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MOTIVIC COHOMOLOGY, LOCALIZED CHERN CLASSES, AND LOCALTERMS

MARTIN OLSSON

Abstract. Let c : C → X×X be a correspondence with C and X quasi-projective schemesover an algebraically closed field k. We show that if u` : c∗1Q` → c!2Q` is an action definedby the localized Chern classes of a c2-perfect complex of vector bundles on C, where ` is aprime invertible in k, then the local terms of u` are given by the class of an algebraic cycleindependent of `. We also prove some related results for quasi-finite correspondences. Theproofs are based on the work of Cisinski and Deglise on triangulated categories of motives.

Contents

1. Introduction 1

2. Motivic categories and the six operations 4

3. Chern classes 9

4. Local Chern classes 12

5. Local terms for motivic actions 14

6. Beilinson motives 17

7. Assumption 5.2 over an algebraically closed field 21

8. Application: local terms for actions given by localized Chern classes 28

9. Application: quasi-finite morphisms and correspondences 29

References 39

1. Introduction

The motivation for this work comes from our study of local terms arising from actions ofcorrespondences defined by local Chern classes of complexes of vector bundles in [18]. Thepurpose of the present paper is to elucidate the motivic nature of these local terms using themachinery developed by Cisinski and Deglise in [4].

The basic problem we wish to address is the following. Fix an algebraically closed field k ofcharacteristic p (possibly 0), and let S denote the category of finite type separated k-schemes.Let c : C → X × X be a correspondence with C,X ∈ S . A c2-perfect complex E· on Cdefines for any prime ` invertible in k an action u` : c∗1Q` → c!

2Q`, and therefore by the generalmachinery of SGA 5 a class Tr(u`) ∈ H0(Fix(c),ΩFix(c)), where Fix(c) denotes the scheme of

1

2 MARTIN OLSSON

fixed points Fix(c) := C×c,X×X,∆XX and ΩFix(c) is the `-adic dualizing complex (see [13, III,

§4] for further discussion). Recall from loc. cit. that for any proper connected componentZ ⊂ Fix(c) the local term of u` is given by the proper pushforward of the restriction of Tr(u`)to Z, and consequently in good situations can be used via the Grothendieck-Lefschetz traceformula [13, III, 4.7] to calculate the trace of the induced action of u` on global cohomology.

On the other hand, H0(Fix(c),ΩFix(c)) is the `-adic Borel-Moore homology of Fix(c) andthere is a cycle class map

cl` : A0(Fix(c))→ H0(Fix(c),ΩFix(c)),

where A0(Fix(c)) denotes the group of 0-cycles on Fix(c) modulo rational equivalence.

The main result about local terms in this paper is the following:

Theorem 1.1 (Theorem 8.7). There exists a zero-cycle Σ ∈ A0(Fix(c))Q such that for anyprime ` invertible in k the class Tr(u`) is equal to cl`(Σ).

As we explain, this theorem is a fairly formal consequence of a suitable theory of derivedcategories of motives and six operations for such categories. The fact that such a theoryexists is due to Cisinski and Deglise [4]. They developed a notion of triangulated motiviccategories with a six operations formalism realizing a vision of Beilinson. Roughly speakingsuch a category is a fibered category M over S such that for every X ∈ S the fiber M (X) isa monoidal triangulated category and for every morphism f : X → Y in S we have functors

f!, f∗ : M (X)→M (Y ), f ∗, f ! : M (Y )→M (X)

satisfying the usual properties. In addition there should be a suitable notion of Chern classes.Already in this context we can define localized Chern classes of complexes of vector bundlesas well as analogous of the classes Tr(u`), which are functorial in M . In particular, we canconsider the category MB of Beilinson motives defined in [4, §14]. This category not onlyhas a good six functor formalism, but is also closely related to algebraic cycles as one wouldexpect from a good motivic theory.

The connection with cycles (discussed in more detail in sections 4 and 6) is established bydeveloping the basic theory of Borel-Moore homology, discussed in the etale setting in [18], toa rather general context of triangulated motivic categories with a six operations formalism.Let M be such a motivic category, and for quasi-projective X ∈ S let ΩM

X ∈ M (X)(or sometimes we just write ΩX if the reference to M is clear) denote f !1Spec(k), where1Spec(k) ∈M (Spec(k)) is the unit object for the monoidal structure and f : X → Spec(k) isthe structure morphism. For an integer i the i-th M -valued Borel-Moore homology of X isdefined to be

HMi,BM(X) := Ext−2i

M (X)(1X ,ΩMX (−i)),

where the notation ΩMX (−i) denotes a suitable Tate twist of ΩM

X . Then there is a naturalcycle class map

(1.1.1) Ai(X)Q → HMi,BM(X),

where Ai(X)Q denotes the i-th Chow homology groups (as defined in [11, §1.8]) tensored withQ.

Theorem 1.2 (Special case of 6.2). If M is the motivic category MB of Beilinson motivesand X is quasi-projective then the map (1.1.1) is an isomorphism for all i.

MOTIVIC COHOMOLOGY, LOCALIZED CHERN CLASSES, AND LOCAL TERMS 3

The idea behind the proof of 1.1 is to lift the construction of local terms to the category ofBeilinson motives MB, where the local term is by 1.2 given by an algebraic cycle, and thenshow that the etale realizations of the motivic local term is equal to Tr(u`).

The proof of 1.1 can essentially be phrased as saying that actions arising from c2-perfectcomplexes are motivic. In general it seems a difficult question to prove that a given actionof a correspondence is motivic. There is one other case, however, where one can fairly easilydetect if an action is motivic. Namely, for a quasi-finite morphism f : Y → X there isa natural necessary condition for a section u` ∈ H0(Y, f !Q`) to be the etale realization ofa morphism u : 1Y → f !1X in the triangulated category of Beilinson motives over Y . Intheorem 9.4 we show that this condition is also sufficient. This also has global consequences.In particular, a special case of theorem 9.24 is the following:

Theorem 1.3. Let k be an algebraically closed field and let X/k be a separated Deligne-Mumford stack. Let f : X → X be a finite morphism (as a morphism of stacks). Then thealternating sum of traces ∑

i

(−1)itr(f ∗|H i(X,Q`))

is in Q and independent of `.

Remark 1.4. Following standard conventions we usually write tr(f ∗|RΓ(X,Q`)) for thealternating sum of traces

∑i(−1)itr(f ∗|H i(X,Q`)).

Remark 1.5. It is tempting to try to prove 1.3 using Fujiwara’s theorem and naive localterms as in [14, 3.5 (c)]. The cohomology RΓ(X,Q`) is dual to the compactly supportedcohomology RΓc(X,ΩX) of the dualizing complex ΩX of X, and the dual operator to f ∗ isthe map

f∗ : RΓc(X,ΩX)→ RΓc(X,ΩX)

induced by the map f∗ΩX → ΩX arising from the identification f! ' f∗ (since f is proper) andadjunction. In the finite field case, one can then apply Fujiwara’s theorem to the complex ΩX

with this action of the correspondence (id, f) : X → X×X to relate the trace on RΓc(X,ΩX)to the so-called naive local terms of this action on ΩX . However, the calculation of these naivelocal terms of ΩX is not immediate and they are not formally rational and independent of `.

Remark 1.6. Since the trace appearing 1.3 is in Z` it follows that the alternating sum oftraces is in Z[1/p], where p is the characteristic of k. In fact, notice that since RΓ(X,Z`) isa perfect complex we can define tr(u∗|RΓ(X,Z`)) ∈ Z`, which by the above is an element ofZ[1/p] which reduces mod ` to tr(u∗|RΓ(X,F`)), thereby yielding `-independence for mod `traces as well.

Remark 1.7. One might hope more generally to use the techniques of this paper to studymotivic local terms with Z coefficients to obtain cycles in A0(Fix(c)) before tensoring with Q.However, the theory at present seems restricted to Q-coefficients as the six operations on asuitable triangulated category of motives is not known to exist integrally. Work in preparationby Cisinski and Deglise on integral motives may, however, lead to integral results.

Remark 1.8. In this paper we discuss etale cohomology and local terms defined in the etaletheory. However, with a suitable theory of p-adic local terms and p-adic realization functorsone would also get rationality of p-adic local terms and compatility with the etale local terms.

4 MARTIN OLSSON

Remark 1.9. Theorem 1.3 has also been obtained by Bondarko using variant motivic meth-ods [3, Discussion following 8.4.1].

Remark 1.10. Many of the foundational results obtained in this paper hold not just over afield but over more general base schemes and we develop the theory in greater generality. Forthe applications to local terms, however, it suffices to work over an algebraically closed field.

1.11. Acknowledgements. The author is grateful to Doosung Park for suggesting that thework of Cisinski and Deglise should imply 1.1, and for comments of Cisinski and Degliseon a preliminary draft. We also thank the referee for a number of helpful suggestions andcorrections. The author was partially supported by NSF CAREER grant DMS-0748718 andNSF grant DMS-1303173.

2. Motivic categories and the six operations

Let B be a regular separated scheme of finite dimension, and let S denote the category offinite type separated B-schemes.

2.1. Recall from [4, Section 1] that a triangulated premotivic category M is a fibered categoryover S satisfying the following five conditions (a good summary is given in [5, A.1.1]):

(PM1) For every S ∈ S the fiber category M (S) is a well-generated (in the sense of [16])triangulated category with a closed monoidal structure.

(PM2) For every morphism f : X → Y in S the functor (well-defined up to unique isomor-phism)

f ∗ : M (Y )→M (X)

is triangulated, monoidal, and admits a right adjoint f∗.(PM3) For every smooth morphism f : X → Y in S the functor f ∗ : M (Y ) → M (X)

admits a left adjoint f].(PM4) For every cartesian square with p smooth

Yq //

g

∆

X

f

T

p // S

there is a canonical isomorphism of functors

Ex(∆∗] ) : q]g∗ ' f ∗p].

(PM5) For every smooth morphism p : T → S, M ∈ M (T ), and N ∈ M (S) there is acanonical isomorphism

Ex(p∗] ,⊗) : p](M ⊗T p∗N) ' p](M)⊗S N.

Remark 2.2. Note that for any category S we can talk about a triangulated fibered categoryover S . By this we mean a fibered category p : M → S satisfying axioms (PM1) and (PM2).

Remark 2.3. In [4, Section 1.4] there is a notion of a premotivic triangulated category overa general base category, but the above suffices for our purposes.


2.4. For every X ∈ S , the monoidal structure on M (X) gives a unit object 1X ∈ M (X).For a smooth morphism f : X → S in S define MS(X) ∈M (S) to be f](1X). Because thepullback functor f ∗ is monoidal we have f ∗1S = 1X and therefore by adjunction a morphism

MS(X) = f]f∗1S

f]f∗→id// 1S,

which we denote by aX/S.

A Tate motive for M is a cartesian section τ : S → M with τ(S) fitting into a distin-guished triangle

τ(S)[−2] // MS(P1S)

aP1S/S// 1S // τ(S)[−1]

functorial in S. We usually write just 1S(1) for τ(S).

2.5. We can consider various other natural axioms on a triangulated premotivic categorywith a Tate object:

(Semi-separation) For any finite surjective radical morphism f : X → Y the functorf ∗ : M (Y )→M (X) is conservative.

(Homotopy axiom) For every S ∈ S the map

aA1S/S

: MS(A1S)→ 1S

is an isomorphism.

(Stability property) The Tate motive 1S(1) is ⊗-invertible. In this case we get motives1S(n) for all n ∈ Z, and for any F in M and integer n we can define F (n).

2.6. Given a triangulated premotivic category M with a Tate motive satisfying the stabilityproperty we define motivic cohomology, a bigraded cohomology theory on S , by

H i,nM (S) := ExtiM (S)(1S, 1S(n)).

2.7. A morphism between two triangulated premotivic categories M and M ′ is a cartesianfunctor ϕ∗ : M →M ′ such that the following hold:

(i) For every S ∈ S the functor ϕ∗S : M (S)→M ′(S) is a triangulated monoidal functorwhich admits a right adjoint ϕS∗.

