MOTIVIC COHOMOLOGY, LOCALIZED CHERN CLASSES, AND …molsson/Chernandmotives-1.0.pdf · A Tate motive for M is a cartesian section ˝ : S !M with ˝(S) tting into a dis-tringuished

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 3

3. Chern classes 6

4. Local Chern classes 9

5. Local terms for motivic actions 11

6. Beilinson motives 13

7. Application: local terms for actions given by localized Chern classes 19

8. Application: quasi-finite morphisms and correspondences 19

References 26

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 [16]. 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 offixed points Fix(c) := C×c,X×X,∆X

X and ΩFix(c) is the `-adic dualizing complex (see [12, III,§4] for further discussion). Recall from loc. cit. that for any proper connected component

1

2 MARTIN OLSSON

Z ⊂ 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 [12, 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 7.4). 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]. In this case, the MB-version of H0(Fix(c),ΩFix(c)) is simply the group A0(Fix(c))Q and the MB-version of Tr(u`)is the cycle appearing in 1.1. Using [5], we then get 1.1 by passing to etale realizations.

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 8.2 we show that this condition is also sufficient. This also has global consequences.In particular, a special case of theorem 8.18 is the following:

Theorem 1.2. 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 1.3. Since the trace appearing 1.2 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 of

MOTIVIC COHOMOLOGY, LOCALIZED CHERN CLASSES, AND LOCAL TERMS 3

Z[1/p] which reduces mod ` to tr(u∗|RΓ(X,F`)), thereby yielding `-independence for mod `traces as well.

Remark 1.4. 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 1.5. 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.

Remark 1.6. Theorem 1.2 has also been obtained by Bondarko using variant motivic meth-ods [3, Discussion following 8.4.1].

Remark 1.7. 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.

1.8. 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 Deglise on apreliminary draft. The author was partially supported by NSF CAREER grant DMS-0748718and NSF 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 [15])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].

4 MARTIN OLSSON

(PM5) For every smooth morphism p : T → S, M ∈M (T ), and N ∈M (S) then 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)–(PM3)and (PM4).

2.3. 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 dis-tringuished triangle

τ(S) // MS(P1S)

aP1S/S// 1S // τ(S)

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

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

(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.

2.5. 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) := HomM (S)(1S, 1S(n)[i]).

2.6. 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 2.7. Similarly we can consider the 2-category of triangulated fibered categories overany base category S .


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

s, t : Ar(S )→ S

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:

(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∗.

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

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

HomX(f!L,K) ' f∗HomY (L, f !K),

and

f !HomX(L,M) ' HomY (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.

6 MARTIN OLSSON

(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→ HomX(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)) ' HomX(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)).

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

Remark 2.9. This is stronger than what is in [4, 2.4.45] but we will not need their slightlyweaker notion.

Remark 2.10. The relative purity property follows from property 2.8 (3), but we state itexplicitly for later use.

Remark 2.11. 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.

3. 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.

3.1. 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) := HomM (X)(1X , 1X(m)[n]).

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].


Combining this with the adjunction i∗i! → idX we get a morphism

H0,0(Z)→ H2,1(X).

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

3.2. 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 3.3. 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.

3.4. 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 ).

8 MARTIN OLSSON

3.5. Let X ∈ S be a scheme, let E be a vector bundle on X. Fix a sequence of integers(p1, . . . , pm) 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 pi. 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 pi on F .

Consider the polynomial ring A∗,∗M (X)[Ti,ji ]1≤i≤m,1≤ji≤pm with variable Ti,ji of bidegree(ji, ji), and let J be the bigraded ideal in this ring generated by the homogeneous elements(note that cp(E) ∈ Ap,pM (X))

cp(E)−∑

i1+···+im=p

T1,i1 · · ·Tm,im , p ≥ 1.

There is a map of bigraded rings

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

Proposition 3.6. The map α induces an isomorphism

A∗,∗M (X)[Ti,ji ]/J ' A∗,∗M (F).

Proof. This follows from the argument of [12, 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 3.7. Let S ∈ S be an object and E a locally free sheaf of rank r on S with associatedprojective bundle p : PE → S. Let c1 ∈ H2,1

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

⊕r−1i=0 1S(−i)[−2i]→ p∗1PE

in M (S) is an isomorphism.

Proof. Notice that we have p] ' p!(r − 1)[2(r − 1)] ' p∗(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 [6, 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 cp(E) − σp, where σp is the p-th symmetric function in the Uk. Wethen have 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

(Up1+···+pi−1+s)1≤s≤pi .


