Tuesday, February 15, 14:00
Giovanna Le Gros, Generalisations of Bass' Theorem P over commutative rings.
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.