K-theory of Azumaya algebras over schemes

Abstract

Let $X$ be a connected, noetherian scheme and $\mathcal{A}$ be a sheaf of Azumaya algebras on $X$ which is a locally free ${\mathcal{O}}_X$-module of rank $a$. We show that the kernel and cokernel of $K_i(X)\to K_i(\mathcal{A})$ are torsion groups with exponent $a^m$ for some $m$ and any $i\ge 0$, when $X$ is regular or $X$ is of dimension $d$ with an ample sheaf (in this case $m\le d+1$). As a consequence, $K_{i}(X,\mathbb Z/m)\cong K_{i}(\mA,\mathbb Z/m)$, for any $m$ relatively prime to $a$.

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