Hearts of t-structures which are Grothendieck or module categories (1409.6639v1)

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.