 
Summary: Arch. Math., Vol. 62, 401407 (1994) 0003889X/94/62050401 $ 2.90/0
9 1994 Birkh/iuser Verlag, Basel
On the global dimension of multiplicative Weyl algebras
By
ELI ALJADEFF and YUVAL GINOSAR
0. Introduction. Let k be any field and let k [x~ 1..... x+ i] be the Laurent polynomial
ring in n indeterminates. It is isomorphic to the group ring kG where G is the free abelian
group of rank n. Given an element ~ ~ H2(G, k*) (G acting trivially on k*) we can
construct the twisted group ring k" G. As a kvector space, it is isomorphic to the group
ring kG and ff {u~},~ is a basis, we define the multiplication by the rule
uxi uxj =f(xl, xi)u~x j where f: G x G ~ k* is a 2cocycle representing ~. An element
~ H 2(G, k*) can be interpreted as a system of (~)commutators
~i~ = u~,u~j.u~l, u~ 1 = f (xi, xfl f 1 (xj, xl).
We will abuse the notation and simple write
~ij = xlxjx71x; ~
In this paper we are concerned with the global dimension of these algebras. Recall from
[2] two basic results on the global dimension of twisted group rings:
(1) gl.dim k" G < gl.dim kG (= n)
(2) (monotonicity): if H is a subgroup of G then
gl.dim k~H < gl.dim k~G, where ~ = res~a.