(ii) For every smooth morphism p : T → S in S there is a canonical isomorphism

Ex(p], ϕ∗) : p]ϕ

∗T → ϕ∗Sp].

In fact triangulated premotivic categories form a 2-category in which the above morphisms arethe 1-morphisms, and 2-morphisms are given by morphisms of cartesian functors ε : ϕ∗ → ψ∗

compatible with the structures in (i) and (ii).

Remark 2.8. Similarly we can consider the 2-category of triangulated fibered categories overany base category S .

2.9. Let S be as above, and let Ar(S ) be the category of morphisms in S . We have twofunctors

s, t : Ar(S )→ S

6 MARTIN OLSSON

given by the source and target respectively. For a triangulated premotivic category M overS let M s (resp. M t) denote s∗M (resp. t∗M ), a triangulated fibered category over Ar(S ).

A six functor formalism for M consists of the following data (see [4, A.5] for more details):

(1) 2-functors f 7→ f∗ and f 7→ f! from M s →M t and f 7→ f ∗ and f 7→ f ! from M t toM s such that for every f : X → Y ∈ Ar(S ) the functors f∗ and f ∗ are as previouslydefined, and f! is left adjoint to f !.

(2) There exists a morphism of 2-functors α : f! → f∗ which is an isomorphism if f isproper.

(3) For any smooth morphism f : X → S in S of relative dimension d there are iso-morphisms pf : f] → f!(d)[2d] and p′f : f ∗ ' f !(−d)[−2d]. These are given by iso-morphisms of 2-functors on the category of smooth morphisms of relative dimensiond.

(4) For every cartesian square

Y ′q //

g

∆

X ′

f

Y

p // X

there are natural isomorphisms of functors

p∗f! ' g!q∗,

g∗q! ' p!f∗.

In the case when f , and hence also g, is proper the induced isomorphism

g!q! α // g∗q

! ' p!f∗α−1// p!f!

is the map induced by the adjunction

q! id→f!f!// q!f !f!

' // g!p!f!.

(5) For every f : Y → X there are natural isomorphisms

Ex(f ∗! ,⊗) : (f!K)⊗X L ' f!(K ⊗Y f ∗L),

H omX(f!L,K) ' f∗H omY (L, f !K),

and

f !H omX(L,M) 'H omY (f ∗L, f!M).

(Loc) Let X ∈ S be an object, i : Z → X a closed imbedding, and let j : U → X be thecomplementary open set. Then there exists a map of functors ∂ : i∗i

∗ → j!j![1] such

for every F ∈M(X) the induced triangle

j!j!F // F // i∗i

∗F∂ // j!j

!F [1]

is distinguished, where the first two maps are those induced by adjunction.


Finally Deglise and Cisinski consider purity and duality properties:

(Relative Purity) For a closed immersion i : Z → X of smooth separated B-schemes thereis a canonical isomorphism

1Z(−c)[−2c] ' i!(1X),

where c is the codimension of Z in X.

(Duality) For X ∈ S with structure morphism f : X → B we write ΩMX (or just ΩX if no

confusion seems likely to arise) for f !1B ∈M (X). Define DX : M (X)op →M (X) to be thefunctor M 7→H omX(M,ΩM

X ).

(a) For every M ∈M (X) the natural map

M → DX(DX(M))

is an isomorphism.(b) For every X and M,N ∈M (X) we have a canonical isomorphism

DX(M ⊗DX(N)) 'H omX(M,N).

(c) For every f : Y → X in S , M ∈ M (X), and N ∈ M (Y ) we have natural isomor-phisms

DY (f ∗(M)) ' f !(DX(M)),

f ∗DX(M) ' DY (f !(M)),

DX(f!(N)) ' f∗(DY (N)),

f!(DY (N)) ' DX(f∗(N)).

These isomorphisms interchange the base change isomorphisms in (4).

We say that a triangulated premotivic category M is a triangulated motivic category overS if all of the above conditions hold.

Remark 2.10. This is stronger than what is in [4, 2.4.45] but we will not need their slightlyweaker notion.

Remark 2.11. The relative purity property follows from property 2.9 (3), but we state itexplicitly for later use.

Remark 2.12. If R is a ring we can also consider a notion of an R-linear triangulated motiviccategory over S . By definition this means that each M (X) is anR-linear symmetric monoidaltriangulated category, and that all the above structure respects this R-linear structure.

Remark 2.13. It is shown in [4, 2.1.9] that our assumptions on M (in particular 2.9 (2) and(4) and the semi-separation) imply that for any finite surjective radicial morphism f : X → Ythe pullback functor f ∗ : M (Y ) → M (X) is an equivalence of categories. Since f∗ is rightadjoint to f ∗ it is also an equivalence and for any K ∈M (Y ) the adjunction map K → f∗f

∗Kis an isomorphism. In particular, the adjunction map 1Y → f∗1X is an isomorphism.

Since this point is crucial for what follows we sketch for the convenience of the reader theproof given in [4, 2.1.9] of the statement that f ∗ : M (Y ) → M (X) is an equivalence for ffinite surjective radicial.

8 MARTIN OLSSON

In the case when f is also a closed immersion the square

Y Y

f

Y

f // X

is cartesian and by 2.9 (2) and (4) the adjunction map f ∗f∗ → id is an isomorphism. Thisimplies that the adjunction id → f∗f

∗ is also an isomorphism as this can be verified afterapplying f ∗, since f is semi-separated, and the induced map f ∗ → f ∗f∗f

∗ is a section of theisomorphism induced by the adjunction f ∗f∗ → id.

For the general case consider the commutative diagram

Yδ

$$

id

))id

Y ×X Yg1 //

g2

Y

f

Y

f // X,

where δ is the diagonal imbedding and g1 (resp. g2) is the first (resp. second) projection. Sincef is finite surjective and radicial the morphism δ is a closed imbedding which is surjectiveand radicial. By the first case considered the functor δ∗ : M (Y ×X Y )→M (Y ) is thereforean equivalence with quasi-inverse δ∗. Since the compositions δ∗ g∗i are isomorphic to theidentity functor on M (Y ) it follows that g∗i : M (Y ) → M (Y ×X Y ) is also an equivalencewith quasi-inverse gi∗ for i = 1, 2. This also implies that gi∗ is a quasi-inverse for δ∗.

Now to verify that the morphism of functors id → f∗f∗ is an isomorphism it suffices by

semi-separation to show that the induced map f ∗ → f ∗f∗f∗ is an isomorphism. For this note

by 2.9 (2) and (4) and the fact that δ∗ is an equivalence we have

f ∗f∗f∗ ' g1∗g

∗2f∗ ' g1∗δ∗δ

∗g∗2f∗ ' f ∗

and composing f ∗ → f ∗f∗f∗ with this isomorphism we get the identity map f ∗ → f ∗. This

also implies that the adjunction map f ∗f∗ → id is an isomorphism as we have

f ∗f∗ ' g1∗g∗2 ' g1∗δ∗δ

∗g2∗ ' id.

Now the fact that the adjunctions id → f∗f∗ and f ∗f∗ → id are isomorphisms implies that

f ∗ is an equivalence of categories.

Remark 2.14. Assume that B is the spectrum of a field k and let X ∈ S be an object.Then, using for example Noether normalization, we can find a nonempty open subset U ⊂ Xof some pure dimension d and a factorization of the structure morphism

Ua // V

b // Y // Spec(k),

where a is etale, b is finite radicial, and Y is smooth of relative dimension d over k. Combiningthe preceding remark with property (3) we find that if u : U → Spec(k) is the structure


morphism then u!1Spec(k) ' 1U(d)[2d]. More generally, for any other object T ∈ S we get afactorization of the second projection pr2 : U × T → T as

U × T → V × T → Y × T → T,

from which it follows that if f : U × T → Spec(k) and g : T → Spec(k) are the structuremorphisms, then f !1Spec(k) ' pr∗2g

!1Spec(k)(d)[2d].

We will use this in various devissage arguments that follow.

3. Chern classes

3.1. The key ingredient in our study of etale Borel-Moore homology in [19] is the theoryof local Chern classes. In order to develop a good theory of Borel-Moore homology in themotivic setting we need to define local Chern classes also in this more general setup. Ourapproach to local Chern classes follows the method of Iversen [15] which uses the calculationof the cohomology of certain relative flag varieties. In this section we summarize the basictheory of Chern classes in the motivic setting and explain the calculations (3.7 and 3.10) ofcohomology that are necessary for Iversen’s method. In the following section we then explainhow to define local Chern classes.

As in the previous section let B be a regular separated scheme of finite dimension, and letS denote the category of finite type separated B-schemes.

3.2. Fix an R-linear triangulated motivic category M over S . For n,m ∈ Z let

Hn,mM : S op → ModR

be the functor sending X ∈ S to Hn,mM (X) := ExtnM (X)(1X , 1X(m)).

Let Pic (resp. Vec, K0) be the functor on S sending X to the Picard group Pic(X) of X(resp. the set of isomorphism classes of finite rank vector bundles on X, the Grothendieckgroup of vector bundles on X). A pre-orientation on M is a morphism of functors

c1 : Pic→ H2,1M .

Let X ∈ S be an object smooth over B and let i : Z → X be a Cartier divisor smoothover B. By relative purity, we get a canonical isomorphism

i!1X ' 1Z(−1)[−2].

Applying a shift, Tate twist, and i∗, this isomorphism defines an isomorphism

i∗1Z → i∗i!1X(1)[2],

which upon composition with the adjunction i∗i!1X → 1X gives a morphism

i∗1Z → 1X(1)[2].

Applying HomM (X)(1X ,−) we get a map

H0,0M (Z)→ H2,1

M (X).

We say that a pre-orientation c1 is an orientation if this map sends the identity class inH0,0

M (Z) to c1(OX(Z)) for every such closed imbedding i : Z → X.

For the remainder of this section we fix an orientation c1 on M .

10 MARTIN OLSSON

3.3. A theory of Chern classes for M is a collection of morphisms of functors

cn : Vec→ H2n,nM , n ≥ 0

such that the following conditions hold:

(i) c0 is the constant 1 and c1 is the given orientation.(ii) (Vanishing) For a vector bundle E on X of rank r we have ci(E) = 0 for i > r.(iii) (Commutativity) For vector bundles E and F on X ∈ S and integers i, j ∈ Z we

have

ci(E) · cj(F ) = cj(F ) · ci(E).

(iv) (Whitney sum) For a short exact sequence of vector bundles on X ∈ S

0→ E ′′ → E → E ′ → 0

we have

ck(E) =∑i+j=k

ci(E′′)cj(E

′).

Remark 3.4. When R is a Q-algebra, we can define as usual the Chern character whichdefines a morphism of functors

ch : K0 →∏n

H2n,nM .

as well as Todd classes.

3.5. Assume given a theory of Chern classes for M . The classical computations of cohomologyfor flag varieties can then be carried out in our cohomology theory as well. Let us brieflyrecall the statement and construction. For X ∈ S define

An,mM (X) := H2n,mM (X),

and set

A∗,∗M (X) := ⊕n,m∈ZAn,mM (X).

Then A∗,∗M (X) is a bigraded ring. For F ∈M (X) define

An,mM (X,F ) := Ext2nM (X)(1X , F (m)),

and set

A∗,∗M (X,F ) := ⊕n,mAn,mM (X,F ),

a module over A∗,∗M (X). The main case of interest is when F = 1X(m)[2n] for some n andm, in which case A∗,∗M (X,F ) is a free module of rank 1 over A∗,∗M (X) with generator in degreeA−n,−mM (X,F ).

3.6. Let X ∈ S be a scheme, let E be a vector bundle on X. Fix a sequence of integers(r1, . . . , rm) and let p : F → X be the flag variety classifying flags

0 = F0 ⊂ F1 ⊂ · · · ⊂ Fm = E

such that the rank of Fi/Fi−1 is equal to ri. Over F there is a universal flag F u· on p∗E. Set

Ei := F ui /F

ui−1 (i = 1, . . .m), so Ei is a locally free sheaf of rank ri on F .


