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STRATIFYING SYSTEMS FOR EXACT CATEGORIES

Part of: Homological algebra Abelian categories

Published online by Cambridge University Press:  23 July 2018

VALENTE SANTIAGO
VALENTE SANTIAGO*
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, C.P. 04510, Mexico, D.F. Mexico, Mexico E-mail: [email protected]
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Abstract

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In this paper, we develop the theory of stratifying systems in the context of exact categories as a generalisation of the notion of stratifying systems in module categories, which have been studied by different authors. We prove that attached to a stratifying system in an exact category $(\mathcal{A},\mathcal{E})$ there is an standardly stratified algebra B such that the category $\mathscr{F}$F(Θ), of F-filtered objects in the exact category $(\mathcal{A},\mathcal{E})$ is equivalent to the category $\mathscr{F}$(Δ) of Δ-good modules associated to B. The theory we develop in exact categories, give us a way to produce standardly stratified algebras from module categories by just changing the exact structure on it. In this way, we can construct exact categories whose bounded derived category is equivalent to the bounded derived category of an standardly stratified algebra. Finally, applying the relative homological algebra developed by Auslander–Solberg, we can construct examples of stratifying systems that are not a stratifying system in the classical sense, so our approach really produces new stratifying systems.

MSC classification

Primary: 18E10: Exact categories, abelian categories
Secondary: 18G25: Relative homological algebra, projective classes
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 501 - 521
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

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