aa r X i v : . [ m a t h . AG ] S e p VIRTUAL FUNDAMENTAL CLASSES OFDERIVED STACKS I
ADEEL A. KHAN
Abstract.
We construct the ´etale motivic Borel–Moore homology ofderived Artin stacks. Using a derived version of the intrinsic normalcone, we construct fundamental classes of quasi-smooth derived Artinstacks and demonstrate functoriality, base change, excess intersection,and Grothendieck–Riemann–Roch formulas. These classes also satisfy ageneral cohomological B´ezout theorem which holds without any transver-sity hypotheses. The construction is new even for classical stacks andas one application we extend Gabber’s proof of the absolute purity con-jecture to Artin stacks.
Introduction 11. The intrinsic normal bundle 61.1. Stacks 61.2. Vector bundle stacks 61.3. Normal bundle stacks 71.4. Deformation to the normal bundle stack 82. Motivic Borel–Moore homology of derived stacks 92.1. Definition and examples 92.2. Basic operations 122.3. Basic compatibilities 142.4. Properties 153. Fundamental classes 163.1. Construction 173.2. Properties 183.3. Comparison with Behrend–Fantechi 203.4. Non-transverse B´ezout theorem 213.5. Grothendieck–Riemann–Roch 223.6. Absolute purity 24Appendix A. The six operations for derived Artin stacks 26A.1. Derived algebraic spaces 26A.2. Derived algebraic stacks 28References 34
Introduction
In this paper we revisit the foundations of the theory of virtual funda-mental classes using the language of derived algebraic geometry.
Date : 2019-09-03.
ADEEL A. KHAN
Quasi-smoothness.
Let X be a smooth algebraic variety of dimension m over a field k . Any collection of regular functions f , . . . , f n ∈ Γ ( X , O X ) determines a quasi-smooth derived subscheme Z = Z ( f , . . . , f n ) of X. Itsunderlying classical scheme Z cl is the usual zero locus, but Z admits a perfect2-term cotangent complex of the form L Z = (O ⊕ n Z → Ω X ∣ Z ) whose virtual rank encodes the virtual dimension d = m − n . Every quasi-smooth derived Artin stack Z is given by this construction, locally on somesmooth atlas.To any such Z , the main construction of this paper assigns a virtualfundamental class [ Z ] vir . More generally, for any quasi-smooth morphism f ∶ X → Y of derived Artin stacks, we define a relative virtual fundamentalclass [ X / Y ] vir . The normal bundle stack.
We begin in Sect. 1 by introducing a derivedversion of the intrinsic normal cone of Behrend–Fantechi [BF]. For anyquasi-smooth morphism f ∶ X → Y of derived Artin stacks, this is a vectorbundle stack N X /Y over X . When X and Y are classical 1-Artin stacks and f is a local complete intersection morphism that is representable by Deligne–Mumford stacks, then N X /Y is the relative intrinsic normal cone defined in[BF, Sect. 7]. If f is not representable by Deligne–Mumford stacks, thenN X /Y is only a 2-Artin stack. The key geometric construction, which is jointwith D. Rydh, is called “deformation to the normal bundle stack”. For anyquasi-smooth morphism f ∶ X → Y it provides a family of quasi-smoothmorphisms parametrized by A , with generic fibre f ∶ X → Y and specialfibre the zero section 0 ∶ X → N X /Y . Motivic Borel–Moore homology theories.
In Sect. 2 we construct ´etalemotivic Borel–Moore homology theories on derived Artin stacks. If SH ( S ) denotes Voevodsky’s stable motivic homotopy category over a scheme S, anyobject F ∈ SH ( S ) gives rise to relative Borel–Moore homology groupsH BM s ( X / S , F ( r )) ∶ = Hom SH ( S ) ( S ( r )[ s ] , f ∗ f ! ( F )) , bigraded by integers r, s ∈ Z (where ( r ) denotes the Tate twist), where X isa locally of finite type S-scheme with structural morphism f ∶ X → S. It wasobserved in [De2] that as X and S vary, these groups behave just like a bivari-ant theory in the sense of [FM] except that they are bigraded. Appropriatechoices of the coefficient F give rise to bivariant versions of such theoriesas motivic cohomology, algebraic cobordism, ´etale cohomology with finiteor adic coefficients, and singular cohomology. Using the extension of SHto derived schemes constructed in [Kh1], we also obtain derived extensionsof all these bivariant theories. Moreover, for coefficients F satisfying ´etaledescent, these bivariant theories extend further to derived Artin stacks (thisis done by extending the ´etale-local motivic homotopy category SH ´et and itssix operations to derived Artin stacks, see Appendix A). In Subsect. 2.4 we IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 3 demonstrate the expected properties: long exact localization sequences, ho-motopy invariance for vector bundle stacks, and Poincar´e duality for smoothstacks.
Fundamental classes.
Sect. 3 contains our construction of the virtual class [ X / Y ] vir of a quasi-smooth morphism f ∶ X → Y of relative virtual dimension d . Assume F is oriented for simplicity. The idea is that there are canonicalisomorphisms H BM2 d ( X / Y , F ( d )) ≃ H BM2 d ( X cl / Y cl , F ( d )) through which the virtual class corresponds to a more intrinsic fundamentalclass [ X / Y ] ∈ H BM2 d ( X / Y , F ( d )) . The latter is constructed, much as in Ful-ton’s intersection theory, by using deformation to the normal bundle stackto define a specialization mapsp X /Y ∶ H BM s ( Y / S , F ( r )) → H BM s ( N X /Y / S , F ( r )) , see Subsect. 3.1. By homotopy invariance for vector bundle stacks, the targetis identified with H BM s + d ( X / Y , F ( r + d )) , so we get a Gysin map(0.1) f ! ∶ H BM s ( Y / S , F ( r )) → H BM s + d ( X / S , F ( r + d )) . The fundamental class [ X / Y ] is the image of the unit 1 ∈ H BM0 ( Y / Y , F ) ,where we take S = Y .The two key properties of the fundamental class are functoriality andstability under arbitrary derived base change, see Theorems 3.12 and 3.13.We also have excess intersection, self-intersection, and blow-up formulas(Subsect. 3.2). In the sequel we intend to prove analogues of the virtualAtiyah–Bott localization and cosection formulas in this framework. Non-transverse B´ezout theorem.
The fundamental classes satisfy a co-homological B´ezout theorem that holds without any transversity hypotheses(Subsect. 3.4). For schemes, it can be stated in the Chow group as follows.Let X be a smooth quasi-projective scheme over a field k . Let f ∶ Y → Xand g ∶ Z → X be quasi-smooth projective morphisms of derived schemes ofrelative virtual dimensions − d and − e , respectively. Then the intersectionproduct of the fundamental classes [ Y ] ∈ A d ( X ) and [ Z ] ∈ A e ( X ) is given bythe fundamental class of the derived fibred product:(0.2) [ Y ] ⋅ [ Z ] = [ Y R × X Z ] in A d + e ( X ) .If k is of characteristic zero, this formula completely characterizes theintersection product in A ∗ ( X ) , since by resolution of singularities the Chowgroup is generated by fundamental classes [ Z ] where f ∶ Z → X is a projectivemorphism with Z smooth (so that f is automatically quasi-smooth). ADEEL A. KHAN
Grothendieck–Riemann–Roch.
In Subsect. 3.5 we prove a generaliza-tion of the Grothendieck–Riemann–Roch theorem to derived Artin stacks.For a locally noetherian derived Artin stack X , denote by G ( X ) the G-theoryof X , i.e., the Grothendieck group of coherent sheaves on X . Let f ∶ X → Y be a quasi-smooth morphism of derived Artin stacks, locally of finite typeover some regular noetherian base scheme. Then there is a commutativediagram(0.3) G ( Y ) G ( X ) A ∗ ( Y ) Q A ∗ ( X ) Q , f ∗ τ Y τ X Td X/Y ∩ f ! where Td X /Y is the Todd class of the relative cotangent complex L X /Y .In particular, if X is a quasi-smooth derived Artin stack over a field, thisgives the following formula for the fundamental class of X in A ∗ ( X ) Q :(0.4) [ X ] = Td − X ∩ τ X ( O X ) . Through the canonical isomorphisms A ∗ ( X ) Q ≃ A ∗ ( X cl ) Q and G ( X ) ≃ G ( X cl ) , this becomes the formula [ X ] vir = ( Td vir X ) − ∩ ( ∑ i ∈ Z ( − ) i ⋅ τ X cl ( π i ( O X ))) in A ∗ ( X cl ) Q , relating the virtual class [ X ] vir ∈ A ∗ ( X cl ) Q with the K-theoreticfundamental class in G ( X cl ) . The virtual Todd class Td vir X is the Todd classof the perfect complex L X ∣ X cl on X cl . This extends the formula predicted byKontsevich in the case of schemes [Ko, 1.4.2]. Absolute purity.
Our construction of fundamental classes is interestingeven when we restrict to classical algebraic geometry; in this case quasi-smoothness translates to being a local complete intersection morphism (whichneed not admit a global factorization through a regular immersion andsmooth morphism). For example, we get Gysin maps for proper lci mor-phisms between Artin stacks in ´etale cohomology and mixed Weil cohomol-ogy theories such as Betti and de Rham cohomology. In terms of the sixoperations, if f ∶ X → Y is an lci morphism of virtual dimension d betweenArtin stacks, then the fundamental class can be viewed as a canonical mor-phism(0.5) f ∗ F ( d )[ d ] → f ! ( F ) for any coefficient F . In the context of ´etale cohomology, such morphismswere constructed previously by Gabber [Fu], [ILO, Exp. XVI] in the caseof schemes and assuming the existence of a global factorization of f . Thustaking F to be the ´etale motivic cohomology spectrum Λ ´et (with coefficientsin Λ = Z / n Z , n invertible on Y ) gives a generalization of Gabber’s construc-tion. In Subsect. 3.6 we prove that the morphism (0.5) is invertible when F = Λ ´et and X and Y are regular Artin stacks. This extends Gabber’sproof of the absolute purity conjecture to Artin stacks (and drops the globalfactorization hypothesis in the case of schemes). IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 5
Related work.
The yoga of fundamental classes in motivic bivariant theo-ries was developed in [De2] and [DJK]. This paper extends these construc-tions on one hand from classical to derived algebraic geometry, and on theother hand from schemes to algebraic stacks (at least for ´etale coefficients).The notion of perfect obstruction theory introduced by K. Behrend andB. Fantechi [BF] is a useful approximation to a quasi-smooth derived struc-ture on a scheme or Deligne–Mumford stack, and actually suffices for the con-struction of virtual fundamental classes on Deligne–Mumford stacks. Thisconstruction was done in [BF] in Chow groups, and has been refined toalgebraic cobordism and other Borel–Moore homology theories recently byM. Levine [Le2] and Y.-H. Kiem and H. Park [KP]. Our construction agreeswith the P.O.T. approach when both are defined (see Subsect. 3.3), but itis worth noting that a quasi-smooth derived Artin stack typically has a 3-term cotangent complex, so that the P.O.T. formalism does not apply (infact, there is no associated intrinsic normal cone in the world of classical1-stacks).Virtual fundamental classes have been studied using the language of de-rived algebraic geometry previously in the setting of algebraic cobordismby P. Lowrey and T. Sch¨urg [LS]. They were also studied using the olderlanguage of dg-schemes by I. Ciocan-Fontanine and M. Kapranov [CK] inrational Chow groups and G-theory. These approaches only work for derived schemes and also require other unpleasant hypotheses such as existence of acharacteristic zero base field and embeddings into smooth ambient schemes.The B´ezout formula (0.2) mentioned above was inspired by a similar formulaannounced by J. Lurie [Lu] in Betti cohomology.Classical Borel–Moore homology was recently extended to Artin stacksby M. Kapranov and E. Vasserot [KV], for the purpose of defining a coho-mological Hall algebra whose underlying vector space is the Borel–Moorehomology of the moduli stack of coherent sheaves on a surface. Our for-malism gives a streamlined approach to the construction of this algebra,whose multiplicative structure arises from the quasi-smooth structure onthe moduli stack. Moreover it shows that the same structure exists on theBorel–Moore homology with coefficients in any ´etale motivic spectrum.
