TY - JOUR
T1 - Torsion classes and t-structures in higher homological algebra
AU - Jørgensen, Peter
N1 - Publisher Copyright:
© The Author(s) 2015.
PY - 2016
Y1 - 2016
N2 - 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 formalized by Jasso in the theory of n-abelian categories. There is also a derived version of n-homological algebra, formalized 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 M(Λ) and C (Λ) associated to an n-representation finite algebra Λ, 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 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.
AB - 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 formalized by Jasso in the theory of n-abelian categories. There is also a derived version of n-homological algebra, formalized 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 M(Λ) and C (Λ) associated to an n-representation finite algebra Λ, 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 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.
UR - https://www.scopus.com/pages/publications/84981344688
U2 - 10.1093/imrn/rnv265
DO - 10.1093/imrn/rnv265
M3 - Journal article
AN - SCOPUS:84981344688
SN - 1073-7928
VL - 2016
SP - 3880
EP - 3905
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 13
ER -