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 [12, p. 310] one gets that α is an isomorphism as well.

Remark 3.8. By the splitting principle and using 3.7, a theory of Chern classes is unique ifit exists.

Corollary 3.9. Let X ∈ S , let E1, . . . , Es be vector bundles on X, and let v1, . . . , vs beintegers ≥ 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 setA∗,∗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) Let r : A∗,∗M (M on X) → A∗,∗M (M) be the map induced by the adjunction i∗i

! → id.

Then if cM on X· (K ·) ∈

∏i≥1A

i,iM (M on X) denotes the vector of the cM on X

i (K ·)then

r(cM on X· (K ·)) + 1 =

∏i

c·(K2i)c·(K

2i−1)−1.

Using 3.6 and the argument of Iversen [14] one obtains:

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

10 MARTIN OLSSON

4.3. In the case when R is a Q-algebra one can introduce as in [14, §1] the localized Cherncharacter

chM on X(K ·) ∈∏i

Ai,iM (M on X).

By the argument of [14] 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 ) := ExtnM (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 [16, 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).

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

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 isomor-phism is induced by the isomorphism Hom(pr∗2ΩY ,ΩX×Y ) ' pr!

21Y , the second isomorphismis by adjunction, and the third isomorphism is by base change. The map εX×Y is the mapcorresponding under these isomorphisms to the adjunction map f!f

!1Spec(k) → 1Spec(k).

Assumption 5.2. Assume that the map εX×Y is an isomorphism.

5.3. Note that there is a natural map

1X ΩY → HomX×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 ).

Lemma 5.4. The map ρX×Y is an isomorphism.

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

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

Applying HomX×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].

12 MARTIN OLSSON

Our map ρX×Y is compatible with this triangle, in the sense that we get a commutativediagram

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].

From this and induction on the dimension of Y it follows that it suffices to consider the casewhen Y is smooth of some dimension d, in which case the result is immediate.

5.5. Fix a correspondence c : C → X ×X with C,X ∈ S and fixed point scheme Fix(c) :=C ×c,X×X,∆X

X. For an action (a morphism in M (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 .

Therefore we can also view u as an element of H0M (C, c!(ΩX 1X)). Let ∆ : X → X ×X be

the diagonal morphism and note that there is a 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

∆Xoo

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 .

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

R : M → N

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

RX : M (X)→ N (X)

is a triangulated monoidal functor, and assume further that R 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 R induces for every X ∈ S a map (which we abusively also denote simply by R)

R : HM∗,BM(X)→ HN

∗,BM(X).


Let c : C → X × X be a correspondence as in 5.5, 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 R is compatible with the six operations that

R(TrM (u)) = TrN (R(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 finite type separated 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 (whichin particular implies that 5.2 holds).

Proposition 6.2. For X ∈ S quasi-projective over B the map 4.6.1 induces for every i anisomorphism

Aδ+i(X)Q ' HMBi,BM(X),

where the left side refers to Chow homology groups, tensor Q, as defined in [11, §1.8] (or inthe case when B is the spectrum of a field [10, §1.3]).

Proof. To properly defined MB requires a lot of preparatory material, for which we refer to[4, §14]. With notation as in [4, 14.1.2], we have a map of ring spectra

(6.2.1) HB → KGLQ.

One definition of the category of Beilinson motives (see [4, 14.2.9]) is as the homotopy categoryHo(HB −mod). The map 6.2.1 therefore induces a morphism of motivic categories

MB ⊂ Ho(HB −mod)→ Ho(KGLQ −mod).

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

(6.2.2) · · · → 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 [17, Proof of Theoreme 8].Recall from [17, 7.2 and Theoreme 8 (v)] that for any X ∈ S the filtration F· on K0(X)Qdefined by the Riemann-Roch isomorphism K0(X)Q ' A∗(X)Q (so Fj/Fj+1 ' Aj(X)Q) isinduced by operations φk on K0(X)Q. Following the proof of [17, Theoreme 8 (ii) ] we getthe long exact sequence

· · · → Km(X)→ Km(M)→ Km(U)→ Km−1(X)→ · · · ,

14 MARTIN OLSSON

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

(6.2.3) · · · → grjKm(X)→ grjKm(M)→ grjKm(U)→ · · · .The fact that the map from K-theory to the cohomology of Beilinson motives is defined bythe map of ring spectra 6.2.1 and the description of the Adam’s operations as coming fromthe decomposition in [4, 14.1.1 (K5)] implies that we get an induced map from the long exactsequence 6.2.3 to the long exact sequence 6.2.2. We therefore obtain a commutative diagramwith exact rows

grδ+iK1(M) //

grδ+iK1(U) //

grδ+iK0(X)

// grδ+iK0(M) //

grδ+iK0(U)

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).

