Séminaires : Séminaire d'Algèbre

Higher homological algebra was introduced by Iyama in the late 2000's. His point-of-view was that some classical results by Auslander and Auslander--Reiten were somehow 2-dimensional and should have n-dimensional equivalents. This new theory quickly attracted a lot of attention, with many authors generalising classical notions to the setting of higher homological algebra. Examples of such generalisations are the introduction of n-abelian categories by Jasso, n-angulated categories by Geiss--Keller--Oppermann, and n-torsion classes by Jørgensen.Recently, it was shown by Kvamme and, independently, by Ebrahimi and Nasr-Isfahani, that every small n-abelian category is the n-cluster-tilting subcategory of an abelian category. In this talk we will focus on the relation between n-torsion classes in an n-abelian category MM and (classical) torsion classes of the abelian category AA in which MM is embedded. By considering functorially finite torsion classes, this will allow us to relate n-torsion classes with maximal tau_n-rigid objects in MM.Some of the results presented in this talk are part of a joint work by J. Asadollahi, P. Jørgensen, S. Schroll, H. Treffinger. The rest corresponds to an ongoing project by J. August, J. Haugland, K. Jacobsen, S. Kvamme,Y. Palu and H. Treffinger.