Subject: Question : Ext group for non-projective resolutions ?
Date: Fri, 30 May 2003 13:55:17 -0700
From: Timo Hanke <thanke@math.ucsd.edu>
To: dmd1@lehigh.edu

I would like to ask the following question on the topology mailing list.

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Hello.

I'm using an object that could be named something like the "Ext-group
for non-projective resolutions" (see below for a definition) and am
wondering if anybody knows of an investigation of this object in the
literature or whether there is a common name and notation for it ? I was
unable to find much discussion of non-projective resolutions in the
standard references on homological algebra.

Definition :
Let M be a fixed module. Consider a resolution

  X: 0 -> K -> P -> A -> 0

of A, where P is not necessarily projective, and define

  Ext(X,M) := Hom(K,M)/Hom(P,M).

Of course, if P is projective then Ext(X,M)=Ext^1(A,M), but in general
Ext(X,M) is a proper subgroup of Ext^1(A,M). There are interesting
non-projective (!) cases where the groups are equal. The one I am
particularly interested in is the one of G-modules (G a finite group)
with $P$ a so-called "permutation lattice" and M "H^1-trivial". On the
other hand, if the groups are not equal, it is interesting what the
subgroups Ext(X,M) in Ext^1(A,M) actually are and what their relation is
for different X.

Are these things covered somewhere in the literature ?
Thank you very much for your help,

        Timo Hanke.