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Lifting of recollements and gluing of partial silting sets

Part of: Abelian categories

Published online by Cambridge University Press:  07 June 2021

Manuel Saorín  and
Alexandra Zvonareva
Manuel Saorín
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Aptdo. 4021, 30100 Espinardo, Murcia, Spain (msaorinc@um.es)
Alexandra Zvonareva
Affiliation:
Universität Stuttgart, Institut für Algebra und Zahlentheorie, Pfaffenwaldring 57, D-70569, Stuttgart, Germany (alexandra.zvonareva@mathematik.uni-stuttgart.de)
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Abstract

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

Keywords

Recollementssilting setscompactly generated triangulated categoriest-structuresTTF triples

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 18E10: Exact categories, abelian categories
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 209 - 257
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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