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Weil transfer of algebraic cycles. (English) Zbl 1047.14004

Summary: Let \(L/F\) be a finite separable field extension of degree \(n\), \(X\) a smooth quasi-projective \(L\)-scheme, and \(R(X)\) the Weil transfer of \(X\) with respect to \(L/F\). The map \(Z\to R(Z\)) of the set of simple cycles \(Z\subset X\) extends in a natural way to a map \(Z(X)\to Z(R(X))\) on the whole group of algebraic cycles \(Z(X)\). This map factors through the rational equivalence of cycles and induces this way a map of the Chow groups \(\text{CH}(X)\to \text{CH}(R(X))\), which, in its turn, produces a natural functor of the categories of Chow correspondences \(\mathcal{CV}(L)\to \mathcal{CV}(F)\). Restricting to the graded components, one has a map \(Z^\ast (X)\to Z_n^\ast (R(X))\), which produces a functor of the categories of degree 0 Chow correspondences \(\mathcal{CV}^0(L)\to\mathcal{CV}^0(F)\), a functor of the categories of the Grothendieck-Chow motives \(\mathcal M(L)\to\mathcal M(F)\), as well as functors of several other classical motivic categories.

MSC:

14C25 Algebraic cycles
14C15 (Equivariant) Chow groups and rings; motives
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