A classification theorem for $t$-structures

TL;DR

This paper provides a classification theorem for certain $t$-structures in triangulated categories, introduces the $t$-tree construction, and applies these results to tilting theory and module examples.

Contribution

It introduces a classification theorem for $t$-structures with bounded cohomologies, and develops the $t$-tree technique, extending tilting theory and applications.

Findings

Classification of $t$-structures with bounded cohomologies.

Construction of the $t$-tree as a generalization of torsion pair filtrations.

Applications to $n$-tilting objects and module categories.

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, the $t$-structures whose hearts have at most $n$ fixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the $t$-tree, a new technique which generalises the 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 Smal{\o} [HRS96]. 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 [KeV88b], [Ke07].