G-stable support τ-tilting modules

Abstract

Motivated by \tau-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a nite-dimensional algebra \mathrm{\Lambda }with action by a nite group G; we introduce the notion of G-stable support \tau-tilting modules. Then we establish bijections among G-stable support \tau-tilting modules over \Lambda; G-stable two-term silting complexes in the homotopy category of bounded complexes of nitely generated projective \mathrm{\Lambda }-modules, and G-stable functorially nite torsion classes in the category of nitely generated left \mathrm{\Lambda }-modules. In the case when \mathrm{\Lambda } is the endomorphism of a G-stable cluster-tilting object T over a Hom-nite 2-Calabi-Yau triangulated category \ell with a G-action, these are also in bijection with G-stable cluster-tilting objects in \ell : Moreover, we investigate the relationship between stable support \tau-tilitng modules over \mathrm{\Lambda } and the skew group algebra \mathrm{\Lambda }G:

Keywords

G-stable support\tau-tilting modules / G-stable two-term silting complexes / G-stable functorially nite torsion classes / G-stable cluster-tilting objects / bijection / skew group algebras

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2016 Higher Education Press and Springer-Verlag Berlin Heidelberg