Transfer functors on \(k\)-algebras (Q2570680)

The notion of a presheaf with transfer plays a crucial role in Voevodsky's construction of a triangulated category \(DM(k)\) of motives over a field \(k\) and a functor from the category of smooth schemes over \(k\) to \(DM(k)\). In this paper the author introduces the notion of a transfer functor \(F\) from normal algebras over a field \(k\). The first ingredient is the category \({\mathcal C}or_k\) whose objects are normal \(k\)-algebras of finite type over \(k\) and morphisms are \(\mathbb Z\)-linear combinations of elementary correspondences from an algebra \(A\) to an algebra \(B\). An elementary correspondence is a prime ideal \(P\) in \(A\otimes _kB\) such that \(P\cap B = 0\) and the inclusion \(B\subset A\otimes_k B/P\) is finite. If \(\text{char\,} k = 0\) then \({\mathcal C}or_k(A,B)\) is the group of finite correspondences from \(A\) to \(B\). When \(A\) and \(B\) are smooth then \({\mathcal C}or_k(A,B)\) coincides with the group \({\mathcal C}or_k(\text{Spec}\,B,\text{Spec\,}A)\) of finite correspondences considered by \textit{V. Voevodsky} [``Triangulated categories of motives over a field'', Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009)]. A transfer functor is an additive functor from \({\mathcal C}or_k\) to abelian groups. Among several examples of transfer functors, like the etale cohomology \(H^*_{\text{et}}(\text{Spec\,}A,{\mathcal F})\), where \(\mathcal F\) is an etale sheaf with transfers, and the Picard group, a remarkable one is given by a Hecke functor. This is defined as a contravariant additive functor \(M\) from the category \({\mathcal C}or_G\), \(G\) a group, to abelian groups. Here \({\mathcal C}or_G\) denotes the category of finite \(G\)-sets and equivariant multivalued functions. The interesting case is when \(G ={\mathcal G}al(\overline k/k)\): then \({\mathcal C}or_G\) is a subcategory of \({\mathcal C}or_k\) and the restriction of any transfer functor to \({\mathcal C}or_G\) is a Hecke functor. Conversely any Hecke functor \(M\) induces a transfer functor by the formula \(M(A)= M(L^s_A)\), where \(L_A\) is the integral closure of \(k\) in the algebra \(A\) and \(L^s_A\subset L_A\) is the separable closure of \(k\) in \(A\).