Cluster tilting modules and noncommutative projective schemes

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 ${\mathsf{tails}\,} A$.Comment: 16 page

Graded Twisted Calabi–yau Algebras Are Generalized Artin–schelter Regular

“…Condition (b) of Theorem 5.2 was also shown by Minamoto-Mori to be a consequence of their definition of AS regularity over in [33, Proposition 3.5]; our theorem gives another proof of this. We also note that there is a notion of ASF-regular algebra in [33], defined in terms of graded local cohomology, which Ueyama recently showed [46, Corollary 2.11] to be equivalent for noetherian algebras A to the condition of being AS regular over . It seems possible that a suitable “invertible bimodule twist” of this property could be equivalent to the generalized AS regular property, at least for noetherian algebras, but we do not pursue that possibility here.…”

Auslander’s Theorem for Group Coactions on Noetherian Graded Down-Up Algebras

“…Mori and Ueyama prove that if the Auslander map is an isomorphism, then A G has a graded isolated singularity if and only if A#G/I is finite-dimensional [21,Theorem 3.10]. Examples of graded isolated singularities are of particular interest, since when A G has a graded isolated singularity, the category of graded CM A G -modules has several nice properties (see [25]).…”