Acknowledgments.
During the very long gestation period of this paper,I benefited from helpful discussions with Denis-Charles Cisinski, Fr´ed´ericD´eglise, Marc Hoyois, Fangzhou Jin, Marc Levine, Mauro Porta, CharanyaRavi, Marco Robalo, and especially David Rydh. Thanks to the organizers ofthe June 2019 summer school “New perspectives in Gromov-Witten theory”in Paris which made some of the above conversations possible and where Iwas inspired to finally write up these results. Thanks to the Institute forAdvanced Study which hosted me in July 2019 while the first draft of thispaper was being finished.
ADEEL A. KHAN The intrinsic normal bundle
Stacks.
In this paper, we define a stack to be a “higher stack” in thesense of [HS]. That is, it is a functorR ↦ X ( R ) assigning to any commutative ring R an ∞ -groupoid X ( R ) of R-valuedpoints, and satisfying hyperdescent with respect to the ´etale topology (inthe sense of ∞ -category theory, see e.g. [To2, p. 183]).We say X is 0 -Artin if it is (representable by) an algebraic space. Wedefine k -Artin stacks inductively, following [To1, § k ⩾
0, a morphism f ∶ X → Y is k -representable if forevery k -Artin Y ′ and every morphism Y ′ → Y , the fibred product X × Y Y ′ is k -Artin. An k -representable morphism f is smooth if for every scheme Y,every morphism Y → Y , and every smooth atlas X → X × Y Y, the compositeX → X × Y Y → Y is a smooth morphism of schemes. A stack X is ( k + ) -Artin if its diagonal X → X × X is representable by k -Artin stacks, and thereexists a scheme X and a morphism X → X (automatically k -representable)which is smooth and surjective. The morphism X → X is called a smoothatlas for X .An k -Artin stack X always takes values in k -groupoids: for every com-mutative ring R, the ∞ -groupoid X ( R ) is k -truncated. We say a stack is Artin if it is k -Artin for some k ⩾
0. Artin stacks in this sense form an ∞ -category, whose full subcategory spanned by 1-Artin stacks is equivalentto the ( , ) -category of Artin stacks in the usual sense.Now replace the category of commutative rings by its nonabelian derived ∞ -category, i.e., the ∞ -category of simplicial commutative rings. This isthe natural target for derived functors on the nonabelian category of com-mutative rings, such as the derived tensor product. A simplicial commuta-tive ring R has an underlying ordinary commutative ring π ( R ) as well as π ( R ) -modules π i ( R ) . We say R is discrete if π i ( R ) = i > ≃ π ( R ) ); the discrete simplicial commutative rings span a full subcate-gory equivalent to the ordinary category of commutative rings. The notionsof ´etale and smooth homomorphism admit natural extensions to simplicialcommutative rings. See [SAG, Chap. 25] or [To1, § derived stack X is a functor R ↦ X ( R ) , assigning an ∞ -groupoid ofR-points to every simplicial commutative ring R, that satisfies ´etale hyper-descent. Derived k -Artin and Artin stacks are defined following the patternoutlined above, see e.g. [To1, § Vector bundle stacks.
Let X be a derived Artin stack and E a per-fect complex on X of Tor-amplitude [ − k, ] , for some integer k ⩾ −
1. Theassociated vector bundle stack π ∶ V X ( E [ − ]) → X IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 7 is the moduli stack of co-sections of E [ − ] . That is, for any affine derivedscheme S over X , the ∞ -groupoid of X -morphisms S → V X ( E [ − ]) is natu-rally equivalent to the ∞ -groupoid of O S -linear morphisms of perfect com-plexes E [ − ]∣ S → O S .Since E [ − ] is perfect of Tor-amplitude [ − k − , ] , V X ( E [ − ]) is a smooth ( k + ) -Artin derived stack over X of relative dimension − d , where d is thevirtual rank of E . See [To2, Subsect. 3.3, p. 201].1.3. Normal bundle stacks.
The normal bundle stack is a derived versionof the relative intrinsic normal cone of [BF].A morphism f ∶ X → Y of derived Artin stacks is quasi-smooth if it islocally of finite presentation and the relative cotangent complex L X /Y is ofTor-amplitude ( −∞ , ] . Note that we use homological grading: this meansthat, for every discrete quasi-coherent sheaf E on X , we have π i ( L X /Y ⊗ L O X E ) = i >
1. If X = X and Y = Y are derived schemes, this is equivalent tothe following condition: Zariski-locally on X, f factors through a smoothmorphism M → Y and a morphism X → M which exhibits X as the de-rived zero-locus of some functions on M [KR, 2.3.14]. If f ∶ X → Y is k -representable, then it is quasi-smooth if and only if for every derived schemeY, every morphism Y → Y , and every smooth atlas X → X × Y Y, the com-posite X → X × Y Y → Y is a quasi-smooth morphism of derived schemes.The relative virtual dimension of a quasi-smooth morphism f ∶ X → Y isvd ( X / Y ) ∶ = rk ( L X /Y ) , the virtual rank (Euler characteristic) of the relative cotangent complex.Let f ∶ X → Y be a k -representable quasi-smooth morphism. The cotan-gent complex L X /Y is perfect of Tor-amplitude [ − k, ] , so the associatedvector bundle stack V X ( L X /Y [ − ]) is a smooth ( k + ) -Artin stack of rela-tive virtual dimension − vd ( X / Y ) . Definition 1.1.
Let f ∶ X → Y be a quasi-smooth morphism of derived Artinstacks. The normal bundle stack is the vector bundle stack N X /Y = V X ( L X /Y [ − ]) → X . If f is a closed immersion, then L X /Y [ − ] is of Tor-amplitude [ , ] , andthe normal bundle stack is just the normal bundle. If f is smooth, then L X /Y [ − ] is of Tor-amplitude ( −∞ , − ] , and the normal bundle stack is theclassifying stack of the tangent bundle T X /Y . If f factors through a closedimmersion i ∶ X → Y ′ and a smooth morphism p ∶ Y ′ → Y , then the normalbundle stack is the quotientN X /Y = [ N X /Y ′ / i ∗ T Y ′ /Y ] . Proposition 1.2.
ADEEL A. KHAN (i)
The construction N X /Y → X is stable under derived arbitrary base changein X . That is, for any homotopy cartesian square of derived Artin stacks X ′ Y ′ X Y f ′ f with f quasi-smooth, there is a canonical isomorphism N X /Y R × X X ′ → N X ′ /Y ′ of derived Artin stacks over X ′ . (ii) Suppose given a commutative square
X Y
X Y ip qf with f quasi-smooth, p and q smooth surjections with X and Y schematic,and i a quasi-smooth closed immersion. Then N X /Y is the quotient of thegroupoid N ˇC ( X /X ) ● / ˇC ( Y /Y) ● ∶ = [ ⋯ →→→ N X × R X X / Y × R Y Y ⇉ N X / Y ] , i.e., the geometric realization of this simplicial diagram.Proof. The first claim follows from the fact that the cotangent complex isstable under derived base change [Lu, Prop. 3.2.10]. The second follows fromthe fact that the cotangent complex satifies descent for smooth surjections[Bh, Cor. 2.7]. (cid:3)
Deformation to the normal bundle stack.
For any quasi-smoothmorphism f ∶ X → Y , there is a canonical A -deformation to the zero sec-tion 0 ∶ X → N X /Y , generalizing the classical construction of Verdier. Thisconstruction is joint with D. Rydh.
Theorem 1.3.
Let f ∶ X → Y be a quasi-smooth morphism of derived Artinstacks. (i)
There exists a quasi-smooth derived Artin stack D X /Y over Y × A , and aquasi-smooth morphism X × A → D X /Y over Y × A . The fibre over G m = A ∖ { } is the quasi-smooth morphism X × G m → Y × G m and the fibre over { } is the quasi-smooth morphism ∶ X → N X /Y . (ii) The construction D X /Y → Y is stable under arbitrary derived base change in Y . IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 9
In the case where f is a closed immersion, D X /Y was already constructedin [KR, Thm. 4.1.13]. For a general quasi-smooth morphism with a presen-tation as in Proposition 1.2(ii), it can be described as the quotient of thegroupoid D ˇC ( X /X ) ● / ˇC ( Y /Y) ● ∶ = [ ⋯ →→→ D X × R X X / Y × R Y Y ⇉ D X / Y ] . Without choosing a presentation, it can be described simply as the Weilrestriction D
X /Y = Res
Y/Y× A ( X ) of X along Y = Y × { } → Y × A . Details will be provided elsewhere.2. Motivic Borel–Moore homology of derived stacks
In this section we construct, given a “coefficient” F over a derived Artinstack S , a (relative) Borel–Moore homology theory with coefficients in F .The main example is F = Q S , the rational motivic cohomology spectrum.The construction requires a formalism of six operations on derived Artinstacks such as that developed in Appendix A.2.1. Definition and examples.
Let S be a derived Artin stack and let F ∈ SH ´et ( S ) be an ´etale motivic spectrum (see Appendix A). Definition 2.1.
For a derived Artin stack X locally of finite type over S with structural morphism f ∶ X → S , we define Borel–Moore homology withcoefficients in F by the formula (2.2) H BM s ( X / S , F ( r )) = Hom SH ´et (S) ( S ( r )[ s ] , f ∗ f ! F ) , r, s ∈ Z where S ∈ SH ´et ( S ) is the monoidal unit. Similarly we define cohomologywith coefficients in F by (2.3) H s ( X , F ( r )) = Hom SH ´et (S) ( S , f ∗ f ∗ F ( r )[ s ]) for any derived Artin stack X over S . The observation that H s ( X , F ( r )) = H BM − s ( X / X , F ( − r )) (by adjunction)allows us to pass freely from Borel–Moore homology statements to theircohomological counterparts, which is why we generally stick with the formerperspective. For an immersion i ∶ Y → X , we have also cohomology withsupport:(2.4) H s Y ( X , F ( r )) = H BM − s ( X / Y , F ( − r )) . Remark 2.5.
The Borel–Moore homology groups H BM s ( X / S , F ( r )) onlydepend on the homotopy category (underlying triangulated category) of thestable ∞ -category SH ´et ( S ) . A more refined object is the spectrum (in thesense of homotopical algebra) R Γ BM ( X / S , F ( r )) ∶ = Maps SH ´et (S) ( S ( r ) , f ∗ f ! F ) , defined using the spectral enrichment of SH ´et ( S ) . The groups H BM s ( X / S , F ( r )) are the homotopy groups π s R Γ BM ( X / S , F ( r )) . Similarly, there is a coho-mology spectrum R Γ ( X , F ( r )) ∶ = Maps SH ´et (S) ( S ( r ) , f ∗ f ∗ F ) . Remark 2.6.
