Abstract

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.

Abstract

We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a ‘nice’ set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a weighted noncommutative regular projective curve of genus 0.