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
Suppose that A is an abelian category whose derived category D(A) has Hom sets and arbitrary (small) coproducts, let T be a (not necessarily classical) (n-)tilting object of A and let H be the heart of the associated t-structure on D(A). We show that there is a triangulated equivalence of unbounded derived categories D(H) ∼ = −→ D(A) which is compatible with the inclusion functor H ֒→ D(A). The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has Hom sets and arbitrary products.
Tilting modules, generalising the notion of progenerator, furnish equivalences between pieces of module categories. This paper is dedicated to study how much these pieces say about the whole category. We will survey the existing results in the literature, introducing also some new insights.
Dedicated to Alberto Facchini on the occasion of his sixtieth birthday.Abstract. Firstly we provide a technique to move torsion pairs in abelian categories via adjoint functors and in particular through Giraud subcategories. We apply this point in order to develop a correspondence between Giraud subcategories of an abelian category C and those of its tilt H(C) i.e., the heart of a t-structure on D b (C) induced by a torsion pair.
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