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Cohomological theory of presheaves with transfers. (English) Zbl 1019.14010
Voevodsky, Vladimir et al., Cycles, transfers, and motivic homology theories. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 143, 87-137 (2000).
Let \(k\) be a field and \(Sm/k\) the category of smooth schemes over \(k\).
In this paper, the author studies contravariant functors from the category \(Sm/k\) to additive categories equipped with so-called transfer maps. More precisely, he studies functors \(F:(Sm/k)^{op}\to A\), together with a family of morphisms \(\Phi_{X/S}(Z):F(X)\to F(S)\) given for any smooth relative curve \(X@>f>>S\) over a smooth \(k\)-scheme \(S\) and any relative divisor \(Z\) on \(Z\) over \(S\) which is finite over \(S\). If those transfer maps \(\Phi_{X/S}(F)\) satisfy some natural functorial properties, then such a collection of data is called a “pretheory” over \(S\). A pretheory \(F\) is called homotopy invariant if, for any smooth scheme \(X\) over \(S\), the equality \(F(X\times \mathbb{A}^1)= F(X)\) holds.
The main results of this paper concern the construction and behavior of pretheories, their associated homotopy invariant pretheories, and their applications to other (co-)homology theories related to the (conjectural) motivic cohomology theory in the sense of A. Beilinson.
For the entire collection see [Zbl 1021.14006].

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
18C10 Theories (e.g., algebraic theories), structure, and semantics
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
55N35 Other homology theories in algebraic topology
14F42 Motivic cohomology; motivic homotopy theory