\(K\)-theory of algebraic tori and toric varieties
Let \(F\) be a field. Write \({\mathcal V}ar(F)\) for the category of all varieties over \(F\), \({\mathcal {PV}}ar(F)\) for the full subcategory of smooth projective varieties over \(F\), and \({\mathcal A}lg (F)\) for the category of separable \(F\)-algebras. For \(X\in {\mathcal V}ar(F)\) and \(A\in {\mathcal A}lg (F)\) one has the exact category \({\mathcal P}(X,A)\) 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. Similarly one has the abelian category \({\mathcal M}(X,A)\) of left \(A\otimes_F{\mathcal O}_X\)-modules which are coherent \({\mathcal O}_X\)-modules and morphisms of \(A\otimes_F{\mathcal O}_X\)-modules. One can define the \(K\)-groups \(K_n(X,A):=K_n ({\mathcal P}(X,A))\) and \(K'_n(X,A):=K_n ({\mathcal M}(X,A))\), with the usual properties such as inverse image, direct image, base change, products, projection formula, pairings. For \(X\in {\mathcal {PV}}ar(F)\) and \(A\in {\mathcal A}lg(F)\) consider the pairs \((X,A)\) with morphisms given by \(\text{Mor}((X,A),(Y,B))=K_0(X\times Y,A^{\text{op}}\otimes_FB)\), where \(A^{\text{op}}\) is the opposite \(F\)-algebra. One obtains the motivic category \({\mathcal C}= {\mathcal C}_F\) with involution and tensor structure. One has functors \(i:{\mathcal {PV}}ar(F)\to {\mathcal C}\) taking \(X\) to \((X,F)\), and \(j:{\mathcal A}lg(F)\to{\mathcal C}\) taking \(A\) to \((\text{Spec}(F),A)\). For any \(V\in{\mathcal{PV}}ar(F)\) the composition \({\mathcal{PV}}ar(F)@>j>>{\mathcal C}@>{\mathbf K}_n^V>>{\mathcal A}b\) coincides with the functor \(X\mapsto K'_n(V\times X)\), and for any \(V\in{\mathcal V}ar(F)\) the composition \({\mathcal A}lg(F)@>j>>{\mathcal C}@>{\mathbf K}_n^V>>{\mathcal A}b\) coincides with \(A\mapsto K'_n(V,A)\). Let \(S\) be a reductive algebraic group over \(F\) and let \(\pi:U\rightarrow S\) be an \(S\)-torsor over the smooth projective variety \(X\). Assume there is an \(S\)-equivariant open embedding of \(U\) into an affine space on which \(S\) acts linearly. Now the main theorem of the paper says: There exists a separable \(F\)-algebra \(A\) and morphisms \(u:X\rightarrow A\) and \(v:A\rightarrow X\) in \({\mathcal C}\) such that \(v\circ u=\text{Id}_X\). In other words, \(X\) is a `direct summand' of \(A\) in \({\mathcal C}\). Thus the groups \(K_n(X)\) become direct summands of the \(K_n(A)\). For an algebraic torus \(T\) defined over \(F\) one defines a toric \(T\)-variety \(X\) as a normal geometrically irreducible variety \(X\) defined over \(F\) if there is an open orbit being a principal homogeneous space over \(T\). \(X\) is called a toric \(T\)-model if the open orbit has a rational point. An application of the main theorem to this situation gives the result: For a smooth projective toric model of an algebraic torus defined over \(F\) there exists a commutative separable \(F\)-algebra \(A\) and elements \(u,v\in K_0(X,A)\) such that \(X@>u>> A@>v>> X\) is \(\text{Id}_X\) in \({\mathcal C}\) and \(u\) is represented by a class in \(\text{Pic}(X\otimes_FA)\). Several applications of this result are deduced, e.g., \(CH^2(X)\) is finitely generated, and \((d-1)!A_0(X)=0\) (\(d\) is the dimension of \(X\)), etc. Using a splitting principle for morphisms in \({\mathcal C}\) one can deduce the general result: For a smooth projective toric variety \(X\) over \(F\) there exists a separable \(F\)-algebra \(A\) and morphisms \(u:X\rightarrow A\) and \(v:A\rightarrow X\) in \({\mathcal C}\) such that \(v\circ u=\text{Id}_X\). As a corollary one finds that \(K_0(X)\) is a free abelian group of finite rank and \(K_0(X_{sep})\) is a direct summand of a finitely generated permutation \(\text{Gal}(F_{sep}/F)\)-module. Furthermore, the group \(CH^2(X)\) is finitely generated. For an algebraic torus \(T\) one can describe the group \(K_0(T)\) in terms of generators and relations. Let \(L/F\) be a finite splitting field extension of \(T\). \({\mathcal A}(T)\) will be the abelian group with generators the isomorphism classes \(\langle P\rangle\) of invertible sheaves \(P\in\text{Pic}(T_E)\) for all intermediate extensions \(F\subset E\subset L\), and relations are given by \(\langle P\otimes_{E_1}E_2\rangle= (E_2:E_1)\cdot\langle P\rangle\) for any \(F\)-homomorphism \(\phi:E_1\rightarrow E_2\) of subfields in \(L\) and any \(P\in\text{Pic}(T_E)\). Then \({\mathcal A}(T)\) is finitely generated. One has a homomorphism \(\alpha:{\mathcal A}(T)\rightarrow K_0(T)\), \(\langle P\rangle\mapsto(i_{E/F})_*[P]\), \(P\in\text{Pic}(T_E)\), with \(i_{E/F}:T_E\rightarrow T\) the natural morphism. The following theorem is proved: \(\alpha\) is an isomorphism. Several corollaries ensue, e.g., \(K_0(T)^{(1)}\) is finite (where \(K_0(T)={\mathbb{Z}}\cdot 1\oplus K_0(T)^{(1)}\)), \(CH^2(T)\) is finite.
scientific article; zbMATH DE number 1062412