Consider the polynomial ring A∗,∗M (X)[Ti,ji ]1≤i≤m,1≤ji≤ri with variable Ti,ji of bidegree (ji, ji),and let J be the bigraded ideal in this ring generated by the homogeneous elements (notethat ct(E) ∈ At,tM (X))

ct(E)−∑

i1+···+im=t

T1,i1 · · ·Tm,im , t ≥ 1.

There is a map of bigraded rings

α : A∗,∗M (X)[Ti,ji ]→ A∗,∗M (F), Ti,ji 7→ cji(Ei).

Proposition 3.7. The map α induces an isomorphism

A∗,∗M (X)[Ti,ji ]/J ' A∗,∗M (F).

Proof. This follows from the argument of [13, Expose VII, §5]. Let us just indicate thenecessary modifications here.

Let f : D → X be the flag variety classifying flags of type (1, 1, . . . , 1) on E. There is anatural map g : D → F realizing D as the fiber product D1 ×F D2 · · · ×F Dm, where Di isthe variety over F classifying full flags in Ei.

Lemma 3.8. Let S ∈ S be an object and E a locally free sheaf of rank r on S with associatedprojective bundle q : PE → S. Let c1 ∈ H2,1

M (PE) denote the first Chern class of the universalquotient, which induces a map 1S(−1)[−2]→ q∗1PE. Then the map induced by summing themaps ci1

⊕r−1i=0 1S(−i)[−2i]→ q∗1PE

in M (S) is an isomorphism.

Proof. Notice that we have q] ' q!(r − 1)[2(r − 1)] ' q∗(r − 1)[2(r − 1)], so the desiredisomorphism can also be written as an isomorphism MS(PE) ' ⊕r−1

i=0 1S(i)[2i]. This is shownin [7, Theorem 3.2].

Consider the algebra A∗,∗M (X)[Uk]k=1,...,r (with the Uk of bidegree (1, 1)) and the ideal JDgenerated by elements ct(E)−σt, where σt is the t-th symmetric function in the Uk. We thenhave a map

αD : A∗,∗M (X)[Uk]/JD → A∗,∗M (D), Uk 7→ c1(Lk),

where Lk is the k-th universal quotient on D. Factoring f : D → X as a sequence of projectivebundles one sees that the map αD is an isomorphism. Now let

θ : A∗,∗M (X)[Ti,ji ]/J → A∗,∗M (X)[Uk]/JD

be the map induced by the map sending Ti,ji to the ji-th elementary symmetric polynomialin the variables

(Ur1+···+ri−1+s)1≤s≤ri .

We then have a commutative diagram

A∗,∗M (F) // A∗,∗M (D)

A∗,∗M (X)[Ti,ji ]/J

α

OO

θ // A∗,∗M (X)[Uk]/JD.

αD

OO

Analyzing this as in [13, p. 310] one gets that α is an isomorphism as well.

12 MARTIN OLSSON

Remark 3.9. A theory of Chern classes with c1 equal to a given orientation is unique if itexists. This follows from the usual argument, as discussed for example in [11, Remark 3.2.1],using the splitting principle and 3.8,

Corollary 3.10. Let X ∈ S , let E1, . . . , Es be vector bundles on X, and let v1, . . . , vsbe integers ≥ 0. Let Gi denote the Grassmanian of vi-planes in Ei, and let Pi denote theuniversal vi-sub-bundle of Ei|Gi

. Then the A∗,∗M (X)-algebra A∗,∗M (∏

iGi) is generated by thehomogeneous components of the elements pr∗i c·(P

i).

Proof. This follows from the above description and factoring∏

iGi → X through a sequenceof Grassman bundles.

4. Local Chern classes

We continue with the notation of the preceding section.

4.1. For a closed imbedding i : X →M in S , define

An,mM (M on X) := Ext2nM (X)(1X , i

!1M(m)),

and set

A∗,∗M (M on X) := ⊕n,m∈ZAn,mM (M on X).

This is a bigraded module over A∗,∗M (X). A theory of local Chern classes consists of an as-signment to every bounded complex K · of locally free sheaves on M with support in Xcohomology classes

cM on Xi (K ·) ∈ Ai,iM (M on X)

satisfying the following properties:

(i) (Pullback) If f : M ′ → M is a morphism and i′ : X ′ → M ′ denotes f−1(X) thenf ∗cM on X

i (K ·) ∈ Ai,iM (M ′ on X ′) is equal to cM′ on X′

i (f ∗K ·).(ii) Applying HomM (M)(1M ,−) to the adjunction maps i∗i

!1M(m)[2n] → 1M(m)[2n] weget a morphism r : A∗,∗M (M on X)→ A∗,∗M (M). If cM on X

· (K ·) ∈∏

s≥1As,sM (M on X)

denotes the vector of the cM on Xs (K ·) then we require

r(cM on X· (K ·)) + 1 =

∏s

c·(K2s)c·(K

2s−1)−1.

Using 3.7 and the argument of Iversen [15] one obtains:

Proposition 4.2. Suppose given a theory of Chern classes for M . Then a theory of localChern classes for M exists and is unique.

4.3. In the case when R is a Q-algebra one can introduce as in [15, §1] the localized Cherncharacter

chM on X(K ·) ∈∏s

As,sM (M on X).

By the argument of [15] this satisfies the following properties:


(i) (Functoriality) If f : M ′ →M is a morphism and i′ : X ′ →M ′ denotes f−1(X) then

f ∗chM on X(K ·) = chM′ on X′(f ∗K ·).

(ii) r(chM on X(K ·)) = ch(K ·).(iii) (Decalage) chM on X(K ·[1]) = −chM on X(K ·).(iv) For complexes K · and L· on M supported on X we have

chM on X(K · ⊕ L·) = chM on X(K ·) + chM on X(L·).

(v) (Multiplicativity) Let K · (resp. L·) be a complex on M supported on Z (resp. V ).Then

chM on (Z∩V )(K · ⊗ L·) = chM on Z(K ·) · chM on V (L·).

4.4. More generally, for a morphism f : X → Y in S we define A∗,∗M (f : X → Y ) by theformula

An,mM (f : X → Y ) := Ext2nM (X)(1X , f

!1Y (m)).

Note that for a factorization of f

(4.4.1) X

f

i // M

g // Y

with i an imbedding and g smooth of relative dimension d we have f !1Y (m) ' i!g!1Y (m) 'i!1Y (m+ d)[2d], whence a canonical isomorphism

An,mM (f : X → Y ) ' An+d,m+dM (i : X →M).

4.5. For a quasi-projective morphism f : X → Y in S , one has the Grothendieck groupof f -perfect complexes defined as in [18, 3.10]. Moreover, the same argument is in loc. cit.shows that there is a transformation

τXY : K(f -perfect complexes on X)→ ⊕iAi,iM (f : X → Y ).

This transformation is defined by choosing a factorization of f as in (4.4.1) and sending acomplex K · to

td(i∗TM/Y ) · chXM(K ·) ∈ A∗,∗M (X →M) ' A∗−d,∗−dM (X → Y ).

4.6. In particular, for X ∈ S , quasi-projective over B, we can consider An,mM (X → B), withX → B the structure morphism. In this case we define

HMi,BM(X) := A−i,−iM (X → B),

called the i-th M -valued Borel-Moore homology of X (or just i-th Borel-Moore homology ifthe reference to M is clear). In this case the Grothendieck group of f -perfect complexes issimply the Grothendieck group of coherent sheaves on X (since B is regular) so we get a map

(4.6.1) τX : K(Coh(X))→ ⊕iHMi,BM(X).

Remark 4.7. In the case when B is the spectrum of a field, the map (4.6.1) can also beviewed as a cycle class map, using the identification of K(Coh(X)) with Chow groups (tensorQ).

14 MARTIN OLSSON

Remark 4.8. Our construction of the map τXY above for a morphism f : X → Y in Suses the existence of a factorization through a smooth morphism, and therefore necessitatesimposing quasi-projectivity hypotheses. It seems likely that another construction exists whichgeneralizes to more general morphisms.

5. Local terms for motivic actions

Let k be a field and let S denote the category of finite type separated k-schemes. Fix aring R and let M be an R-linear triangulated motivic category.

5.1. Let X, Y ∈ S be two objects. For F ∈M (X) and G ∈M (Y ) let F G ∈M (X × Y )denote pr∗1F ⊗X×Y pr∗2G. There is a map

εX×Y : ΩX ΩY → ΩX×Y

defined as follows. We have isomorphisms

(5.1.1) HomX×Y (ΩX ΩY ,ΩX×Y ) ' HomX×Y (pr∗1ΩX , pr!21Y )

' HomY (pr2!pr∗1ΩX , 1Y )

' HomY (g∗f!f!1Spec(k), g

∗1Spec(k)),

where f : X → Spec(k) (resp. g : Y → Spec(k)) is the structure morphism, the first iso-morphism is induced by the isomorphism H omX×Y (pr∗2ΩY ,ΩX×Y ) ' pr!

21Y (coming from 2.9(Duality) (c) which identifies pr!

2(−) with H omX×Y (pr∗2DY (−),ΩX×Y )) and the adjunctionisomorphism

HomX×Y (ΩX ΩY ,ΩX×Y ) ' HomX×Y (pr∗1ΩX ,H omX×Y (pr∗2ΩY ,ΩX×Y )),

the second isomorphism is by adjunction, and the third isomorphism is by base change.The map εX×Y is the map corresponding under these isomorphisms to the adjunction mapf!f

!1Spec(k) → 1Spec(k).

Assumption 5.2. Assume that the map εX×Y is an isomorphism for all X, Y ∈ S .

Remark 5.3. For any F ∈M (X) the adjunctions used in (5.1.1) define an isomorphism

HomX×Y (F ΩY ,ΩX×Y ) ' HomY (g∗f!F, g∗1Spec(k))

functorial in F .

5.4. Fix X, Y ∈ S . Note that there is a natural map

1X ΩY →H omX×Y (ΩX 1Y ,ΩX ΩY )

which, with the above identification of ΩX ΩY with ΩX×Y , gives a map

ρX×Y : 1X ΩY → D(ΩX 1Y ).


5.5. Consider a closed imbedding i : Z → Y with complement j : U → Y , and let i :X ×Z → X ×Y and j : X ×U → X ×Y be the inclusions defined by base change. We thenhave a distinguished triangle (using assumption 5.2)

i∗(ΩX ΩZ)→ ΩX ΩY → j∗(ΩX ΩU)→ i∗(ΩX ΩZ)[1].

Applying H omX×Y (ΩX 1Y ,−) to this triangle we get a distinguished triangle

i∗D(ΩX 1Z)→ D(ΩX 1Y )→ j∗D(ΩX 1U)→ i∗D(ΩX 1Z)[1],

and a diagram

(5.5.1) i∗(1X ΩZ)

ρX×Z

// 1X ΩY

ρX×Y

// j∗(1X ΩU) //

ρX×U

i∗(1X ΩZ)[1]

ρX×Z

i∗D(ΩX 1Z) // D(ΩX 1Y ) // j∗D(ΩX 1U) // i∗D(ΩX 1Z)[1],

where the horizontal rows are distinguished triangles.

Lemma 5.6. The diagram (5.5.1) is commutative.

Proof. Let ε′X×Y (resp. ε′′X×Y ) denote the element of

HomX×Y (pr∗1ΩX , pr!21Y ) (resp. HomY (pr2!pr∗1ΩX , 1Y ))

corresponding to εX×Y under the isomorphisms in (5.1.1), and let α : f!f!1k → 1k denote

the adjunction morphism, where to ease notation we write 1k for 1Spec(k). Similarly defineε′X×Z , ε′′X×Z , ε′X×U , and ε′′X×U . Then ρX×Y (resp. ρX×Z , ρX×U) is Verdier dual to ε′X×Y(resp. ε′X×Z , ε′X×U). Let gZ : Z → Spec(k) and gU : U → Spec(k) denote the restrictions ofg : Y → Spec(k). For any F ∈M (X×Y ) we have have by 2.9 (Loc) a distinguished triangleon Y

j!j∗F → F → i∗i

∗F → j!j∗F [1]

functorial in F . In particular, we get a morphism of distinguished triangles

(5.6.1) j!g∗Uf!f

!1k

j!g∗Uα

// g∗f!f!1k

g∗α

// i∗g∗Zf!f

!1k //

i∗g∗Zα

j!g∗Uf!f

!1k[1]

j!g∗Uα

j!g∗U1k // g∗1k // i∗g

∗Z1k // j!g

∗U1k[1].

