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

, not coming from morphisms of schemes but also from finite correspondences from X to Y A presheaf F with transfers is said to be

be algebraic schemes (i.e., separated and of finite type over a field) and suppose

Then an elementary correspondence is an irreducible closed subscheme

be the free abelian group generated by elementary correspondences from X to Y; elements of

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

of smooth algebraic schemes as a subcategory in the following sense: there is a faithful functor

With the product of schemes taken as the monoid operation, the category

The basic notion underlying all of the different theories are presheaves with transfers.

These are defined as presheaves with transfers such that the restriction to any scheme

There is a similar definition for Nisnevich sheaf with transfers, where the Etale topology is switched with the Nisnevich topology.

One of the basic examples of presheaves with transfers are given by representable functors.

Another class of elementary examples comes from pointed schemes

There is a splitting coming from the structure morphism

There is a representable functor associated to the pointed scheme

[3]This is analogous to the smash product in topology since

is homotopy invariant if the projection morphism

There is a construction associating a homotopy invariant sheaf[2] for every presheaf with transfers

is the universal homotopy invariant presheaf with transfers associated to

These give the motivic cohomology groups defined by

restrict to a complex of Zariksi sheaves of

These results can be found in the fourth lecture of the Clay Math book.

This case requires more work, but the end result is a quasi-isomorphism between

This gives the two motivic cohomology groups

which is found using splitting techniques along with a series of quasi-isomorphisms.

The details are in lecture 15 of the Clay Math book.