Silting and cosilting classes in derived categories

Abstract

An important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Moreover, cotilting classes are precisely the resolving and definable subcategories of the module category whose Extorthogonal class has bounded injective dimension.In this article, we prove a derived counterpart of the statements above in the context of silting theory. Silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules. They give rise to subcategories of the derived category, called silting and cosilting classes, which are part of both a tstructure and a co-t-structure. We characterise these subcategories: silting classes are precisely those which are intermediate and Ext-orthogonal classes to a set of compact objects, and cosilting classes are precisely the cosuspended, definable and co-intermediate subcategories of the derived category.Theorem. Let A be a ring, D(A) the derived category of A-Mod and V a full subcategory of D(A).(1) V is a silting class if and only if V is intermediate and V = S ⊥ >0 for a set S of compact objects.(2) V is a cosilting class if and only if V is cosuspended, co-intermediate and definable.Moreover, the first statement induces a bijection between co-intermediate and cosuspended subcategories of K b (A-proj) and silting complexes up to equivalence.This statement summarises Theorems 3.6 and 3.14. For the terminology used, we refer to the relevant sections. However, the parallel with the results concerning tilting and cotilting modules is evident.The proof of our main theorem involves some module categories built from the category of complexes. In fact, we reduce (co)silting problems in the derived category to (co)tilting problems in these module categories. This allows us to use the finite-type characterisation of tilting classes to conclude the compact generation of silting co-t-structures. Moreover, we use the fact that cotilting modules are pure-injective to conclude that so are cosilting complexes, from which we deduce the definability of cosilting classes.Finally, note that for the special case of two-term silting and cosilting complexes, the theorem above specialises to certain classification results recently obtained in the context of silting and cosilting modules. More precisely, part (1) translates to saying that a torsion class in the module category arises from a silting module if and only if it is divisible with respect to a set of maps between finitely generated projective modules (see [23, Theorem 6.3]). Part (2) of the theorem is, in this context, equivalent to stating that a torsionfree class in the module category arises from a cosilting module if and only if it is definable (see [2, Corollary 3.9] and the references therein).The structure of the paper is as follows. In Section 2, we set up the aforementioned module categories built from the category of complexes, and we show a useful correspondence between (...

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

“…The notion of a cosilting complex was introduced in [ZW17] as a dualization of the (big) silting complexes of [AHMV15]. Combining the recent works [MV18] and [Lak18] gives a useful characterization of t-structures induced by cosilting complexes. Before that, we need to recall a couple of definitions.…”

Silting Objects

“…We now return to cosilting complexes. It is shown in , dually to Theorem , that every cosilting complex can be interpreted as a cotilting module in a suitable module category. Observe that both in module categories and in triangulated categories, pure‐injectivity of an object can be characterised by the following factorisation property: For every set , the summation map factors through the canonical map .…”

Torsion pairs in silting theory

“…Dually, a cosilting object C of D(A) lying in K b (Inj(A)) gives rise to a cosuspended TTF triple, that is, the cosilting t-structure ( ⊥ ≤0 C, ⊥ >0 C) admits a right adjacent co-t-structure with coheart Prod(C). For this dual statement, we refer to forthcoming work in [25].…”

“…For partial results in this direction we refer to [29,Proposition 4.2] and [11,Corollary 2.5]. In a forthcoming paper ( [25]), it will be proved that cosilting complexes in derived module categories are always pure-injective and give rise to definable subcategories as above. We do not know, however, if the same holds true for arbitrary cosilting objects in compactly generated triangulated categories.…”