Now observe that the natural morphism of distinguished triangles

j!pr!X×U,21U //

pr!X×Y,21Y

id

// i∗pr!X×Z,21Z //

j!pr!X×U,21U [1]

pr!X×Y,2j!1U // pr!

X×Y,21Y // pr!X×Y,2i∗1Z

// pr!X×Y,2j!1U [1]

16 MARTIN OLSSON

is an isomorphism, as the two middle vertical arrows are isomorphisms (using 2.9 (4)). Byadjunction the commutativity of (5.6.1) then implies that the diagram

j!pr∗X×U,1ΩX

j!ε′X×U

// pr∗X×Y,1ΩX

ε′X×Y

// i∗pr∗X×Z,1ΩX

i∗ε′X×Z

// j!pr∗X×U,1ΩX [1]

j!ε′X×U

j!pr!X×U,21U // pr!

X×Y,21Y // i∗pr!X×Z,21Z // j∗pr!

X×U,21U [1]

commutes, and finally dualizing this diagram we obtain that (5.5.1) commutes.

Lemma 5.7. For any X, Y ∈ S the map ρX×Y is an isomorphism.

Proof. Consider a closed imbedding i : Z → Y with complement j : U → Y equidimensionalof some dimension d and admitting a factorization of its structure morphism as in 2.14. Leti : X × Z → X × Y and j : X × U → X × Y be the inclusions defined by base change.Considering the morphism of distinguished triangles (5.5.1) it then suffices to show the resultfor the pair (X,Z) and (X,U), which by induction on the dimension of Y reduces the proofto the case when Y admits a factorization as in 2.14. In this case ΩY (rest. ΩX×Y ) is equal,up to a shift and Tate twist, to 1Y (rest. ΩX 1Y ) and the result is immediate.

5.8. Fix a correspondence c : C → X × X with C,X ∈ S and quasi-projective, and letFix(c) := C ×c,X×X,∆X

X denote the fixed point scheme. For an action (a morphism inM (C))

u : c∗11X ' 1C → c!21X ,

we define Trc(u) ∈ HM0,BM(Fix(c)) as follows. Note that we have

c!(ΩX 1X) ' DCc∗(1X ΩX) ' DCc

∗2ΩX ' c!

21X ,

where the first isomorphism uses 5.7. Therefore we can also view u as an element ofH0

M (C, c!(ΩX 1X)). Let ∆ : X → X ×X be the diagonal morphism and note that there isa natural map

ΩX 1X → ∆∗ΩX .

Applying this map and using the base change isomorphism for the cartesian diagram

C

c

Fix(c)δoo

c′

X ×X X

∆oo

we get a morphism

c!(ΩX 1X)→ c!∆∗ΩX ' δ∗c′!ΩX ' δ∗ΩFix(c).

Applying HomM (C)(1C ,−) to this map we get a map

TrM : HomM (C)(c∗11X , c

!21X)→ HM

0,BM(Fix(c)).

If no confusion seems likely we write simply Tr for TrM .


5.9. Let N be a second motivic category satisfying the assumption 5.2, and let

T : M → N

be a morphism of fibered categories such that for every X ∈ S the morphism on fibers

TX : M (X)→ N (X)

is a triangulated monoidal functor, and assume further that T is compatible with the opera-tions f∗, f

∗, f!, f! for morphisms f : X → Y in S , Tate twists, and internal Hom. Note that

then T induces for every quasi-projective X ∈ S a map (which we abusively also denotesimply by T )

T : HM∗,BM(X)→ HN

∗,BM(X).

Let c : C → X × X be a correspondence as in 5.8, and let u : c∗11X,M → c!21X,M be an

action in M (C), where we write 1X,M for the unit object in M (X). Then it follows from theconstruction and the fact that T is compatible with the six operations that

T (TrM (u)) = TrN (T (u))

in HN0,BM(Fix(c)).

6. Beilinson motives

In this section B is a regular excellent scheme of finite Krull dimension δ, and S denotesthe category of quasi-projective B-schemes.

6.1. For R = Q, there are several equivalent constructions of triangulated motivic categories.The one most convenient for us in this paper is the category of constructible Beilinson motivesdefined in [4, §14] which we will denote by MB.

The main properties of this category that we will need are the following 6.2 and 6.5.

Proposition 6.2. For X ∈ S the map (4.6.1) induces for every i an isomorphism

Aδ+i(X)Q ' HMBi,BM(X),

where the left side refers to Chow homology groups, tensor Q, as defined in [12, §1.8] (or inthe case when B is the spectrum of a field [11, §1.3]).

Proof. To properly define MB requires a lot of preparatory material, for which we refer to [4,§14]. One definition of the category of Beilinson motives (see [4, 14.2.9]) is as the homotopycategory Ho(HB − mod), where HB is a cofibrant cartesian commutative monoid in thesymmetric monoidal fibered model category of Tate spectra over the category of schemes;see [4, 7.2.10, 5.3.18]. The category MB is defined as the subcategory of Ho(HB −mod) ofconstructible objects [4, 15.1.1].

Let KGLQ denote the absolute ring spectrum defined in [4, 14.1.1]. Then as in [4, 13.3.3](which is an integral version) Ho(KLGQ−mod) forms a motivic category over S in the senseof loc. cit. (which is slightly weaker than the definition used in this paper).

If X ∈ S is a regular scheme, then by [4, 13.3.2.1] we have a canonical functorial isomor-phism

ExtnHo(KGLQ−mod(X))(KGLQ(X), KGLQ(X)(m)) ' K2n−m(X)Q,

18 MARTIN OLSSON

where K2n−m(X)Q denotes algebraic K-theory.

By [4, 14.2.17] there is a map of ring spectra

(6.2.1) HB → KGLQ,

which induces a morphism of motivic categories

ϕ∗ : Ho(HB −mod)→ Ho(KGLQ −mod).

The functor ϕ∗ is the extension of scalars and the functor ϕ∗ (the right adjoint to ϕ∗, whichexists by 2.7 (i)) is the forgetful functor. By [4, 14.2.17 (3)] there is also a morphism

(6.2.2) π0 : ϕ∗KGLQ → HB

in Ho(HB −mod) such that the composition HB → ϕ∗KLGQ → HB is the identity map.

Fix a closed imbedding i : X → M , with M smooth of constant dimension d over B, andlet j : U →M be the complement of X. Taking cohomology of the distinguished triangle

i∗i!1M(d+ a)[2d]→ 1M(d+ a)[2d]→ j∗1U(d+ a)[2d]

we get a long exact sequence

(6.2.3) · · · → HsMB

(X,ΩX(a))→ Hs+2d,d+aMB

(M)→ Hs+2d,d+aMB

(U)→ · · · .

To compare this with K-theory we following the argument of [21, Proof of Theoreme 8].For X ∈ S let K ′m(X)Q denote the K-theory of the category of coherent sheaves on X tensorQ. Recall from [21, 7.2 and Theoreme 8 (v)] that for any X ∈ S the filtration F· on K ′0(X)Qdefined by the Riemann-Roch isomorphism K ′0(X)Q ' A∗(X)Q (so Fj/Fj+1 ' Aj(X)Q) isinduced by operations φk on K ′0(X)Q. Following the proof of [21, Theoreme 8 (ii) ] we getthe long exact sequence

· · · → K ′m(X)→ K ′m(M)→ K ′m(U)→ K ′m−1(X)→ · · · ,

and as in loc. cit. this sequence is compatible with the Adams operations and induces anexact sequence on the associated graded pieces

(6.2.4) · · · → grjK′m(X)→ grjK

′m(M)→ grjK

′m(U)→ · · · .

The fact that the map from K-theory to the cohomology of Beilinson motives is definedby the map π0 (6.2.2) and the description of the Adam’s operations as coming from thedecomposition in [4, 14.1.1 (K5)] implies that we get an induced map from the long exactsequence (6.2.4) to the long exact sequence (6.2.3) (using also the compatibility [4, 13.4.1(K6)]. We therefore obtain a commutative diagram with exact rows(6.2.5)

grδ+iK′1(M) //

a

grδ+iK′1(U) //

b

grδ+iK′0(X)

c

// grδ+iK′0(M) //

d

grδ+iK′0(U)

e

H2(d−i)−1,d−iMB

(M) // H2(d−i)−1,d−iMB

(U) // HMBi,BM(X) // H

2(d−i),d−iMB

(M) // H2(d−i),d−iMB

(U).

In general, if M → B is smooth of relative dimension d then for any integer s ≥ 0 the map

grδ+iK′s(M)→ H

2(d−i)−s,d−iMB

(M)


is an isomorphism. This follows from [4, 14.2.14], which identifies H2(d−i)−s,d−iMB

(M) with

grd−iγ Ks(M) (associated graded of the γ-filtration), and [21, 7.2 (vi)] which shows that

grd−iγ Ks(M) = gri+δK′s(M).

Therefore the maps a, b, d, and e in (6.2.5) are isomorphisms, which implies that the mapc is also an isomorphism.

We therefore get an induced isomorphism αX : Aδ+i(X)Q ' HMBi,BM(X). Let βX : Aδ+i(X)Q →

HMBi,BM(X) be the map defined by (4.6.1). We expect that αX = βX but this does not follow

immediately from the construction. Nonetheless we get that βX is an isomorphism as follows.

Observe that if z : Z → X is a closed subscheme, then the diagram

Aδ+i(Z)Q

αZ

z∗ // Aδ+i(X)Q

αX

HMBi,BM(Z)

z∗ // HMBi,BM(X)

commutes by construction, and similarly for βX . Furthermore, using [4, 13.6.4] we know thatαX and βX agree in the case when X is regular.

Fix a sequence of closed subschemes

∅ = Xn ⊂ Xn−1 ⊂ Xn−2 ⊂ · · · ⊂ X1 ⊂ X0 = X

such that for every t the complement Vt := Xt −Xt+1 is regular. Let Ft ⊂ Aδ+i(X)Q (resp.

Gt ⊂ HMBi,BM(X)) denote the image of Aδ+i(Xt)Q (resp. HMB

i,BM(Xt)). Then αX and βX respectthese filtrations.

We claim that αX and βX induced the same maps on the graded vector spaces associatedto these filtrations. Since αX is an isomorphism this implies that βX induces an isomorphismon the associated graded, which in turn implies that βX is an isomorphism.

To see that αX and βX agree on the associated graded vector spaces, we proceed byinduction on the length n of the sequence of closed subschemes. The base case n = 1 followsfrom the case when X itself is regular.

For the inductive step we assume that the result holds for each Xt, t > 0, with the sequenceof closed subschemes

∅ = Xn ⊂ Xn−1 ⊂ · · ·Xt.

Since Ft (resp. Gt) for t > 0 is the image of the corresponding step of the filtration on

Aδ+i(Xn−1)Q (resp. HMBi,BM(Xn−1)) it follows from the inductive hypothesis that αX and βX

induce the same map

Ft/Ft+1 → Gt/Gt+1

for t > 0. Thus it suffices to show that αX and βX induce the same map

Aδ+i(X)Q → Coker(HMBi,BM(X1)→ HMB

i,BM(X)).

This follows from noting that the composition of this map (for either αX or βX) with theinclusion

Coker(HMBi,BM(X1)→ HMB

i,BM(X)) → HMBi,BM(V0)

20 MARTIN OLSSON

is equal to the restriction map Aδ+i(X)Q → Aδ+i(V0) followed by αV0 (which agrees withβV0).

Remark 6.3. If X ∈ S is regular we have, by [4, 14.2.14], an isomorphism

Hq,pMB

(X) ' grpγK2p−q(X)Q.

This isomorphism implies various vanishing results for motivic cohomology. We will need twocases in what follows:

(i) (p < 0) If d is the dimension of X then by [21, Theoreme 7.2 (vi)] we have

grpγK2p−q(X)Q = grd−pK2p−q(X)Q,

which in the notation of [21, 7.4] is equal to H2d−q(X, d− p). If p < 0 then d− p > din which case this group is 0 by [21, Theoreme 8 (i)].

