3. Cohomological Theory of Presheaves with Transfers

Abstract

Letkbe a field and$Sm/k$be the category of smooth schemes overk.In this paper we study contravariant functors from the category$Sm/k$to additive categories equipped withtransfer maps.More precisely we consider contravariant functors$F:{(Sm/k)^{op}} \to A$together with a family of morphisms${\phi _{{X {\left/ {} \right.} S}}}(Z):F(X) \to F(S)$given for any smooth curve$X \to S$ over a smooth schemeSoverkand a relative divisorZonXoverSwhich is finite overS.If these maps satisfy some natural properties (see definition 3.1) such a collection of data is called apretheory over k.Some examples of pretheories are