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 ) → F ( X ) {\displaystyle 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 A 1 {\displaystyle \mathbb {A} ^{1}} -homotopy invariant if F ( X ) ≃ F ( X × A 1 ) {\displaystyle F(X)\simeq F(X\times \mathbb {A} ^{1})} for every X. For example, Chow groups as well as motivic cohomology groups form presheaves with transfers.
Finite correspondence
Let X , Y {\displaystyle X,Y} be algebraic schemes (i.e., separated and of finite type over a field) and suppose X {\displaystyle X} is smooth. Then an elementary correspondence is an irreducible closed subscheme W ⊂ X i × Y {\displaystyle W\subset X_{i}\times Y} , X i {\displaystyle X_{i}} some connected component of X, such that the projection Supp ( W ) → X i {\displaystyle \operatorname {Supp} (W)\to X_{i}} is finite and surjective. Let Cor ( X , Y ) {\displaystyle \operatorname {Cor} (X,Y)} be the free abelian group generated by elementary correspondences from X to Y; elements of Cor ( X , Y ) {\displaystyle \operatorname {Cor} (X,Y)} are then called finite correspondences. The category of finite correspondences, denoted by C o r {\displaystyle Cor} , is the category where the objects are smooth algebraic schemes over a field; where a Hom set is given as: Hom ( X , Y ) = Cor ( X , Y ) {\displaystyle \operatorname {Hom} (X,Y)=\operatorname {Cor} (X,Y)}
and where the composition is defined as in intersection theory: given elementary correspondences α {\displaystyle \alpha } from X {\displaystyle X} to Y {\displaystyle Y} and β {\displaystyle \beta } from Y {\displaystyle Y} to Z {\displaystyle Z} , their composition is:
β ∘ α = p 13 , ∗ ( p 12 ∗ α ⋅ p 23 ∗ β ) {\displaystyle \beta \circ \alpha =p_{{13},*}(p_{12}^{*}\alpha \cdot p_{23}^{*}\beta )}
where ⋅ {\displaystyle \cdot } denotes the intersection product and p 12 : X × Y × Z → X × Y {\displaystyle p_{12}:X\times Y\times Z\to X\times Y} , etc. Note that the category C o r {\displaystyle Cor} is an additive category since each Hom set Cor ( X , Y ) {\displaystyle \operatorname {Cor} (X,Y)} is an abelian group. This category contains the category Sm {\displaystyle {\textbf {Sm}}} of smooth algebraic schemes as a subcategory in the following sense: there is a faithful functor Sm → C o r {\displaystyle {\textbf {Sm}}\to Cor} that sends an object to itself and a morphism f : X → Y {\displaystyle f:X\to Y} to the graph of f {\displaystyle f} . With the product of schemes taken as the monoid operation, the category C o r {\displaystyle Cor} is a symmetric monoidal category.
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