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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 $$K_0$$
##### Keywords:
Weil transfer; motivic category
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