Let S = S be a derived algebraic space and X a locally offinite type derived Artin stack over S. The formula (A.4) implies that theBorel–Moore spectra R Γ BM ( X / S , F ( r )) can be computed by the homotopylimit(2.7) R Γ BM ( X / S , F ( r )) = lim ←Ð u R Γ BM ( X / S , F ( r + d u ))[ − d u ] over the ∞ -category of smooth morphisms u ∶ X → X with X a scheme,where d u is the relative dimension of u . Similarly, the cohomology spectrumis computed as the homotopy limit(2.8) R Γ ( X , F ( r )) = lim ←Ð u R Γ ( X , F ( r )) . Alternatively, we can fix a smooth atlas X → X and use (A.3) to write R Γ BM ( X / S , F ( r )) as the homotopy limit or totalization of the cosimplicialdiagram(2.9) R Γ BM ( X / S , F ( r + d ))[ − d ] ⇉ R Γ BM ( X R × X X / S , F ( r + d ))[ − d ] →→→ R Γ BM ( X R × X X R × X X / S , F ( r + d )[ − d ]) →→→→ ⋯ where d = vd ( X / X ) , and again similarly for R Γ ( X , F ( r )) . Example 2.10.
Let Q denote the rational motivic cohomology spectrumover Spec ( Z ) (see [CD1, Chap. 14], [Sp]). It satisfies ´etale (hyper)descent[CD2, Prop. 2.2.10], so for any derived Artin stack S we may define Q S asits inverse image along S →
Spec ( Z ) . The groupsH BM s ( X / S , Q ( r )) , H s ( X , Q ( r )) are simply called the (rational) motivic Borel–Moore homology and mo-tivic cohomology groups. If S is the spectrum of a field k and X = X isa quasi-projective classical scheme, then motivic Borel–Moore homology iscomputed as the cohomology of Bloch’s cycle complex; in particularH
BM2 n ( X / Spec ( k ) , Q ( n )) = A n ( X ) Q . For X a classical scheme locally of finite type over a field k , it is com-puted by the Zariski hypercohomology of the same complex. More precisely, R Γ BM ( X / Spec ( k ) , Q ( r )) is the Zariski localization of Bloch’s cycle complex.Thus for X an Artin stack locally of finite type over k , H BM ∗ ( X / Spec ( k ) , Q ( r )) is computed according to the formula (2.9) by the ´etale hypercohomology ofthe complex lim ←Ð [ m ]∈ ∆ z r + md ( X R × X ⋯ R × X X , ∗ ) Q [ − md ] where there are m terms in the fibred product. These are thus the sameas the rational higher Chow groups defined by Joshua [Jo], and they agreewith the rationalization of Kresch’s Chow groups [Kr]. IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 11
Example 2.11.
Integrally, we can take the ´etale motivic cohomology spec-trum Z ´et . More generally for every commutative ring Λ, let Λ ´et denote the´etale hyperlocalization of the Λ-linear motivic cohomology spectrum andwrite Λ ´et S for its inverse image to any derived Artin stack S (along the struc-tural morphism S →
Spec ( Z ) ). The resulting groups are called ´etale motivicBorel–Moore homology and ´etale motivic cohomology (or “Lichtenbaum mo-tivic cohomology”), respectively:H BM s ( X / S , Λ ´et ( r )) , H s ( X , Λ ´et ( r )) . Rationally these give back the groups just defined above since Q S alreadysatisfies ´etale hyperdescent. With finite coefficients Λ = Z / n Z it follows fromRemark 2.6 and [CD2, Thm. 4.5.2] that these agree with ´etale Borel–Moorehomology and ´etale cohomology [La, Ol, LO1], over classical Artin stackswith n invertible. Taking Z ∧ ℓ, S to be the ℓ -adic completion of Z S as in [CD2,Subsect. 7.2], for a prime ℓ , we also recover ℓ -adic Borel–Moore homologyand cohomology, respectively. Example 2.12.
Let MGL denote Voevodsky’s algebraic cobordism spec-trum. If X is a smooth algebraic space over a perfect field k , then the coho-mology groups H n ( X , MGL ( n )) are computed for n ⩾ k is of characteristic zero, thenthey are identified with Levine–Morel’s algebraic cobordism Ω n ( X ) , andmoreover the Borel–Moore homology groups H BM2 n ( X , MGL ( n )) are identi-fied with Ω n ( X ) for all n ∈ Z , also for X singular [Le1].Let MGL ´et denote the ´etale hyperlocalization of MGL. For a derivedArtin stack S , let MGL ´et S denote the inverse image along the structuralmorphism S →
Spec ( Z ) . This gives ´etale algebraic cobordism and bordismgroups for derived Artin stacksH BM s ( X / S , MGL ´et ( r )) , H s ( X , MGL ´et ( r )) . If X is smooth over a perfect field k , then H n ( X , MGL ´et ( n )) is computedusing the construction of Remark 2.6 by the same presheaf of spectra men-tioned above (for n ⩾ Q already satisfies ´etale hyperdescent (MGL Q ≃ MGL ´et Q ) and is identified withMGL Q ≃ Q [ c , c , . . . ] , where c i is a generator of bidegree ( i, i ) , by [NSØ]. We thus define MGL Q , S as the inverse image of MGL Q for any derived Artin stack S . There arecanonical mapsH BM s ( X / S , MGL ´et ( r )) → H BM s ( X / S , MGL Q ( r )) → H BM s ( X / S , Q ( r )) for all X locally of finite type over S . Example 2.13.
Let KGL ´et S denote the ´etale hyperlocalization of the alge-braic K-theory spectrum. Assuming that S is a regular (classical) stack, suchas the spectrum of Z or a field, the Borel–Moore homology represented byKGL ´et S coincides with ´etale hypercohomology with coefficients in G-theory, and the proper covariance and smooth Gysin maps are compatible with therespective intrinsic operations in G-theory [Ji, Cor. 3.3.7]. Note that in thiscase the formula of Remark 2.6 simplifies since there are Bott periodicityisomorphisms KGL ( n )[ n ] ≃ KGLfor all n ∈ Z . Remark 2.14.
If we restrict to derived schemes or algebraic spaces, thenwe are allowed to take coefficients that do not satisfy ´etale descent, suchas the integral motivic cohomology spectrum Z or the algebraic cobordismspectrum MGL. Indeed, for derived algebraic spaces the formalism of six op-erations is already available before imposing ´etale descent (see Subsect. A.1).The basic operations discussed in the next section will also carry over tothat setting. Moreover, the fundamental class can still be defined at leastfor smoothable quasi-smooth morphisms (Variant 3.11).2.2. Basic operations.
The formalism of six operations (see Appendix A)immediately yields the following structure on Borel–Moore homology groups.Here F is any coefficient defined over S , though for simplicity we assume that F is multiplicative (a motivic ring spectrum) and oriented . In particularthere is a unit element 1 ∈ H BM0 ( X / X , F ) = H ( X , F ) induced by the unit η F ∶ S → F .2.2.1. Proper direct image. If f ∶ X → Y is a representable proper morphismof derived Artin stacks locally of finite type over S , then there are functorialdirect image homomorphisms f ∗ ∶ H BM s ( X / S , F ( r )) → H BM s ( Y / S , F ( r )) . These are induced by the co-unit f ∗ f ! = f ! f ! → id. If F satisfies h-descent,e.g., F is Q or MGL Q , then by Theorem A.7 this extends to arbitraryproper morphisms f ∶ X → Y as long as X and Y are Deligne–Mumford (seeExample A.8 for some milder assumptions that work).2.2.2. Smooth contravariance. If f ∶ X → Y is a smooth morphism of relativedimension d between derived Artin stacks locally of finite type over S , thenthere are functorial Gysin homomorphisms f ! ∶ H BM s ( Y / S , F ( r )) → H BM s + d ( X / S , F ( r + d )) . These are compatible with proper direct images by a base change formula.They are induced by the co-trace transformation id → f ∗ Σ −L X/Y f ! , righttranspose of the purity equivalence Σ L X/Y f ∗ = f ! (Theorem A.13). This essentially amounts to admitting a theory of Chern classes. The constructionsalso work for non-oriented spectra [DJK], but are more notationally complex due to thenecessity of grading by K-theory classes instead of just pairs of integers. We are onlyinterested in oriented examples here.
IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 13
Change of base. If f ∶ T → S is a morphism of derived Artin stacksand X is a derived Artin stack locally of finite type over S, then there arechange of base homomorphisms f ∗ ∶ H BM s ( X / S , F ( r )) → H BM s ( X T / T , F ( r )) , where X T = X × R S T is the derived fibred product. More generally, for anycommutative square Y TX S ∆ f which is cartesian on underlying classical stacks, there are homomorphisms f ∗ ∆ ∶ H BM s ( X / S , F ( r )) → H BM s ( Y / T , F ( r )) . These are induced by the unit map id → f ∗ f ∗ and the base change formula(Theorem A.5). Remark 2.15.
Note that f ∗ always denotes contravariant functoriality inthe base (change of base homomorphisms, 2.2.3), while f ! denotes contravari-ant functoriality in the source (Gysin homomorphisms, 2.2.2). Potentiallythe notation also clashes with that of the six operations (Appendix A), butthere should be no risk of confusion.2.2.4. Top Chern class.
Let E be a finite locally free sheaf of rank r on aderived Artin stack X over S . Then there is a top Chern class (Euler class) c r ( E ) ∈ H r ( X , F ( r )) . This is induced by the Euler transformation id → Σ E (Construction A.16).There is a general theory of Chern classes c i ( E ) (when F is oriented), as in[De1, Sect. 2.1], but we will not need it here.2.2.5. Composition product.
Given a derived Artin stack T locally of finitetype over S and a derived Artin stack X locally of finite type over T , thereis a pairing ○ ∶ H BM s ( X / T , F ( r )) ⊗ H BM s ′ ( T / S , F ( r ′ )) → H BM s + s ′ ( X / S , F ( r + r ′ )) . This comes from the multiplication map m ∶ F ⊗ F → F , see [DJK, 2.2.7(4)]for details.Special cases of the composition product are cap and cup products:2.2.6.
Cap product.
Given a derived Artin stack X locally of finite type over S , there is a pairing(2.16) ∩ ∶ H s ( X , F ( r )) ⊗ H BM s ′ ( X / S , F ( r ′ )) → H BM s ′ − s ( X / S , F ( r ′ − r )) . Cup product.
Given a derived Artin stack X over S , there is a pairing(2.17) ∪ ∶ H s ( X , F ( r )) ⊗ H s ′ ( X , F ( r ′ )) → H s + s ′ ( X , F ( r + r ′ )) . From now on, whenever we consider a Borel–Moore homology groupH BM s ( X / S , F ( r )) , we will implicitly assume that X is locally of finite typeover S (so that the exceptional inverse image functor f ! exists, see Sub-sect. A.2).2.3. Basic compatibilities.
The operations on Borel–Moore homology aresubject to the following compatibilities, direct analogues of the axioms of abivariant theory in the sense of Fulton–MacPherson [FM, Sect. 2.2].2.3.1.
Change of base and composition product.
Suppose given a commuta-tive diagram X T XY T YT S gf where the squares are cartesian. Then for classes α ∈ H BM r ( X / Y , F ( s )) , β ∈ H BM r ′ ( Y / S , F ( s ′ )) , we have f ∗ ( α ○ β ) = g ∗ ( α ) ○ f ∗ ( β ) in H BM s + s ′ ( X T / T , F ( r + r ′ )) .2.3.2. Change of base and direct image.
Suppose given a commutative dia-gram X T XY T YT S h ′ hf where the squares are cartesian. Then for any class α ∈ H BM r ( X / S , F ( s )) ,we have f ∗ h ∗ ( α ) = h ′∗ f ∗ ( α ) in H BM s ( Y T / T , F ( r )) .2.3.3. Direct image and composition product (on the right).
