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 , not coming from morphisms of schemes but also from finite correspondences from X to Y
A presheaf F with transfers is said to be -homotopy invariant if for every X.
For example, Chow groups as well as motivic cohomology groups form presheaves with transfers.
Finite correspondence
Let be algebraic schemes (i.e., separated and of finite type over a field) and suppose is smooth. Then an elementary correspondence is an irreducible closed subscheme , some connected component of X, such that the projection is finite and surjective. Let be the free abelian group generated by elementary correspondences from X to Y; elements of are then called finite correspondences.
The category of finite correspondences, denoted by , is the category where the objects are smooth algebraic schemes over a field; where a Hom set is given as: and where the composition is defined as in intersection theory: given elementary correspondences from to and from to , their composition is:
where denotes the intersection product and , etc. Note that the category is an additive category since each Hom set is an abelian group.
This category contains the category of smooth algebraic schemes as a subcategory in the following sense: there is a faithful functor that sends an object to itself and a morphism to the graph of .
With the product of schemes taken as the monoid operation, the category is a symmetric monoidal category.
Sheaves with transfers
The basic notion underlying all of the different theories are presheaves with transfers. These are contravariant additive functorsand their associated category is typically denoted , or just if the underlying field is understood. Each of the categories in this section are abelian categories, hence they are suitable for doing homological algebra.
Etale sheaves with transfers
These are defined as presheaves with transfers such that the restriction to any scheme is an etale sheaf. That is, if is an etale cover, and is a presheaf with transfers, it is an Etale sheaf with transfers if the sequenceis exact and there is an isomorphismfor any fixed smooth schemes .
Nisnevich sheaves with transfers
There is a similar definition for Nisnevich sheaf with transfers, where the Etale topology is switched with the Nisnevich topology.
Examples
Units
The sheaf of units is a presheaf with transfers. Any correspondence induces a finite map of degree over , hence there is the induced morphismshowing it is a presheaf with transfers.
Representable functors
One of the basic examples of presheaves with transfers are given by representable functors. Given a smooth scheme there is a presheaf with transfers sending .
Representable functor associated to a point
The associated presheaf with transfers of is denoted .
Pointed schemes
Another class of elementary examples comes from pointed schemes with . This morphism induces a morphism whose cokernel is denoted . There is a splitting coming from the structure morphism , so there is an induced map , hence .
Representable functor associated to A1-0
There is a representable functor associated to the pointed scheme denoted .
Smash product of pointed schemes
Given a finite family of pointed schemes there is an associated presheaf with transfers , also denoted from their Smash product. This is defined as the cokernel ofFor example, given two pointed schemes , there is the associated presheaf with transfers equal to the cokernel ofThis is analogous to the smash product in topology since where the equivalence relation mods out .
Wedge of single space
A finite wedge of a pointed space is denoted . One example of this construction is , which is used in the definition of the motivic complexes used in Motivic cohomology.
Homotopy invariant sheaves
A presheaf with transfers is homotopy invariant if the projection morphism induces an isomorphism for every smooth scheme . There is a construction associating a homotopy invariant sheaf for every presheaf with transfers using an analogue of simplicial homology.