Giovanna Le Gros, Generalisations of Bass' Theorem P over commutative rings.

Abstract:

Perfect rings were introduced and characterised by Bass in his pivotal 1960 paper. In Theorem P of this paper, Bass gives both a homological and ring-theoretic characterisation of these rings, moreover finding a connection between approximation theory in the module category over the ring and the finitistic dimensions of a ring. In particular, for a commutative ring R, R is perfect (that is, every R-module has a projective cover) if and only if the big finitistic dimension of R is zero.

In this talk we will discuss some natural generalisations of this theory, in particular considering the rings over which the class of modules of projective dimension at most one is covering, and some partial results in this direction in the case of commutative rings. This study is related to Enochs' Conjecture, that is that a covering class is necessarily closed under direct limits, in the specific case of the class of modules of projective dimension at most one. Time permitting, we will also discuss some characterisations of 1-tilting cotorsion pairs which provide minimal approximations over commutative rings.

This talk is based on work in progress with Silvana Bazzoni.