Presheaf with transfers

In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences (defined below) to the category of abelian groups (in category theory, “presheaf” is another term for a contravariant functor).

When a presheaf F with transfers is restricted to the subcategory of smooth separated schemes, it can be viewed as a presheaf on the category with extra maps $F(Y)\to F(X)$ , not coming from morphisms of schemes but also from finite correspondences from X to Y

A presheaf F with transfers is said to be ${\mathbb {A}}^{1}$ -homotopy invariant if $F(X)\simeq F(X\times \mathbb {A} ^{1})$ for every X.

For example, a Chow group as well as motivic cohomology are presheaves with transfers.

Finite correspondence

Let $X,Y$ be algebraic schemes (i.e., separated and of finite type over a field) and suppose $X$ is smooth. Then an elementary correspondence is a closed subvariety $W\subset X_{i}\times Y$ , $X_{i}$ some connected component of X, such that the projection $W\to Y$ is finite and surjective. Let $\operatorname {Cor} (X,Y)$ be the free abelian group generated by elementary correspondences from X to Y; elements of $\operatorname {Cor} (X,Y)$ are then called finite correspondences.

The category of finite correspondences, denoted by $Cor$ , is the category where the objects are smooth algebraic schemes over a field; where a Hom set is given as: $\operatorname {Hom} (X,Y)=\operatorname {Cor} (X,Y)$ and where the composition is defined as in intersection theory: given elementary correspondences $\alpha$ from $X$ to $Y$ and $\beta$ from $Y$ to $Z$ , their composition is:

\beta \circ \alpha =p_{{13},*}(p_{12}^{*}\alpha \cdot p_{23}^{*}\beta )

where $\cdot$ denotes the intersection product and $p_{12}:X\times Y\times Z\to X\times Y$ , etc. Note that the category $Cor$ is an additive category since each Hom set $\operatorname {Cor} (X,Y)$ is an abelian group.

This category contains the category ${\textbf {Sm}}$ of smooth algebraic schemes as a subcategory in the following sense: there is a faithful functor ${\textbf {Sm}}\to Cor$ that sends an object to itself and a morphism $f:X\to Y$ to the graph of $f$ .

With the product of schemes taken as the monoid operation, the category $Cor$ is a symmetric monoidal category.

References

Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284

External links

https://ncatlab.org/nlab/show/sheaf+with+transfer

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Presheaf with transfers

Finite correspondence

See also

References

External links