ∈
Mod A we have Γ A ( F ) = Γ( F ) as R -modules. We will often abuse notation by writing Γ = Γ A .10. Finiteness conditions
Here are some finiteness conditions we can impose.
Definition 10.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1.(1) An object F of C is said to be of finite type if F ( F ) is a finitely generated A -module.(2) An ring object A of C is said to be of finite type if F ( A ) is a finitelygenerated A -algebra.(3) A module object F over a ring object A of C is said to be of finite type if F ( F ) is of finite type over F ( A ). Note that the ring objects in this definition need not be commutative. A noncom-mutative algebra S over A is finitely generated if it is isomorphic to a quotient ofthe free algebra A h x , . . . , x n i for some n .11. Adequacy
The notion of adequacy , which is our analogue of geometric reductivity, can beformulated in a variety of different ways.
Definition 11.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axiom:(N) The ring A is Noetherian. Definition 11.2.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. An object F of C is called locally finite if it is a filtered colimit F = colim F i of finite typeobjects F i such that also F ( F ) = colim F ( F i ). Definition 11.3.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe axiom:(L) Every object F of C is locally finite. Lemma 11.4.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. A quotientof a locally finite object of C is locally finite. If axioms (N) and (D) hold, then asubobject of a locally finite object is locally finite and the subcategory of locally finiteobjects is abelian.Proof. Suppose that
F → Q is surjective and that F is locally finite. Write F =colim F i of finite type objects F i such that also F ( F ) = colim F ( F i ). Set Q i =Im( F i → Q ). We claim that Q = colim i Q i and that F ( Q ) = colim F ( Q i ). Thelast statement follows from exactness of F and the fact that colimits commute withimages in Mod A . If β i : Q i → G is a compatible system of maps to an object of C ,then composing with the surjections F i → Q i gives a compatible system of mapsalso, whence a morphism β : F → G . But F ( β ) factors through F ( F ) → F ( Q ) andhence is zero on F (Ker( F → Q ). Because F is faithful and exact we see that β factors as Q → G as desired.Suppose that
J → F is injective, that F is locally finite and that (N) and (D) hold.Write F = colim F i of finite type objects F i such that also F ( F ) = colim F ( F i ).By the argument of the preceding paragraph applied to id F : F → F we mayassume F i ⊂ F i for each i . Set J i = F i ∩ J . Since axiom (N) holds we see thateach J i is of finite type. As F is exact we see that colim F ( J i ) = F ( J ). As axiom(D) holds we know that J ′ = colim J i exists and colim F ( J i ) = F ( J ′ ). Hence weget a canonical map J ′ → J which has to be an isomorphism as F is exact andfaithful. This proves that J is locally finite.Assume (N) and (D). Let α : F → G be a morphism of locally finite objects. Wehave to show that the kernel and cokernel of α are locally finite. This is clear bythe results of the preceding two paragraphs. (cid:3) Lemma 11.5.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Assume axiom(D) holds. The tensor product of locally finite objects is locally finite. For any R -module M the object M ⊗ R O is locally finite. If A is a locally finite ring object,then A ⊗ Γ( A ) M is locally finite for any Γ( A ) -module M . NVARIANT RINGS THROUGH CATEGORIES 11
Proof.
This is clear since in the presence of (D), the tensor product commutes withcolimits. (cid:3)
Lemma 11.6.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Assume axiom(S) holds. Consider the following conditions (1) For every surjection of finite type objects
G → F and f ∈ Γ( F ) there existsan n > and a g ∈ Γ( Sym n C ( G )) which maps to f n in Γ( Sym n C ( F )) . (2) For every surjection
G → O with G of finite type and f ∈ Γ( O ) there existsan n > and a g ∈ Γ( Sym n C ( G )) which maps to f n in Γ( O ) . (3) For every surjection of weakly commutative ring objects
