Torsion pairs in silting theory

TL;DR

This paper investigates the conditions under which the heart of (co)silting t-structures in compactly generated triangulated categories is a Grothendieck category, linking purity assumptions and definability to categorical properties.

Contribution

It establishes a characterization of when the heart of a (co)silting t-structure is Grothendieck, relating purity, definability, and algebraic conditions in triangulated categories.

Findings

Heart of (co)silting t-structure is Grothendieck iff purity condition holds.

In algebraic categories, all nondegenerate compactly generated t-structures have Grothendieck hearts.

Conditions connect the structure of the t-structure to purity and definability properties.

Abstract

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we further assume our triangulated category to be algebraic, it follows that the heart of any nondegenerate compactly generated t-structure is a Grothendieck category.