(ii) (X affine p = 0 and q < 0) In this case we have

Hq,pMB

(X) ' gr0γK−q(X).

The vanishing of this group is a known special case of Beilinson-Soule vanishing [21,2.9].

The second property of the category of Beilinson motives that we will need is the followingresult, which will enable us to use de Jong’s results on equivariant alterations [8].

6.4. Let Y be a quasi-projective B-scheme and G a finite group acting on Y . Let X ′ denotethe coarse moduli space of the quotient stack [Y/G] and let π : X ′ → X be a finite surjectiveradicial morphism. Let p : Y → X be the projection and define (p∗ΩY )G as in [4, 3.3.21], so(−)G denotes derived G-invariants. There is a natural morphism p∗ΩY → ΩX (dual to theadjunction morphism 1X → p∗1Y ) which induces a morphism

(6.4.1) h : (p∗ΩY )G → ΩX .

Proposition 6.5. The map (6.4.1) is an isomorphism.

Proof. Let i : Z → X be a closed imbedding with complement j : U → X such that thefollowing hold:

(1) U is everywhere dense in X.(2) If YU denotes U ×X Y then Ured and YU,red are regular, and the map YU,red → Ured is

flat.

Let YZ denote Y ×XZ and let X ′Z denote the coarse moduli space of [YZ/G]. The formationof the coarse moduli space does not in general commute with base change. It is still true,however, that YZ → X ′Z → Z satisfies the assumptions in 6.4. Consider the induced map ofdistinguished triangles

(i∗pZ∗ΩYZ )G //

hZ

(p∗ΩY )G //

h

(j∗pU∗ΩYU )G //

hU

(i∗pZ∗ΩYZ )G[1]

hZ

i∗ΩZ// ΩX

// j∗ΩU// i∗ΩZ [1].


By induction we may assume that the map (pZ∗ΩYZ )G → ΩZ is an isomorphism, and since theformation of derived G-invariants commutes with pushforward we may therefore assume thathZ is an isomorphism. It therefore suffices to consider the case when X and Y are regular ofthe same dimension, where it follows from [4, 3.3.35 and 14.3.3] and the following lemma.

Lemma 6.6. Let X ∈ S be regular of dimension d. Then ΩX ' 1X(d− δ)[2(d− δ)].

Proof. Since X is quasi-projective over B we can find a locally free sheaf E of finite rank r+1on B and an imbedding i : X → PE over B. Then i is a regular imbedding of codimensionr+δ−d. We have ΩX ' i!1PE(r)[2r] so it suffices to show that i!1PE ' 1X(d−δ−r)[2(d−δ−r)].This follows from absolute purity for Beilinson motives [4, 14.4.1] (see also [5, A.2.8]).

An immediate corollary is the following:

Corollary 6.7. Let Y ∈ S be regular of dimension d, and let G be a finite group acting onY . Let X := Y/G be the coarse moduli space of the corresponding Deligne-Mumford stack[Y/G]. Then ΩX ' 1X(d− δ)[2(d− δ)].

In particular, if f : Z → X is a morphism in S then we have Ai,iMB(Z → X) ' Ad−i(Z)Q.

Proof. The first statement follows immediately from 6.5 and 6.6. For the second statementobserve that we have

Ai,iMB(Z → X) = H2i(Z, f !1X(i))

= H2i(Z, f !ΩX(δ − d+ i)[2(δ − d)]) using ΩX ' 1X(d− δ)[2(d− δ)]= H2(i+δ−d)(Z,ΩZ(i+ δ − d))

= HMB−i−δ+d,BM(Z)

' Ad−i(Z)Q by 6.2.

Remark 6.8. Our proofs above use imbeddings into smooth schemes and therefore requireimposing quasi-projectivity assumptions. It seems plausible that they could be generalizedto statements without the quasi-projectivity assumption.

7. Assumption 5.2 over an algebraically closed field

7.1. In this section we restrict attention to the setting when B is the spectrum of an alge-braically closed field. In this case, proposition 6.5 also implies that assumption 5.2 holds forMB. The proof of this is a bit more intricate and it is useful to consider the following variantstatements and intermediate results.

7.2. Fix X, Y ∈ S and a closed imbedding i : Z → X. Let j : U → X be the complementof Z and let

Z × Y i // X × Y U × Y? _joo

denote the base changes to Y . For F ∈ MB(X) let FU denote the restriction to U , andconsider the following conditions:

(AZX(F )) The natural map (j∗FU) ΩY → j∗(FU ΩY ) is an isomorphism.

(BZX(F )) The natural map (i!F ) ΩY → i!(F ΩY ) is an isomorphism.

22 MARTIN OLSSON

(CX) The map εX×Y : ΩX ΩY → ΩX×Y is an isomorphism.

Lemma 7.3. Properties AZX(F ) and BZX(F ) are equivalent.

Proof. Indeed the maps in question fit into a morphism of distinguished triangles

i∗(i!F ΩY ) //

F ΩY// (j∗FU) ΩY

//

i∗(i!F ΩY )[1]

i∗i!(F ΩY ) // F ΩY

// j∗(FU ΩY ) // i∗i!(F ΩY )[1].

Lemma 7.4. Let p : X ′ → X be a proper morphism and let p : X ′ × Y → X × Y denote thebase change of p to Y . Then for any F ∈M (X ′) and G ∈M (Y ) the natural map

(7.4.1) (p∗F ) G→ p∗(F G)

is an isomorphism.

Proof. Indeed since p is proper we have

(p∗F′) G = (pr∗1p∗F

′)⊗ pr∗2G' (p∗pr′∗1 F

′)⊗ pr∗2G by 2.9 (4)' p∗(pr′∗1 F

′ ⊗ p∗pr∗2G) by 2.9 (5)= p∗(F

′ G),

where we write

pr′1 : X ′ × Y → X ′, pr1 : X × Y → X, pr2 : X × Y → Y,

for the projections.

Lemma 7.5. Let p : X ′ → X be a proper morphism and let p : X ′ × Y → X × Y denote thebase change of p to Y . Then the diagram

p∗ΩX′ ΩYa //

tp⊗id

p∗(ΩX′ ΩY )p∗εX′×Y// p∗ΩX′×Y

tp

ΩX ΩY

εX×Y // ΩX×Y

commutes, where the map tp (resp. tp) denotes the adjunction map p∗ΩX′ → ΩX (resp.p∗ΩX′×Y → ΩX×Y ) and a is as in (7.4.1).

Proof. It follows from the definitions that under the isomorphism

HomX×Y (p∗ΩX′ ΩY ,ΩX×Y ) ' HomY (g∗f!p∗ΩX′ , g∗1Spec(k))

defined in 5.3, the composition tp (p∗εX′×Y ) a corresponds to the map

g∗f!p∗ΩX′ ' g∗f!p∗p!f !1Spec(k)

' // g∗(f p)!(f p)!1Spec(k)// g∗1Spec(k),

where the last map is induced by the adjunction (f p)!(f p)! → id.


The map εX×Y tp id corresponds to the map

g∗f!p∗ΩX′ ' g∗f!p∗p!f !1Spec(k)

p∗p!→id// g∗f!f!1Spec(k)

f!f!→id// g∗1Spec(k),

so the lemma follows from the fact that the adjunction maps are associative in the sense thatthe following diagram of functors and adjunction maps

f!p∗p!f !

'

p∗p!→id // f!f!

(f p)!(f p)! // id

commutes.

Lemma 7.6. The diagram

i!ΩX ΩY

'

α // i!(ΩX ΩY )i!εX×Y // i!ΩX×Y

'

ΩZ ΩY

εZ×Y // ΩZ×Y

commutes, where α denotes the map occurring in BZX(ΩX) and the vertical arrows are induced

by the natural isomorphism i!ΩX ' ΩZand i!ΩX×Y ' ΩZ×Y .

Proof. Let fZ : Z → Spec(k) denote the structure morphism of Z.

The composition

(7.6.1) i!ΩX ΩYα // i!(ΩX ΩY )

i!εX×Y // i!ΩX×Y

is adjoint to the map

i∗(i!ΩX ΩY ) ' (i∗i

!ΩX) ΩYi∗i!→id// ΩX ΩY

εX×Y // ΩX×Y ,

from which it follows that under the isomorphism of 5.3

HomZ×Y (i!ΩX ΩY ,ΩZ×Y ) ' HomY (g∗fZ!i!ΩX , g

∗1Spec(k))

the composition (7.6.1) corresponds to the map g∗fZ!i!ΩX → g∗1Spec(k) induced by the com-

position

fZ!i!ΩX ' f!i∗i

!ΩXi∗i!→id// f!f

!1Spec(k)f!f

!→id// 1Spec(k).

By associativity of the adjunction maps as in the proof of 7.5 this composition is equal tocomposition

fZ!i!ΩX ' fZ!ΩZ ' fZ!f

!Z1Spec(k)

fZ!f!Z→id// 1Spec(k).

which implies the lemma.

Lemma 7.7. Let p : X ′ → X be a proper morphism, let i′ : Z ′ → X ′ be the preimage of Z,and let F ′ ∈MB(X ′) be an object. Then BZ′

X′(F′) implies BZ

X(p∗F′).

24 MARTIN OLSSON

Proof. Let p : X ′ × Y → X × Y (resp. pZ : Z ′ → Z, pZ : Z ′ × Y → Z × Y ) denote themorphism induced by p so we have isomorphisms

(7.7.1) i!p∗ ' pZ∗i′!, i!p∗ ' pZ∗i

′!,

and let

α′ : i′!F ′ ΩY → i′!(F ′ ΩY ) (resp. α : (i!p∗F′) ΩY → i!((p∗F

′) ΩY ))

denote the map in BZ′

X′(F′) (resp. BZ

X(p∗F )). It then suffices to show that the diagram

pZ∗(i′!F ′ ΩY )

pZ∗α′// pZ∗i

′!(F ′ ΩY )

(7.7.1)

(pZ∗i′!F ′) ΩY

(7.4.1)

OO

i!p∗(F′ ΩY )

(i!p∗F′) ΩY

(7.7.1)

OO

α // i!((p∗F′) ΩY )

(7.4.1)

OO

commutes, since all the vertical morphisms are isomorphisms. The adjoint of the composition


(7.4.1)// pZ∗(i

′!F ′ ΩY )pZ∗α

′// pZ∗i

′!(F ′ ΩY )(7.7.1)

// i!p∗(F′ ΩY )

is the composition

i∗((pZ∗i′!F ′) ΩY )

(7.4.1)// (i∗pZ∗i

′!F ′) ΩY

i∗pZ∗'p∗i′∗// (p∗i′∗i′!F ′) ΩY

i′∗i′!→id

uu(p∗F

′) ΩY

(7.4.1)// p∗(F

′ ΩY ).

The adjoint of the composition


(7.7.1)−1

// (i!p∗F′) ΩY

α // i!((p∗F′) ΩY )

(7.4.1)// i!p∗(F

′ ΩY ),

on the other hand, is equal to the composition

i∗((pZ∗i′!F ′) ΩY )

(7.4.1)// (i∗pZ∗i

′!F ′) ΩYpZ∗i

′!'i!p∗// (i∗i!p∗F

′) ΩY

i∗i!→id

uu(p∗F

′) ΩY

(7.4.1)// p∗(F

′ ΩY ).

It therefore suffices to show that the diagram of functors

i∗pZ∗i′!i∗pZ∗'p∗i

′∗//

pZ∗i′!'i!p∗

p∗i′∗i′!

i′∗i′!→id

i∗i

!p∗i∗i!→id // p∗


commutes. Since i, p, and pZ are proper this diagram is identified using the isomorphism αin 2.9 (2) with the diagram

i!pZ!i′!i!pZ!'i′!p!//

pZ!i′!'i!p!

p!i′!i′!

i′!i′!→id

i!i

!p!i!i

!→id // p!,

where the isomorphism pZ!i′! ' i!p! is induced by adjunction by our assumptions in 2.9 (4),

and this diagram commutes as all the maps are induced by the natural adjunctions.

