Presheaves & sheaves with transfers
Projection formula and composition of correspondences
17A.11
Let \(f:X\rightarrow Y\) be a morphism of smooth schemes. Suppose given a cycle \(\mathscr{X}\) on \(X\), whose support is finite over \(Y\), and a cycle \(\mathscr{Y}\) on \(Y\) which is in good position for \(\mathscr{X}\). Then \(f_*\mathscr{X}\) and \(\mathscr{Y}\) intersect properly, and the projection formula holds:
\[ f_*(\mathscr{X}\cdot f^*\mathscr{Y})=f_*\mathscr{X}\cdot\mathscr{Y}. \]
This holds especially when computing compositions of correspondences, namely
\[ X\xrightarrow{\Gamma_f} Y\xrightarrow{W} Z \]
can be calculated as
\[ XZ\xrightarrow{\Gamma_f\times id_Z}XYZ\xrightarrow{p_{XZ}} XZ. \]
Some examples
1.10
(Excercise) \(Cor_k(k,X)\) is the group of zero-cycles in \(X\). If \(W\) is a finite correspondence from \(\mathbb{A}^1\) to \(X\) and \(s,t:k\rightarrow\mathbb{A}^1\) are \(k\)-points, then the zero-cycles
Presheaf with transfers
Remember we have defined a category \(Cor_k\) with objects being smooth schemes over \(k\) and morphisms being finite correspondences. A presheaf with transfers is a contravariant additive functor \(F:Cor_k\rightarrow Ab\), remembering that \(Cor_k\) is an additive category. The category of presheaves with transfers (over \(k\)) is denoted by \(PST\) (\(PST(k)\)).
So if \(F\) is a presheaf with transfers, there is a natural pairing
\[ Cor(X,Y)\otimes F(Y)\rightarrow F(X). \]
2.3
(Theorem) The category \(PST(k)\) is abelian and has enough injectives and projectives.
Examples
2.8
(Definition) If \(X\) is a smooth scheme over \(k\) we let \(\mathbb{Z}_{tr}(X)\) denote the presheaf with transfers represented by \(X\), or equivalantly \(\mathbb{Z}_{tr}(X)(U)=Cor_k(U,X)\). By Yoneda lemma it’s obvious that
\[ Hom_{PST}(\mathbb{Z}_{tr}(X), F)=F(X). \]
It follows that \(\mathbb{Z}_{tr}(X)\) is a projective object in \(PST(k)\).