Realisation functors in tilting theory

Abstract

Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras.

Discreteness of silting objects and t-structures in triangulated categories

“…Proof (b) was shown in Remark . By (ST2), is a silting object in the sense of [, Definition 4.1]. Thus (c) follows from [, Proposition 4.3].…”

“…Let us prove (a). By [, Proposition 4.3], is projective in . Let .…”

“…Later such a correspondence was studied in the setting of finite-dimensional algebras in [50], in the setting of non-positive differential graded (dg) algebras with finite-dimensional total cohomology in [67] and in the setting of homologically smooth non-positive dg algebras with finite-dimensional zeroth cohomology in a forthcoming paper by Keller and Nicolás. In recent years the connection between silting theory and the theory of t-structures (and co-t-structures) has received much attention [4,7,14,16,38,55,62], and silting theory has been playing an increasingly important role in the study of triangulated categories of algebraic origin, due to such a connection as well as its connection to cluster-tilting theory [6,24,25,34,41] and in a forthcoming paper by Keller and Nicolás, to stability spaces [20,64] and to the theory of universal localisation of rings [8,54].…”

Partial silting objects and smashing subcategories

“…In order to prove this result, we build on some recent work on (partial) silting objects in triangulated categories with coproducts (see [36,37]). We show that partial silting objects, suitably defined, give rise to smashing subcategories and we study the torsion pairs associated with them.…”

A note on cosilting modules

“…Silting complexes, introduced by Keller and Vossieck [13], are also important tools in order to study t-structures of the bounded derived categories, and they are a generalization of tilting complexes. This topic is intensively studied by many authors (see, for instance, [2,10,14,16,18]). In [5], the authors introduced a concept of silting module as a common generalization of tilting modules over an arbitrary ring and support τ -tilting modules over a finite-dimensional algebra (see [1]) and they study important properties of these modules.…”