Lemma 7.8. Let

F1 → F2 → F3 → F1[1]

be a distinguished triangle in MB(X). If BZX(F1) and BZ

X(F2) hold then so does BZX(F3).

Proof. This follows from noting that the morphisms in the properties BZX(Fi) fit into a mor-

phism of distinguished triangles

i!F1 ΩY

// i!F2 ΩY

// i!F3 ΩY

// i!F1 ΩY [1]

i!(F1 ΩY ) // i!(F2 ΩY ) // i!(F3 ΩY ) // i!(F1 ΩY )[1].

Lemma 7.9. If X ∈ S be smooth over k.

(i) Property CX holds.

(ii) If i : Z → X is a closed subscheme and CZ holds then BZX(ΩX) holds.

Proof. Let d be the dimension of X (a locally constant function on X). Then by 2.9 (3)applied to the morphisms X → Spec(k) and X × Y → Y , we have ΩX ' 1X(d)[2d] andΩX×Y ' 1X ΩY (d)[2d], and from its definition the map εX×Y is equal to the isomorphismobtained from these identifications implying (i).

For (ii) observe that since pr2 : X × Y → Y is smooth we have

i!(ΩX ΩY ) ' i!pr!2ΩY ' ΩZ×Y ,

and it follows from 7.6 that the map

ΩZ ΩY ' (i!ΩX) ΩY → i!(ΩX Ω) ' ΩZ×Y

arising in BZX(ΩX) is equal to εZ×Y . This implies (ii).

Lemma 7.10. Let X ∈ S be a scheme, and suppose that for every nowhere dense closedsubscheme i : Z → X the properties CZ and BZ

X(ΩX) hold. Then property CX also holds.

Proof. Let j : U → X be an everywhere dense open subscheme with Ured smooth over k, andlet i : Z → X be the complementary closed subscheme (with the reduced structure). Fromthe distinguished triangle

i∗ΩZ → ΩX → j∗ΩU → i∗ΩZ [1]

26 MARTIN OLSSON

and its variant for X × Y we get a commutative diagram

i∗ΩZ ΩY//

a

c

%%

ΩX ΩY// j∗ΩU ΩY

b

i∗(i!(ΩX ΩY ))

d

// ΩX ΩY//

f

j∗(ΩU ΩY )

e

i∗ΩZ×Y // ΩX×Y // j∗ΩU×Y

By property BZX(ΩX) the map labelled a is an isomorphism, and by 7.9 (ii) and 7.3 the map

e is also an isomorphism. Now by property CZ the map c is an isomorphism whence the mapd is also an isomorphism. From this it follows that the map f is an isomorphism as well.

Lemma 7.11. Let i : Z → X be a closed imbedding. The properties CZ and CX implyBZX(ΩX).

Proof. Indeed we have(i!ΩX) ΩY ' ΩZ ΩY ' ΩZ×Y ,

where the second isomorphism is by property CZ . Similarly we have

i!(ΩX ΩY ) ' i!(ΩX×Y ) ' ΩZ×Y ,

where the first isomorphism is by property CX . Under these identifications the map occurringin property BZ

X(ΩX) is identified with the identity map on ΩZ×Y .

7.12. Let p : P → X be a proper morphism, and fix a distinguished triangle in MB(X)

1X → p∗1P → F → 1X [1].

Assume there exists a closed imbedding i : Z → X with everywhere dense complementj : U → X such that the restriction pU : PU → U of p to U is finite radicial and surjective. LetpZ : PZ → Z be the restriction of p to Z. Let FZ denote a cone of the morphism 1Z → pZ∗1PZ

so we can find a morphism of distinguished triangles in MB(X)

1X

a

// p∗1P

b

// Fc

// 1X [1]

a

i∗1Z // i∗pZ∗1PZ

// i∗FZ // i∗1Z [1],

where the maps labelled a and b are the adjunction maps.

Lemma 7.13. The map c : F → i∗FZ is an isomorphism.

Proof. Considering the distinguished triangle

j!j∗F → F → i∗i

∗F → j!j∗F [1]

it suffices to show that j∗F = 0 and that the map i∗F → FZ is an isomorphism. The firststatement follows from the fact that the morphism 1X → p∗1P is an isomorphism over U ,since p is finite radicial and surjective over U (see 2.13), and the second statement followsfrom the fact that the base change map i∗p∗1P → pZ∗1PZ

is an isomorphism.


Remark 7.14. Lemma 7.13 holds over a general base B, with the same proof.

Theorem 7.15. Let i : Z → X be a closed imbedding in S . Then properties AZX(1X),BZX(1X), AZX(ΩX), BZ

X(ΩX), and CX hold.

Proof. By induction on the dimension d of X we may assume that we have X ∈ S ofdimension d and that the theorem is true for every object of S of dimension < d. To verifythe theorem for X it then suffices by 7.3 and 7.10 to show that for every i : Z → X theproperties BZ

X(1X) and BZX(ΩX) hold.

By [8, 7.3] we can find a proper morphism p : P → X with P smooth, quasi-projective, andequipped with an action of finite group G over X, such that if p : P → X is the coarse moduli

space of the stack [P /G] then p is generically on X finite surjective and radicial. By 6.7 this

implies that ΩP ' 1P (d)[2d], where d is the dimension of P (a locally constant function).This also implies that ΩP×Y ' 1P (d)[2d] ΩY since by 6.5 we have an isomorphism

ΩP×Y ' (p∗ΩP×Y )G ' (p∗(1P (d)[2d] ΩY ))G ' (p∗p∗(1P (d)[2d] ΩY ))G ' 1P (d)[2d] ΩY ,

where the last isomorphism is by [4, 3.3.35] and the second isomorphism uses 7.9.

This also gives the following description of the map

εP×Y : ΩP ΩY → ΩP×Y .

Let π : P → P be the projection, and let π : P ×Y → P ×Y be the base change to Y . Thenby 7.5 the diagram

π∗(ΩP ΩY )

εP×Y // π∗ΩP×Y

ΩP ΩY

εP×Y // ΩP×Y

commutes, where the vertical maps are as in 7.5, and by 6.5 this identifies the map εP×Y withthe map on G-invariants

(π∗(ΩP ΩY ))GεP×Y // (π∗ΩP×Y )G.

In particular since εP×Y is an isomorphism by 7.9 (i) property CP holds.

From this we also get that for every closed t : T → P properties BTP (ΩP ) and BT

P (1P ) hold.Indeed since P is obtained as the coarse moduli space of a smooth Deligne-Mumford stack,for any connected component Pi of P the intersection T ∩Pi is either all of Pi or of dimension< d. We can therefore apply 7.11 to get BT

P (ΩP ) and since 1P ' ΩP (−d)[−2d] (by 6.7) thisalso implies BT

P (1P ).

Let Q be a cone of the morphism 1X → p∗1P . By 7.7 we then have properties BZX(p∗1P )

and BZX(p∗ΩP ). Lemma 7.8 then implies that to verify property BZ

X(1X) it suffices to verifythe property BZ

X(Q). Dualizing we also have a distinguished triangle

DX(Q)→ p∗ΩP → ΩX → DX(Q)[1],

and to verify property BZX(ΩX) it suffices to verify property BZ

X(DX(Q)).

Let α : T → X be a nowhere dense closed subscheme such that the restriction of p to thecomplement of T is finite and radicial. Let ZT (resp. PT , Z ′T ) denote Z ∩ T (resp. T ×X P ,

28 MARTIN OLSSON

ZT×XP ). Then by 7.13 we have Q = α∗QT for some QT ∈MB(T ) fitting into a distinguishedtriangle

1T → pT∗1PT→ QT → 1T [1].

Dualizing we also get a distinguished triangle

DT (QT )→ pT∗ΩPT→ ΩT → DT (QT )[1].

By the induction hypothesis and applying 7.8 we conclude that BZTT (QT ) and BZT

T (DT (QT ))hold, and therefore by 7.7 properties BZ

X(Q) and BZX(DX(Q)) also hold.

8. Application: local terms for actions given by localized Chern classes

Let k be an algebraically closed field, and let MB denote the motivic category of Beilinsonmotives over k.

8.1. For a prime ` invertible in k there is constructed in [5, 7.2.24] an etale realization functor

T` : MB → DMc,`

where for X ∈ S the fiber DMc,`(X) is isomorphic to the idempotent completion Dbc(X,Q`)

]

of the triangulated category Dbc(X,Q`). Here the idempotent completion is defined as in [2].

This realization functor is compatible with the six operations and Chern classes. Note alsothat by [2, 1.4] the functor

Dbc(X,Q`)→ Db

c(X,Q`)]

is fully faithful.

8.2. Let f : X → Y be a morphism of quasi-projective schemes in S . Let An,met,` (X → Y )

denote H2n(X, f !Q`(m)), and let Hi,BM,`(X) denote the i-th `-adic Borel-Moore homology ofX. These groups were considered in [18, 3.1 and 2.2] (with different notation). In [18, 4.2and 2.10] there is constructed maps, which are special cases of the constructions in 4.5 and4.6,

τXY,` : K(f -perfect complexes on X)→ ⊕iAi,iet,`(X → Y ), cl` : Ai(X)Q → Hi,BM,`(X),

where Ai(X)Q denotes the Chow groups of X tensor Q. By 4.5 and 4.6 we also have maps

τXY : K(f -perfect complexes on X)→ ⊕iAi,iMB(X → Y ), cl` : Ai(X)Q → HMB

i,BM(X),

The realization functor also defines maps, which we somewhat abusively also denote by T`,

T` : Ai,iMB(X → Y )→ Ai,iet,`(X → Y ), T` : HMB

i,BM(X)→ Hi,BM,`(X).

Lemma 8.3. The diagrams

(8.3.1) K(f -perfect complexes on X)τXY

tt

τXY,`

**

⊕iAi,iMB(X → Y )

T` // ⊕iAi,iet,`(X → Y )


and

(8.3.2) Ai(X)Qcl

xx

cl`

&&HMBi,BM(X)

T` // Hi,BM,`(X)

commute.

Proof. The commutativity of (8.3.2) is a special case of the commutativity of (8.3.1) takingY = Spec(k). To see the commutativity of (8.3.1) observe that for f a closed imbeddingthe composition of τXY with the realization functor T` defines a theory of local Chern classestaking values in ⊕iAi,iet,`(X → Y ), which is compatible with the standard etale Chern classes(since T` is compatible with Chern classes). By the uniqueness part of 4.2 this implies thatit agrees with the local Chern classes defined in [18, 3.11]. From this it follows that (8.3.1)commutes.

Remark 8.4. Lemma 8.3 is stated and proven for the `-adic realization functor, but can begeneralized to a statement for morphisms of motivic categories as in 5.9.

Corollary 8.5. Let c : C → X × X be a correspondence with C and X quasi-projectiveschemes, and let u : c∗11X → c!

21X be an action in MB(C). Then there exists an algebraiccycle Σ ∈ A0(Fix(c))Q such that for any prime ` invertible in k we have Tr(u`) = cl`(Σ),where u` : c∗1Q` → c!

2Q` is the `-adic realization of u.

Proof. Let Σ ∈ A0(Fix(c))Q be the class corresponding to TrMB(u) under the isomorphism

HMB0,BM(Fix(c)) ' A0(Fix(c))Q given by 6.2. By 7.15 the assumption 5.2 is satisfied so by 5.9

we have Tr(u`) = T`(cl(Σ)), which by the commutativity of (8.3.2) is equal to cl`(Σ).

8.6. Let c : C → X ×X be a correspondence with C and X quasi-projective schemes, andlet E be a c2-perfect complex on C. We then get an action u` : c∗1Q` → c!

2Q` from the classτCX,`(E) ∈ H0(C, c!

2Q`) defined in [18, 4.2].

Theorem 8.7. There exists a cycle Σ ∈ A0(Fix(c))Q, independent of `, such that Trc(u`) ∈H0(Fix(c),ΩFix(c)) is equal to cl`(Σ).