It therefore suffices to show that if M → B is smooth of relative dimension d then for anyinteger s ≥ 0 the map

grδ+iKs(M)→ H2(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 [17, 7.2 (vi)] which shows that

grd−iγ Ks(M) = gri+δKs(M).

We therefore get an induced isomorphism Ai(X)Q ' HMBi,BM(X). The fact that this isomor-

phism is induced by the map 4.6.1 follows from the uniqueness in 4.2

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 [17, Theoreme 7.2 (vi)] we have

grpγK2p−q(X)Q = grd−pK2p−q(X)Q,

which in the notation of [17, 7.4] is equal to H2d−q(X, d− p). If p < 0 then d− p > din which case this group is 0 by [17, 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 [17,2.9].

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 [7].

6.4. Let Y be a quasi-projective k-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].


There is a natural morphism p∗ΩY → ΩX (dual to the adjunction 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. There is an induced map ofdistinguished triangles

(i∗pZ∗ΩYZ )G //

hZ

(p∗ΩY )G //

h

(j∗pU∗ΩYU )G //

hU

(i∗pZ∗ΩZ)G[1]

hZ

i∗ΩZ// ΩX

// j∗ΩU// i∗ΩZ ,

where we use the fact that the formation of homotopy fixed points commutes with push-forward. By induction we may assume that hZ is an isomorphism. It therefore suffices toconsider the case when X and Y are regular of the 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 and quasi-projective over B. ThenΩX ' 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 a regular quasi-projective B-scheme of dimension d, and let Gbe a finite group acting on Y . Let X := Y/G be the coarse moduli space of the correspondingDeligne-Mumford stack [Y/G]. Then ΩX ' 1X(d− δ)[2(d− δ)].

In particular, if f : Z → X is a morphism then we have Ai,iMB(Z → X) ' Ai−d+δ(Z)Q.

Proof. The first statement follows immediately from 6.5. The second statement follows fromthis and 6.2.

6.8. For the remainder of this section we assume that B is the spectrum of a field. In thiscase, proposition 6.5 also implies that assumption 5.2 holds for MB. The proof of this is a bitmore intricate and it is useful to consider the following variant statements and intermediateresults.

16 MARTIN OLSSON

6.9. 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.

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

Lemma 6.10. 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∗F ) ΩY

//

i∗(i!F ΩY )[1]

i∗i!(F ΩY ) // F ΩY

// j∗(F ΩY ) // i∗i!(F ΩY )[1].

Lemma 6.11. Let f : 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(f∗F′).

Proof. Let f : X ′ × Y → X × Y (resp. fZ : Z ′ → Z, fZ : Z ′ × Y → Z × Y ) denote themorphism induced by f . Since f is proper we have isomorphisms

i!f∗ ' fZ∗i′!, i!f∗ ' fZ∗i

′!.

From this and the projection formula for proper morphisms 2.8 (4) we get isomorphisms

(i!f∗F′) ΩY ' fZ∗((i

′!F ′) ΩY )

andi!(f∗F

′ ΩY ) ' fZ∗(i′!(F ′ ΩY ))

which identifies the map in BZX(f∗F

′) with the pushforward of the map in BZ′

X′(F′). From

this the lemma follows.

Lemma 6.12. LetF1 → 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 6.13. 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]

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. Since Ured is smooth the map e

is trivially an isomorphism. Now by property CZ the map c is an isomorphism whence themap d is also an isomorphism. From this it follows that the map f is an isomorphism aswell.

Lemma 6.14. 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 .

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

1X → p∗1E → F → 1X [1].

Assume there exists a closed imbedding i : Z → X with everywhere dense complementj : U → X such that the restriction pU : EU → U of p to U is finite radicial and surjective.Let pZ : EZ → Z be the restriction of p to Z. Let FZ denote a cone of the morphism1Z → pZ∗1EZ

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

1X

a

// p∗1E

b

// Fc

// 1X [1]

a

i∗1Z // i∗pZ∗1EZ

// i∗FZ // i∗1Z [1],

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

Lemma 6.16. The map c : F → i∗FZ is an isomorphism.