A → B in C with A locally finite, and any f ∈ Γ( B ) , there exists an n > and an element g ∈ Γ( A ) such that g f n in Γ( B ) .We always have (1) ⇒ (2) and (1) ⇒ (3) . If axiom (N) holds, then (2) ⇒ (1) . Ifaxiom (D) holds, then (3) ⇒ (1) . Furthermore, consider the following variations (1’) For every surjection of objects
G → F and f ∈ Γ( F ) there exists an n > and a g ∈ Γ( Sym n C ( G )) which maps to f n in Γ( Sym n C ( F )) . (2’) For every surjection
G → O and f ∈ Γ( O ) there exists an n > and a g ∈ Γ( Sym n C ( G )) which maps to f n in Γ( O ) . (3’) For every surjection of weakly commutative ring objects
A → B in C , andany f ∈ Γ( B ) , there exists an n > and an element g ∈ Γ( A ) such that g f n in Γ( B ) .If axiom (L) holds, then (1) ⇔ (1 ′ ) , (2) ⇔ (2 ′ ) , and (3) ⇔ (3 ′ ) .Proof. It is clear that (1) implies (2). Assume (N) + (2) and let us prove (1).Consider
G → F and f as in (1). Let H = G × F O . Then H → O is surjective,and F ( H ) = F ( G ) × F ( F ) A . By assumption (N) this implies that F ( H ) is a finite A -module.Let us prove that (1) implies (3). Let A → B and f be as in (3). Write A = colim i G i as a directed colimit such that F ( A ) = colim i F ( G i ) and such that each G i is offinite type. Think of f ∈ Γ( B ) ⊂ F ( B ). Then for some i there exists a ˜ f ∈ F ( G i )which maps to f . Set G = G i , set F = Im( G i → B ). The map G → F is surjective.Since F is exact we see that f ∈ F ( F ) ⊂ F ( B ). Hence, as Γ is left exact weconclude that f ∈ Γ( F ) as well. Thus property (1) applies and we find an n > g ∈ Γ(Sym n C ( G )) which maps to f n in Γ(Sym n C ( F )). Since A and B are ringobjects we obtain a canonical diagram G ⊗ n / / (cid:15) (cid:15) F ⊗ n (cid:15) (cid:15) A / / B Since A and B are weakly commutative this produces a commutative diagramSym n C ( G ) / / (cid:15) (cid:15) Sym n C ( F ) (cid:15) (cid:15) A / / B Hence the element g ∈ Γ(Sym n C ( G )) maps to the desired element of Γ( A ). If (D) holds, then given
G → F as in (1) we can form the map of “symmetric”algebras Sym ∗C ( G ) −→ Sym ∗C ( F )and we see that (3) implies (1).The final statement is clear. (cid:3) We do not know of an example of Situation 2.1 where axiom (D) does not hold. Onthe other hand, we do know cases where (S) does not hold, namely, the categoryof comodules over a general bialgebra. Hence we take property (3) of the lemmaabove as the defining property, since it also make sense in those situations.
Definition 11.7.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axiom:(A) For every surjection of weakly commutative rings A → B in C with A locallyfinite, and any f ∈ Γ( B ), there exists an n > g ∈ Γ( A )such that g f n in Γ( B ).A much stronger condition is the notion of goodness , which is our analogue of linearreductivity. It can hold even in geometrically interesting situations. Definition 11.8.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. We introducethe following axiom:(G) The functor Γ is exact.12. Preliminary results
Let A be a weakly commutative ring object of C . This implies that Γ( A ) ⊂ F ( A )is a commutative ring. Let I ⊂ Γ( A ) be an ideal. Assuming the axiom (D) we havethe object A ⊗ Γ( A ) I (see Lemma 8.3) and a canonical map(12.1) A ⊗ Γ( A ) I −→ A . Namely, this is the adjoint to the map I → Γ( A ). Applying F to the the map(12.1) gives the obvious map F ( A ) ⊗ Γ( A ) I → F ( A ). The image of (12.1) will bedenoted A I in the sequel. We have F ( A I ) = F ( A ) I by exactness of the functor F .For an ideal I of a commutative ring B we set I ∗ = { f ∈ B | ∃ n > , f n ∈ I n } . Note that it is not clear (or even true) in general that I ∗ is an ideal. (Our notation isnot compatible with notation concerning integral closure of ideals in algebra texts.We will only use this notation in this section.) Lemma 12.1.
Assume that we are in Situation 2.1 and that axiom (D) holds. Let A be a locally finite, weakly commutative ring object of C . Let I ⊂ Γ( A ) be an ideal.Consider the ring map ϕ : Γ( A ) /I −→ Γ( A / A I ) . (1) If the axiom (G) holds, ϕ is an isomorphism. (2) If the axiom (A) holds, then (a) the kernel of ϕ is contained in I ∗ Γ( A ) /I ; in particular it is locallynilpotent, and (b) for every element f ∈ Γ( A / A I ) there exists an integer n > and anelement g ∈ Γ( A ) /I which maps to f n via ϕ . NVARIANT RINGS THROUGH CATEGORIES 13
Proof.
