Seminar (Ring Theory)

Giovanna Le Gros (Università di Padova) delivered the talk:

Minimal approximations and 1-tilting cotorsion pairs over commutative rings


Minimal approximations of modules, or covers and envelopes of modules, were introduced as a tool to approximate modules by classes of modules which are more manageable. For a class C of R-modules, the aim is to characterise the rings over which every module has a C-cover or C-envelope. Moreover A-precovers and B-preenvelopes are strongly related to the notion of a cotorsion pair (A,B).

In this talk we are interested in the particular case that (P_1,B) is the cotorsion pair generated by the modules of projective dimension at most one (denoted P_1) over commutative rings. More precisely, we investigate over which rings these cotorsion pairs admit covers or envelopes. Furthermore, we interested in Enochs’ Conjecture in this setting, that is if P_1 is covering necessarily implies that it is closed under direct limits. The investigation of the cotorsion pair (P_1,B) splits into two cases: when the cotorsion pair is of finite type and when it is not. In this talk I will outline some results for the case that the cotorsion pair is of finite type, where we consider more generally a 1-tilting cotorsion pair over a commutative ring.