K-Theory of Azumaya Algebras over Schemes

Abstract

Let X be a connected, noetherian scheme and 𝒜 be a sheaf of Azumaya algebras on X, which is a locally free 𝒪 _X -module of rank a. We show that the kernel and cokernel of K _i (X) → K _i (𝒜) are torsion groups with exponent a ^m for some m and any i ≥ 0, when X is regular or X is of dimension d with an ample sheaf (in this case m ≤ d + 1). As a consequence, K _i (X, ℤ/m) ≅ K _i (𝒜, ℤ/m), for any m relatively prime to a.

Key Words:

• Azumaya algebra
• K-theory

2000 Mathematics Subject Classification:

• Primary 14F22
• Secondary 16K20

ACKNOWLEDGMENT

The first author acknowledges the support of EPSRC first grant scheme EP/D03695X/1 and Queen's University PR grant.

Notes

Communicated by V. A. Artamonov.