Independence of ℓ and traces on cohomology

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 $\ell ”$ properties of étale cohomology of varieties in positive characteristic. Although still partly conjectural, there have been many recent advances in the development of the foundations of a motivic theory. In this paper, we explain how these recent advances in the theory of motives enable one to prove certain expected independence of $\mathrm{\ell }$ properties for operators acting on étale cohomology.

Abstract

Let k be an algebraically closed field, and let $f:X\mathrm{\to }X$ be an endomorphism of a separated scheme of finite type over k. We show that for any $\ell$ invertible in k, the alternating sum of traces ${\displaystyle {\sum }_i{\left(-1\right)}^i\mathrm{tr}\left(f^{*}|H^i\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right)\right)}$ of pullback on étale cohomology is a rational number independent of $\ell .$ This is deduced from a more general result for motivic sheaves.

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 $\ell$-adic étale cohomology for any prime $\ell$ invertible in 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 $\ell ”$ results for étale cohomology from this formalism of six operations and our work in [2].

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 $\mathcal{M}\left(X\right)$ denote the triangulated category of constructible Beilinson motives over 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

$$c\mathrm{=}\left(c_1,c_2\right)\mathrm{:}C\to X\times X$$

be a correspondence over k, and let $F\in \mathcal{M}\left(X\right)$ be an object equipped with a morphism

$$u:c_1^{*}F\mathrm{\to }{c}_2^{\mathrm{!}}F$$

in $\mathcal{M}\left(C\right).$ For any $\ell$ invertible in k, we have a realization functor (see [3, 7.2.24])

$$R_{\ell }:\mathcal{M}\left(X\right)\to D_c^b\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right)$$

to the bounded derived category of complexes of ${\mathrm{\mathbb{Q}}}_{\ell }$ -sheaves on the étale site of X with constructible cohomology. The map u induces a map

$$u_{\ell }:c_1^{*}{R}_{\ell }\left(F\right)\to c_2^{!}{R}_{\ell }\left(F\right)$$

in $D_c^b\left(C,{\mathrm{\mathbb{Q}}}_{\ell }\right).$

If $c_2$ is proper, then the map $u_{\ell }$ induces an endomorphism

$$u_{\ell }^{*}:R\mathrm{\Gamma }\left(X,R_{\ell }\left(F\right)\right)\to R\mathrm{\Gamma }\left(X,R_{\ell }\left(F\right)\right)$$

defined as the composite

where the map $c_{2*}{c}_2^{!}\to \text{id}$ is defined using the properness of $c_2.$

The main result of the paper is the following:

Theorem 1.2. Let $c:C\to X\times X$ be a correspondence with $c_2$ proper, and let $F\in \mathcal{M}\left(X\right)$ be an object with a map $u:c_1^{*}F\to c_2^{!}F$ in $\mathcal{M}\left(C\right).$ Then for any $\ell$ invertible in k, the alternating sum of traces

$${\displaystyle \sum _i{\left(-1\right)}^i\text{tr}\left(u_{\ell }^{*}|H^i\left(X,R_{\ell }\left(F\right)\right)\right)}$$ [1.2.1]

is in $\mathrm{\mathbb{Q}}$ and independent of $\ell .$

1.3. Let $f:X\to X$ be an endomorphism of a finite type separated k-scheme X and consider the correspondence

$${\mathrm{\Gamma }}_f:=\left(f,\text{id}\right):X\to X\times X.$$

Taking $F=1_X$ and $u:f^{*}{1}_X\to 1_X$ the standard isomorphism we have $R_{\ell }\left(F\right)={\mathrm{\mathbb{Q}}}_{\ell },$ and in this case, the sum 1.2.1 reduces to the trace on cohomology

$${\displaystyle \sum _i{\left(-1\right)}^i\text{tr}\left(f^{*}|H^{*}\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right)\right)}.$$

Theorem 1.2 therefore implies the following:

