Summary: JOURNAL OF ALGEBRA 121, 99-116 (1989)
Crossed Products, Co homology,
and Equivariant Projective Representations
ELI ALJADEFF AND SHMUEL ROSSET
Tel-Aviv Universiry, Ramat-Aviv 69978, Israel
Communicated by Nathan Jacobson
Received December 1, 1986
Let k be a commutative ring and K a commutative k algebra. If G is a
group acting on K via a homomorphism t: G + Aut, (K), which may (in
general) have a nontrivial kernel, then the multiplicative group K* of
invertible elements of K is a G module. The elements of the "galois"
cohomology group H'(G, K*) give rise to the well known Crossed Product
Construction (see [ 11). It is defined as follows. Let a EH2(G, K*) and let
f: G x G + K* be a cocycle representing a, i.e., a = [f 1. The crossed
product, given t and a, is a k algebra, denoted by K;G. As left K module it
is LIdEGKu,, while the product is defined by the rule
(XU,)(yu,)=xa(y)f(a,r)u,, (x,y EK,6,r EG).
It is easily verified that this is an associative k algebra, in fact it is even a
K" algebra (where p is the fixed ring), and-up to isomorphism of
algebras-does not depend on the choice of representing cocycle.