Torsion Classes and t-Structures in Higher Homological Algebra

Peter Jørgensen
2015 International mathematics research notices  
Higher homological algebra was introduced by Iyama. It is also known as n-homological algebra where n 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
more » ... 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 M (Λ) and C (Λ) associated to an n-representation finite algebra Λ, 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 M (Λ) and intermediate t-structures in C (Λ) which is a category one can reasonably view as the n-derived category of M (Λ). We hint at the link to n-homological tilting theory.
doi:10.1093/imrn/rnv265 fatcat:ayvlzun3mnbtnpgs7i2a2zcdja