Cluster tilting modules and noncommutative projective schemes

TL;DR

This paper explores the connection between noncommutative projective schemes and cluster tilting modules, establishing conditions under which their associated algebras are equivalent and possess desirable regularity properties.

Contribution

It proves that under specific conditions, the endomorphism algebra of a cluster tilting module yields a noetherian AS-regular algebra equivalent to the original scheme.

Findings

The endomorphism algebra is two-sided noetherian and AS-regular.

The noncommutative projective schemes are equivalent under certain cluster tilting conditions.

The results apply to AS-Gorenstein algebras with finite global dimension.

Abstract

In this paper, 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 $d\geq 2$ and ${\mathsf{tails}\,} A$ the noncommutative projective scheme associated to $A$. If $\operatorname{gldim}({\mathsf{tails}\,} A)< \infty$ and $A$ has a $(d-1)$-cluster tilting module $X$ satisfying that its graded endomorphism algebra is $\mathbb N$-graded, then the graded endomorphism algebra $B$ of a basic $(d-1)$-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 ${\mathsf{tails}\,} A$.