Corollary 1.4. If $f:X\to X$ is an endomorphism of a finite type separated k-scheme, then for any $\ell$ invertible in k,

$${\displaystyle \sum _i{\left(-1\right)}^i\text{tr}\left(f^{*}|H^i\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right)\right)}$$ [1.4.1]

is in $\mathrm{\mathbb{Q}}$ and independent of $\ell .$

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 $f:X\to X$ is proper, one can also consider the correspondence

$${\mathrm{\Gamma }}_f^t:\mathrm{=}\left(\text{id},f\right):X\to X\times X,$$

the motivic dualizing complex ${\mathrm{\Omega }}_X^{\mathcal{M}}$ discussed in [2, 2.9 (Duality)], and the canonical isomorphism

$$u:{\mathrm{\Gamma }}_f^{t\ast }{\mathrm{\Omega }}_X^{\mathcal{M}}={\mathrm{\Omega }}_X^{\mathcal{M}}\to f^{!}{\mathrm{\Omega }}_X^{\mathcal{M}}.$$

For any $\ell$ invertible in k, the realization $R_{\ell }\left({\mathrm{\Omega }}_X^{\mathcal{M}}\right)$ is the $\ell$-adic dualizing complex ${\mathrm{\Omega }}_X$ and $R\mathrm{\Gamma }\left(X,R_{\ell }\left({\mathrm{\Omega }}_X^{\mathcal{M}}\right)\right)$ is dual to the compactly supported étale cohomology $R{\mathrm{\Gamma }}_c\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right).$ The endomorphism of $R{\mathrm{\Gamma }}_c\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right)$ obtained by this identification is the pullback map (defined because f is proper)

$$f^{*}:R{\mathrm{\Gamma }}_c\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right)\to R{\mathrm{\Gamma }}_c\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right).$$

From 1.2, we therefore obtain:

Corollary 1.8. Let $f:X\to X$ be a proper endomorphism of a separated finite type k-scheme X. Then the alternating sum of traces

$${\displaystyle \sum _i{\left(-1\right)}^i\text{tr}\left(f^{*}|H_c^i\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right)\right)}$$ [1.8.1]

is in $\mathrm{\mathbb{Q}}$ and independent of $\ell .$

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 $\mathrm{\mathbb{Z}}\left[1/p\right],$ where p is the characteristic of k, because the traces are rational numbers that lie in ${\mathrm{\mathbb{Z}}}_{\ell }$ for all $\ell \ne p.$ Using suitable p-adic realizations, one might be able to prove that these traces are in $\mathrm{\mathbb{Z}},$ which is expected, but we do not pursue this direction in the present paper.

2. Motivic Characteristic Classes

2.1. Let X and Y denote finite type separated k-schemes, and let $p:X\times Y\to X$ (resp. $q:X\times Y\to Y)$ be the projection. Recall [2, 7.15] that there is a natural isomorphism

$${\mathit{ϵ}}_{X\times Y}:{\mathrm{\Omega }}_X^{\mathcal{M}}\mathrm{⊠}{\mathrm{\Omega }}_Y^{\mathcal{M}}\to {\mathrm{\Omega }}_{X\times Y}^{\mathcal{M}}.$$

This isomorphism induces for any $A\in \mathcal{M}\left(X\right)$ and $B\in \mathcal{M}\left(Y\right)$ a map

$${\alpha }_{A,B}:D_X\left(A\right)\mathrm{⊠}B\to \mathcal{R}\hspace{0.17em}Ho{m}_{X\times Y}\left(p^{*}A,q^{!}B\right),$$

where $D_X\left(A\right)$ denotes the Verdier dual $\mathcal{R}\hspace{0.17em}ho{m}_X\left(A,{\mathrm{\Omega }}_X^{\mathcal{M}}\right)$ of A. Indeed, giving such a map is by adjunction equivalent to giving a map