18 MARTIN OLSSON

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∗1E is an isomorphism over U ,and the second statement follows from the fact that the base change map i∗p∗1E → pZ∗1EZ

is an isomorphism.

Theorem 6.17. 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 E ∈ S of dimension < d. To verifythe theorem for X it then suffices by 6.10 and 6.13 to show that for every i : Z → X theproperties BZ

X(1X) and BZX(ΩX) hold.

By [7, 7.3] we can find a proper morphism p : E → X with E smooth and equipped withan action of finite group G over X, such that if p : E → X is the coarse moduli space of

the stack [E/G] then p is generically on X finite surjective and radicial. By 6.5 this implies

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

(p∗ΩE×Y )G ' (p∗(1E(d)[2d] ΩY ))G ' (p∗p∗(1E(d)[2d] ΩY ))G ' 1E(d)[2d] ΩY ,

where the last isomorphism is by [4, 3.3.35]. In particular, property CE holds.

This implies that for every closed t : T → E properties BTE(ΩE) and BT

E(1E) hold. Indeedsince E is obtained as the coarse moduli space of a smooth Deligne-Mumford stack, for anyconnected component Ei of E the intersection T ∩Ei is either all of Ei or of dimension < d.We can therefore apply 6.14.

Let Q be a cone of the morphism 1X → p∗1E. By 6.11 we then have properties BZX(p∗1E)

and BZX(p∗ΩE). Lemma 6.12 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∗ΩE → 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. ET , Z ′T ) denote Z ∩ T (resp. T ×X E,ZT×XE). Then by 6.16 we have Q = α∗QT for some QT ∈MB(T ) fitting into a distinguishedtriangle

1T → pT∗1ET→ QT → 1T [1].

Dualizing we also get a distinguished triangle

DT (QT )→ pT∗ΩET→ ΩT → DT (QT )[1].

By the induction hypothesis and applying 6.12 we conclude that BZTT (QT ) and BZT

T (DT (QT ))hold, and therefore by 6.11 properties BZ

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


7. 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.

7.1. For a prime ` invertible in k there is constructed in [5, 5.9.21] an etale realization functor

R` : 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. Combining this with 7.2 we obtain the following:

Corollary 7.2. Let c : C → X×X be a correspondence and let u : c∗11X → c!21X be an action

in MB(C). Then there exists an algebraic cycle 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. In fact the algebraic cycle Σ is given by TrMB(u) ∈ HMB0,BM(Fix(c)) ' A0(Fix(c))Q.

7.3. We apply this to correspondences as follows. Let c : C → X ×X be a correspondencewith C and X quasi-projective schemes, and let E be a c2-perfect complex on C. We thenget an action u` : c∗1Q` → c!

2Q` from the class τCX (E) ∈ H0(C, c!2Q`) defined in [16, 4.2].

Theorem 7.4. 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. Since the `-adic realization functor is compatible with Chern classes 7.1, it is alsocompatible with localized Chern classes, by 4.2. Therefore the action u` is induced by amorphism in MB(C) and the result follows from 7.2.

8. 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.

8.1. 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.

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

f−1(U) =∐i∈I

Vi,

20 MARTIN OLSSON

where Vi ⊂ Yi is a dense open and Vi,red is regular of the same dimension of its image in U .We then have a canonical isomorphism

H0(f−1(U), f !Q`,U) ' QI` ,

and therefore we obtain an inclusion

(8.1.1) 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`).

Theorem 8.2. (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 8.3. If the weight vector w(u`) lies in QI we say that u` has rational weight vector.

The proof of 8.2 occupies the following (8.4)–(8.13).

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

8.5. By [8, 5.15] (in the case when B is the spectrum of a field one can also use [7, 7.3]) we

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

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

F

q

g // E

p

Y

f // X.

Let ν` : Q` → g!Q` be the map which on a connected component F dominating an irreduciblecomponent Yi of Y is given by w(u)i divided by the number of irreducible components of Fwhich dominate Yi. Here we are using the fact that since E is the coarse moduli space of aregular Deligne-Mumford stack we have ΩE ' Q`(d − δ)[2(d − δ)], where d (resp. δ) is thedimension of E (resp. B), as in 6.7.

Lemma 8.6. The diagram

(8.6.1) Q`,Y//

u`

q∗Q`,F

p∗ν`

f !Q`,X// f !p∗Q`,E

commutes.

