Structure and cancellation theorems for graded Azumaya algebras

Abstract

Let R be a commutative ring with 1. A. Roy and R. Shridharan [7] have proved that if R is a semilocal ring then for Azumaya R-algebras A, B and C the “cancellation law” A ORB -A OR C 3 B = C holds. In this paper we prove cancellation theorems for Z/22-graded Azumaya algebras. In the course of proving cancellation theorems we develop some structure theorems for Z/2Z-graded Azumaya algebras over a connected semilocal ring (i.e. a semilocal ring with only idempotents 0 and 1). In what follows R will denote a commutative ring (trivially graded by Z/22) and 0 will stand for tensor product over R. All graded objects will be graded by Z/22. If S is a subset of a graded object, hS will denote the set of homogeneous elements of S and &Y = degree of x for x # 0 in hS. Our notations and definitions for graded algebras, graded modules and their homomorphisms, opposite algebras, graded separable algebras and graded Azumaya algebras are the same as in [8]. Also definitions for graded tensor product products and graded centres of graded algebras are the same as in [8], but we denote them by 6 and Z respectively. We always assume modules to be left unless otherwise specified.