The surjectivity of ϕ in (1) is immediate from axiom (G). The ring object A / A I is weakly commutative (by Lemma 9.2). Hence (2b) is implied by axiom(A).Suppose that f ∈ Γ( A ) maps to zero in Γ( A / A I ). This means that f ∈ Γ( A I ).Choose generators f s ∈ I , s ∈ S for I . Consider the ring map A [ x s ; s ∈ S ] −→ B = M I n A which maps x s to f s ∈ Γ( I A ), see Remark 8.4. This is a surjection of ring objectsof C . Hence if (G) holds, then we see that f is in the image of L s ∈ S Γ( A ) → Γ( A I ),i.e., f is in Γ( A ) I and injectivity in (1) holds. For the rest of the proof assume(A). Clearly the polynomial algebra A [ x s ; s ∈ S ] is weakly commutative and locallyfinite. Hence (A) implies there exists an n > g ∈ Γ( A [ x s ; s ∈ S ])which maps to f n in the summand Γ( A I n ) of Γ( B ). Hence we may also assumethat g is in the degree n summandΓ( M | J | = n A x J )of Γ( A [ x s ; s ∈ S ]). Now, note that there is a ring map B → A and that thecomposition A [ x s ; s ∈ S ] −→ B −→ A in degree n maps Γ( L | J | = n A x J ) into Γ( A ) I n , because x s maps to f s . Hence f n ∈ I n . This finishes the proof. (cid:3) Let A be a weakly commutative ring object of C . Let Γ( A ) → Γ ′ be a homomor-phism of commutative rings. Write Γ ′ = Γ( A )[ x s ; s ∈ S ] /I . Assume axiom (D)holds. Then we see that we have the equality A ⊗ Γ( A ) Γ ′ = A [ x s ; s ∈ S ] / ( A [ x s ; s ∈ S ]) I where the polynomial algebra is as in Remark 8.4 and the tensor product as inLemma 8.3. The reason is that there is an obvious map (from right to left) andthat we have F ( A ⊗ Γ( A ) Γ ′ ) = F ( A ) ⊗ Γ( A ) Γ ′ = F ( A )[ x s ; s ∈ S ] / ( F ( A )[ x s ; s ∈ S ]) I by the properties of the functor F and the results mentioned above. Hence A ⊗ Γ( A ) Γ ′ is a weakly commutative ring object (see Lemma 9.2). Note that if A is locallyfinite, then so is A ⊗ Γ( A ) Γ ′ , see Lemma 11.5. Lemma 12.2.
Assume that we are in Situation 2.1 and that axiom (D) holds. Let A be a ring object. (1) Assume that also axiom (G) holds. If M is a left Γ( A ) -module, then theadjunction map ϕ : M −→ Γ( A ⊗ Γ( A ) M ) is an isomorphism. (2) Assume the axiom (A) holds, and that A is locally finite and weakly commu-tative. Let Γ( A ) → Γ ′ be a commutative ring map. Consider the adjunctionmap ϕ : Γ ′ −→ Γ( A ⊗ Γ( A ) Γ ′ )(a) the kernel of ϕ is locally nilpotent, and (b) for every element f ∈ Γ( A ⊗ Γ( A ) Γ ′ ) there exists an integer n > andan element g ∈ Γ ′ which maps to f n via ϕ .Proof. For (1), since both functors
A ⊗ Γ( A ) − and Γ commute with arbitrary directsums, the map ϕ is an isomorphism when M is free. Furthermore, since A ⊗ Γ( A ) − is right exact and Γ is exact, the general case follows. For (2), the map is anisomorphism when Γ ′ is a polynomial algebra (since we are assuming all functorscommute with direct sums). And the general case follows from this, the discussionabove the lemma and Lemma 12.1. (cid:3) Lemma 12.3.
