# Seminario MALGA Padova Verona - Moduli Algebre Anelli: “When is the heart of a t-structure a Grothendieck category?”

## Martedì 17 Ottobre 2017, ore 15:00 - Sala Riunioni VII piano - Simone Virili

ARGOMENTI: Seminars

Martedì 17 Ottobre 2017 alle ore 15:00 in Sala Riunioni VII piano, Simone Virili (University of Murcia) terrà un seminario nell'ambito del Seminario Padova-Verona MALGA - Moduli Algebre Anelli dal titolo “When is the heart of a t-structure a Grothendieck category?”.

Abstract
Let $\mathcal{D}$ be a triangulated category endowed with a $t$-structure $t = (\mathcal{U}, \Sigma \mathcal{V})$ and denote by $\mathcal{H} := \mathcal{U} \cap \Sigma \mathcal{V}$ its heart. In this seminar I will report on some recent results, obtained in collaboration with Manuel Saorín and Jan Šťovíček, partially answering the following well-known question:
Under what conditions on $\mathcal{D}$ and $t$ can we say that $\mathcal{H}$ is a Grothendieck category?
We will concentrate on the case when $\mathcal{D}$ is the base of a stable derivator.
In this generality we will see that, under very natural hypotheses on $t$, direct limits in $\mathcal{H}$ are exact. Furthermore, when $\mathcal{D} = ho(\mathcal{G})$ is the homotopy category of a suitable model structure on a Grothendieck category $\mathcal{G}$, $\mathcal{H}$ has also a set of generators. This last case includes derived categories of Grothendieck categories and of small dg categories.
Joint Work with Manuel Saorín and Jan Šťovíček.