Hearts of t-structures which are Grothendieck or module categories

TL;DR

This thesis investigates conditions under which the heart of a t-structure in a triangulated category is a Grothendieck or module category, focusing on specific examples like torsion pairs and compactly generated t-structures.

Contribution

It provides new criteria linking torsion pairs and support filtrations to the heart being Grothendieck or module categories, especially in derived categories of rings.

Findings

AB5 condition implies closure under direct limits for torsion-free class

Characterization of hereditary torsion pairs with module category hearts

Identification of hearts as Grothendieck or module categories in derived categories of rings

Abstract

This thesis deals with the general problem of determining when the heart $\mathcal{H}$ of a t-structure in a triangulated category $\mathcal{D}$ is a Grothendieck or a module category. As preliminaries, we study Grothendieck conditions AB3-AB5 for $\mathcal{H}$ in a very general setting. We then concentrate on two familiar examples of smashing t-structures. First, we consider that $\mathcal{D}=\mathcal{D}(\mathcal{G})$ is the (unbounded) derived category of a Grothendieck category $\mathcal{G}$ and that the t-structure is the one associated to a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal{G}$, usually known as Happel-Reiten-Smal$\emptyset$ t-structure. In the second example studied, we assume that $\mathcal{D}=\mathcal{D}(R)$ is the derived category of a commutative Noetherian ring $R$ and that the t-structure is compactly generated. On what concern the Happel-Reiten-Smal$\emptyset$ example, we show that if $\mathcal{H}=\mathcal{H}_\mathbf{t}$ is AB5, then $\mathcal{F}$ is closed under taking direct limits in $\mathcal{G}$. Moreover, the converse is true, even implying that $\mathcal{H}_\mathbf{t}$ is a Grothendieck category, for a wide class of torsion pairs in $\mathcal{G}$ which includes the hereditary, tilting and cotilting ones. When $\mathcal{G}=R-\text{Mod}$ is a module category, we are able to identify the hereditary torsion pairs $\mathbf{t}$ in $R-\text{Mod}$ for which $\mathcal{H}_\mathbf{t}$ is a module category. When $R$ is a commutative noetherian ring, we show that all compactly generated t-structures in $\mathcal{D}(R)$ whose associated filtration by supports is left bounded have a heart $\mathcal{H}$ which is a Grothendieck category. This is used to identify all compactly generated t-structures in $\mathcal{D}(R)$ whose heart is a module category.