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Graded singularity category of Gorenstein algebras with levelled Beilinson algebras

Banff International Research Station for Mathematical Innovation and Discovery

Our goal is to find conditions on a noetherian AS-regular algebra $A$ and an idempotent $e\in A$ for which the graded singularity category $\mathsf{Sing}^{\mathsf{gr}}(eAe)$ admits a tilting object. Of particular interest is the situation in which $A$ is a graded skew-group algebra $S\#G$, where $S$ is the polynomial ring in $n$ variables and $G < SL(n,k)$ is finite, and $e = \frac{1}{|G|}\sum_{g\in G} g$, so that $eAe\cong S^G$. A tilting object was found by Amiot, Iyama and Reiten in the case where $A$ has Gorenstein parameter $1$. Generalizing the work of Iyama and Takahashi, Mori and Ueyama obtained a tilting object in $\mathsf{Sing}^{\mathsf{gr}}(S^G)$, provided that $S$ is a noetherian AS-regular Koszul algebra generated in degree $1$ and $G$ has homological determinant $1$. In this talk, we will discuss certain silting objects and then specialise to the setting in which the Beilinson algebra is a levelled algebra, giving a generalisation of the result of Mori and Ueyama.

Mathematics; Associative rings and algebras; Algebraic geometry; Representation theory

video/mp4

Author affiliation: Norwegian University of Science and Technology

2020-03-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73676

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/