#### Vol. 289, No. 2, 2017

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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 $\mathsf{tails}\phantom{\rule{0.3em}{0ex}}A$ the noncommutative projective scheme associated to $A$. If $gldim\left(\mathsf{tails}\phantom{\rule{0.3em}{0ex}}A\right)<\infty$ and $A$ has a $\left(d-1\right)$-cluster tilting module $X$ with the property that its graded endomorphism algebra is $ℕ$-graded, then the graded endomorphism algebra $B$ of a basic $\left(d-1\right)$-cluster tilting submodule of $X$ is a two-sided noetherian $ℕ$-graded AS-regular algebra over ${B}_{0}$ of global dimension $d$ such that $\mathsf{tails}\phantom{\rule{0.3em}{0ex}}B$ is equivalent to $\mathsf{tails}\phantom{\rule{0.3em}{0ex}}A$.

##### 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