Silting theory in triangulated categories with coproducts

By Pedro Nicolás, Manuel Saor'in, A. Zvonareva. Journal of Pure and Applied Algebra, 2015

66 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/713a969dffa70d658b945287e92de740cd34d501

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Tilting–Cotilting Correspondence (2017, background): “This puts several recent generalizations of the Brenner–Butler correspondence (see for instance [13, 15, 56, 74, 81]) into a unified framework.”