Triangulated subcategories of extensions, stable t-structures, and triangles of recollements

TL;DR

This paper investigates conditions under which subcategories of extensions in triangulated categories are themselves triangulated, and introduces methods for constructing stable t-structures and triangles of recollements.

Contribution

It provides new criteria for extension subcategories to be triangulated and offers a systematic approach to constructing triangles of recollements in triangulated categories.

Findings

Established conditions for extension subcategories to be triangulated.

Developed tools for constructing stable t-structures.

Reproduced known triangles of recollements using new methods.

Abstract

In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X*Y. We give conditions for X*Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y), (Y,Z), (Z,X). We easily recover some triangles of recollements known from the literature.