nLab sheaf with transfer

Idea

Given some category (site) $S$ of test spaces, suppose one fixes some category $Corr_p(S)$ of correspondences in $S$ equipped with certain cohomological data on their correspondence space. Then a sheaf with transfer on $S$ is a contravariant functor on $Corr_p(S)$ such that the restriction along the canonical embedding $S \to Corr_p(S)$ makes the resulting presheaf a sheaf.

Traditionally this is considered for $S$ the Nisnevich site and $Corr_p(S)$ constructed from correspondences equipped with algebraic cycles as discussed at pure motive, (e.g. Voevodsky 00, 2.1 and def. 3.1.1).

The idea is that, looking at it the other way around, the extension of a sheaf to a sheaf with transfer defines a kind of Umkehr map/fiber integration by which the sheaf is not only pulled back along maps, but also pushed forward, hence “transferred” (this concept of course makes sense rather generally in cohomology, see e.g. Piacenza 84, 1.1).

The derived categories of those abelian sheaves with transfers for the Nisnevich site with are A1-homotopy invariant provides a model for motives known as Voevodsky motives or similar (Voevodsky 00, p. 20).

Definition

On this page we construct the category of Nisnevich sheaves with transfer over a base scheme $S$ which is assumed to be regular and noetherian, following Déglise. Nisnevich sheaves with transfer play an important role in the theory of mixed motives.

1. Let $S$ be a base scheme which is regular and noetherian. Let $Sm_S$ denote the category of schemes that are smooth and of finite type over $S$. For $X,Y \in \Sm_S$, one defines the group of finite $S$-correspondences $C_S(X, Y)$ as the free abelian group generated by closed integral subschemes $Z \subset X \times_S Y$ such that the projection $Z \to X$ is finite and equidimensional?. This defines an additive category $Sm_S^{cor}$ whose objects are the same as $Sm_S$ and whose morphisms are finite $S$-correspondences.

2. Note that there is a canonical functor $\gamma_S : \Sm_S \to \Sm_S^cor$ which acts on a morphism by taking the finite correspondence induced by its graph.

3. Recall that the fibred product over $S$ defines a symmetric monoidal structure on $\Sm_S$. One can check that this also induces a symmetric monoidal structure on $\Sm_S^{cor}$. The functor $\gamma_S$ respects these structures.

4. Let $P_S$ (resp. $P_S^{tr}$) denote the category of abelian presheaves on $\Sm_S$ (resp. on $\Sm_S^{cor}$). Objects of $P_S^{tr}$ are called presheaves with transfer on $S$. We will write $L_S[X]$ for the presheaf with transfers represented by an object $X \in \Sm_S$.

5. Let $N_S$ denote the category of sheaves on $\Sm_S$ with respect to the Nisnevich topology. Let $N_S^{tr} \subset P_S^{tr}$ be the full subcategory spanned by presheaves $F : \Sm_S^{cor} \to Ab$ whose composition with $\gamma_S : \Sm_S \to \Sm_S^{cor}$ is a sheaf on $Sm_S$ (with respect to the Nisnevich topology). Objects of $N_S^{tr}$ are called Nisnevich sheaves with transfer.

6. Proposition. The category $N_S^{tr}$ is cocomplete and Grothendieck abelian with an essentially small set of generators given by the objects $L_S[X]$ for $X \in \Sm_S$.

7. This allows one to define a symmetric monoidal structure on $N_S^{tr}$, which is in fact the unique one for which the functor $\Sm_S \to \N_S^{tr}$ is symmetric monoidal and $- \otimes_S^{tr} -$ preserves colimits. Given $F$ and $G$ in $N_S^{tr}$, we can write

   $$F = \colim_{X \to F} L_S[X] ; \qquad G = \colim_{Y \to G} L_S[Y]$$

   where the colimits are taken over the family of morphisms $L_S[X] \to F$ (resp. $L_S[Y] \to G$. Then we define

   $$F \otimes_S^{tr} G = \colim_{X \to F ; Y \to G} L_S[X \times_S Y].$$

8. This symmetric monoidal structure is further closed: one defines

   $$\underline{\Hom}_{N_S^{tr}} (F,G) (X) = \Hom_{N_S^{tr}} (F \otimes_S^{tr} L_S[X], G)$$

   for all $X \in \Sm_S$.

Related concepts

• Becker-Gottlieb transfer, transfer context

• Mackey functor

References

• Frédéric Déglise, Finite correspondences and transfers over a regular base, pdf.

• Vladimir Voevodsky, section 3.1 of Triangulated categories of motives over a field, K-Theory, 74 (2000) (pdf, web)

• Marc Levine, section 3.2 of Six lectures on motives, ICTP lectures 2006 (pdf)

Relation to Mackey functors:

• Bruno Kahn, Takao Yamazaki, Voevodsky’s motives and Weil reciprocity, Duke Mathematical Journal 162, 14 (2013) 2751-2796 (arXiv:1108.2764)

In the more general context of abelian sheaf cohomology a kind of “transfer” is discussed in

• Robert Piacenza, Transfer in generalized sheaf cohomology, Proceedings of the AMS, Volume 90, Number 4 (1984)