Suppose givena commutative diagram
X YS T f IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 15 with f representable and proper. Then for classes α ∈ H BM s ( X / S , F ( r )) , β ∈ H BM s ′ ( S / T , F ( r ′ )) , we have f ∗ ( α ) ○ β = f ∗ ( α ○ β ) in H BM s + s ′ ( Y / T , F ( r + r ′ )) .2.3.4. Direct image and composition product (on the left).
Suppose given acommutative diagram X ′ Y ′ X YS gf where the square is cartesian. Then for classes α ∈ H BM s ( X / S , F ( r )) and β ∈ H BM s ′ ( Y ′ / Y , F ( r ′ )) , we have β ○ f ∗ ( α ) = g ∗ ( f ∗ ( β ) ○ α ) in H BM s + s ′ ( Y ′ / S , F ( r + r ′ )) .2.4. Properties.
The following two statements follow immediately fromTheorem A.9.
Theorem 2.18 (Localization) . Let i ∶ Z → X be a closed immersion ofderived Artin stacks over S , with open complement j ∶ U → X . Then forevery integer r there is a long exact sequence ⋯ ∂ Ð→ H BM s + ( Z / S , F ( r )) i ∗ Ð→ H BM s + ( X / S , F ( r )) j ! Ð→ H BM s + ( U / S , F ( r )) ∂ Ð→ H BM s ( Z / S , F ( r )) i ∗ Ð→ H BM s ( X / S , F ( r )) j ! Ð→ ⋯ Theorem 2.19 (Derived invariance) . Let X be a derived Artin stack over S . (i) Let i S ∶ S cl → S denote the inclusion of the underlying classical stack.Then the change of base homomorphisms i ∗S ∶ H BM s ( X / S , F ( r )) → H BM s ( X R × S S cl / S cl , F ( r )) are bijective for all r, s ∈ Z . (ii) Let i X ∶ X cl → X denote the inclusion of the underlying classical stack.Then the direct image homomorphisms ( i X ) ∗ ∶ H BM s ( X cl / S , F ( r )) → H BM s ( X / S , F ( r )) are bijective for all r, s ∈ Z . Proposition 2.20 (Homotopy invariance) . Let X be a derived Artin stackover S . For a perfect complex E on X of Tor-amplitude [ − k, ] , where k ⩾ − ,6 ADEEL A. KHAN
Theorem 2.18 (Localization) . Let i ∶ Z → X be a closed immersion ofderived Artin stacks over S , with open complement j ∶ U → X . Then forevery integer r there is a long exact sequence ⋯ ∂ Ð→ H BM s + ( Z / S , F ( r )) i ∗ Ð→ H BM s + ( X / S , F ( r )) j ! Ð→ H BM s + ( U / S , F ( r )) ∂ Ð→ H BM s ( Z / S , F ( r )) i ∗ Ð→ H BM s ( X / S , F ( r )) j ! Ð→ ⋯ Theorem 2.19 (Derived invariance) . Let X be a derived Artin stack over S . (i) Let i S ∶ S cl → S denote the inclusion of the underlying classical stack.Then the change of base homomorphisms i ∗S ∶ H BM s ( X / S , F ( r )) → H BM s ( X R × S S cl / S cl , F ( r )) are bijective for all r, s ∈ Z . (ii) Let i X ∶ X cl → X denote the inclusion of the underlying classical stack.Then the direct image homomorphisms ( i X ) ∗ ∶ H BM s ( X cl / S , F ( r )) → H BM s ( X / S , F ( r )) are bijective for all r, s ∈ Z . Proposition 2.20 (Homotopy invariance) . Let X be a derived Artin stackover S . For a perfect complex E on X of Tor-amplitude [ − k, ] , where k ⩾ − ,6 ADEEL A. KHAN denote by π ∶ V X ( E [ − ]) → X the associated vector bundle stack. Then forevery r, s ∈ Z there is a canonical isomorphism π ! ∶ H BM s ( X / S , F ( r )) → H BM s − d ( V X ( E [ − ])/ S , F ( r − d )) , where d is the virtual rank of E .Proof. The map is induced by the natural transformation f ∗ f ! ( F ) unit ÐÐ→ f ∗ π ∗ π ∗ f ! ( F ) pur π ÐÐ→ f ∗ π ∗ Σ −L π π ! f ! ( F ) = f ∗ π ∗ Σ π ∗ (E) π ! f ! ( F ) , which is invertible by Props. A.10 and A.13. (cid:3) Definition 2.21.
Let X be a derived Artin stack over S . If X is smooth ofrelative dimension d , then there is a relative fundamental class [ X / S ] ∈ H BM2 d ( X / S , F ( d )) defined as the image of the unit by the Gysin map f ! (2.2.2). More explicitly,this class is induced by the morphism S → f ∗ Σ −L X/S f ! ( F ) = f ∗ f ! ( F )( − d )[ − d ] coming by adjunction from the purity isomorphism f ! = Σ L X/S f ∗ (Theo-rem A.13), where f ∶ X → S is the structural morphism.
Remark 2.22.
For X smooth over S as above, the fundamental class [ X / S ] is “classical”, in the sense that it is insensitive to the derived structure. Thatis, under the canonical isomorphism (Theorem 2.19)H BM2 d ( X / S , F ( d )) ≃ H BM2 d ( X cl / S cl , F ( d )) , the class [ X / S ] corresponds to [ X cl / S cl ] , the fundamental class of the mor-phism X cl → S cl . Note that this makes sense because the latter is again asmooth morphism of relative dimension d . This is in contrast to the moregeneral case of quasi-smooth morphisms (Sect. 3). Theorem 2.23 (Poincar´e duality) . Let X be a smooth derived Artin stackover S . Then cap product (2.2.6) with the fundamental class [ X / S ] inducesa canonical isomorphism H s ( X , F ( r )) ∩[X /S] ÐÐÐÐ→ H BM2 d − s ( X / S , F ( d − r )) for all r, s, ∈ Z .Proof. Unraveling definitions, this follows from the fact that the morphism S → f ∗ f ! ( F )( − d )[ − d ] defining [ X / S ] is the “right transpose” of an iso-morphism. See the discussion after [DJK, Def. 2.3.11]. (cid:3) Fundamental classes
We develop some basic tools of intersection theory, namely the specializa-tion and Gysin maps, for Borel–Moore homology of derived Artin stacks. Wefollow the constructions of D´eglise–Jin–Khan [DJK] closely, the main differ-ence being the introduction of the normal bundle stack to handle the casesof quasi-smooth closed immersions and smooth morphisms simultaneously.
IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 17
Construction.
Let S be a derived Artin stack and fix a coefficient F as in Sect. 2. Let f ∶ X → Y be a quasi-smooth morphism of derivedArtin stacks over S , say of relative virtual dimension d . Denote by N X /Y the normal bundle stack (Definition 1.1).
Construction 3.1 (Specialization map) . We define a specialization map(3.2) sp
X /Y ∶ H BM s ( Y / S , F ( r )) → H BM s ( N X /Y / S , F ( r )) for all r, s ∈ Z . First, the localization long exact sequence associated to theclosed immersion Y = Y × { } → Y × A splits into short exact sequences0 → H BM s + ( A Y / S , F ( r )) → H BM s + ( G m, Y / S , F ( r )) ∂ Ð→ H BM s ( Y / S , F ( r )) → . The homomorphism ∂ admits a canonical section(3.3) γ t ∶ H BM s ( Y / S , F ( r )) → H BM s + ( G m, Y / S , F ( r )) , see [DJK, 3.2.2] for details.Let D X /Y be the deformation space (Subsect. 1.4). Let i ∶ N X /Y → D X /Y denote the inclusion of the exceptional fibre and j ∶ Y × G m → D X /Y itscomplement. The associated localization long exact sequence has boundarymap ∂ ∶ H BM s + ( G m, Y / S , F ( r )) → H BM s ( N X /Y / S , F ( r )) , and we define (3.2) as the compositeH BM s ( Y / S , F ( r )) γ t Ð→ H BM s + ( G m, Y / S , F ( r )) ∂ Ð→ H BM s ( N X /Y / S , F ( r )) . Construction 3.4 (Gysin map) . We now construct the Gysin map(3.5) f ! ∶ H BM s ( Y / S , F ( r )) → H BM s + d ( X / S , F ( r + d )) , where f ∶ X → Y is as above. Let π ∶ N X /Y → X denote the projection. TheGysin map (3.5) is the compositeH BM s ( Y / S , F ( r )) sp X/Y
ÐÐÐ→ H BM s ( N X /Y / S , F ( r )) ( π ! ) − ÐÐÐ→ H BM s + d ( X / S , F ( r + d )) . where π ! is the isomorphism of Proposition 2.20. Construction 3.6 (Fundamental class) . The (relative) fundamental class of f ∶ X → Y is the class [ X / Y ] ∶ = f ! ( ) ∈ H BM2 d ( X / Y , F ( d )) which is the image of 1 ∈ H BM0 ( Y / Y , F ) . When f is smooth, this is thefundamental class already defined (see before Theorem 2.23).The (relative) virtual fundamental class is defined to be the unique class [ X / Y ] vir ∈ H BM2 d ( X cl / Y cl , F ( d )) corresponding to [ X / Y ] under the canonical isomorphisms of Theorem 2.19. Remark 3.7.
The Gysin map and fundamental class are essentially inter-changeable data, as we can recover the former via the composition product(2.2.5) with [ X / Y ] : f ! ( x ) = [ X / Y ] ○ x ∈ H BM s + d ( X / S , F ( r + d )) for all x ∈ H BM s ( Y / S , F ( r )) . Remark 3.8 (Purity transformation) . In terms of the six operations, thefundamental class can be interpreted as a canonical natural transformation(3.9) pur f ∶ Σ L X/Y f ∗ → f ! of functors SH ´et ( Y ) → SH ´et ( X ) , where Σ L X/Y is the operation defined in(A.12). Through the orientation of F , this induces a canonical isomorphism(3.10) f ∗ ( F )( d )[ d ] ≃ Σ L X/Y f ∗ ( F ) → f ! ( F ) . See [DJK, Subsects. 2.5, 4.3] for details on this perspective.
Variant 3.11.
Let’s restrict our attention to derived schemes or algebraicspaces. As explained in Remark 2.14, there is a well-behaved theory of Borel–Moore homology with coefficients in any F , not necessarily satisfying ´etaledescent (such as the integral motivic cohomology or algebraic cobordismspectrum). Following the constructions of [DJK, § [ X / Y ] ∈ H BM2 d ( X / Y , F ( d )) for smoothable quasi-smoothmorphisms f ∶ X → Y (where d = vd ( X / Y ) ).First let i ∶ Z → X be a quasi-smooth closed immersion (or quasi-smoothunramified morphism [KR, § N Z / X is a vector bundle (as opposed to a vector bundle stack), and Con-structions 3.1 and 3.4 only involve derived algebraic spaces. Thus we getthe fundamental class [ Z / X ] ∈ H BM2 d ( Z / X , F ( d )) , where d = vd ( Z / X ) . Thesefundamental classes also satisfy the properties asserted in the next section.Now let f ∶ X → Y be a smoothable quasi-smooth morphism of derivedalgebraic spaces, i.e., one that admits a global factorizationX i Ð→ M p Ð→ Ywith p smooth and i a (quasi-smooth) closed immersion. Define the funda-mental class [ X / Y ] ∈ H BM2 d ( X / Y , F ( d )) , where d = vd ( X / Y ) , by [ X / Y ] = [ X / M ] ○ [ M / Y ] . Exactly as in [DJK, § Properties.
We record the basic properties of the fundamental class.These could equivalently be stated for the Gysin maps.