Assume that we are in Situation 2.1 and that axioms (D) and (A)hold. Then for every locally finite, weakly commutative ring object A of C the mapSpec ( F ( A )) −→ Spec (Γ( A )) is surjective.Proof. Let Γ( A ) → K be a ring map to a field. We have to show that the ring F ( A ) ⊗ Γ( A ) K = F ( A ⊗ Γ( A ) K )is not zero. This follows from Lemma 12.2 and the fact that K is not the zeroring. (cid:3) In the following lemma we use the notion of a universally subtrusive morphismof schemes f : X → Y . This means that f satisfies the following valuation liftingproperty: for every valuation ring V and every morphism Spec( V ) → Y there existsa local map of valuation rings V → V ′ and a morphism Spec( V ′ ) → X such that X (cid:15) (cid:15) Spec( V ′ ) o o (cid:15) (cid:15) Y Spec( V ) o o is commutative. It turns out that if f : X → Y is of finite type, and Y is Noetherian,then this notion is equivalent to f being universally submersive . Lemma 12.4.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Let A be a ringobject. Assume that (1) axioms (D) and (A) hold, and (2) A is locally finite and weakly commutative.Then Spec ( F ( A )) → Spec (Γ( A )) is universally subtrusive. If in addition, (3) R → A is finite type, (4) A is of finite type, and (5) Γ( A ) is Noetherian.Then Spec ( F ( A )) → Spec (Γ( A )) is universally submersive.Proof. To show the first part, let Spec( V ) → Spec(Γ( A )) be a morphism where V is a valuation ring with fraction field K . We must show that f : Spec( F ( A ) ⊗ Γ( A ) V ) −→ Spec( V )is subtrusive. Let η ∈ Spec( V ) be the generic point. It suffices to show that theclosure of f − ( η ) in Spec( F ( A ) ⊗ Γ( A ) V ) surjects onto Spec( V ). If we set I = ker( A ⊗ Γ( A ) V −→ A ⊗ Γ( A ) K ) NVARIANT RINGS THROUGH CATEGORIES 15 then F ( I ) is the kernel of F ( A ) ⊗ Γ( A ) V → F ( A ) ⊗ Γ( A ) K and defines the closureof f − ( η ). The ring object ( A ⊗ Γ( A ) V ) / I is weakly commutative and locally finite.By Lemma 12.3,Spec( F (( A ⊗ Γ( A ) V ) / I )) −→ Spec(Γ((
A ⊗ Γ( A ) V ) / I ))is surjective. Axiom (A) applied to the surjection A ⊗ Γ( A ) V → ( A ⊗ Γ( A ) V ) / I implies that Spec(Γ(( A ⊗ Γ( A ) V ) / I )) −→ Spec( V )is integral. Therefore the composition of the two morphisms above is surjective sothat the closure of f − ( η ) surjects onto Spec( V ).The hypotheses in the second part imply that Γ( A ) → F ( A ) is of finite type andΓ( A ) is Noetherian, hence the remark preceding the lemma applies. (cid:3) Below we will use the following algebraic result to get finite generation.
Theorem 12.5.
Consider ring maps R → B → A such that (1) B and R are noetherian, (2) R → A is of finite type, and (3) Spec ( A ) → Spec ( B ) is universally submersive.Then R → B is of finite type.Proof. This is a special case of Theorem 6.2.1 of [Alp10]. It was first discoveredwhile writing an earlier version of this paper. (cid:3)
The main result
The main argument in the proof of Theorem 13.5 is an induction argument. Inorder to formulate it we use the following condition.
Definition 13.1.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Let A be aweakly commutative ring object. Consider the following property of A ( ⋆ ) The ring Γ( A ) is a finite type R -algebra and for every finite type module F over A the Γ( A )-module Γ( F ) is finite. Lemma 13.2.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Let A → B bea surjection of ring objects. Assume (1) R is Noetherian and axiom (A) holds, (2) A is locally finite and weakly commutative, and (3) Γ( B ) is a finitely generated R -algebra.Then Γ( B ) is a finite Γ( A ) -module and there exists a finitely generated R -subalgebra B ⊂ Γ( A ) such that Im (Γ( A ) −→ Γ( B )) = Im ( B −→ Γ( B )) . Proof.
Since A is weakly commutative, so is B . Hence Γ( B ) is a commutative R -algebra. Pick f , . . . , f n ∈ Γ( B ) which generate as an R -algebra. By axiom (A)we can find g , . . . , g n ∈ Γ( A ) which map to f n , . . . , f n n n in Γ( B ) for some n i > B ) is generated by the elements f e . . . f e n n , ≤ e i ≤ n i − B ) is finite over Γ( A ). As a first approximation, let B = R [ g , . . . , g n ] ⊂ Γ( A ). Then the equality of the lemma may not hold, but in any case Γ( A ) is finite over B . Since B is a Noetherian ring, Im(Γ( A ) → Γ( B )) is a finite B -moduleso be choose finitely many generators g n +1 , . . . , g n + m ∈ Γ( A ). Hence by setting B = R [ g , . . . , g n + m ], the lemma is proved. (cid:3) Lemma 13.3.