K-theory of Azumaya algebras over schemes

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 $i\geq 0$, when $X$ is regular or $X$ is of dimension $d$ with an ample sheaf (in this case $m\leq d+1$). As a consequence, $K_{i}(X,\mathbb Z/m)\cong K_{i}(\mA,\mathbb Z/m)$, for any $m$ relatively prime to $a$.

💡 Research Summary

The paper investigates the relationship between the algebraic K‑theory of a connected Noetherian scheme X and that of an Azumaya algebra 𝔄 defined over X. The algebra 𝔄 is assumed to be a locally free 𝒪_X‑module of constant rank a. The central object of study is the natural homomorphism
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