Fitting Ideals and Finite Projective Dimension

Throughout we let (T,m, k) denote a commutative Noetherian local ring with maximal ideal m and residue field k. We let I ⊆ T be an ideal generated by a regular sequence of length c and set R := T/I. In the important paper [A], Avramov addresses the following question. Given a finitely generated R-module M , when does M have finite projective dimension over a ring of the form T/J , where J is generated by part (or all) of a set of minimal generators for I? The paper [A] gives a fairly complete answer to this question that is expressed in terms of the geometry of varieties in affine space defined by annihilators of certain graded modules derived from resolutions over R. In an attempt to understand these ideas more fully, we became interested in the idea that one might answer the question at hand by using data about M (or its syzygies) coming from T , in particular, information gleaned from various Fitting ideals defined over T . The following theorem from section two is one of our main results. We use FittT (M) to denote the Fitting ideal of M .