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 formula_1, not coming from morphisms of schemes but also from finite correspondences from "X" to "Y" 
A presheaf "F" with transfers is said to be formula_2-homotopy invariant if formula_3 for every "X". 
For example, a Chow group as well as motivic cohomology are presheaves with transfers.
Finite correspondence.
Let formula_4 be algebraic schemes (i.e., separated and of finite type over a field) and suppose formula_5 is smooth. Then an elementary correspondence is a closed subvariety formula_6, formula_7 some connected component of "X", such that the projection formula_8 is finite and surjective. Let formula_9 be the free abelian group generated by elementary correspondences from "X" to "Y"; elements of formula_9 are then called finite correspondences.
The category of finite correspondences, denoted by formula_11, is the category where the objects are smooth algebraic schemes over a field; where a Hom set is given as: formula_12
and where the composition is defined as in intersection theory: given elementary correspondences formula_13 from formula_5 to formula_15 and formula_16 from formula_15 to formula_18, their composition is:
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 formula_22 is a presheaf with transfers. Any correspondence formula_23 induces a finite map of degree formula_24 over formula_5, hence there is the induced morphismformula_26showing 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 formula_5 there is a presheaf with transfers formula_28 sending formula_29.
Representable functor associated to a point.
The associated presheaf with transfers of formula_30 is denoted formula_31.
Pointed schemes.
Another class of elementary examples comes from pointed schemes formula_32 with formula_33. This morphism induces a morphism formula_34 whose cokernel is denoted formula_35. There is a splitting coming from the structure morphism formula_36, so there is an induced map formula_37, hence formula_38.
Representable functor associated to A1-0.
There is a representable functor associated to the pointed scheme formula_39 denoted formula_40.
Smash product of pointed schemes.
Given a finite family of pointed schemes formula_41 there is an associated presheaf with transfersformula_42, also denoted formula_43 from their Smash product. This is defined as the cokernel offormula_44For example, given two pointed schemes formula_45, there is the associated presheaf with transfers formula_46 equal to the cokernel offormula_47This is analogous to the smash product in topology since formula_48 where the equivalence relation mods out formula_49.
Wedge of single space.
A finite wedge of a pointed space formula_32 is denoted formula_51. One example of this construction is formula_52, which is used in the definition of the motivic complexes formula_53 used in Motivic cohomology.
Homotopy invariant sheaves.
A presheaf with transfers formula_54 is homotopy invariant if the projection morphism formula_55 induces an isomorphism formula_56 for every smooth scheme formula_5. There is a construction associating a homotopy invariant sheaf for every presheaf with transfers formula_54 using an analogue of simplicial homology.
Simplicial homology.
There is a schemeformula_59giving a cosimplicial scheme formula_60, where the morphisms formula_61 are given by formula_62. That is,formula_63gives the induced morphism formula_64. Then, to a presheaf with transfers formula_54, there is an associated complex of presheaves with transfers formula_66 sendingformula_67and has the induced chain morphismsformula_68giving a complex of presheaves with transfers. The homology invaritant presheaves with transfers formula_69 are homotopy invariant. In particular, formula_70 is the universal homotopy invariant presheaf with transfers associated to formula_54.
Relation with Chow group of zero cycles.
Denote formula_72. There is an induced surjection formula_73 which is an isomorphism for formula_5 projective.
Zeroth homology of Ztr(X).
The zeroth homology of formula_75 is formula_76 where homotopy equivalence is given as follows. Two finite correspondences formula_77 are formula_2-homotopy equivalent if there is a morphism formula_79 such that formula_80 and formula_81.
Motivic complexes.
For Voevodsky's category of mixed motives, the motive formula_82 associated to formula_5, is the class of formula_84 in formula_85. One of the elementary motivic complexes are formula_53 for formula_87, defined by the class offormula_88For an abelian group formula_89, such as formula_90, there is a motivic complex formula_91. These give the motivic cohomology groups defined byformula_92since the motivic complexes formula_53 restrict to a complex of Zariksi sheaves of formula_5. These are called the formula_95-th motivic cohomology groups of weight formula_96. They can also be extended to any abelian group formula_89,formula_98giving motivic cohomology with coefficients in formula_89 of weight formula_96.
Special cases.
There are a few special cases which can be analyzed explicitly. Namely, when formula_101. These results can be found in the fourth lecture of the Clay Math book.
Z(0).
In this case, formula_102 which is quasi-isomorphic to formula_31 (top of page 17), hence the weight formula_104 cohomology groups are isomorphic toformula_105where formula_106. Since an open cover
Z(1).
This case requires more work, but the end result is a quasi-isomorphism between formula_107 and formula_108. This gives the two motivic cohomology groupsformula_109where the middle cohomology groups are Zariski cohomology.
General case: Z(n).
In general, over a perfect field formula_110, there is a nice description of formula_111 in terms of presheaves with transfer formula_112. There is a quasi-ismorphismformula_113henceformula_114which is found using splitting techniques along with a series of quasi-isomorphisms. The details are in lecture 15 of the Clay Math book.