$$\left(D_X\left(A\right)\otimes A\right)\mathrm{⊠}B\to q^{!}B\simeq \mathcal{R}\hspace{0.17em}Ho{m}_{X\times Y}\left(q^{*}{D}_Y\left(B\right),{\mathrm{\Omega }}_X^{\mathcal{M}}\mathrm{⊠}{\mathrm{\Omega }}_Y^{\mathcal{M}}\right),$$

and using adjunction, again giving such a map is equivalent to giving a map

$$\left(D_X\left(A\right)\otimes A\right)\mathrm{⊠}\left(B\otimes D_Y\left(B\right)\right)\to {\mathrm{\Omega }}_X^{\mathcal{M}}\mathrm{⊠}{\mathrm{\Omega }}_Y^{\mathcal{M}}.$$

To define ${\alpha }_{A,B},$ we take the tensor product of the evaluation maps

$$D_X\left(A\right)\otimes A\to {\mathrm{\Omega }}_X^{\mathcal{M}},\hspace{0.17em}B\otimes D_Y\left(B\right)\to {\mathrm{\Omega }}_Y^{\mathcal{M}}.$$

2.2. Let $\mathit{\pi }:X\prime \to X$ be a proper morphism, let $\tilde{\mathit{\pi }}:X\prime \times Y\to X\times Y$ be the base change of π, and let

$$p\prime :X\prime \times Y\to X\prime ,\hspace{1em}q\prime :X\prime \times Y\to Y$$

be the projections. Then it follows from the construction that for $A\prime \in \mathcal{M}(X\prime ),$ the diagram

commutes, where all of the vertical maps are isomorphisms. In particular, if ${\alpha }_{A\prime ,B}$ is an isomorphism, then ${\alpha }_{{\pi }_{\ast }A\prime ,B}$ is also an isomorphism.

2.3. Similarly, if $\mathit{\gamma }:Y\prime \to Y$ is a proper morphism with resulting morphisms

$$\tilde{\gamma }:X\times Y\prime \to X\times Y,\hspace{0.17em}p\prime :X\times Y\prime \to X,\hspace{0.17em}q\prime :X\times Y\prime \to Y\prime ,$$

and if $B\prime \in \mathcal{M}(Y\prime ),$ then the resulting diagram

commutes, where again the vertical morphisms are isomorphisms. It follows that if ${\alpha }_{A,B\prime }$ is an isomorphism, then so is ${\alpha }_{A,{\gamma }_{\ast }B\prime }.$

Proposition 2.4. For any $A\in \mathcal{M}\left(X\right)$ and $B\in \mathcal{M}\left(Y\right),$ the map ${\alpha }_{A,B}$ is an isomorphism.

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 $\mathcal{D}\subset \mathcal{M}\left(Z\right)$ is a thick triangulated subcategory of $\mathcal{M}\left(Z\right)$ containing all objects of the form $f_{\ast }\left(1_{Z\prime }\left(n\right)\right),$ for $f:Z\prime \to Z$ a projective morphism and $n\in \mathrm{\mathbb{Z}},$ then $\mathcal{D}=\mathcal{M}\left(Z\right).$

We will use this observation to reduce the proof of 2.4 to the case when $A=1_X$ and $B=1_Y.$

Lemma 2.5. Let Z be a finite type separated k-scheme. Then for any $A\in \mathcal{M}\left(Z\right)$ and $n\in \mathrm{\mathbb{Z}},$ the natural map

$$D_Z\left(A\right)\left(n\right)\to D_Z\left(A\left(-n\right)\right)$$ [2.5.1]

adjoint to the evaluation map

$$D_Z\left(A\right)\left(n\right)\otimes A\left(-n\right)\to {\mathrm{\Omega }}_Z^{\mathcal{M}}$$

is an isomorphism.