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Let A be a ringobject and let I ⊂ A be a left ideal. Assume (1) R is Noetherian and axiom (A) holds, (2) A is locally finite and weakly commutative, (3) ( ⋆ ) holds for A / I , and (4) there is a quotient A → A ′ such that ( ⋆ ) holds for A ′ and such that I is afinite A ′ -module.Then ( ⋆ ) holds for A .Proof. Since A is weakly commutative and locally finite so are A / I and A ′ . ByLemma 13.2 the rings Γ( A ′ ) and Γ( A / I ) are finite Γ( A )-algebras. Consider theexact sequence 0 → Γ( I ) → Γ( A ) → Γ( A / I ) . By ( ⋆ ) for A ′ we see that Γ( I ) is a finite Γ( A ′ )-module, hence a finite Γ( A )-module.Choose generators x , . . . , x s ∈ Γ( I ) as a Γ( A )-module. By Lemma 13.2 we canfind a finite type R -subalgebra B ⊂ Γ( A ) such that the image of B in Γ( A ′ ) andthe image of B in Γ( A / I ) is the same as the image of Γ( A ) in those rings. Weclaim that Γ( A ) = B [ x , . . . , x s ]as subrings of Γ( A ). Namely, if h ∈ Γ( A ) then we can find an element b ∈ B whichhas the same image as h in Γ( A / I ). Hence replacing h by h − b we may assume h ∈ Γ( I ). By our choice of x , . . . , x s we may write h = P a i x i for some a i ∈ Γ( A ).But since I is a A ′ -module, we can write this as h = P a ′ i x i with a ′ i ∈ Γ( A ′ ) theimage of a i . By choice of B we can find b i ∈ B mapping to a ′ i . Hence we seethat h ∈ B [ x , . . . , x s ] as desired. This proves that Γ( A ) is a finitely generated R -algebra.Let F be a finite type A -module. Set IF equal to the image of the map I ⊗ F → F which is the restriction of the multiplication map of F . Consider the exact sequence0 → IF → F → F / IF → F , and a surjectivemap F ( I ) ⊗ A F ( F ) → F ( IF ) which factors through F ( I ) ⊗ F ( A ) F ( F ) as A isweakly commutative. Since F ( F ) is finite as a F ( A )-module, and F ( I ) is finite asa F ( A ′ )-module, we conclude that F ( IF ) is a finite F ( A ′ )-module, i.e., that IF is a finite A ′ -module. In the same way we see that F / IF is a finite A / I -module.Hence in the exact sequence0 → Γ( IF ) → Γ( F ) → Γ( F / IF )we see that the modules on the left and the right are finite Γ( A )-modules. SinceΓ( A ) is Noetherian by the result of the preceding paragraph we see that Γ( F ) is afinite Γ( A )-module. This conclude the proof that property ( ⋆ ) holds for A . (cid:3) Lemma 13.4.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Let A be a ringobject, and let I ⊂ A be a left ideal. Assume that (1) axioms (N) and (A) hold and R is Noetherian, NVARIANT RINGS THROUGH CATEGORIES 17 (2) A is locally finite, weakly commutative and of finite type, (3) I n = 0 for some n ≥ , and (4) A / I has property ( ⋆ ).Then A has property ( ⋆ ).Proof. We argue by induction on n and hence we may assume that I = 0. Thenwe get an exact sequence 0 → I → A → A / I → . Because (N) holds and A is of finite type we see that F ( A ) is a finitely generated A -algebra hence Noetherian. Thus I is a finite type A -module, and hence also afinite type A / I -module. This means that Lemma 13.3 applies, and we win. (cid:3) Theorem 13.5.
Let ( R → A, C , ⊗ , F, γ, O , µ ) be as in Situation 2.1. Assume (1) R is Noetherian, (2) R → A is of finite type, and (3) the axioms (A) and (D) hold.Then for every finite type, locally finite, weakly commutative ring object A of C property ( ⋆ ) holds.Proof. Let A be a finite type, locally finite, weakly commutative ring object A of C . For every left ideal I ⊂ A the quotient A / I is also a finite type, locally finite,weakly commutative ring object of C . Consider the set {I ⊂ A | ( ⋆ ) fails for A / I} . To get a contradiction assume that this set is nonempty. By Noetherian inductionon the ideal F ( I ) ⊂ F ( A ) we see there exists a maximal left ideal I max ⊂ A suchthat ( ⋆ ) holds for any ideal strictly containing I max but ( ⋆ ) does not hold for I max .Replacing A by A / I max we may assume (in order to get a contradiction) that ( ⋆ )does not hold for A but does hold for every proper quotient of A .Let f ∈ Γ( A ) be nonzero. If Ker( f : A → A ) is nonzero, then we see that we getan exact sequence 0 → ( f ) → A → A / ( f ) → ⋆ ) holds for both A / Ker( f : A → A ) and A / ( f ) and sinceKer( f ) is a finite A / ( f )-module, we can apply Lemma 13.3. Hence we see thatwe may assume that any nonzero element f ∈ Γ( A ) is a