Jorge Vitoria (City U London), TTF-triples in silting theory.

Abstract:

Subcategories of abelian or triangulated categories which are simultaneously torsion and torsion-free (for suitable torsion pairs) are of particular interest in representation theory. A classical theorem of Jans states that such TTF-classes in the category of modules over a ring R correspond bijectively to the set of idempotent ideals of R. In the triangulated setting, it is well-known that the triangulated TTF-classes in a triangulated category T correspond bijectively to recollements of T (up to equivalence).

We consider TTF-classes which are not triangulated but just cosuspended, i.e., subcategories V of T for which there is a (nondegenerate) t-structure (U,V) and a co-t-structure (V,W). Examples of such TTF-triples (U,V,W) arise from compactly generated t-structures or from certain cosilting t-structures in derived module categories. In this talk, we show that in a compactly generated triangulated category, a large class of these TTF-triples can be parametrised by pure-injective silting objects. Moreover, it turns out that the heart of such a cosilting t-structure is a Grothendieck category and, as a consequence, nondegenerate compactly generated t-structures have Grothendieck hearts.