Proof. Let u : c∗11X → c!21X be the morphism in MB(C) defines by τCX (E)0. Then by the

commutativity of (8.3.2) the action u` is the `-adic realization of u. From this and 8.5 theresult follows.

9. Application: quasi-finite morphisms and correspondences

In this section B denotes a regular excellent scheme of dimension ≤ 2, and S is thecategory of finite type separated B-schemes.

9.1. Let ` be a prime invertible on B, and let f : Y → X be a quasi-finite morphism betweenquasi-projective B-schemes. Let u` ∈ H0(Y, f !Q`) be a section. We say that u` is motivic ifthere exists a morphism u : 1Y → f !1X in MB(Y ) such that u` is the `-adic realization of u.

30 MARTIN OLSSON

9.2. The condition that u` be motivic has the following more concrete characterization. Sincef is quasi-finite, f!Q` is a sheaf. For any dense open subscheme j : U → X the adjunctionmap Q`,X → R0j∗Q`,U is injective so the restriction map

HomX(f!Q`,Q`,X)→ HomX(f!Q`, R0j∗Q`,U) ' HomU(f!Q`|U ,Q`,U)

is injective. By adjunction it follows that the restriction map

HomY (Q`,Y , f!Q`,X)→ Homf−1(U)(Q`,f−1(U), f

!Q`,X |f−1(U))

is injective, so the map u` is determined by its restriction to f−1(U). In particular, let Yii∈Ibe the irreducible components of Y which dominate an irreducible component of X via f ,and choose a dense open subscheme U ⊂ X such that Ured is regular and

f−1(U) =∐i∈I

Vi,

where Vi ⊂ Yi is a dense open and Vi,red is regular of the same dimension of its image in U .We then have f !Q`,U ' Q`,f−1(U) and a canonical isomorphism

H0(f−1(U), f !Q`,U) ' QI` .

From this we obtain an inclusion

(9.2.1) H0(Y, f !Q`,X) → QI` .

It follows immediately from the construction that this is independent of the choice of U . Theimage of u` in QI

` will be called the weight vector of u`, and will be denoted w(u`).

Remark 9.3. In the above we do not assume that f is necessarily dominant. The argumentshows that a morphism u` is determined by its restriction to those components of Y whichdominate X.

Theorem 9.4. (i) The section u` is motivic if and only if the weight vector w(u`) lies inQI ⊂ QI

` .

(ii) If u` is the `-adic realization of u : 1Y → f !1X , then for any other prime `′ invertiblein k the `′-adic realization u`′ of u has w(u`′) = w(u`) in QI .

Remark 9.5. If the weight vector w(u`) lies in QI we say that u` has rational weight vector.

The proof of 9.4 occupies the following (9.6)–(9.19).

9.6. Fix a prime ` and an element u` ∈ H0(Y, f !Q`) with weight vector w ∈ QI . We showthat u` is motivic as follows.

9.7. By [9, 5.15] (in the case when B is the spectrum of a field one can also use [8, 7.3]) we

can find a proper morphism p : P → X with P regular and equipped with an action of finite

group G over X, such that if p : P → X is the coarse moduli space of the stack [P /G] then pis generically on X finite surjective and radicial. Next choose a proper surjective genericallyfinite morphism κ : F → Y ×f,X P , with F regular, which fits into a commutative diagram

F

q

g // P

p

Y

f // X.


Let dF (resp. dP ) be the dimension of F (resp. P ), a locally constant function on F (resp.P ). Note that we can also view dP as a locally constant function on F via g. By 6.7 we haveΩP ' Q`(d − δ)[2(d − δ)], and therefore g!Q` ' ΩF (δ − d)[2(δ − d)], which by 6.6 appliedto F is isomorphic to Q`. From this it follows that if π0(F ) denotes the set of connected

components of F then H0(F, g!Q`) ' Qπ0(F )` . Thus giving a map Q` → g!Q` is equivalent to

specifying a function π0(F )→ Q`.

9.8. Under these identifications the map Q` → g!Q` corresponding to the function sendingall elements of π0(F ) to 1 restricts on a connected component Fi of F to the map

[Fi] : Q` → g!Q` ' ΩFi(δ − d)[2(δ − d)]

given by the fundamental class of Fi in H0(Fi,ΩFi(δ − d)[2(δ − d)]) defined in [20, Expose

XVI] (see also [17, 3.1] in the case when B is the spectrum of an algebraically closed field).

9.9. For an irreducible component Yi let Ni be the number of irreducible components of Fwhich dominate Yi, and let ν` : Q` → g!Q` be the map corresponding to the function assigningto a connected component Fj of F dominating Yi the number

w(u)i/(Ni · deg(Fj/P )),

where deg(Fj/P ) denotes the degree of Fj over its image in P , and taking value 0 on connectedcomponents of F not dominating an irreducible component of Y .

9.10. The description of ν` in terms of maps defined by algebraic cycles implies that the mapν` : Q`,F → g!Q`,P is motivic. Indeed observe that the etale realization functor is compatiblewith the purity isomorphisms [4, 14.4.1] (this follows from [7, 4.3] and the compatibility withChern classes), and therefore for each connected component Fi of F the map

[Fi] : Q`,Fi→ g!

iQ`,P

defined by the fundamental class of Fi is the realization of the corresponding map in MB(Fi).Since ν` is obtained by summing these maps multiplied by rational numbers it follows thatthere exists a morphism ν : 1F → g!1P in MB(F ) whose realization is equal to ν`.

Note that for u` we have no such description in terms of cycles which is why the proof of9.4 is more complicated.

Lemma 9.11. The diagram

(9.11.1) Q`,Y//

u`

q∗Q`,F

q∗ν`

f !Q`,X// f !p∗Q`,P

commutes.

Proof. A morphism Q`,Y → f !p∗Q`,P is equivalent by adjunction to a morphism f!Q`,Y →p∗Q`,P . Since f is quasi-finite f!Q`,Y is a sheaf and therefore such a morphism is in turnequivalent to a morphism of sheaves f!Q`,Y → R0p∗Q`,P . From this it follows it follows thata morphism Q`,Y → f !p∗Q`,P is determined by its restriction to the inverse of any dense opensubset of X (using an argument as in 9.2).

32 MARTIN OLSSON

To prove the lemma it therefore suffices, by shrinking on X, to consider the case when Xred

is regular, p : P → X is finite surjective and radicial, g is finite, and Yred regular. Restrictingto a connected component of Y we may further assume that Y is connected. In this casef !p∗Q`,P is isomorphic to Q`,Y and the map

Q`,Yu` // f !Q`,X

// f !p∗Q`,P ' Q`,Y

is given by multiplication by the weight vector w(u) (which after our reductions is simply anelement of Q`).

On the other hand, for a connected component Fi of F let qi : Fi → Y be the restrictionof q. Then by compatibility of the cycle class map with proper pushforward [17, 6.1] (thestatement there is under the assumption that B is the spectrum of an algebraically closedfield but the proof works in general) the composite map

Q`,Y// qi∗Q`,Fi

[Fi] // qi∗g!Q`,P

// f !p∗Q` ' Q`,Y

is equal to multiplication by the degree of Fi over P , where [Fi] is defined as in 9.8. Thereforethe composite map

Q`,Y// q∗Q`,F

q∗ν` // f !p∗Q` ' Q`,Y

is equal to multiplication by∑Fj

w(u)

(N · deg(Fj/P ))· deg(Fj/P ) = w(u)

proving the lemma.

9.12. Fix a distinguished triangle in MB(X)

1X → p∗1P → F → 1X [1],

and let F` denote the `-adic realization of F , so we have a distinguished triangle in Dbc(X,Q`)

Q`,X → p∗Q`,P → F` → Q`,X [1].

Let i : Z → X be a closed imbedding with everywhere dense complement j : U → X suchthat the restriction pU : PU → U of p to U is finite radicial and surjective. Let pZ : PZ → Zbe the restriction of p to Z. Let FZ denote a cone of the morphism 1Z → pZ∗1PZ

so we canfind a morphism of distinguished triangles in MB(X)

1X

a

// p∗1P

b

// Fc

// 1X [1]

a

i∗1Z // i∗pZ∗1PZ

// i∗FZ // i∗1Z [1],

where the maps labelled a and b are the adjunction maps. By 7.13 (and 7.14) the mapc : F → i∗FZ is an isomorphism.

Notation 9.13. If W is a quasi-projective B-scheme and F ∈MB(W ) we write H iMB

(W,F )

for ExtiMB(W )(1W , F ).


Lemma 9.14. Let f : Y → X be a quasi-finite morphism of quasi-projective B-schemes.Then the `-adic realization map

(9.14.1) H iMB

(Y, f !1X)→ H i(Y, f !Q`)

is injective for i ≤ 0, and H iMB

(Y, f !1X) = 0 for i < 0.

Proof. The second statement follows from the first and the fact that the functor f ! : Dbc(X,Q`)→

Dbc(Y,Q`) takes D≥0

c (X,Q`) to D≥0c (Y,Q`) by [1, XVIII, 3.1.7].

Consider first the case when X is the coarse moduli space of a stack of the form [M/G]with M regular of some dimension d and G a finite group acting on M . In this case we haveby 6.7

H iMB

(Y, f !1X) ' H iMB

(Y,ΩY (δ − d)[2(δ − d)]),

and in particular by 6.2

H0MB

(Y, f !1X) ' H−2(d−δ)MB

(Y,ΩY (−(d− δ))) = HMBd−δ,BM(Y ) ' Ad(Y )Q.

Since f is quasi-finite this is canonically isomorphic to the Q-vector space with basis theirreducible components of Y of dimension d. This implies the injectivity of (9.14.1) for i = 0,and also shows that if j : V ⊂ Y is the preimage of a dense open subset in X then therestriction map

H0MB

(Y, f !1X)→ H0MB

(V, f !V 1X)

is injective, where fV : V → X is the restriction of f . Let r : Z → Y be the complement ofV and let fZ : Z → X be the restriction of f . Choose V such that Vred is affine and regularof some dimension e ≤ d. In this case we have f !

V 1X ' 1V (e− d)[2(e− d)] so

H iMB

(V, f !V 1X) ' H

i+2(e−d)MB

(V, 1V (e− d)).

By 6.3 these groups are zero if i < 0. Now from the distinguished triangle

r∗f!Z1X → f !1X → j∗f

!V 1X → r∗f

!Z1X [1]

we get a long exact sequence

· · · → H iMB

(Z, f !Z1X)→ H i

MB(Y, f !1X)→ H i

MB(V, f !

V 1X)→ · · · .

By induction on the dimension of Y (with base case handled by the case when Y is regular)we have H i(Z, f !

Z1X) = 0 for i < 0, and as discussed above we also have H iMB

(V, f !V 1X) = 0

for i < 0. This therefore completes the proof in the case when X is the coarse space of astack [M/G] as above.

For the general case we proceed by induction on the dimension of X. Let p : P → X be asin 9.7, and consider the resulting distinguished triangle

1X → p∗1P → F → 1X [1].

Applying f ! we get a distinguished triangle

f !1X → f !p∗1P → f !F → f !1X [1].

34 MARTIN OLSSON

Let PY denote the fiber product Y ×X P so we have a cartesian square

PY

q

g // P

p

Y

f // X.

By base change, we have f !p∗1P ' q∗g!1P and therefore

H iMB

(Y, f !p∗1P ) ' H iMB

(PY , g!1P ).

By the case considered at the beginning of the proof, it follows that the `-adic realizationmap

H iMB

(Y, f !p∗1P )→ H i(Y, f !p∗Q`)

is injective for i ≤ 0. To prove the lemma it therefore suffices to show that H i(Y, f !F) = 0for i < 0. Let i : Z → X, YZ and FZ be as in 9.12 so we have

H iMB

(Y, f !F) ' H iMB

(YZ , f!ZFZ).

Now consider the distinguished triangle on Z

1Z → pZ∗1PZ→ FZ → 1Z [1],

and the resulting distinguished triangle

f !Z1Z → f !

ZpZ∗1PZ→ f !

