Commutative Algebra Seminar

Title

On liftings of modules of finite projective dimension

Abstract

Suppose R -> S is a surjective map of local Noetherian rings. In this talk I will discuss a notion of lifting S-modules to R-modules along this map. Classically, we say an S-module M lifts to an R-module M’ if M’ is isomorphic to M upon extending scalars via R -> S, and Tor_i(M’, R) = 0 for i > 0. It turns out, it is interesting to consider a much weaker and more geometric notion of lifting modules. Instead of asking the higher Tors to vanish, we require that the lift M’ of M is of the “correct codimension,” that is, dim R - dim M’ = dim S - dim M.

This is mostly joint work with Andrew Soto Levins and partly ongoing joint work with Kesavan Mohana Sundaram, Ben Katz, and Ryan Watson.