From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: homological dimension of a module
Date: 24 Nov 1997 14:45:15 GMT
In article <347689C0.66@univ-rennes1.fr>, wrote:
>Can you give me some examples of concrete computation of homological
>dimension of a module and projective dimension. for example, Z/2Z as a
>Z/4Z module.
I'm not sure I've heard of the concept of "homological dimension of a
module"; more precisely, if I heard it, I would translate it into
"projective dimension". The latter is simply the length of the
shortest projective resolution of the module.
You're not likely to get good examples of the type you describe, since
if M is a Z[G]-module, it can only have finite projective dimension
if that dimension is at most 1, that is, M is either projective or the
quotient of a projective module by a projective submodule. Equivalently,
M is cohomologically trivial. This result is due to Rim.
Certainly if G=Z/4Z and M=Z/2Z, we have H^i(G,M) = Z/2Z for all i,
so this module is not cohomologically trivial; M has infinite projective
dimension.
dave