 
Summary: JOURNAL OF ALGEBRA 121, 99116 (1989)
Crossed Products, Co homology,
and Equivariant Projective Representations
ELI ALJADEFF AND SHMUEL ROSSET
TelAviv Universiry, RamatAviv 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), andup to isomorphism of
algebrasdoes not depend on the choice of representing cocycle.
