Torsion Classes and t-Structures in Higher Homological Algebra

By Peter Jørgensen. 2014

52 citations recorded by the source corpus.

Higher homological algebra was introduced by Iyama. It is also known as $n$-homological algebra where $n \geq 2$ is a fixed integer, and it deals with $n$-cluster tilting subcategories of abelian categories. All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalised by Jasso in the theory of $n$-abelian categories. There is also a derived version of $n$-homological algebra, formalised by Geiss, Keller, and Oppermann in the theory of $( n+2 )$-angulated categories (the reason for the shift from $n$ to $n+2$ is that angulated categories have triangulated categories as the "base case"). We introduce torsion classes and t-structures into the theory of $n$-abelian and $( n+2 )$-angulated categories, and prove several results to motivate the definitions. Most of the results concern the $n$-abelian and $( n+2 )$-angulated categories ${\mathcal M}( \Lambda )$ and ${\mathcal C}( \Lambda )$ associated to an $n$-representation finite algebra $\Lambda$, as defined by Iyama and Oppermann. We characterise torsion classes in these categories in terms of closure under higher extensions, and give a bijection between torsion classes in ${\mathcal M}( \Lambda )$ and intermediate t-structures in ${\mathcal C}( \Lambda )$ which is a category one can reasonably view as the $n$-derived category of ${\mathcal M}( \Lambda )$. We hint at the link to $n$-homological tilting theory.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e522b7177c10885afe87adfce1d2dede3f33e445

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In Algebra and Number Theory

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Citations in the corpus

An introduction to higher Auslander–Reiten theory (2016, background): “The intrinsic properties of d-cluster-tilting subcategories have been investigated in [14, 25, 26].”
Auslander-Gorenstein algebras and precluster tilting (2016, background): “) See for example [1, 19, 20, 26, 28, 29, 30, 31, 34, 36] for further results in higher Auslander–Reiten theory.”