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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].

##### MSC:
 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