Cluster tilting modules and noncommutative projective schemes

Kenta Ueyama

Vol. 289 (2017), No. 2, 449–468

DOI: 10.2140/pjm.2017.289.449

Abstract

We study the relationship between equivalences of noncommutative projective schemes and cluster tilting modules. In particular, we prove the following result. Let $A$ be an AS-Gorenstein algebra of dimension $d\ge 2$ and $\mathsf{tails}A$ the noncommutative projective scheme associated to $A$. If $gldim\left(\mathsf{tails}A\right)<\infty$ and $A$ has a $\left(d-1\right)$-cluster tilting module $X$ with the property that its graded endomorphism algebra is $\mathbb{N}$-graded, then the graded endomorphism algebra $B$ of a basic $\left(d-1\right)$-cluster tilting submodule of $X$ is a two-sided noetherian $\mathbb{N}$-graded AS-regular algebra over $B_0$ of global dimension $d$ such that $\mathsf{tails}B$ is equivalent to $\mathsf{tails}A$.

Keywords

noncommutative projective scheme, cluster tilting module, AS-Gorenstein algebra, AS-regular algebra, ASF-regular algebra

Mathematical Subject Classification 2010

Primary: 16S38

Secondary: 14A22, 16G50, 16W50

Milestones

Received: 13 April 2016

Revised: 6 February 2017

Accepted: 10 February 2017

Published: 19 June 2017