Theorem 3.12 (Functoriality) . Let f ∶ X → Y and g ∶ Y → Z be quasi-smooth morphisms of derived Artin stacks, of relative virtual dimensions d and e , respectively. Then we have [ X / Y ] ○ [ Y / Z ] = [ X / Z ] in H BM2 d + e ( X / Z , F ( d + e )) . Use the double deformation space as in [DJK, Prop. 3.2.19].
IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 19
Theorem 3.13 (Base change) . Suppose given a cartesian square of derivedArtin stacks (3.14) X ′ Y ′ X Y gp qf over S , where f is quasi-smooth. Then there is an equality p ∗ [ X / Y ] = [ X ′ / Y ′ ] ∈ H BM2 d ( X ′ / Y ′ , F ( d )) , where d is the relative virtual dimension of f (and hence of g ). This follows easily from the stability of the deformation space D
X /Y underbase change (Theorem 1.3(ii)). More generally:
Proposition 3.15 (Excess intersection formula) . Suppose given a commu-tative square of derived Artin stacks (3.16) X ′ Y ′ X Y gp ∆ qf over S , where f and g are quasi-smooth. Assume that ∆ is an excess inter-section square , i.e., that it is cartesian on underlying classical stacks andthat the fibre E of the canonical map p ∗ L X /Y [ − ] → L X ′ /Y ′ [ − ] is a locally free O X -module of finite rank. Then there is an equality q ∗ ∆ [ X / Y ] = c r ( E ) ∩ [ X ′ / Y ′ ] ∈ H BM2 d ( X ′ / Y ′ , F ( d )) , where q ∗ ∆ denotes the change of base homomorphism (2.2.3), d = vd ( X / Y ) ,and r = rk ( E ) . Same as the proof of [DJK, Prop. 3.2.8]. We call E the excess sheaf associated to ∆. Note that its rank is r = vd ( X ′ / Y ′ ) − vd ( X / Y ) . Corollary 3.17 (Self-intersection formula) . Let i ∶ X → Y be a quasi-smooth closed immersion of relative virtual codimension n . Consider theself-intersection square X XX Y . ∆ ii We have i ∗ ∆ [ X / Y ] = c n ( N X /Y ) ∈ H BM − n ( X / X , F ( − n )) = H n ( X , F ( n )) , where N X /Y = L X /Y [ − ] is the conormal sheaf. Corollary 3.18 (Key formula) . Let i ∶ X → Y be a quasi-smooth closedimmersion of relative virtual codimension n . Form the blow-up square [KR,Thm. 4.1.5] : D Bl X /Y
X Y , p ∆ qi where D = P X ( N X /Y ) is the virtual exceptional divisor. We have q ∗ ∆ [ X / Y ] = c n − ( E ) ∩ [ D / Bl X /Y ] ∈ H BM − n ( D / Bl X /Y , F ( − n )) , where E is the excess sheaf. Comparison with Behrend–Fantechi.
Let f ∶ X → Y be a quasi-smooth morphism of derived 1-Artin stacks. Assume that f is representableby derived Deligne–Mumford stacks and Y is classical. In this case thevirtual fundamental class [ X / Y ] vir ∈ H BM2 d ( X cl / Y , F ( d )) can also be defined using the approach of Behrend–Fantechi [BF]. Below, wegive a variant of the construction of the Gysin map f ! (3.5) which will visiblyagree with the “virtual pullback” of Manolache [Ma]. By Corollary 3.12 of op. cit. , this will therefore identify our virtual fundamental class [ X / Y ] vir with the construction of Behrend–Fantechi.Let C X cl /Y denote the relative intrinsic normal cone [BF, Sect. 7] of themorphism X cl → X → Y , and let D X cl /Y denote Kresch’s deformation to theintrinsic normal cone [Ma, Thm. 2.31]. There is a commutative diagramC X cl /Y D X cl /Y Y × G m N X /Y D X /Y Y × G ma where the vertical arrows are closed immersions. Using the upper row, oneconstructs just as in (3.2) a specialization mapsp X cl /Y ∶ H BM s ( Y / S , F ( r )) → H BM s ( C X cl /Y / S , F ( r )) . By naturality of the localization triangle with respect to proper covariance(e.g. [DJK, Prop. 2.2.10]), we have an equality a ∗ ○ sp X cl /Y = sp X /Y of morphisms H BM s ( Y / S , F ( r )) → H BM s ( N X /Y / S , F ( r )) . In particular weget f ! = ( π ! ) − ○ sp X /Y = ( π ! ) − ○ a ∗ ○ sp X cl /Y , where π ∶ N X /Y → X is the projection. Now the right-hand side is preciselythe virtual pullback f !N X/Y [Ma, Constr. 3.6] constructed with respect to thevector bundle stack N
X /Y . IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 21
Non-transverse B´ezout theorem.
Let f ∶ Z → X be a morphismof derived Artin stacks over S . Suppose that f is quasi-smooth of relativevirtual dimension d . The fundamental class [ Z / X ] induces a cohomologicalGysin map(3.19) f ! ∶ H r ( Z , F ( s )) → H r − d Z ( X , F ( s − d )) where the target is the cohomology of X with support in Z (2.4). This mapis the compositeH BM − r ( Z / Z , F ( − s )) ○ [Z/X ] ÐÐÐÐ→ H BM − r + d ( Z / X , F ( − s + d )) . Composing further with the Borel–Moore direct image (2.2.1) f ∗ ∶ H BM − r + d ( Z / X , F ( − s + d )) → H BM − r + d ( X / X , F ( − s + d )) , when it exists, gives rise to the Gysin map(3.20) f ! ∶ H r ( Z , F ( s )) → H r − d ( X , F ( s − d )) valued in the cohomology of X . For example, this exists when f is properand representable, or just proper if X is Deligne–Mumford and F = Q orMGL Q .In particular we have a cohomological fundamental class(3.21) [ Z ] = f ! ( ) ∈ H − d ( X , F ( − d )) under these assumptions. For simplicity we’ll state Theorem 3.22 below onlyfor the representable case, but the proof only requires the existence of properdirect images.The following is a generalized cohomological B´ezout theorem, where notransversity assumptions are imposed. Theorem 3.22.
Let X be derived Artin stack over S , and let f ∶ Y → X and g ∶ Z → X be representable proper quasi-smooth morphisms of relativevirtual dimension − d and − e , respectively. Then we have [ Y ] ⋅ [ Z ] = [ Y R × X Z ] ∈ H d + e ( X , F ( d + e )) . Proof.
Consider the homotopy cartesian square
W ZY X pq h gf where W = Y × R X Z . Under the identificationH d + e ( X , F ( d + e )) = H BM − d − e ( X / X , F ( − d − e )) , the desired equality is h ∗ [ W / X ] = g ∗ p ∗ ([ W / Z ] ○ [ Z / X ]) = g ∗ p ∗ ( g ∗ [ Y / X ] ○ [ Z / X ]) = g ∗ ( p ∗ g ∗ [ Y / X ] ○ [ Z / X ]) = g ∗ ( g ∗ f ∗ [ Y / X ] ○ [ Z / X ]) = f ∗ [ Y / X ] ○ g ∗ [ Z / X ] . The first and second equalities follow from the functoriality and base changeproperties of the fundamental class (Theorems 3.12 and 3.13). For the restwe use the formulas (2.3.3), (2.3.2), and (2.3.4), in that order. (cid:3)
The variant for schemes stated in the introduction (0.2) is obtained byapplying this to the integral motivic cohomology spectrum F = Z and usingthe fundamental classes of Variant 3.11.3.5. Grothendieck–Riemann–Roch.
Let F and G be two (multiplica-tive) coefficients over S and φ ∶ F → G a ring morphism. The morphism φ induces a homomorphism φ ∗ ∶ H s ( X , F ( r )) → H s ( X , G ( r )) for every X over S . Given a quasi-smooth morphism of derived Artinstacks f ∶ X → Y , let [ X / Y ] F and [ X / Y ] G denote the fundamental classesformed with respect to F and G , respectively. The following Grothendieck–Riemann–Roch formula compares these two classes in terms of a certainclass Td φ X /Y ∈ H ( X , G ) . Theorem 3.23.
Let f ∶ X → Y be a quasi-smooth morphism of derived Artinstacks over S . Then we have (3.24) φ ∗ ([ X / Y ] F ) = Td φ X /Y ∩ [ X / Y ] G in H BM2 d ( X / Y , G ( d )) , where d = vd ( X / Y ) and where ∩ denotes the cap prod-uct (2.2.6). These immediately gives the usual formulas in Borel–Moore homologyand cohomology:
Corollary 3.25.
Let f ∶ X → Y be a quasi-smooth morphism of derivedArtin stacks over S . Then the square H BM s ( Y / S , F ( r )) H BM s + d ( X / S , F ( r + d )) H BM s ( Y / S , G ( r )) H BM s + d ( X / S , G ( r + d )) f ! φ ∗ φ ∗ Td φ X/Y ∩ f ! commutes. If f is moreover proper, then the square H s ( X , F ( r )) H s − d ( Y , F ( r − d )) H s ( X , G ( r )) H s − d ( Y , G ( r − d )) f ! φ ∗ φ ∗ f ! ( Td φ X/Y ∪− ) IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 23 also commutes.
Let E be a perfect complex of Tor-amplitude [ − k, ] , k ⩾ −
1, on X of vir-tual rank d . We define the Todd class td φ ( E ) . Since capping with the Thomclass th GX ( − E ) defines an isomorphism H ( X , G ) → H d X ( V X ( E [ − ]) , G ( d )) ,there exists a unique class td φ ( E ) ∈ H ( X , G ) × such that the relation(3.26) φ ∗ ( th FX ( − E )) = td φ ( E ) ∩ th GX ( − E ) holds in H d X ( V X ( E [ − ]) , G ( d )) . Exactly as in [De1, Subsect. 5.2], this Toddclass can be described explicitly using the formalism of formal group laws.We set Td φ X /Y = td φ ( L X /Y ) for f ∶ X → Y quasi-smooth.By deformation to the normal bundle stack (Subsect. 1.4), the formula(3.24) reduces to the case where f is the zero section of a vector bundlestack Y = V X ( E [ − ]) , where E is a perfect complex of Tor-amplitude [ − k, ] , k ⩾ −
1. In this case the fundamental class [ X / Y ] F is nothing else than theThom class th FX ( E ) , and similarly for G , so the formula reduces to (3.26).Theorem 3.23 is proven.Let’s make this formula slightly more explicit when φ is the total Cherncharacter. This is a morphism of motivic ring spectrach ∶ KGL → ⊕ i ∈ Z Q ( i )[ i ] in SH ( Spec ( Z )) , which induces an isomorphism KGL Q ≃ ⊕ i ∈ Z Q ( i )[ i ] upon rationalization [Ri], [De1, 5.3.3]. Since Q satisfies ´etale descent, chfactors through the ´etale localization KGL ´et . For any derived Artin stack S , we obtain by inverse image along the structural morphism a canonicalChern character ch ∶ KGL ´et S → ⊕ i ∈ Z Q S ( i )[ i ] in SH ´et ( S ) , which induces an isomorphism KGL ´et Q , S ≃ ⊕ i ∈ Z Q S ( i )[ i ] . Thesource and target admit canonical orientations such that the Todd classTd X /Y is the classical Todd class [De1, 5.3.3]. Suppose that S is the spec-trum of a field k , so that the Borel–Moore homology represented by KGL ´et S coincides with ´etale hypercohomology with coefficients in G-theory, and theproper covariance and Gysin maps are compatible with the respective in-trinsic operations in G-theory (Example 2.13). The Borel–Moore homologyrepresented by Q S coincides with the rational (higher) Chow groups. Underthese identifications the Chern character ch induces canonical homomor-phisms which we denote τ X ∶ H ( X , G ) → A ∗ ( X ) Q . We also write τ X for the composite with the canonical morphism G ( X ) → H ( X , G ) . Corollary 3.25 now yields the formula τ X ( O X ) = Td X ∩ [ X ] in A d ( X ) Q , or equivalently(3.27) [ X ] = Td − X ∩ τ X ( O X ) , where we write simply [ X ] for [ X / Spec ( k )] and similarly for the Toddclass. This is an extension of Kontsevich’s original conjectural formula forthe virtual fundamental class [ X ] vir [Ko, 1.4.2] to Artin stacks.3.6. Absolute purity.