Proof. The category $\mathcal{D}\subset \mathcal{M}\left(Z\right)$ for which the lemma holds is a thick triangulated subcategory, so it suffices to show that it contains objects of the form $f_{*}{1}_{Z\prime }\left(n\right)$ for a projective morphism $f:Z\prime \to Z$ and $n\in \mathrm{\mathbb{Z}}.$ Now observe that under the isomorphisms (using that f is proper)

$$D_Z\left(f_{*}{1}_{Z\prime }\right)\left(n\right)\simeq \left(f_{*}{D}_{Z\prime }\left(1_{Z\prime }\right)\right)\left(n\right),$$

$$D_Z\left(f_{*}{1}_{Z\prime }\left(n\right)\right)\simeq f_{*}{D}_{Z\prime }\left(1_{Z\prime }\left(n\right)\right)$$

the map 2.5.1 is identified with the pushforward of the corresponding map

$$D_{Z\prime }\left(1_{Z\prime }\right)\left(n\right)\to D_{Z\prime }\left(1_{Z\prime }\left(-n\right)\right),$$

which is the natural isomorphism

$$\mathcal{R}\hspace{0.17em}Ho{m}_{Z\prime }\left(1_{Z\prime },{\mathrm{\Omega }}_{Z\prime }^{\mathcal{M}}\right)\left(n\right)\to \mathcal{R}\hspace{0.17em}Ho{m}_{Z\prime }\left(1_{Z\prime }\left(-n\right),{\mathrm{\Omega }}_{Z\prime }^{\mathcal{M}}\right).$$

This implies, in particular, that for any morphism $g:W\to Z$, $A\in \mathcal{M}\left(Z\right),$ and $n\in \mathrm{\mathbb{Z}},$ the natural map

$$\left(g^{!}A\right)\left(n\right)\to g^{!}\left(A\left(n\right)\right)$$

is an isomorphism.

With notation as in 2.1, we then get for any integer $n\in \mathrm{\mathbb{Z}},$ a square

where the vertical isomorphisms are obtained from the preceding identifications. Chasing through the definitions, one finds that this diagram commutes. In particular, ${\alpha }_{A,B\left(n\right)}$ is an isomorphism if, and only if, ${\alpha }_{A\left(-n\right),B}$ is an isomorphism.

In the case when $A=1_X$ and $B=1_Y,$ the map

$${\alpha }_{1_X,1_Y}:{\mathrm{\Omega }}_X^{\mathcal{M}}\mathrm{⊠}{1}_Y\to q^{!}{1}_Y\simeq \mathcal{R}\hspace{0.17em}Ho{m}_{X\times Y}\left(q^{*}{\mathrm{\Omega }}_Y^{\mathcal{M}},{\mathrm{\Omega }}_X^{\mathcal{M}}\mathrm{⊠}{\mathrm{\Omega }}_Y^{\mathcal{M}}\right)$$

is the map denoted ${\rho }_{X\times Y}$ in 5.4 in ref. 2 (with X and Y interchanged) and, in particular, ${\alpha }_{1_X,1_Y}$ is an isomorphism by [2, 5.7]. Because the identification $q^{!}\left(1_Y\left(n\right)\right)\simeq \left(q^{!}{1}_Y\right)\left(n\right)$ identifies the map ${\alpha }_{1_X,1_Y\left(n\right)}$ with the map obtained from ${\alpha }_{1_X,1_Y}$ by tensoring with $1_{X\times Y}\left(n\right),$ it also follows that ${\alpha }_{1_X,1_Y\left(n\right)}$ is an isomorphism for any integer n.

From this, we deduce that ${\alpha }_{A,1_Y}$ is an isomorphism for any $A\in \mathcal{M}\left(X\right).$ Indeed, the collection of $A\in \mathcal{M}\left(X\right),$ for which ${\alpha }_{A,1_Y}$ is an isomorphism, is a thick triangulated subcategory of $\mathcal{M}\left(X\right)$ and by the discussion in 2.2, the map ${\alpha }_{{\pi }_{\ast }{1}_{X\prime }\left(n\right),1_Y}$ is an isomorphism for all proper morphisms $\pi :X\prime \to X$ and all n. Using [3, 6.2.6], it follows that ${\alpha }_{A,1_Y}$ is an isomorphism for all $A\in \mathcal{M}\left(X\right)$ and also that ${\alpha }_{A,1_Y\left(n\right)}$ is an isomorphism for all $A\in \mathcal{M}\left(X\right)$ and $n\in \mathrm{\mathbb{Z}}$ because the maps ${\alpha }_{A\left(-n\right),1_Y}$ are isomorphisms.

