K-theory of Azumaya algebras over schemes

TL;DR

This paper investigates the algebraic K-theory of Azumaya algebras over schemes, showing that the difference between the K-theory of the scheme and the algebra is torsion with bounded exponent under certain conditions.

Contribution

It establishes bounds on the torsion in the kernel and cokernel of the K-theory map for Azumaya algebras over schemes, extending understanding of their algebraic K-theory.

Findings

Kernel and cokernel are torsion groups with exponent dividing a power of a.

K-theory with coefficients modulo m agrees for the scheme and Azumaya algebra when m is coprime to a.

Results apply to regular schemes and schemes with ample sheaves, with explicit bounds on torsion.

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 $i\geq 0$, 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 $a$.