Proof. It suffices to verify this at the generic point of each irreducible component of X, wherethe result is immediate from the construction.


8.7. Fix a distinguished triangle in MB(X)

1X → p∗1E → 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`,E → F` → Q`,X [1].

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

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

1X

a

// p∗1E

b

// Fc

// 1X [1]

a

i∗1Z // i∗pZ∗1EZ

// i∗FZ // i∗1Z [1],

where the maps labelled a and b are the adjunction maps. By 6.16 the map c : F → i∗FZ isan isomorphism.

Lemma 8.8. Let f : Y → X be a quasi-finite morphism of quasi-projective B-schemes. Thenthe `-adic realization map

(8.8.1) 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 have

H iMB

(Y, f !1X) ' H iMB

(Y,ΩY (δ − d)[2(δ − d)]).

By 6.2 we therefore have

H0MB

(Y, f !1X) ' HMd−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 8.8.1 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 i(V, f !V 1X) ' H i+2(e−d)(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]

22 MARTIN OLSSON

we get a long exact sequence

· · · → H i(Z, f !Z1X)→ H i(Y, f !1X)→ H i(V, f !

V 1X)→ · · · .By induction on the dimension of Y (with base case handled by the case when Y is smooth)we have H i(Z, f !

Z1X) = 0 for i < 0, and as discussed above we also have H i(V, f !V 1X) = 0 for

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

For the general case we proceed by induction on the dimension of X. Let p : E → X be asin 8.5, and consider the resulting distinguished triangle

1X → p∗1E → F → 1X [1].

Applying f ! we get a distinguished triangle

f !1X → f !p∗1E → f !F → f !1X [1].

Let EY denote the fiber product Y ×X E so we have a cartesian square

EY

q

g // E

p

Y

f // X.

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

H iMB

(Y, f !p∗1E) ' H iMB

(EY , g!1E).

By the regular case, it follows that the `-adic realization map

H iMB

(Y, f !p∗1E)→ 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) = 0 fori < 0. Let i : Z → X, YZ and FZ be as in 8.7 so we have

H i(Y, f !F) ' H i(YZ , f!ZFZ).

Now consider the distinguished triangle on Z

1Z → pZ∗1EZ→ FZ → 1Z [1],

and the resulting distinguished triangle

f !Z1Z → f !

ZpZ∗1EZ→ f !

ZFZ → f !Z1Z [1]

on YZ . By induction the lemma holds for the quasi-finite morphisms fZ : YZ → Z andgZ : EYZ → EZ . To prove that H i(YZ , f

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

the map on etale cohomology

H0(YZ , f!ZQ`)→ H0(EYZ , g

!ZQ`)

is injective, which can be seen by restricting to a regular dense open subset.

8.9. Returning to the setting of 8.5 and 8.7, 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 .


8.10. Applying i∗ to the diagram 8.6.1 we obtain a commutative diagram

(8.10.1) Q`,YZ

uZ,`

// qZ∗Q`,FZ

pZ∗νZ,`

f !ZQ`,Z

// f !ZpZ∗Q`,EZ

.

The map ν` is motivic being given by the local Chern classes of a g-perfect complex ofsheaves on F . The pullback νZ is therefore also motivic, whence the map pZ∗νZ,` is motivicas well.

Lemma 8.11. The weight vector of uZ,` is rational.

Proof. Let W ⊂ YZ be an irreducible component which dominates an irreducible componentZ ′ ⊂ Z via fZ , and let U ′ ⊂ W be a nonempty regular open subset mapping to a regularopen subset U ⊂ Z ′. Let V ′ ⊂ U ′ ×Z EZ be a nonempty smooth open subset mapping to asmooth open subset V ⊂ EZ . Note that since U ′ is quasi-finite over Z, V ′ is also quasi-finiteover EZ , whence V ′ and V have the same dimension. Let α : U ′ → U (resp. β : V ′ → V ) bethe map induced by fZ (resp. gZ). Then α! ' α∗. By the commutativity of (8.10.1), we thenhave a commutative diagram

(8.11.1) H0(U ′,Q`)tZ∗u //

H0(U ′, α!Q`)

' // H0(U ′, α∗Q`)

can

H0(V ′,Q`)

ε // H0(V ′, β!Q`)' // H0(V ′, β∗Q`),

where if d denotes the dimension of V then ε is induced by an element of Ad(V′)Q = Q · [V ′]

independent of `. It follows that the bottom horizontal composite in (8.11.1) is given bymultiplication by a rational number independent of `, whence the top horizontal compositeis also given by multiplication by a rational number independent of `.

