nLab
sheaf with transfer

Contents

Idea

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

Traditionally this is considered for SS the Nisnevich site and Corr p(S)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 SS 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 SS be a base scheme which is regular and noetherian. Let Sm SSm_S denote the category of schemes that are smooth and of finite type over SS. For X,YSm SX,Y \in \Sm_S, one defines the group of finite SS-correspondences C S(X,Y)C_S(X, Y) as the free abelian group generated by closed integral subschemes ZX× SYZ \subset X \times_S Y such that the projection ZXZ \to X is finite and equidimensional?. This defines an additive category Sm S corSm_S^{cor} whose objects are the same as Sm SSm_S and whose morphisms are finite SS-correspondences.

  2. Note that there is a canonical functor γ S:Sm SSm S cor\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 SS defines a symmetric monoidal structure on Sm S\Sm_S. One can check that this also induces a symmetric monoidal structure on Sm S cor\Sm_S^{cor}. The functor γ S\gamma_S respects these structures.

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

  5. Let N SN_S denote the category of sheaves on Sm S\Sm_S with respect to the Nisnevich topology. Let N S trP S trN_S^{tr} \subset P_S^{tr} be the full subcategory spanned by presheaves F:Sm S corAbF : \Sm_S^{cor} \to Ab whose composition with γ S:Sm SSm S cor\gamma_S : \Sm_S \to \Sm_S^{cor} is a sheaf on Sm SSm_S (with respect to the Nisnevich topology). Objects of N S trN_S^{tr} are called Nisnevich sheaves with transfer.

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

  7. This allows one to define a symmetric monoidal structure on N S trN_S^{tr}, which is in fact the unique one for which the functor Sm SN S tr\Sm_S \to \N_S^{tr} is symmetric monoidal and S tr- \otimes_S^{tr} - preserves colimits. Given FF and GG in N S trN_S^{tr}, we can write

    F=colim XFL S[X];G=colim YGL S[Y] 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]FL_S[X] \to F (resp. L S[Y]GL_S[Y] \to G. Then we define

    F S trG=colim XF;YGL S[X× SY]. 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

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

    for all XSm SX \in \Sm_S.

References

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)

Last revised on June 17, 2014 at 23:35:21. See the history of this page for a list of all contributions to it.