In this subsection we extend Gabber’s proof of theabsolute cohomological purity conjecture [SGA5, Exp. I, 3.1.4] to Artinstacks.
Theorem 3.28 (Absolute purity) . Let f ∶ X → Y be a locally of finite typerepresentable morphism between regular Artin stacks over Z [ n ] , for someinteger n ∈ Z . Let Λ = Z / n Z and denote by Λ ´et the Λ -linear ´etale motiviccohomology spectrum (Example 2.11). Then the purity transformation pur f (3.10) induces a canonical isomorphism (3.29) Λ ´et X ( d )[ d ] → f ! ( Λ ´et Y ) of ´etale motivic spectra over X . It follows from the rigidity theorem of Cisinski–D´eglise [CD2, Thm. 4.5.2]that with finite coefficients, ´etale motivic cohomology agrees with usual ´etalecohomology, so this does recover the classical statement when we restrict toschemes. Actually, even in the case of schemes this statement is new, sinceGabber’s statement [ILO, Exp. XVI, Cor. 3.1.2] requires the schemes toadmit ample line bundles.The new ingredient we use here is the purity transformation (Remark 3.8)which generalizes Gabber’s construction of Gysin maps [ILO, Exp. XVI,2.3]. Neither the statement nor the proof of Theorem 3.28 uses any derivedgeometry, but it is worth recalling that our construction of pur f involves thenormal bundle stack N X /Y , which is a classical 2-Artin stack even when X and Y are classical 1-Artin stacks. Proof of Theorem 3.28.
Let v ∶ Y → Y be a smooth surjection with Yschematic, and form the homotopy cartesian squareX Y X Y . f u vf The upper arrow f is a locally of finite type morphism and X (resp. Y) is aregular algebraic space (resp. regular scheme). In terms of the purity trans-formation, the base change property of the fundamental class (Theorem 3.13) Recall that an Artin stack S is regular if and only if, for every smooth morphismS → S with S a scheme, S is regular. IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 25 translates to the commutativity of the diagram (cf. [DJK, Prop. 2.5.4(ii)]) u ∗ f ∗ ( Λ ´et Y )( d )[ d ] u ∗ f ! ( Λ ´et Y ) f ∗ v ∗ ( Λ ´et Y )( d )[ d ] f !0 v ∗ ( Λ ´et Y ) . pur f Ex ∗ ! pur f The right-hand vertical arrow is the isomorphism induced by the exchangetransformation Ex ∗ ! (Corollary A.15). Therefore, it will suffice to replace f by f and thereby assume that Y = Y is a regular scheme and X = X is aregular algebraic space.We can find an ´etale surjection p ∶ U → X such that U is a (regular)scheme and f ○ p ∶ U → Y is smoothable. The functoriality property of thefundamental class (Theorem 3.12) translates to the commutativity of thediagram (cf. [DJK, Prop. 2.5.4(i)]) p ∗ f ∗ ( Λ ´etY )( d )[ d ] p ∗ f ! ( Λ ´etY ) p ! f ! ( Λ ´etY )( f ○ p ) ∗ ( Λ ´etY )( d )[ d ] ( f ○ p ) ! ( Λ ´etY ) . pur f pur p pur f ○ p Since p is ´etale, the upper right-hand arrow pur p is invertible (Theorem A.13).Therefore, replacing X by U and f by f ○ p ∶ U → Y, we may assume that f ∶ X → Y is a smoothable morphism between regular schemes.Choose a factorization of f through a closed immersion i ∶ X → X ′ anda smooth morphism g ∶ X ′ → Y. Since pur g is invertible by Theorem A.13,applying the functoriality property again shows that we may replace f by i and thereby assume that f = i is a closed immersion between regularschemes.The assertion is that pur i induces an isomorphismΛ ´etX ( d )[ d ] → i ! ( Λ ´etY ) , or equivalently isomorphisms in ´etale motivic cohomology(3.30) H k − c ( X , Λ ´et ( k − c )) → H k X ( Y , Λ ´et ( k )) for all integers k ∈ Z , where c = − d is the codimension of i . In this situa-tion the purity transformation pur i is the same as the one constructed in[DJK, 4.3.1], and by [DJK, 4.4.3] it agrees with the construction of [De2,2.4.6] when applied to the ´etale motivic cohomology spectrum. The latteragrees, through the ridigity equivalence [CD2, Thm. 4.5.2] identifying the´etale motivic cohomology groups in (3.30) with classical ´etale cohomology,with Gabber’s construction in [ILO, Exp. XVI, 2.3] by design. Thus theclaim follows from [ILO, Exp. XVI, Thm. 3.1.1]. (cid:3) Remark 3.31.
The argument applies more generally to show that for an´etale motivic spectrum F , absolute purity holds for locally of finite type rep-resentable morphisms of regular Artin stacks (the analogue of Theorem 3.28)if and only if it holds for closed immersions between regular schemes. Forexample, this also applies to h-motivic cohomology [CD2, Thm. 5.6.2].6 ADEEL A. KHAN
The argument applies more generally to show that for an´etale motivic spectrum F , absolute purity holds for locally of finite type rep-resentable morphisms of regular Artin stacks (the analogue of Theorem 3.28)if and only if it holds for closed immersions between regular schemes. Forexample, this also applies to h-motivic cohomology [CD2, Thm. 5.6.2].6 ADEEL A. KHAN Appendix A. The six operations for derived Artin stacks
In this appendix we extend the six operations to derived Artin stacks.The category of coefficients we use is SH ´et , the ´etale-local motivic homotopycategory, but the construction works for any motivic ∞ -category of coeffi-cients in the sense of [Kh1, Chap. 2] that satisfies ´etale descent. The notionof “motivic ∞ -category of coefficients” is a refinement of that of “motivic tri-angulated category” studied in [CD1], but every example of the latter thatarises in practice can in fact be promoted to a motivic ∞ -category. The ( ∞ , ) -categorical refinement is crucial for the construction below. See [To2,Sect. 2] for a quick introduction to the theory of ∞ -categories.The six operations in the (Nisnevich-local) motivic homotopy categorySH were already constructed by Ayoub and Voevodsky for schemes. Theywere extended to derived schemes by Khan [Kh1]. Below we begin by record-ing the extension from derived schemes to derived algebraic spaces; this isstraightforward and will not come as a surprise to certain readers. It is forthe further extension to derived Artin stacks that we pass to the ´etale-localcategory, so that we can extend the operations essentially “by descent”.A.1. Derived algebraic spaces.Theorem A.1.
The formalism of six operations on SH extends to derivedalgebraic spaces. In particular: (i) For every derived algebraic space X , there is a closed symmetric monoidalstructure on SH ( X ) . In particular, there are adjoint bifunctors ( ⊗ , Hom ) . (ii) For any morphism of derived algebraic spaces f ∶ X → Y , there is an adjunc-tion f ∗ ∶ SH ( Y ) → SH ( X ) , f ∗ ∶ SH ( X ) → SH ( Y ) . The assignments f ↦ f ∗ , f ↦ f ∗ are -functorial. The functor f ∗ is sym-metric monoidal. (iii) For any locally of finite type morphism of derived algebraic spaces f ∶ X → Y ,there is an adjunction f ! ∶ SH ( X ) → SH ( Y ) , f ! ∶ SH ( Y ) → SH ( X ) . The assignments, f ↦ f ! , f ↦ f ! are -functorial. (iv) The operation f ! satisfies the base change and projection formulas against f ∗ . That is, for any cartesian square X ′ Y ′ X Y f ′ p qf there are identifications q ∗ f ! = ( f ′ ) ! p ∗ and f ! ( F ) ⊗ G = f ! ( F ⊗ f ∗ ( G )) IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 27 naturally in F and G . There is a natural transformation α f ∶ f ! → f ∗ ,functorial in f , which is invertible if f is proper. (v) Let i ∶ X → Y be a closed immersion of derived algebraic spaces, with opencomplement j . Then the operation i ∗ = i ! induces a fully faithful functor i ∗ ∶ SH ( X ) → SH ( Y ) whose essential image is the kernel of j ∗ . In particular, if i induces anisomorphism on underlying reduced classical stacks, then i ∗ is an equivalence. (vi) Let X be a derived algebraic space and E a locally free sheaf on X . If p ∶ V X ( E ) → X denotes the associated vector bundle, then the unit trans-formation id → p ∗ p ∗ is invertible. (vii) There is a canonical map of presheaves of E ∞ -group spaces on the site ofderived algebraic spaces, (A.2) K ( − ) → Aut ( SH ( − )) , from the algebraic K-theory of perfect complexes to the ∞ -groupoid of auto-equivalences of SH . For a perfect complex E on a derived algebraic space X ,we let Σ E denote the induced auto-equivalence of SH ( X ) , and Σ − E = Σ E ∨ itsinverse. If E is locally free, then we have Σ E = s ∗ p ! , Σ − E = s ! p ∗ , where p ∶ V X ( E ) → X is the projection of the associated vector bundle and s ∶ X → V X ( E ) the zero section. (viii) Let f ∶ X → Y be a smooth morphism between derived algebraic spaces. Thenthere is a purity equivalence pur f ∶ Σ L X / Y f ∗ = f ! which is natural in f . This was proven in [Kh1] for derived schemes so I only describe the mod-ifications that need to be made for derived algebraic spaces. The idea isthat derived algebraic spaces are
Nisnevich -locally affine (see e.g. the proofof [Kh2, Prop. 2.2.13]), which is good enough since SH satisfies Nisnevichdescent. Thus in Chap. 0, one needs to replace “Zariski” by “Nisnevich” inPropositions 5.3.5 and 5.6.2 (the proofs don’t change). In the proof of Propo-sition 6.3.4, one needs to replace the reference to [Con07] by [CLO], whereNagata compactifications are constructed for classical algebraic spaces. Theonly modification necessary in Chap. 1 is that the proof of Proposition 2.2.9needs to be replaced by the proof of [Kh2, Prop. 2.2.13]. This extends theproof of the localization theorem [Kh1, Chap. 1, Thm. 7.4.3] to derived al-gebraic spaces. Chap. 2 then goes through mutatis mutandis to give the sixoperations on derived algebraic spaces. In fact, item (v) below implies that it suffices to assume that the square is cartesianon underlying classical schemes.