nonzero divisor on A . Inparticular, Γ( A ) is a domain.Again, assume that f ∈ Γ( A ) is nonzero. Consider the sequence0 → A f −→ A → A /f A → → Γ( A ) f −→ Γ( A ) → Im(Γ( A ) → Γ( A /f A )) → R -algebra which is finite overΓ( A ), see Lemma 13.2. Hence any ideal I ⊂ Γ( A ) containing f maps to a finitelygenerated ideal in it. This implies that Γ( A ) is Noetherian.Next, we claim that for any finite type A -module F the module Γ( F ) is a finiteΓ( A )-module. Again we can do this by Noetherian induction applied to the set {G ⊂ F is an A -submodule such that finite generation fails for Γ( F / G ) } . In other words, we may assume that F is a minimal counter example in the sensethat any proper quotient of F gives a finite Γ( A )-module. Pick s ∈ Γ( F ) nonzero(if Γ( F ) is zero, we’re done). Let A · s ⊂ F denote the image of A → F which ismultiplying against s . Now we have0 → A · s → F → F / A · s → → Γ( A · s ) → Γ( F ) → Γ( F / A · s )By minimality we see that the module on the right is finite over the Noetherianring Γ( A ). On the other hand, the module on the left is Γ( A / I ) for the ideal I = Ker( s : A → F ). If I = 0 then this is Γ( A ) and therefore finite, and if I 6 = 0then this is a finite Γ( A )-module by Lemma 13.2 and minimality of A . Hence weconclude that the middle module is finite over the Noetherian ring Γ( A ) which isthe desired contradiction.Finally, we show that Γ( A ) is of finite type over R which will finish the proof.Namely, by Lemma 12.4 the morphism of schemesSpec( F ( A )) −→ Spec(Γ( A ))is universally submersive. We have already seen that Γ( A ) is a Noetherian ring.Thus Theorem 12.5 kicks in and we are done. (cid:3) Remark . We note that the proof of Theorem 13.5 can be simplified if theaxiom (G) is also satisfied. In fact, if axiom (G) holds in addition to the conditions(1) - (3) of Theorem 13.5, then for every finite type, weakly commutative (but notnecessarily locally finite) ring object A , property ( ⋆ ) holds. Lemma 12.2 impliesthat for any ideal I ⊆ Γ( A ), I = IF ( A ) ∩ Γ( A ); therefore Γ( A ) is Noetherian.We can then apply Theorem 12.5 to conclude that Γ( A ) is a finite type R -algebra.Furthermore, a simple noetherian induction argument shows that for every finitetype module F over A the Γ( A )-module Γ( F ) is finite type.14. Quasi-coherent sheaves on algebraic stacks
Let S = Spec( R ) be an affine scheme. Let X be a quasi-compact algebraic stackover S . Let p : T → X be a smooth surjective morphism from an affine scheme T = Spec( A ). Lemma 14.1.
In the situation above, the category QCoh ( O X ) endowed with itsnatural tensor product, pullback functor F : QCoh ( O X ) → QCoh ( O T ) = Mod A and structure sheaf O = O X is an example of Situation 2.1. The functor Γ :
QCoh ( O X ) → Mod R is identified with the functor of global sections F 7−→ Γ( X , F ) . Axioms (D), (C), and (S) hold. If X is noetherian (eg. X is quasi-separated and A is Noetherian), then axiom (L) holds.Proof. The final statement is [LMB00, Prop 15.4]. The rest is clear. (cid:3)
The following definition reinterprets the adequacy axiom (A).
NVARIANT RINGS THROUGH CATEGORIES 19
Definition 14.2.
Let X be an quasi-compact algebraic stack over S = Spec( R ).We say that X is adequate if for every surjection A → B of quasi-coherent O X -algebras with A locally finite and f ∈ Γ( X , B ), there exists an n > g ∈ Γ( X , A ) such that g f n in Γ( X , B ). Lemma 14.3.
Let X be an quasi-compact algebraic stack over S = Spec ( R ) . Thefollowing are equivalent: (1) X is adequate. (2) For every surjection of finite type O X -modules G → F and f ∈ Γ( X , F ) ,there exists an n > and a g ∈ Γ( X , Sym n G ) such that g f n in Γ( X , Sym n F ) .If X is noetherian, then the above are also equivalent to: (3) For every surjection
G → O with G of finite type and f ∈ Γ( X , O X ) , thereexists an n > and a g ∈ Γ( X , Sym n G ) such that g f n in Γ( X , O X ) . (1’) For every surjection
A → B of quasi-coherent O X -algebras and f ∈ Γ( X , B ) ,there exists an n > and a g ∈ Γ( X , A ) such that g f n in Γ( X , B ) . (2’) For every surjection of O X -modules G → F and f ∈ Γ( X , F ) , there existsan n > and a g ∈ Γ( X , Sym n G ) such that g f n in Γ( X , Sym n F ) . (3’) For every surjection
G → O and f ∈ Γ( X , O X ) , there exists an n > anda g ∈ Γ( X , Sym n G ) such that g f n in Γ( X , O X ) .Proof. This is Lemma 11.6. (cid:3)
Corollary 14.4.
