Silting theory in triangulated categories with coproducts

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 restricts to: i) a bijection between equivalence classes of self-small partial silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products; ii) a bijection between equivalence classes of classical silting objects and nondegenerate smashing and co-smashing t-structures whose heart is a module category.We describe the objects in the aisle of the t-structure associated to a partial silting set T as Milnor (or homotopy) colimits of sequences of morphisms with successive 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 in 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 H ֒→ D extends to a triangulated equivalence D(H) ∼ −→ D which restricts to bounded levels. * The authors thank Chrysostomos Psaroudakis, Jorge Vitória, Francesco Mattiello and Luisa Fiorot for their careful reading of two earlier versions of the paper and for their subsequent comments and suggestions which helped us a lot. We also thank JanŠťovíček for telling us about Lemma 7. Finally, the authors deeply thank the referee for the careful reading of the paper and for her/his comments and suggestions. Nicolás and Saorín are supported by research projects from the Spanish Ministerio de Economía y Competitividad (MTM2016-77445-P) and from the Fundación 'Séneca' of Murcia (19880/GERM/15), with a part of FEDER funds. Zvonareva is supported by the RFFI Grant 16-31-60089. The authors thank these institutions for their help. Zvonareva also thanks the University of Murcia for its hospitality during her visit, on which this research started.

Compactly generated t-structures in the derived category of a commutative ring

“…In the setting of the derived category of a ring, t-structures proved to be an indispensable tool in various general instances of tilting theory, replacing the role played by the ordinary torsion pairs in more traditional tilting frameworks (see e.g. [PV18], [NSZ18], [AHMV17], and [MV18]).…”

Partial silting objects and smashing subcategories

“…In [36], partial silting objects were defined as those T satisfying conditions (PS1) and (PS2). Inspired by the definition of partial tilting modules proposed in [15,Definition 1.4], our definition of partial silting objects includes the extra condition (PS3).…”

“…It is a direct consequence of Lemma 3.3 that a partial silting object T in T is silting if and only if T ⊥ Z = 0. The above notion of a silting object appeared already in [37] and [36]. Examples of silting objects are silting complexes in the unbounded derived category of modules over a ring (see [6] and [40]).…”

“…In this article, we study smashing subcategories via silting theory. This approach is motivated by some recent work indicating that localisations of categories and rings are intrinsically related to silting objects (see [6,27,36,37]). In particular, it was shown in [27] that universal localisations of rings in the sense of [39] are always controlled by a (not necessarily finitely generated) partial silting module.…”

“…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.…”