Only item (vii) requires further explanation, as the map (A.2) encodesmuch more coherence of the assignment
E ↦ Σ E than was constructed in[Kh1]. On the site of classical schemes, such a map is constructed in [BH,Subsect. 16.2]. It factors through homotopy invariant K-theory KH [BH,Rem. 16.11]. By right Kan extension, the map KH → Aut ( SH ) extendsuniquely to the site of classical algebraic spaces. By derived nil-invarianceof KH and SH, see [Kh3, Subsect. 5.4] and [Kh1, Thm. 7.4.3] respectively,we obtain a unique extension of this map to the site of derived algebraicspaces, and we define (A.2) to be the composite K → KH → Aut ( SH ) .Strictly speaking, this only gives the operation f ! for separated morphismsof finite type. Using Zariski descent and the homotopy coherence of thesix functor formalism, one extends this to locally of finite type morphisms.Indeed, the coherence of the data in (iii) and (iv) can be encoded usingthe formalism of Gaitsgory–Rozenblyum [GR, Part III] (as done in [Kh1,Chap. 2, Thm. 5.1.2]) or that of Liu–Zheng [LZ] (as done in [Ro, Sect. 9.4]);the two formalisms are almost equivalent, as explained in [GR, Part III, 1.3].Then an easy application of the “DESCENT” program [LZ, Thm. 4.1.8]gives the desired extension.A.2. Derived algebraic stacks.
We begin with the presheaf of ∞ -categoriesX ↦ SH ( X ) , f ↦ f ∗ on the site of derived algebraic spaces. This is a Nisnevich sheaf, and assuch is right Kan-extended from the site of derived schemes or even affinederived schemes.Now let SH ´et denote its ´etale localization. In other words, we force SH ´et to satisfy descent for ˇCech covers in the ´etale topology. We then take itsright Kan extension to the site of derived Artin stacks. This is thus theunique extension of SH ´et to an ´etale sheaf on derived Artin stacks.We can be more explicit. If X is a derived Artin stack and p ∶ X → X isa smooth surjection with X a derived algebraic space, then p is a coveringin the ´etale topology so the ∞ -category SH ´et ( X ) fits into a homotopy limitdiagram of ∞ -categories(A.3) SH ´et ( X ) p ∗ Ð→ SH ´et ( X ) ⇉ SH ´et ( X R × X X ) →→→ SH ´et ( X R × X X R × X X ) →→→→ ⋯ . More canonically, SH ´et ( X ) is identified with the homotopy limit(A.4) SH ´et ( X ) = lim ←Ð SH ´et ( X ) taken over the ∞ -category Lis X of all smooth morphisms u ∶ X → X withX schematic. Roughly speaking, objects F ∈ SH ´et ( X ) may be viewed ascollections ( u ∗ F ) u , indexed over ( u ∶ X → X ) ∈ Lis X , compatible up tocoherent homotopies. In particular, the family of functors u ∗ is conservativeas u varies in Lis X . Theorem A.5. (i)
For every derived Artin stack X , there is a closed symmetric monoidal struc-ture on SH ( X ) . In particular, there are adjoint bifunctors ( ⊗ , Hom ) . IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 29 (ii)
For any morphism of derived Artin stacks f ∶ X → Y , there is an adjunction f ∗ ∶ SH ´et ( Y ) → SH ´et ( X ) , f ∗ ∶ SH ´et ( X ) → SH ´et ( Y ) . The assignments f ↦ f ∗ , f ↦ f ∗ are -functorial. (iii) For any locally of finite type morphism of derived Artin stacks f ∶ X → Y ,there is an adjunction f ! ∶ SH ´et ( X ) → SH ´et ( Y ) , f ! ∶ SH ´et ( Y ) → SH ´et ( X ) . The assignments f ↦ f ! , f ↦ f ! are -functorial. (iv) The operation f ! satisfies the base change and projection formulas against g ∗ , and f ! satisfies base change against g ∗ . If f is representable by derivedDeligne–Mumford stacks, then there is a natural transformation α f ∶ f ! → f ∗ ,functorial in F . If f is -representable and proper, then α f is invertible. On the site of derived algebraic spaces, we may view SH ´et as a presheafvalued in the ∞ -category of presentably symmetric monoidal ∞ -categoriesand symmetric monoidal left-adjoint functors. Since the forgetful functorto (large) ∞ -categories preserves limits [HTT, Prop. 5.5.3.13], the rightKan extension can be performed either way without changing the under-lying presheaf of ∞ -categories. In particular, we find that SH ´et ( X ) is apresentably symmetric monoidal ∞ -category for every derived Artin stack X and that f ∗ is a symmetric monoidal left-adjoint functor for every mor-phism f . We let ⊗ denote the monoidal product, Hom the internal hom,and f ∗ the right adjoint of f ∗ .Similarly, if we restrict the presheaf SH ´et to smooth morphisms betweenderived algebraic spaces, then it takes values in presentable ∞ -categoriesand right adjoint functors (as follows from Theorem A.1(viii)). By [HTT,Thm. 5.5.3.18] its right Kan extension to derived Artin stacks will havethe same property; that is, f ∗ admits a left adjoint f ♯ for every smoothmorphism f of derived Artin stacks.Let SH !´et denote the ´etale sheaf on the site of derived algebraic spaces,and locally of finite type morphisms, given byX ↦ SH ´et ( X ) , f ↦ f ! and take its right Kan extension to derived Artin stacks. For every X thereis then a canonical equivalenceΘ X ∶ SH ´et ( X ) → SH !´et ( X ) determined by the property that u ! ( Θ X ( F )) = Σ L X /X u ∗ ( F ) for all u ∶ X → X in Lis X . For any morphism f ∶ X → Y , we define f ! ∶ SH ´et ( Y ) → SH ´et ( X ) by f ! = Θ − X ○ f ! ○ Θ Y . More concretely, f ! is determined From Theorem A.9 below it follows that the base change formula applies also tosquares that are only cartesian on underlying classical stacks. by the fact that for any commutative squareX Y
X Y f u vf with u and v smooth and f a morphism of derived algebraic spaces, wehave u ∗ f ! ( F ) = Σ f ∗ (L Y /Y ) − L X /X f !0 ( v ∗ F ) for all F ∈ SH ´et ( Y ) , or equivalently(A.6) Σ L X /X u ∗ f ! ( F ) = f !0 Σ L Y /Y v ∗ ( F ) . Moreover, these identifications are subject to a homotopy coherent systemof compatibilities as f varies.On SH !´et , the operation f ! automatically admits a left adjoint f ! for ev-ery morphism f . Indeed, the right Kan extension can be computed in the ∞ -category of presentable ∞ -categories and right adjoint functors (as theforgetful functor preserves limits [HTT, Thm. 5.5.3.18]). This induces anoperation f ! ∶ SH ´et ( X ) → SH ´et ( Y ) by f ! = Θ − Y ○ f ! ○ Θ X , so that f ! u ♯ Σ − L X /X = v ♯ Σ − L Y /Y ( f ) ! for all commutative squares as above.As mentioned in Subsect. A.1, all the data in Theorem A.5 can be encodedusing the formalism of either Gaitsgory–Rozenblyum [GR, Part III] or Liu–Zheng [LZ]. In the former case, one may apply [GR, Chap. 8, Thm. 6.1.5] (cf.[GR, Chap. 5, Thm. 3.4.3], [RS, Sect. 2.2]) to glue together the required datafrom its restriction to algebraic spaces (already constructed in Theorem A.1),via an ( ∞ , ) -categorical right Kan extension. Alternatively, we apply the“DESCENT” program of [LZ, Thm. 4.1.8], just as in [LZ, Subsect. 5.4].Under certain assumptions the identification f ! = f ∗ can be extended tonon-representable proper morphisms: Theorem A.7.
Let f ∶ X → Y be a morphism of derived Artin stacks that isrepresentable by derived Deligne–Mumford stacks. Assume that there existsa finite surjection g ∶ Z → X with Z an algebraic space. For F ∈ SH ´et ( X ) ,consider the morphism α f ∶ f ! ( F ) → f ∗ ( F ) induced by the natural transformation α f (Theorem A.5 (iv) ). If f is properand F satisfies descent for finite surjections, then this morphism is invertible.In particular, this applies to the rational motivic cohomology spectrum Q X (Example 2.10), the rational algebraic cobordism spectrum MGL Q , X (Exam-ple 2.12), or more generally any MGL Q , X -module.Proof. Since F satisfies descent along the ˇCech nerve of g ∶ Z → X , it willsuffice to show that α f ∶ f ! ( h ∗ h ∗ F ) → f ∗ ( h ∗ h ∗ F ) IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 31 is invertible for every finite surjection h ∶ W → X with W an algebraic space.Since h and f ○ h are 0-representable and proper, α h and α f ○ h are invertibleby Theorem A.5(iv). Therefore the claim follows from the functoriality of α f in f . It applies to Q X because the latter satisfies descent for the h topology[CD2, Cor. 5.5.5]. (cid:3) Example A.8.
Note that X admits a finite cover by an algebraic space ifand only if the classical stack X cl does. This is the case for example if X cl has quasi-finite separated diagonal [Ry, Thm. B], or if X cl has quasi-finitediagonal and is of finite type over a noetherian scheme [EHKV, Thm. 2.7].In particular this holds if X cl is a Deligne–Mumford stack. Theorem A.9 (Localization) . Let i ∶ X → Y be a closed immersion ofderived Artin stacks, with open complement j . Then the operation i ∗ = i ! induces a fully faithful functor i ∗ ∶ SH ´et ( X ) → SH ´et ( Y ) whose essential image is the kernel of j ∗ . In particular, if i induces anisomorphism on underlying reduced classical stacks, then i ∗ is an equivalence.Proof. For fully faithfulness it suffices to show that the co-unit i ∗ i ∗ → idis invertible. After base change along a smooth atlas v ∶ Y → Y with Yschematic, we get a closed immersion i ∶ X → Y and an induced atlas u ∶ X → X . It suffices to show the co-unit becomes invertible after applying u ∗ on the left, in which case it is identified with i ∗ ( i ) ∗ u ∗ ( F ) → u ∗ ( F ) ,by the base change formula (Theorem A.5(iv)). This is invertible by thelocalization theorem for derived schemes ([Kh1, Chap. 1, Cor. 7.4.9]).Since X × Y ( Y ∖ X ) is empty, the base change formula shows that j ∗ i ∗ = F ∈ SH ´et ( Y ) satisfies j ∗ ( F ) =
0, then the unitmap
F → i ∗ i ∗ ( F ) is invertible. By descent we reduce again to the schematiccase which is [Kh1, Chap. 1, Cor. 7.4.7]. (cid:3) Thanks to David Rydh for the idea of the inductive argument in the proofbelow.
