Commutative Algebra Seminar

If R is a local ring of dimension d and x = x_1, ..., x_d is a maximal regular sequence, then R/(x) is an R-module of finite length and finite projective dimension. There exist at least three open questions which stipulate that such quotients of regular sequences are the "simplest" or "smallest" modules amongst all modules of finite length and finite projective dimension. I will talk about some evidence supporting this philosophy. I will also sketch a proof from a joint work with Josh Pollitz where we solve the Loewy length version of this problem over strict Cohen-Macaulay rings.