Silting Theory in triangulated categories with coproducts

TL;DR

This paper develops a comprehensive theory of silting and tilting objects in triangulated categories with coproducts, linking them to t-structures, and extends tilting theory to general abelian categories, with applications to derived equivalences.

Contribution

It introduces noncompact silting and tilting sets in triangulated categories, establishing bijections with t-structures and extending tilting theory to broad abelian contexts.

Findings

Equivalence between partial silting sets and certain t-structures.

Description of t-structure aisles as homotopy colimits.

Extension of tilting theory to general AB3 abelian categories.

Abstract

We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection with t-structures generated by their co-heart whose heart has a generator, and in case D is compactly generated, this bijection restricts to one between equivalence classes of self-small partially silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products. We describe the objects in the aisle of the t-structure associated to a partial silting set T as the Milnor (aka homotopy) colimit of sequences of morphisms with succesive cones in Sum(T)[n]. We use this fact to develop a theory of tilting objects in very general AB3 abelian categories, a setting and its dual on which we show the validity of several well-known results of tilting and cotilting theory of modules. Finally, we show that if T is a bounded tilting set in a compactly generated algebraic triangulated category D and H is the heart of the associated t-structure, then the inclusion of H in D extends to a triangulated equivalence between the derived category D(H) of H and the ambient triangulated category D which restricts to bounded levels.