Proposition A.10 (Homotopy invariance) . Let X be a derived Artin stackand E a perfect complex on X of Tor-amplitude [ − k, ] , for k ⩾ − . If π ∶ V X ( E [ − ]) → X denotes the associated vector bundle stack, then theunit transformation id → π ∗ π ∗ is invertible.Proof. First assume that E is of Tor-amplitude [ , ] , so that π is a vectorbundle. By descent we may assume that X is schematic, in which case theclaim holds almost by construction (see [Kh1, Chap. 2, Subsect. 3.2]).If E is of Tor-amplitude [ , ] , then π is the projection of the classifyingstack of the vector bundle V X ( E ) → X , and the canonical section σ ∶ X → V X ( E [ − ]) is a smooth surjection. The composite of the two unit mapsid → π ∗ π ∗ → π ∗ σ ∗ σ ∗ π ∗ = id is the identity, so will suffice to show thatthe unit id → σ ∗ σ ∗ is invertible. Since σ is a smooth surjection it suffices moreover to show that σ ! → σ ! σ ∗ σ ∗ is invertible. By the base change formulafor the square V X ( E ) XX V X ( E [ − ]) , pp σσ we reduce to showing that the unit map id → p ∗ p ∗ is invertible. This holdsby the Tor-amplitude [ , ] case already proven above. Repeating the sameargument inductively shows the case of Tor-amplitude [ − k, − k ] for all k ⩾ [ − k, ] , we argue by induction on k to reduce to the k = − X , wemay find a surjection E [ − k ] → E with E locally free. If E ′ is the fibre ofthis map, then E ′ [ ] is then of Tor-amplitude [ − ( k − ) , ] , so by indutiveassumption we know that the claim holds for π ′ ∶ V X ( E ′ ) → X (i.e., thatid → ( π ′ ) ∗ ( π ′ ) ∗ is invertible). There is a commutative diagram X V X ( E ′ ) X V X ( E [ − ]) V X ( E [ − k − ]) X , σ ′ σ π ′ τ σ ρ π π where the square is cartesian. As E [ − k − ] is of Tor-amplitude [ − k − , − k − ] , we already know that the unit id → ( π ) ∗ ( π ) ∗ is invertible byabove. It remains to show that id → ρ ∗ ρ ∗ is invertible, which can be doneafter applying σ !0 on the left. By the base change formula this follows frominvertibility of the unit id → ( π ′ ) ∗ ( π ′ ) ∗ . (cid:3) The canonical map (A.2) of Theorem A.1(vii) also extends to the site ofderived Artin stacks:(A.11) K ( − ) → Aut ( SH ´et ( − )) . Indeed as the target satisfies ´etale descent, the map factors through ´etale K-theory K ´et and arises via right Kan extension from derived algebraic spaces.We thus also have the (invertible) operations(A.12) Σ E ∶ SH ( X ) → SH ( X ) for E ∈ Perf ( X ) . Theorem A.13 (Purity) . Let f ∶ X → Y be a smooth morphism of derivedArtin stacks. Then there is a purity equivalence pur f ∶ Σ L X/Y f ∗ = f ! which is natural in f .Proof. This follows immediately from the characterization of f ! given in theproof of Theorem A.5. (cid:3) IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 33
Example A.14.
Let E be a perfect complex of Tor-amplitude [ − k, ] , k ⩾ −
1, on a derived Artin stack X . Then V X ( E [ − ]) is a smooth Artin stackover X . Let π ∶ V X ( E [ − ]) → X denote the projection and σ ∶ X → V X ( E [ − ]) the canonical section. By purity (Theorem A.13) one has theformulas Σ E = σ ! π ∗ , Σ − E = σ ∗ π ! . Similarly if E is of Tor-amplitude [ , ] (= locally free), thenΣ E = s ∗ p ! , Σ E[ ] = s ! p ∗ , where p ∶ V X ( E ) → X and s ∶ X → V X ( E ) denote the projection and zerosection, respectively. Corollary A.15.
Suppose given a commutative square X ′ Y ′ X Y f ′ p qf of derived Artin stacks which is cartesian on underlying classical stacks. If f is representable and locally of finite type, there is a natural transformation Ex ∗ ! ∶ p ∗ f ! → ( f ′ ) ! q ∗ . If either f or q is smooth, then Ex ∗ ! is invertible.Proof. The natural transformation is defined as the composite p ∗ f ! unit ÐÐ→ p ∗ f ! q ∗ q ∗ ≃ p ∗ p ∗ ( f ′ ) ! q ∗ counit ÐÐÐ→ ( f ′ ) ! q ∗ where the isomorphism in the middle is the base change formula, obtained bypassage to right adjoints from the base change formula (Theorem A.5(iv)).The second statement follows from Theorem A.13. (cid:3) Construction A.16 (Euler transformation) . Let E be a locally free sheafon a derived Artin stack X . There is a natural transformation(A.17) eul E ∶ id → Σ E of auto-equivalences of SH ´et ( X ) . More generally for any surjection φ ∶ E → E ′ of finite locally free sheaves, there is a natural transformationΣ φ ∶ Σ E → Σ E ′ constructed as follows. Consider the commutative triangle V X ( E ′ ) V X ( E ) X iq p and let t and s be the respective zero sections. Then Σ φ is the composite t ∗ q ! = t ∗ i ! p ! Ex ∗ ! ÐÐ→ t ∗ i ∗ p ! = s ∗ p !4 ADEEL A. KHAN under the identifications s ∗ p ! = Σ E and t ∗ q ! = Σ E ′ (Example A.14), whereEx ∗ ! ∶ i ! → i ∗ is the exchange transformation (Corollary A.15) for the self-intersection square of the closed immersion i . References [BF] K. Behrend, B. Fantechi,
The intrinsic normal cone , Invent. Math. (1997),no. 1, 45–88.[BH] T. Bachmann, M. Hoyois,
Norms in motivic homotopy theory , arXiv:1711.03061(2017).[Bh] B. Bhatt,
Completions and derived de Rham cohomology . arXiv:1207.6193(2012).[CD1] D.-C. Cisinski, F. D´eglise,
Triangulated categories of mixed motives .Springer Monogr. Math., to appear. arXiv:0912.2110 (2012).[CD2] D.-C. Cisinski, F. D´eglise, ´Etale motives . Compos. Math. (2016), no. 3,556–666.[CK] I. Ciocan-Fontanine, M. Kapranov,
Virtual fundamental classes via dg-manifolds . Geom. Topol. (2009), no. 3, 1779–1804.[CLO] B. Conrad, M. Lieblich, M. Olsson, Nagata compactification for algebraic spaces .J. Inst. Math. Jussieu (2012), no. 4, 747–814.[Ci] D.-C. Cisinski, Higher category theory and homotopical algebra .Camb. Stud. Adv. Math. (2019).[DJK] F. D´eglise, F. Jin, A. A. Khan,
Fundamental classes in motivic homotopy theory .arXiv:1805.05920 (2018).[De1] F. D´eglise,
Orientation theory in arithmetic geometry . K-Theory—Proceedings of the International Colloquium, Mumbai 2016 (2018),239–247.[De2] F. D´eglise,
Bivariant theories in motivic stable homotopy , Doc. Math. (2018),997–1076.[EHKSY] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo, M. Yakerson, Modules overalgebraic cobordism . arXiv:1908.02162 (2019).[EHKV] D. Edidin, B. Hassett, A. Kresch, A. Vistoli,
Brauer groups and quotient stacks .Amer. J. Math. (2001), no. 4, 761–777.[FM] W. Fulton, R. MacPherson,
Categorical framework for the study of singularspaces. . Mem. Amer. Math. Soc. (1981), no. 243.[Fu] K. Fujiwara, A proof of the absolute purity conjecture (after Gabber) . Algebraicgeometry 2000, Azumino (Hotaka), 153–183, Adv. Stud. Pure Math. (2002).[GR] D. Gaitsgory, N. Rozenblyum, A study in derived algebraic geometry. Vol. I.Correspondences and duality . Math. Surv. Mono. (2017).[HS] A. Hirschowitz, C. Simpson,
Descente pour les n-champs .arXiv:math.AG/9807049 (2001).[HTT] J. Lurie,
Higher topos theory , Ann. Math. Stud. (2009).[ILO] L. Illusie, Y. Laszlo, F. Orgogozo (eds.),
Travaux de Gabber sur l’uniformisationlocale et la cohomologie ´etale des sch´emas quasi-excellents , S´eminaire `a l’´EcolePolytechnique 2006–2008. With the collaboration of F. D´eglise, A. Moreau,V. Pilloni, M. Raynaud, J. Riou, B. Stroh, M. Temkin, W. Zheng. Ast´erisque , Soc. Math. France (2014).[Ji] F. Jin,
Algebraic G-theory in motivic homotopy categories . arXiv:1806.03927(2018).[Jo] R. Joshua,
Higher intersection theory on algebraic stacks. I , K-Theory (2002),no. 2, 133–195.[KP] Y.-H. Kiem, H. Park, Virtual intersection theories . arXiv:arXiv:1908.03340(2019).[KR] A. A. Khan, D. Rydh,
Virtual Cartier divisors and blow-ups . arXiv:1802.05702(2018).
IRTUAL FUNDAMENTAL CLASSES OF DERIVED STACKS I 35 [KV] M. Kapranov, E. Vasserot,
The cohomological Hall algebra of a surface andfactorization cohomology , arXiv:arXiv:1901.07641 (2019).[Kh1] A. A. Khan,
Motivic homotopy theory in derived algebraic geometry
The Morel–Voevodsky localization theorem in spectral algebraic ge-ometry , Geom. Topol , to appear. arXiv:1610.06871 (2016).[Kh3] A. A. Khan,
Descent by quasi-smooth blow-ups in algebraic K-theory .arXiv:1810.12858 (2018).[Ko] M. Kontsevich,
Enumeration of rational curves via torus actions , The modulispace of curves (Texel Island, 1994), 335–368, Progr. Math., (1995).[Kr] A. Kresch,
Cycle groups for Artin stacks , Invent. Math. (1999), no. 3, 495–536.[LO1] Y. Laszlo, M. Olsson,
The six operations for sheaves on Artin stacks. I. Finitecoefficients . Publ. Math. Inst. Hautes ´Etudes Sci, no. (2008), 109–168.[LO2] Y. Laszlo, M. Olsson,
The six operations for sheaves on Artin stacks. II. Adiccoefficients . Publ. Math. Inst. Hautes ´Etudes Sci, no. (2008), 169–210.[LS] P. Lowrey, T. Sch¨urg,
Derived algebraic cobordism . J. Inst. Math. Jussieu (2016), no. 2, 407–443.[LZ] Y. Liu, W. Zheng, Enhanced six operations and base change theorem for higherArtin stacks . arXiv:1211.5948 (2012).[La] G. Laumon,
Homologie ´etale , Expos´e VIII in S´eminaire de g´eom´etrie analytique(Ecole Norm. Sup., Paris, 1974-75), Ast´erisque , 163–188. Soc. Math.France (1976).[Le1] M. Levine,
Comparison of cobordism theories . J. Algebra (2009), no. 9,3291–3317.[Le2] M. Levine,
The intrinsic stable normal cone . arXiv:arXiv:1703.03056 (2017).[Lu] J. Lurie,
Derived algebraic geometry
Virtual pull-backs.
J. Algebraic Geom. (2012), no. 2, 201–245.[NSØ] N. Naumann, M. Spitzweck, P. A. Østvær, Motivic Landweber exactness .Doc. Math. (2009), 551–593.[Ol] M. Olsson, Borel–Moore homology, Riemann–Roch transformations, and localterms , Adv. Math. (2015), 56–123.[RS] T. Richarz, J. Scholbach,
The intersection motive of the moduli stack of shtukas .arXiv:1901.04919 (2019).[Ri] J. Riou,
Algebraic K -theory, A -homotopy and Riemann-Roch theorems ,J. Topol. (2010), no. 2, 229–264.[Ro] M. Robalo, Th´eorie homotopique motivique des espaces non-commutatifs ,Ph.D. thesis (2014). https://webusers.imj-prg.fr/ marco.robalo/these.pdf.[Ry] D. Rydh,
Noetherian approximation of algebraic spaces and stacks . J. Alge-bra (2015), 105–147.[SAG] J. Lurie,
Spectral algebraic geometry .[SGA5] L. Illusie (ed.),
Cohomologie l -adique et fonctions L. S´eminaire de G´eometrieAlg´ebrique du Bois-Marie 1965–1966 (SGA 5). Lecture Notes in Mathemat-ics , Springer (1977).[Sp] M. Spitzweck,
A commutative P -spectrum representing motivic cohomologyover Dedekind domains . M´em. Soc. Math. Fr. (2018).[To1] B. To¨en, Simplicial presheaves and derived algebraic geometry .Simplicial methods for operads and algebraic geometry, 119–186,Adv. Courses Math. CRM Barcelona (2010).[To2] B. To¨en,
Derived algebraic geometry , EMS Surv. Math. Sci. 1 (2014), 153–240. [email protected]¨at f¨ur MathematikUniversit¨at Regensburg6 ADEEL A. KHAN