A splitting principle and the algebraic K-theory of some homogeneous varieties
For a scheme \(X\) and a sheaf \({\mathcal A}\) of algebras let \({\mathcal M}(X,{\mathcal A})\) (resp.\ \({\mathcal M}(X,{\mathcal A}^{\text{op}})\)) denote the exact category of coherent left (resp.\ right) \({\mathcal A}\)-modules, and write \(G_ i(X,{\mathcal A})\) for the associated \(K\)-groups. Similarly, replacing coherent \({\mathcal A}\)-modules with locally free \({\mathcal O}_ X\)-modules with \({\mathcal A}\)-action, one has the \(K\)-groups \(K_ i(X,{\mathcal A})\). If \(X\) is faithfully flat of finite type over \(\mathbb{Z}\), and if \({\mathcal H}\) is a geometrically connected split reductive faithfully flat group scheme over \(\mathbb{Z}\) with right action on \(X\), then a projective \({\mathbb{Z}}\)- representation \(\rho_ i\), \(i=1,\ldots, n\), of \(\mathcal H\) and a 1-cocycle \(c\in H^ 1_{\acute{e}t}(S,{\mathcal H})\), \(S\) a finite-dimensional noetherian scheme, give rise to a twisted form \(X_ c\) of \(X\) and Azumaya algebras \({\mathcal A}_ c(\rho_ i)\). The splitting principle says that under suitable conditions, there is an isomorphism \(\displaystyle{\phi_ S:\coprod_{i=1}^ nG_ *(S,A_ c(\rho_ i))[-d(i)]{\buildrel\sim\over\longrightarrow}G_ *(X_ c)}\). Next, let \(\tilde{H}\) be a geometrically connected split reductive group scheme faithfully flat over \(\mathbb{Z}\) with free abelian \(\pi_ 1(\tilde{H}_{{\mathbf C},an})\). Fix a maximal torus \(\tilde{T}\) and a parabolic group subscheme \(\tilde{P}\) containing \(\tilde{T}\). Let \(\tilde{Z}\) be the center of \(\tilde{H}\), and write \(H=\tilde{H}/\tilde{Z}\), \(P=\tilde{P}/\tilde{Z}\), \({\mathcal F}=\tilde{H}/\tilde{P}=H/P\). From work of Atiyah, Hirzebruch, Grothendieck and Serre, it follows that one has a homomorphism \(\displaystyle{ \alpha_ S(\tilde{H},\tilde{P}):R(\tilde{P})\otimes G_ *(S)\longrightarrow G_ *({\mathcal F}_ S)}\), for any scheme \(S\), where \({\mathcal F}_ S={\mathcal F}\times S\) and \(R(\tilde{P})\) is the Grothendieck ring of the exact category of \(\mathbb{Z}\)-representations of \(\tilde{P}\). If now \(\{a_ i\}_{i\in I}\) is a free basis of the \(R(\tilde{H})\)-module \(R(\tilde{P})\), then it is shown that for finite-dimensional noetherian \(S\), \(\displaystyle{\coprod_{i\in I}G_ *(S){\buildrel\sim\over\longrightarrow}G_ *({\mathcal F}_ S)}\). With notation as above, an element \(c\in H^ 1_{\acute{e}t}(S,\underline{H})\) gives rise to a twisted form \({\mathcal F}_ c\) of \({\mathcal F}\). For any character \(\chi\in X(\tilde{Z})\) and representation \((W_{\chi},\mu_{\chi})\) over \({\mathbb{Z}}\) of \(\tilde{P}\), let \({\mathcal A}_ c(\chi)=\text{End}_ c(W_{\chi})\). Then, if the classes of representations \((W_{\chi_ i},\mu_{\chi_ i})\), \(i\in I\), give a free basis of the \(R(\tilde{H})\)-module \(R(\tilde{P})\) one gets an isomorphism \(\displaystyle{\coprod_{i\in I}G_ *(S,{\mathcal A}_ c(\chi_ i)) {\buildrel\sim\over\longrightarrow}G_ *({\mathcal F}_ c)}\) \((\dagger)\). Next a result of M. Levine which says that for a geometrically connected faithfully flat split reductive rank \(n\) group scheme \(H\) over \({\mathbb{Z}}\) one can give an explicit isomorphism \[ \coprod_{1\leq i_ 1<\cdots< i_ r\leq n}G_ *(S)[-r]{\buildrel\sim\over\longrightarrow}G_ *(H_ S), \] is extended to the case of an interior form of such a group \(H\) with free fundamental group; similarly for torsors for such groups. The paper closes with some comments on \((\dagger)\). As a corollary the following result is stated: For a \({\mathbb{Z}}\)-representation \(\rho\) of \(\tilde{H}\) as above, the order of the class \([{\mathcal A}_ c(\rho)]\) in \(H^ 2_{\acute{e}t}(S,\underline{\mathbf G}_ m)\) divides \(\#(X(\tilde{Z}))\). In particular, if \(\tilde{Z}=\{1\}\), then \(\tilde{H}=H\), \(\tilde{P}=P\) and the canonical homomorphism \(\displaystyle{R(\tilde{P})\otimes_{R(\tilde{H})}G_ *(S)\longrightarrow G_ *({\mathcal F}_ c)}\) is an isomorphism for each \(c\in H^ 1_{\acute{e}t}(S,\underline{H})\) and each parabolic \(\tilde{P}\subseteq\tilde{H}\). Several examples are given, some of the most interesting in terms of the \(K_ *\) rather than the \(G_ *\) functor, in the introduction.
scientific article; zbMATH DE number 404565