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# Independence of ℓ and traces on cohomology

Edited by Janos Kollar, Princeton University, Princeton, NJ, and accepted by the Editorial Board March 7, 2016 (received for review November 9, 2015)

## Significance

The theory of motives is one of the central themes in modern arithmetic geometry and number theory. It provides a theoretical framework for understanding a wide variety of questions in the subject and, in particular, predicts certain “independence of

## Abstract

Let *k* be an algebraically closed field, and let *k*. We show that for any *k*, the alternating sum of traces

One of the main invariants of a variety *X* over an algebraically closed field *k* is its cohomology. In principle, one has infinitely many cohomology theories to work with, including *k*, crystalline cohomology if the characteristic of *k* is positive, and Betti cohomology in characteristic 0. Grothendieck’s conjectural theory of motives predicts that, in some sense, the choice of cohomology theory is irrelevant and that these different cohomology theories are realizations of a single, more fundamental object.

Although the existence of an abelian category of motives remains conjectural, there has been significant progress in recent years realizing a vision of Beilinson of triangulated categories of motives. Building on fundamental work of Voevodsky and others, Cisinski and Déglise developed in [1] a formalism of six operations for triangulated categories of motives. In this paper, we explain how to deduce various “independence of

Throughout the paper, we work over an algebraically closed field *k* and consider only finite type separated *k*-schemes. For such a scheme *X*, let *X* as defined in [1, § 14] A summary of the basic six operations formalism for this category can be found in [2, § 2 and § 6].

## 1. Statement of Main Results

1.1. Let*k*, and let *k*, we have a realization functor (see [3, 7.2.24])*X* with constructible cohomology. The map *u* induces a map

If

where the map

The main result of the paper is the following:

**Theorem 1.2.** *Let* *be a correspondence with* *proper, and let* *be an object with a map* *in* *Then for any* *invertible in k**, the alternating sum of traces**is in* *and independent of*

**1.3.** Let *k*-scheme *X* and consider the correspondence

Taking **1.2.1** reduces to the trace on cohomology

**Corollary 1.4.** *If* *is an endomorphism of a finite type separated k-scheme, then for any* *invertible in k*,*is in* *and independent of*

**Remark 1.5.** Corollary 1.4 answers a question posed in [4, 3.5 (c)].

**Remark 1.6.** Corollary 1.4 has also been obtained independently by Bondarko [5, Discussion following 8.4.1].

**1.7.** If *k*, the realization *f* is proper)

**Corollary 1.8.** *Let* *be a proper endomorphism of a separated finite type k*-*scheme X. Then the alternating sum of traces**is in* *and independent of*

**Remark 1.9.** Corollary 1.8 can also be obtained by first reducing to the case when *k* is a finite field and then using Fujiwara’s theorem [6, 5.4.5] as discussed in [4, 3.5 (c)].

**Remark 1.10.** The traces **1.4.1** and **1.8.1** in fact lie in *p* is the characteristic of *k*, because the traces are rational numbers that lie in *p*-adic realizations, one might be able to prove that these traces are in

## 2. Motivic Characteristic Classes

**2.1.** Let *X* and *Y* denote finite type separated *k*-schemes, and let *A*. Indeed, giving such a map is by adjunction equivalent to giving a map**2.2.** Let

commutes, where all of the vertical maps are isomorphisms. In particular, if

**2.3.** Similarly, if

commutes, where again the vertical morphisms are isomorphisms. It follows that if

**Proposition 2.4.** For any

*Proof.* The key ingredient in the proof is the following fact (see [3, 6.2.6]): If *Z* is a finite type separated *k*-scheme and

We will use this observation to reduce the proof of 2.4 to the case when

**Lemma 2.5.** *Let Z be a finite type separated k-scheme. Then for any* *and* *the natural map**adjoint to the evaluation map**is an isomorphism*.

*Proof.* The category *f* is proper)**2.5.1** is identified with the pushforward of the corresponding map

This implies, in particular, that for any morphism

With notation as in 2.1, we then get for any integer

where the vertical isomorphisms are obtained from the preceding identifications. Chasing through the definitions, one finds that this diagram commutes. In particular,

In the case when *X* and *Y* interchanged) and, in particular, *n*.

From this, we deduce that *n*. Using [3, 6.2.6], it follows that

Now consider the collection of *A* and *B* as desired.

**2.6.** Let*P* denote

By [3, A.1.10 (5)], we have for

we get a morphism*characteristic class* of *u*.”

**Remark 2.7.** If *P* is quasi-projective, then it is shown in 6.2 in ref. 2 that there is a canonical isomorphism*P* tensor

**2.8.** As in [2, 5.9], the formation of characteristic classes is compatible with morphisms of motivic categories. In particular for *k*, we have the étale realization functor [3, 7.2.24]

## 3. Proof of 1.2

**3.1.** By Nagata’s theorem, we can find a commutative diagram

where *j* are dense open imbeddings and *k*. Because

Let *u* extends uniquely to a morphism

Because the realization functors

From this, it follows that it suffices to prove 1.2 in the case when *X* and *C* are proper over *k*.

**3.2.** In this case, the Grothendieck–Lefschetz trace equation [7, III, 4.7] gives

## Acknowledgments

The author was partially supported by National Science Foundation (NSF) Grant DMS-1303173 and a grant from the Simons Foundation. Part of this work was done during a visit to the Institut des Hautes Études Scientifiques (IHES) that was partially funded by NSF Grant 1002477.

## Footnotes

- ↵
^{1}Email: molsson{at}math.berkeley.edu.

Author contributions: M.O. wrote the paper.

The author declares no conflict of interest.

This article is a PNAS Direct Submission. J.K. is a guest editor invited by the Editorial Board.

## References

- ↵.
- Cisinski D-C,
- Déglise F

- ↵
- ↵.
- Cisinski D-C,
- Déglise F

*Compositio Mathematica*152(3):556–666 - ↵
- ↵.
- Bondarko M

- ↵
- ↵.
- Grothendieck A

*Cohomologie l-adic et Fonctions L*], Lectures Notes in Math (Springer, Berlin), Vol 589. French

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