Let X be an algebraic stack finite type over an affine noetherianscheme Spec ( R ) . Suppose X is adequate. Let A be a finite type O X -algebra. Then Γ( X , A ) is finitely generated over R and for every finite type A -module F , the Γ( X , A ) -module Γ( X , F ) is finite.Proof. This is Theorem 13.5. (cid:3)
Bialgebras, modules and comodules
In this section we discuss how modules and comodules over a bialgebra form anexample of our abstract setup. If A is a commutative ring, recall that a bialgebra H over A is an A -module H endowed with maps ( A → H, H ⊗ A H → A, ǫ : H → A, δ : H → H ⊗ A H ). Here H ⊗ A H → H and A → H define an unital A -algebra structureon H , the maps δ and ǫ are unital A -algebra maps. Moreover, the comultiplication µ is associative and ǫ is a counit.Let H be a bialgebra over A . A left H -module is a left module over the R -algebrastructure on H ; that is, there is a A -module homomorphism H ⊗ A M → M sat-isfying the two commutative diagrams for an action. A left H -comodule M is an R -module homomorphism σ : M → H ⊗ A M satisfying the two commutative dia-gram for a coaction. See [Kas95, Chapter 3] and [Mon93, Chapter 1] for the basicproperties of H -modules and H -comodules. Definition 15.1.
Let A be a commutative ring. Let H be a bialgebra over A .(1) Let Mod H be the category of left H -modules. It is endowed with theforgetful functor to A -modules, the tensor product( M, N ) M ⊗ A N where H acts on M ⊗ A N via the comultiplication, and the object O givenby the module A where H acts via the counit. (2) Let Comod H be the category of left H -comodules. It is endowed with theforgetful functor to A -modules, the tensor product( M, N ) M ⊗ A N where comodule structure on M ⊗ A N comes from the multiplication in H ,and the object O given by the module A where H acts via the A -algebrastructure H . Lemma 15.2.
Let R → A be a map of commutative rings. Let H be a bialgebraover A . (1) The category Mod H with its additional structure introduced in Definition15.1 is an example of Situation 2.1. The functor Γ :
Mod H → Mod R isidentified with the functor of invariants M M H = { m ∈ M | h · m = ǫ ( h ) m } . Axioms (D), (I) and (S) hold. Axiom (C) holds if H is cocommutative. (2)
The category Comod H with its additional structure introduced in Definition15.1 is an example of Situation 2.1. The functor Γ :
Comod H → Mod R isidentified with the functor of coinvariants M M H = { m ∈ M | σ ( m ) = 1 ⊗ m } where σ : M → H ⊗ A M indicates the coaction of M . Axiom (D) holds.Axiom (C) holds if H is commutative.Proof. The first two statements in both part (1) and (2) are clear. It also clearthat axiom (D) holds in both cases. Arbitrary direct products exist in the categoryMod H , which is axiom (I), and so by Lemma 7.2 axiom (S) holds. The finalstatement concerning axiom (C) is straightforward, see [Mon93, Section 1.8]. (cid:3) Adequacy for a bialgebra
Let R → A be map of commutative rings. Let H be a bialgebra over A . Let M be an H -module. We can identify Sym nH M := Sym n Mod H M of axiom (S) with the H -module M ⊗ A · · · ⊗ A M | {z } n /M ′ where M ′ is the submodule generated by elements h · (( · · ·⊗ m i ⊗· · ·⊗ m j ⊗· · · ) − ( · · ·⊗ m j ⊗· · ·⊗ m i ⊗· · · )) for h ∈ H and m , . . . , m n ∈ M . And Sym H M := L n Sym nH M is the largest H -module quotient of the tensor algebra on M which is commutative.An H -algebra is an H -module C which is an algebra over the algebra structure on H such that A → C and C ⊗ A C → C are H -module homomorphisms. We say that C is commutative if C is commutative as an algebra. An H -module M is locallyfinite if it is the filtered colimit of finite type H -modules.The following definition reinterprets adequacy axiom (A) for the category Mod H . Definition 16.1.
Let R → A be map of commutative rings. Let H be a bialgebraover A . We say that H is adequate if for every surjection of commutative H -algebras C → D in Mod H with C locally finite, and any f ∈ D H , there exists an n > g ∈ C H such that g f n in D H . NVARIANT RINGS THROUGH CATEGORIES 21
Lemma 16.2.