ZFZ → f !Z1Z [1]

on YZ . By induction the lemma holds for the quasi-finite morphisms fZ : YZ → Z andgZ : PYZ → PZ . To prove that H i

MB(YZ , f

!ZFZ) = 0 for i < 0 it therefore suffices to show that

the map on etale cohomology

(9.14.2) H0(YZ , f!ZQ`)→ H0(PYZ , g

!ZQ`)

is injective. For this, observe that since fZ is quasi-finite it suffices to show injectivity afterrestricting to a dense open of Z (using the argument of 9.2). Thus it suffices to proveinjectivity when Zred and YZ,red are regular, connected, and of the same dimension, in whichcase H0(YZ , f

!ZQ`) ' Q`. Similarly using 9.2 the group H0(PYZ , g

!ZQ`) injects into a vector

space QI` , where I is the set of irreducible components of PYZ which dominate an irreducible

component of PZ . From this the result follows as there exists an irreducible component ofPYZ which dominates an irreducible component of P and also YZ , and the map Q` → QI

`

induced by (9.14.2) is nonzero on the factor corresponding to such a component.

9.15. Returning to the setting of 9.7 and 9.12, fix also a distinguished triangle in MB(Y )

1Y → q∗1F → G → 1Y [1],

and let G` ∈ Dbc(Y,Q`) be the `-adic realization of G. Let i : YZ → Y denote f−1(Z), let

qZ : FZ → YZ denote the pullback of q, and let GZ denote a cone of 1YZ → qZ∗1FZ. Then we

have G ' i∗GZ by 7.13 (and 7.14).


9.16. Applying i∗ to the diagram (9.11.1) and using base change morphisms we obtain acommutative diagram

(9.16.1) Q`,YZ

uZ,`

// qZ∗Q`,FZ

qZ∗νZ,`

f !ZQ`,Z

// f !ZpZ∗Q`,PZ

.

As noted in 9.10 the map ν` is motivic. The pullback νZ` is therefore also motivic, hencethe map pZ∗νZ,` is motivic as well.

Lemma 9.17. The weight vector of uZ,` is rational.

Proof. Let W ⊂ YZ be an irreducible component which dominates an irreducible componentW ⊂ Z via fZ , and let W ⊂ W be a nonempty regular open subset mapping to a regularopen subset W

⊂ W . Let V ⊂ PZ ×Z W be a nonempty connected regular open subsetmapping to a regular open subset V ⊂ PZ . Note that since W is quasi-finite over Z, V isalso quasi-finite over PZ , and therefore V and V have the same dimension. Let α : W → W

and β : V → V denote the projections, so we have a commutative diagram

W _

α // W _

V //

β

PZ ×Z W

99

W _

// W _

YZ // Z

V // PZ .

55

Then α!Q` ' α∗Q` and β!Q` ' β∗Q`. By the commutativity of (9.16.1), we then have acommutative diagram

(9.17.1) H0(W ,Q`)

uZ,` // H0(W , α!Q`)

' // H0(W , α∗Q`)

H0(V,Q`)

ε // H0(V, β!Q`)' // H0(V, β∗Q`),

where the map ε is induced by the composite map

Q`,YZ// qZ∗Q`,FZ

pZ∗νZ,`// f !ZpZ∗Q`,PZ

' prPZ×ZYZ ,2∗pr!PZ×ZYZ ,1

Q`,PZ.

The composition of the top horizontal line in (9.17.1) is given by multiplication by the W -component of the weight vector of uZ,`, so to prove this is a rational number it suffices toshow that the composition of the bottom horizontal line in (9.17.1) is given by multiplicationby a rational number. Since the map pZ∗νZ,` is motivic, the bottom horizontal line is, byinduction, multiplication by an element of H0(V,Q`) ' Q` in the image of H0

MB(V, 1V ) ' Q,

which implies the result.

36 MARTIN OLSSON

9.18. By induction we can find uZ : 1YZ → f !Z1Z in MB(YZ) with `-adic realization uZ,`. Let

νZ : 1FZ→ g!

Z1PZdenote a morphism in MB(FZ) inducing νZ,`. By 9.14 the induced diagram

1YZ

u

// qZ∗1FZ

q∗νZ

f !Z1Z // f !

ZpZ∗1PZ

commutes since this holds for the `-adic realizations. Let ρZ : GZ → f !ZFZ be a morphism

filling in the diagram

1YZ

uZ

// qZ∗1FZ

q∗νZ

// GZρZ

// 1YZ [1]

uZ

f !Z1Z // f !

ZpZ∗1PZ// f !ZFZ // f !

Z1Z [1],

and let ρ : G → f !F be the morphism obtained by applying i∗ to ρZ and using the isomor-phisms 7.13 and 9.15. Then the diagram

q∗1F

q∗ν

// Gρ

f !p∗1P // f !F

commutes. Indeed this can be verified after applying i∗ where the result follows from theconstruction. We can therefore find a morphism λ : 1Y → f !1X so that we have a morphismof distinguished triangles

1Y //

λ

q∗1F

ν

// Gρ

// 1Y [1]

λ

f !1X // f !p∗1P // f !F // f !1X [1].

The `-adic realization of λ is then equal to u`, as this can be verified over a regular denseopen of X where it follows from 9.11. This completes the proof of the “if” part of statement(i).

9.19. To see the “only if” part of statement (i) as well as statement (ii) in 9.4 it sufficesto define the weight vector of u without passing to realizations. For this choose U ⊂ X asin 9.2 so that f−1(U) =

∐i Vi with each Vi,red regular. By 6.6 the restriction of f ! to Vi is

isomorphic to 1Vi , and in particular

H0MB

(Vi, f!1X) ' Q.

We then get a map

H0MB

(Y, f !1X)→∏i

H0MB

(Vi, f!1X) ' QI ,

where the last isomorphism uses purity. By compatibility of the `-adic realization functorwith the purity isomorphisms, as discussed in 9.16, the image of u under this map is equal tothe weight vector of the `-adic realization u`. This completes the proof of 9.4.


Remark 9.20. The proof (in particular 9.14) shows that if u` is motivic, then the morphismu : 1Y → f !1X in MB(X) inducing u` is unique.

9.21. We apply this to correspondences as follows. Assume now that B is the spectrum ofan algebraically closed field k, and let c : C → X × X be a correspondence with X and Cquasi-projective schemes, and c2 quasi-finite.

Theorem 9.22. (i) Let w ∈ QI be a vector and assume that for some prime `0 invertible ink there exists an action u`0 : c∗1Q`0 → c!

2Q`0 with w(u`0) = w. Then there exists an algebraiccycle Σ ∈ A0(Fix(c))Q such that for any prime ` invertible in k and action u` : c∗1Q` → c!

2Q`

with weight vector w we have Trc(u`) = cl`(Σ) in H0(Fix(c),ΩFix(c)).

(ii) Let ` be a prime invertible in k and let u` : c∗1Q` → c!2Q` be an action with rational

weight vector. Then for every proper component Γ ⊂ Fix(c) the local term ltΓ(Q`,X , u`) is inQ.

(iii) Let ` and `′ be two primes invertible in k (possibly equal), and let u` ∈ H0(C, c!2Q`,X)

and u`′ ∈ H0(C, c!2Q`′,X) be sections with weight vectors w(u) and w(u′) in QI and equal.

Then for every proper component Γ ⊂ Fix(c), we have equality of rational numbers

ltΓ(Q`, u`) = ltΓ(Q`′ , u`′).

Proof. Statements (ii) and (iii) follow from (i). Statement (i) follows from 9.4 which impliesthat there exists a morphism u : c∗11X → c!

21X in MB(C) such that for any prime ` invertiblein k the `-adic realization u` : c∗1Q` → c!

2Q` of u has weight vector w.

9.23. Global consequences.

Theorem 9.24. Let c : C → X × X be a correspondence over an algebraically closed fieldwith C and X Deligne-Mumford stacks, and assume c2 is finite and c1 is quasi-finite.

(i) If u : c∗1Q` → c!2Q` is an action with rational weight vector w(u), then the trace

tr(u|RΓ(X,Q`)) of the induced action of u on RΓ(X,Q`) is in Q.

(ii) If ` and `′ are two primes and u : c∗1Q` → c!2Q` and u′ : c∗1Q`′ → c!

2Q`′ are actions withrational weight vectors and w(u) = w(u′), then tr(u|RΓ(X,Q`)) = tr(u′|RΓ(X,Q`′)).

Remark 9.25. The action of u on RΓ(X,Q`) is defined as the composite map

RΓ(X,Q`)c∗1 // RΓ(C,Q`)

u // RΓ(C, c!2Q`)

α // RΓ(X, c2!c!2Q`)

c2!c!2→id// RΓ(X,Q`),

where the map labelled α is the isomorphism induced by the isomorphism c2∗ ' c2!, usingthat c2 is proper (in the stack case this isomorphism is given by [19, 5.1]).

Proof of 9.24. By spreading out and using the generic base change theorem it suffices toconsider the case when k is the algebraic closure of a finite field.

Fix a model c : C0 → X0 × X0 for c over a finite field Fq ⊂ k such that all irreduciblecomponents of C are defined over Fq. Then any map c∗1Q` → c!

2Q` is defined over Fq, and inparticular commutes with Frobenius. For n ≥ 0 let

c(n) : C → X ×X

38 MARTIN OLSSON

be the correspondence given by (c1, FnX c2), where FX : X → X is the base change to k of

the q-power Frobenius morphism on X0.

If u : c∗1Q` → c!2Q` is an action, let u(n) : c

(n)∗1 Q` → c

(n)!2 Q` be the action obtained by

composing u with the n iterates of the canonical isomorphism F !XQ` → Q`. Then as in [14,

3.5 (c)] to prove (i) it suffices to show that there exists n0 such that for n ≥ n0 we havetr(u(n)|RΓ(X,Q`)) ∈ Q for all n ≥ n0, and to prove (ii) it suffices to show that there existsn0 such that for all n ≥ n0 we have an equality of rational numbers

tr(u(n)|RΓ(X,Q`)) = tr(u′(n)|RΓ(X,Q`′)).

Let d : C → X ×X be the transpose of c given by (c2, c1). For n ≥ 0 let (n)d : C → X ×Xdenote the correspondence (F n

Xc2, c1), so (n)d is the transpose of c(n). Let v : d∗1ΩX → d!2ΩX

denote the transpose of u, and for n ≥ n0 let (n)v : (n)d∗1ΩX → (n)d!2ΩX denote the map

obtained by n iterates of the isomorphism F ∗XΩX → ΩX .

By Fujiwara’s theorem [10, 5.4.5] and its variant for Deligne-Mumford stacks (see [19, 1.26];the necessary compactification results can be found in [6, 1.2.1]) there exists an integer n0

such that for all n ≥ n0 the following hold:

(i) The fixed points Fix((n)d) = Fix(c(n)) consists of a finite set of isolated points.(ii) We have

tr((n)v|RΓc(X,ΩX)) =∑

y∈Fix((n)d)

lty(ΩX ,(n)v).

(iii) If U → X is an etale morphism and dU : CU → U ×U denotes the pullback of c alongU × U → X × X, and if vU : d∗U1ΩU → d!

U2ΩU denotes the pullback of v, then forevery y′ ∈ Fix((n)dU) = U ×X Fix((n)d) mapping to y ∈ Fix((n)d) we have

lty′(ΩU ,(n)dU) = lty(ΩX ,

(n)d).

Now since (n)v is adjoint to u(n) we have

tr((n)v|RΓc(X,ΩX)) = tr(u(n)|RΓ(X,Q`)),

and by [13, III, 5.1.6] we have

lty(ΩX ,(n)v) = lty(Q`, u

(n)).

It follows that for n ≥ n0

tr(u(n)|RΓ(X,Q`)) =∑

y∈Fix(c(n))

lty(Q`, u(n)).

It suffices to show that each local term belongs to Q and is independent of `. Now by (iii),the local term lty(Q`, u

(n)) can be computed after replacing X by an etale covering, whichreduces the computation to the case when X is quasi-projective. Combining this with 9.22 weget the theorem (note that if w(u)i denotes the component of the weight vector correspondingto an irreducible component Ci ⊂ C then qdim(Ci)w(u)i = w(u(n))i).


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