Journal Article

Torsion Classes and t-Structures in Higher Homological Algebra

Peter Jørgensen

in International Mathematics Research Notices

Volume 2016, issue 13, pages 3880-3905
Published in print January 2016 | ISSN: 1073-7928
Published online September 2015 | e-ISSN: 1687-0247 | DOI: http://dx.doi.org/10.1093/imrn/rnv265
Torsion Classes and t-Structures in Higher Homological Algebra

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Higher homological algebra was introduced by Iyama. It is also known as [math]-homological algebra where [math] is a fixed integer, and it deals with [math]-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with [math] objects. This was recently formalized by Jasso in the theory of [math]-abelian categories. There is also a derived version of [math]-homological algebra, formalized by Geiss, Keller, and Oppermann in the theory of [math]-angulated categories (the reason for the shift from [math] to [math] is that angulated categories have triangulated categories as the “base case”). We introduce torsion classes and t-structures into the theory of [math]-abelian and [math]-angulated categories, and prove several results to motivate the definitions. Most of the results concern the [math]-abelian and [math]-angulated categories [math] and [math] associated to an [math]-representation finite algebra [math], as defined by Iyama and Oppermann. We characterize torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in [math] and intermediate t-structures in [math] which is a category one can reasonably view as the [math]-derived category of [math]. We hint at the link to [math]-homological tilting theory.

Journal Article.  10564 words.  Illustrated.

Subjects: Mathematics ; Pure Mathematics

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