Now consider the collection of $B\in \mathcal{M}\left(Y\right)$ for which the map ${\alpha }_{A,B}$ is an isomorphism for all $A\in \mathcal{M}\left(X\right).$ Again, this is a thick triangulated subcategory of $\mathcal{M}\left(Y\right)$ and by the discussion in 2.3, and the already known case of the ${\alpha }_{A,1_Y\left(n\right)}$’s, it contains all objects of the form ${\gamma }_{\ast }{1}_{Y\prime }\left(n\right)$ for $\gamma :Y\prime \to Y$ proper. Using [3, 6.2.6], once again, it follows that ${\alpha }_{A,B}$ is an isomorphism for all A and B as desired.

2.6. Let

$$c=\left(c_1,c_2\right):C\to X\times X$$

be a correspondence, and let P denote $C{\times }_{c,X\times X,\mathrm{\Delta }}X,$ so we have a cartesian square

By [3, A.1.10 (5)], we have for $F\in \mathcal{M}\left(X\right),$

$$c^{!}\mathcal{R}\hspace{0.17em}Ho{m}_{X\times X}\left({\text{pr}}_1^{*}F,{\text{pr}}_2^{!}F\right)\simeq \mathcal{R}\hspace{0.17em}Ho{m}_C\left(c_1^{*}F,c_2^{!}F\right).$$

Combining this with 2.4, we get a map

$${\text{Hom}}_{\mathcal{M}\left(C\right)}\left(c_1^{*}F,c_2^{!}F\right)\to c^{!}\left(D_X\left(F\right)\mathrm{⊠}F\right).$$

Composing with the map

we get a morphism

$$\text{Tr}:{\text{Hom}}_{\mathcal{M}\left(C\right)}\left(c_1^{*}F,c_2^{!}F\right)\to {\text{Hom}}_{\mathcal{M}\left(P\right)}\left(1_P,{\mathrm{\Omega }}_P^{\mathcal{M}}\right).$$

The image of a map $u:c_1^{*}F\to c_2^{!}F$ under this map is called the “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

$$A_0{\left(P\right)}_{\mathrm{\mathbb{Q}}}\simeq {\text{Hom}}_{\mathcal{M}\left(P\right)}\left(1_P,{\mathrm{\Omega }}_P^{\mathcal{M}}\right),$$

where $A_0{\left(P\right)}_{\mathrm{\mathbb{Q}}}$ denotes the Chow group of 0-cycles on P tensor $\mathrm{\mathbb{Q}}.$

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

$$R_{\ell }:\mathcal{M}\left(X\right)\to D_c^b\left(X,{\mathrm{\mathbb{Q}}}_{\ell }\right).$$

If $F\in \mathcal{M}\left(X\right)$ is an object and $u:c_1^{*}F\to c_2^{!}F$ is a morphism in $\mathcal{M}\left(C\right)$ with realization

$$u_{\ell }:c_1^{*}{R}_{\ell }\left(F\right)\to c_2^{!}{R}_{\ell }\left(F\right)$$

in $D_c^b\left(C,{\mathrm{\mathbb{Q}}}_{\ell }\right)$, then the corresponding characteristic class $\text{Tr}\left(u_{\ell }\right)\in H^0\left(P,{\mathrm{\Omega }}_P\right)$ defined as in [7, III, 4.1] is the image of $\text{Tr}\left(u\right)$ under the realization map

$${\text{Hom}}_{\mathcal{M}\left(P\right)}\left(1_P,{\mathrm{\Omega }}_P^{\mathcal{M}}\right)\to H^0\left(P,{\mathrm{\Omega }}_P\right).$$