Let R → A be map of commutative rings. Let H be a bialgebra over A . The following are equivalent: (1) H is adequate. (2) For every surjection of finite type H -modules N → M and f ∈ M H , thereexists an n > and a g ∈ ( Sym nH N ) H such that g f n in ( Sym nH M ) H .If A is Noetherian, then the above are also equivalent to: (3) For every surjection of finite type H -modules N → A and f ∈ A , thereexists an n > and a g ∈ ( Sym nH N ) H such that g f n in A .Proof. This is Lemma 11.6. (cid:3)
Corollary 16.3.
Let R → A be a finite type map of commutative rings where R isNoetherian. Let H be an adequate bialgebra over A . Let C be a finitely generated,locally finite, commutative H -algebra. Then C H is a finitely generated R -algebraand for every finite type C -module M, the C H -module M H is finite.Proof. This is Theorem 13.5. (cid:3)
Remark . If R = A = k where k is a field, then [KT08] define a Hopf algebra H over k to be geometrically reductive if any finite dimensional H -module M andany non-zero homomorphism of H -modules N → k there exist n > nH ( N ) H → k is non-zero. By Lemma 16.2, H is geometrically reductive if andonly if H is adequate.In [KT08, Theorem 3.1], Kalniuk and Tyc prove that with the hypotheses of theabove corollary and with R = A = k is a field, C H is finitely generated over k .17. Coadequacy for a bialgebra
Let R → A be map of commutative rings. Let H be a bialgebra over A . An H -coalgebra is an H -comodule C which is an algebra over the algebra structure on H such that A → C and C ⊗ A C → C are H -comodule homomorphisms; C is commutative if C is commutative as an algebra. An H -comodule M is locally finite if it is the filtered colimit of finite type H -comodules.Here we reinterpret the adequacy axiom (A) for the category Comod H . Definition 17.1.
Let R → A be map of commutative rings. Let H be a bialgebraover A . We say that H is coadequate if for every surjection of commutative H -coalgebras C → D with C locally finite, and any f ∈ D H , there exists an n > g ∈ C H such that g f n in D H .Recall that we only know that axiom (S) holds for Comod H when H is commutative. Lemma 17.2.
Let R → A be map of commutative rings. Let H be a commutativebialgebra over A . The following are equivalent: (1) H is adequate. (2) For every surjection of finite type H -modules N → M and f ∈ M H , thereexists an n > and a g ∈ ( Sym nH N ) H such that g f n in ( Sym nH M ) H .If A is Noetherian, then the above are also equivalent to: (3) For every surjection of finite type H -modules N → A and f ∈ A , thereexists an n > and a g ∈ ( Sym nH N ) H such that g f n in A .Proof. This is Lemma 11.6. (cid:3)
Corollary 17.3.
Let R → A be a finite type of commutative rings where R isNoetherian. Let H be an adequate bialgebra over A . Let C be a finitely generated,locally finite, commutative H -coalgebra. Then C H is a finitely generated R -algebraand for every finite type C -module M , the C H -module M H is finite.Proof. This is Theorem 13.5. (cid:3)
References [Alp08] Jarod Alper. Good moduli spaces for Artin stacks. math.AG/0804.2242 , 2008.[Alp10] Jarod Alper. Adequate moduli spaces and geometrically reductive group schemes. math.AG/1005.2398 , 2010.[BFS92] Heloisa Borsari and Walter Ferrer Santos. Geometrically reductive Hopf algebras.
J.Algebra , 152(1):65–77, 1992.[Hab75] W. J. Haboush. Reductive groups are geometrically reductive.
Ann. of Math. (2) ,102(1):67–83, 1975.[Kas95] Christian Kassel.
Quantum groups , volume 155 of
Graduate Texts in Mathematics .Springer-Verlag, New York, 1995.[KT08] Marta Kalniuk and Andrzej Tyc. Geometrically reductive Hopf algebras and their in-variants.
J. Algebra , 320(4):1344–1363, 2008.[LMB00] G´erard Laumon and Laurent Moret-Bailly.
Champs alg´ebriques , volume 39 of
Ergeb-nisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveysin Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series ofModern Surveys in Mathematics] . Springer-Verlag, Berlin, 2000.[MFK94] D. Mumford, J. Fogarty, and F. Kirwan.
Geometric invariant theory , volume 34 of
Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics andRelated Areas (2)] . Springer-Verlag, Berlin, third edition, 1994.[Mon93] Susan Montgomery.
Hopf algebras and their actions on rings , volume 82 of
CBMSRegional Conference Series in Mathematics . Published for the Conference Board of theMathematical Sciences, Washington, DC, 1993.[Nag64] Masayoshi Nagata. Invariants of a group in an affine ring.
J. Math. Kyoto Univ. , 3:369–377, 1963/1964.[Ses77] C. S. Seshadri. Geometric reductivity over arbitrary base.
Advances in Math. , 26(3):225–274, 1977.
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