A classification theorem for t-structures

Abstract

We give a classification theorem for a relevant class of t-structures in triangulated categories, which includes, in the case of the derived category of a Grothendieck category, a large class of t-structures whose hearts have at most nfixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the t-tree, a new technique which generalizesthe filtration induced by a torsion pair. At last we apply our results in the tilting context generalizing the 1-tilting equivalence proved by Happel, Reiten and Smalø. The last section provides applications to classical n-tilting objects, examples of t-trees for modules over a path algebra, and new developments on compatible t-structures

t-Structures for relative D-modules and t-exactness of the de Rham functor

“…Remark 1.4. In the work [8] and [28] the authors propose a generalization of the previous result. In [8,Theorem 2.14 and 4.3] the authors proved that, under some technical hypotheses, given any pair of t-structures D, D satisfying the condition:…”

“…x are exact and the t-structure π on D −dX α coh (O S ) is stable by truncation with respect to the standard t-structure. So, in analogy with Remark 2.13, the tstructure π is left p-compatible and, according to Lemma 2.3 and to [8,Theorem 4.3], it can be recovered from p via an iterated right tilting procedure of length ℓ.…”

“…We conclude by the previous Theorem 2.11 that the t-structure Π is left P -compatible (cf. Remark 1.4) and so, according to Lemma 2.3 and to [8,Theorem 4.3], it can be recovered from P via an iterated right tilting procedure of length ℓ.…”

On a property of $t$-structures generated by non-classical tilting modules

“…In [FMT14] the authors generalise the Happel-Reiten-Smalø result and recover the torsion torsion-free decomposition, passing from classical 1-tilting objects to classical n-tilting objects. Given a Grothendieck category G with a classical n-tilting object T , denoted by T = (T ≤0 , T ≥0 ) the t-structure on D(G) generated by T and by D = (D ≤0 , D ≥0 ) the natural t-structure, the pair (D, T ) is (right) n-tilting, that is: 1) D ≤−n ⊆ T ≤0 ⊆ D ≤0 , and this relation is "strict"; 2) it is (right) filterable, that is, for any i ∈ Z the intersection D ≥i ∩T ≥0 is a co-aisle of a t-structure; 3) if H T denotes the heart of T , then G ∩H T is a cogenerating class of G, that is, each object of G embeds in an object of G ∩ H T .…”

Derived equivalences induced by nonclassical tilting objects

“…The first question was answered in the affirmative by Xiao Wu Chen [Ch10], for any 1-tilting object in an arbitrary abelian category. As for the second question, an affirmative answer in the case of a classical ntilting object in a Grothendieck category, both in the bounded and unbonded ed situations, follows from [FMT14]. When T is a nonclassical tilting object, the second question has an affirmative answer at the bounded level, in essentially any AB3 abelian category, due to a stronger recent result of Psaroudakis and Vitória (see [PV15][Corollary 5.2]).…”