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\(K\)-theory of the Weil transfer functor. (English) Zbl 0969.19003
Let \(F\) be a field and denote by \({\mathcal A} lg(F)\) the category of separable \(F\)-algebras. Denote by \({\mathcal V}ar(F)\) the category of all quasi-projective schemes over \(F\) and by \({\mathcal {PV}}ar(F)\) the full subcategory of smooth projective schemes. For \(X\in {\mathcal V} ar(F)\) and \(A\in {\mathcal A} lg(F)\), let \({\mathcal P}(X,A)\) be the exact category of left \(A\otimes_F\mathcal O_X\)-modules which are locally free \(\mathcal O_X\)-modules of finite rank and morphisms of \(A\otimes_F\mathcal O_X\)-modules. One denotes the \(K\)-groups by \(K_n(X,A)=K_n({\mathcal P}(X,A))\). One defines, after A. S. Merkurjev and I. A. Panin [“\(K\)-theory of algebraic tori and toric varieties”, \(K\)-Theory 12, No. 2, 101-143 (1997; Zbl 0882.19002)], the motivic category \({\mathcal C}= {\mathcal C}_F\) with objects the pairs \((X,A)\) with \(X\in {\mathcal {PV}}ar(F)\) and \(A\in {\mathcal A}lg(F)\), and morphisms \(\text{Mor}_{\mathcal C}((X,A),(Y,B))=K_0(X\times Y,A^{\text{op}}\otimes_FB)\). There are functors \(i:{\mathcal {PV}}ar(F)\rightarrow {\mathcal C}_F\) and \(j: {\mathcal A}lg(F)\rightarrow {\mathcal C}_F\), relating the \(K''\)’s (in the usual sense, defined via coherent, instead of locally free, modules) of schemes and algebras.
Let \(L/F\) be a Galois field extension with Galois group \(G=\text{Gal}(L/F)\) and \(n=[L:F]\). For an \(L\)-linear space \(V\) and an element \(\sigma \in G\) one writes \({}^{\sigma}V\) for the \(L\)-space \(V\) with \(L\)-action given by \(x\cdot v=\sigma^{-1}(x)\cdot v\) for \(x\in L\), \(v\in V\). One also writes \(V^{\otimes G}=\bigotimes_{\sigma\in G}^{\sigma}V\), where the tensor product is taken over \(L\). The group \(G\) acts on \(V^{\otimes G}\) by permutations of tensor factors. One defines the Weil transfer of the \(L\)-linear space \(V\) as the \(F\)-linear space \(RV=R_{L/F}V=(V^ {\otimes G})^G\). Let \(\pi:X\rightarrow\text{Spec}(L)\) be a scheme over \(L\). Then for \(\sigma\in G\) one may endow \(X\) with another structure over \(L\) by \(\sigma\circ\pi:X\rightarrow\text{Spec}(L)\), thus defining the \(L\)-scheme \({}^{\sigma}X\). Define \(X^{\times G}=\prod_ {\sigma\in G}^{\sigma}X\), then \(X^{\times G}\) comes equipped with a \(G\)-scheme structure. One may now define the Weil transfer of the quasi-projective \(L\)-scheme \(X\) as the quasi-projective \(F\)-scheme \(RX=R_{L/F}X=(X^{\times G})/G\). For an \(\mathcal O_X\)-module \(M\) one defines its Weil transfer as the \(\mathcal O_{RX}\)-module \(RM=R_{L/F}M= (p_*M^{\otimes G})^G\), where \(p:X^{\times G}\rightarrow RX\) is the canonical projection. One has the well-known property that for \(X\in {\mathcal V}ar(L)\) and an arbitrary \(F\)-scheme \(Y\), \(\text{Mor}_F(Y,RX)=\text{Mor}_L(Y_L,X)\).
The final step is to define a Weil transfer functor \(R:{\mathcal C}_L\rightarrow {\mathcal C}_F\). This can be done by defining \(R(X,A)=(RX,RA)\), and then verifying that this implies the suitable properties on morphisms.
The paper closes with several examples: (i) A separable \(L\)-algebra \(A\) presented as a product of simple algebras; (ii) the more specific case of an \(F\)-algebra \(A\); (iii) the case where \(n=[L:F]\) equals a prime \(p\), i.e., \(G\) has no non-trivial subgroups; (iv) the effect of Weil transfer on the isomorphisms \(K_n(X)\simeq K_n(A)\) for a scheme \(X\) and an algebra \(A\). Here the functors \(i\) and \(j\) turn up again.

MSC:
19E08 \(K\)-theory of schemes
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19D99 Higher algebraic \(K\)-theory
19A99 Grothendieck groups and \(K_0\)
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