Weil Transfer of Algebraic Cycles

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 7→ R(Z) of the set of simple cycles Z ⊂ X extends in a natural way to a map Z(X) → 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 CH(X) → CH(R(X)), which, in its turn, produces a natural functor of the categories of Chow correspondences CV(L) → CV(F ). Restricting to the graded components, one has a map Z∗(X) → Zn·∗(R(X)), which produces a functor of the categories of degree 0 Chow correspondences CV(L) → CV(F ), a functor of the categories of the Grothendieck Chow-motives M(L) → M(F ), as well as functors of several other classical motivic categories.