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

νZ : 1FZ→ g!

Z1EZdenote a morphism in MB(FZ) inducing νZ,`. By 8.8 the induced diagram

1YZ

u

// q∗1FZ

q∗νZ

f !Z1Z // f !

ZpZ∗1EZ

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

filling in the diagram

1YZ

uZ

// q∗1FZ

q∗νZ

// GZρZ

// 1YZ [1]

uZ

f !Z1Z // f !

ZpZ∗1EZ// f !ZFZ // f !

Z1Z [1],

24 MARTIN OLSSON

and let ρ : G → f !F be the morphism obtained by applying i∗ to ρZ . Then the diagram

q∗1F

q∗ν

// Gρ

f !p∗1E // 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∗1E // 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. This completes the proof of the “if” part of statement (i).

8.13. To see the “only if” part of statement (i) as well as statement (ii) in 8.2 it suffices todefine the weight vector of u without passing to realizations. For this choose U ⊂ X as in8.15 so that f−1(U) =

∐i Vi with each Vi,red regular. We then get a map

H0MB

(Y, f !1X)→∏i

H0MB

(Vi, f!1X) ' QI ,

where the last isomorphism uses purity. The image of u under this map is the weight vector.This completes the proof of 8.2.

Remark 8.14. The proof (in particular 8.8) shows that if u` is motivic, then the morphismu : 1Y → f !1X in MB(X) inducing u` is unique.

8.15. 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. Let u : c∗1Q` → c!

2Q` be an action of c on Q`,which we can also view as a global section u ∈ H0(C, c!

2Q`,X).

Theorem 8.16. Fix a weight vector w ∈ QI , and assume that for some prime ` invertible ink there exists an action u` : c∗1Q` → c!

2Q` with w(u) = w.

(i) There exists an algebraic cycle Σ ∈ A0(Fix(c))Q such that for any prime ` invert-ible in k and action u` : c∗1Q` → c!

2Q` with weight vector w we have Trc(u`) = cl(Σ) inH0(Fix(c),ΩFix(c)).

(ii) If u has rational weight vector, then for every proper component Γ ⊂ Fix(c) the localterm ltΓ(Q`,X , u) is in Q.

(iii) If ` and `′ are two primes different from p, and u ∈ H0(C, c!2Q`,X) and u′ ∈ H0(C, c!

2Q`′,X)are sections with weight vectors w(u) and w(u′) in QI and equal, then for every proper com-ponent 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 8.2 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.

8.17. Global consequences.

Theorem 8.18. Let c : C → X×X be a correspondence over the algebraic closure k of a finitefield with 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 tr(u|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`′)).

Proof. 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 ×Xbe the correspondence given by (c1, F

nX c2), where FX : X → X is the base change to k of

the relative Frobenius 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 [13,

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 [9, 5.4.5] there exists an integer n0 such that for all n ≥ n0 thefollowing 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`)),

26 MARTIN OLSSON

and by [12, 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)).

Now by (iii), the local term lty(Q`, u(n)) can be computed after replacing X by an etale

covering, which reduces the computation to the case when X is quasi-projective. Combiningthis with 8.16 we get the theorem (note that if w(u)i denotes the component of the weightvector corresponding to an irreducible component Ci ⊂ C then qdim(Ci)w(u)i = w(u(n))i).

References

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(1998).[11] W. Fulton, Rational equivalence on singular varieties, Publ. Math. IHES 45 (1975), 147–167.[12] A. Grothendieck, Cohomologie l-adic et Fonctions L, Lectures Notes in Math 589 (1977).[13] L. Illusie, Miscellany on traces in `-adic cohomology: a survey, Jpn. J. Math 1 (2006), 107–136.

[14] B. Iversen, Local Chern classes, Ann. Sci. Ecole Norm. Sup. 9 (1976), 155–169.[15] H. Krause, On Neeman’s well generated triangulated categories, Doc. Math. 6 (2001), 121–126.[16] M. Olsson, Borel-Moore homology, intersection products, and local terms, preprint (2014).[17] C. Soule, Operations en K-theorie algebrique, Canad. J. Math. 37 (1985), 488–550.[18] Y. Varshavsky, Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara, Geom.

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