On a theorem of Hazrat and Hoobler (Q2839345)

Let \(X\) be a \(d\)-dimensional scheme of finite type over a field \(k\) and let \({\mathcal A}\) be an Azumaya algebra on \(X\) of constant degree \(n\). Let \(K_i(X;{\mathcal A})\) denote the \(K\)-theory of left \({\mathcal A}\)-modules which are locally free coherent \({\mathcal O}_X\)-modules of finite rank and \(G_j(X)\) the \(K\)-theory of left \({\mathcal A}\)-modules which are coherent \({\mathcal O}_X\)-modules. This paper contains the proof of the following resultNEWLINENEWLINE Theorem 1. Let \(B_{{\mathcal A}}: G_i(X)\to G_i(X;{\mathcal A})\) and \(B^K_{{\mathcal A}}: K_i(X)\to K_i(X;{\mathcal A})\) be the homomorphisms induced by the functor \({\mathcal F}\mapsto{\mathcal A}\otimes_{{\mathcal O}_X}{\mathcal F}\). Then the kernel and cokernel of \(B_{{\mathcal A}}\) are torsion groups of exponents dividing \(n^{2d+2}\). If \(X\) is regular the same result holds for \(B^K_{{\mathcal A}}\).NEWLINENEWLINE As a corollary one gets that the base extension homomorphism NEWLINE\[NEWLINEB_{{\mathcal A}}: G_*(X)\otimes_{{\mathbb{Z}}}\mathbb{Z}[1/n]\to G_*(X;{\mathcal A})\otimes_{{\mathbb{Z}}}\mathbb{Z}[1/n]NEWLINE\]NEWLINE is an isomorphism, when \({\mathcal A}\) is an Azumaya algebra of constant degree \(n\) over a scheme \(X\) of finite type over \(k\).NEWLINENEWLINE Theorem 1 extends similar results proved by \textit{R. Hazrat} and \textit{R. T. Hoobler} [Commun. Algebra 41, No. 4, 1268--1277 (2013; Zbl 1300.14023)], where the exponents are \(n^{2d(d+ 1)+2}\) for the kernel of \(B_{{\mathcal A}}\) and \(n^{4d+2}\) for the cokernel. For \(B_{{\mathcal A}}\) Hazrat and Hoobler gave the same result as in Theorem 1, under the assumption that \(X\) has an ample line bundle.