Cluster tilting modules and noncommutative projective schemes

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 $d\ge 2$ and $\mathsf{t}\mathsf{a}\mathsf{i}\mathsf{l}\mathsf{s}A$ the noncommutative projective scheme associated to $A$. If $\mathrm{gldim}(\mathsf{t}\mathsf{a}\mathsf{i}\mathsf{l}\mathsf{s}A)<\mathrm{\infty }$ 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 $\mathsf{t}\mathsf{a}\mathsf{i}\mathsf{l}\mathsf{s}B$ is equivalent to $\mathsf{t}\mathsf{a}\mathsf{i}\mathsf{l}\mathsf{s}A$.