3. Proof of 1.2

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

where $\tilde{j}$ and j are dense open imbeddings and $\overline{C}$ and $\overline{X}$ are proper over k. Because $c_2$ is proper, by assumption, this square is cartesian. Modifying $\overline{C}$ along $\overline{C}-C,$ which does not change the property that the square is cartesian, we can further arrange that the map $c_1:C\to X$ extends to a morphism ${\overline{c}}_1:\overline{C}\to \overline{X}.$

Let $\overline{F}\in \mathcal{M}\left(\overline{X}\right)$ denote $j_{\ast }F.$ Then using the canonical base change isomorphism ${\overline{c}}_2^{!}{j}_{*}\simeq {\tilde{j}}_{\ast }{c}_2^{!},$ we have

$${\overline{c}}_2^{!}\overline{F}\simeq {\tilde{j}}_{*}{c}_2^{!}F,$$

so giving a morphism

$${\overline{c}}_1^{*}\overline{F}\to {\overline{c}}_2^{!}\overline{F}$$

is equivalent to giving a morphism $c_1^{\ast }F\to c_2^{!}F$ and u extends uniquely to a morphism $\overline{u}:{\overline{c}}_1^{*}\overline{F}\to {\overline{c}}_2^{!}\overline{F}.$

Because the realization functors $R_{\ell }$ commute with the six operations, we have

$$R_{\ell }\left(\overline{F}\right)=j_{*}{R}_{\ell }\left(F\right).$$

We therefore get an isomorphism

$$R\mathrm{\Gamma }\left(\overline{X},R_{\ell }\left(\overline{F}\right)\right)\simeq R\mathrm{\Gamma }\left(X,R_{\ell }\left(F\right)\right)$$

and

$${\displaystyle \sum _i{\left(-1\right)}^i\text{tr}\left({\overline{u}}_{\ell }^{*}|H^i\left(\overline{X},R_{\ell }\left(\overline{F}\right)\right)\right)}={\displaystyle \sum _i{\left(-1\right)}^i\text{tr}\left(u_{\ell }^{*}|H^i\left(X,R_{\ell }\left(F\right)\right)\right)}.$$

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

$${\displaystyle \sum _i{\left(-1\right)}^i\text{tr}\left({\overline{u}}_{\ell }^{*}|H^i\left(X,R_{\ell }\left(F\right)\right)\right)}={\displaystyle \sum _{Z\in {\pi }_0\left(P\right)}{\displaystyle {\int }_Z\text{Tr}\left(u_{\ell }\right)}},$$

where on the right, the sum is over the connected components of the fixed point locus $P:=C{\times }_{X\times X,{\mathrm{\Delta }}_X}X$, $\text{Tr}\left(u_{\ell }\right)$ is the characteristic class of $u_{\ell }$ defined in [7, III, 4.1], and

$${\displaystyle {\int }_Z:H^0\left(Z,{\mathrm{\Omega }}_Z\right)\to {\overline{\mathrm{\mathbb{Q}}}}_{\ell }}$$

is the pushforward map induced by adjunction. By 2.8, there is a class $\text{Tr}\left(u\right)\in {\text{Hom}}_{\mathcal{M}\left(P\right)}\left(1_P,{\mathrm{\Omega }}_P^{\mathcal{M}}\right)$ such that for all $\ell ,$ we have $\text{Tr}\left(u_{\ell }\right)=R_{\ell }\left(\text{Tr}\left(u\right)\right).$ In particular, we find that ${\displaystyle {\int }_Z\left(\text{Tr}\left(u_{\ell }\right)\right)}$ is equal to the corresponding pushforward of $\text{Tr}\left(u\right)$ to ${\text{Ext}}_{\mathcal{M}\left(k\right)}^0\left(1_k,1_k\right)=\mathbb{Q}.$ This proves 1.2.