Stable Categories of Graded Maximal Cohen-Macaulay Modules over Noncommutative Quotient Singularities

Tilting objects play a key role in the study of triangulated categories. A famous result due to Iyama and Takahashi asserts that the stable categories of graded maximal Cohen-Macaulay modules over quotient singularities have tilting objects. This paper proves a noncommutative generalization of Iyama and Takahashi's theorem using noncommutative algebraic geometry. Namely, if $S$ is a noetherian AS-regular Koszul algebra and $G$ is a finite group acting on $S$ such that $S^G$ is a "Gorenstein isolated singularity", then the stable category ${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$ of graded maximal Cohen-Macaulay modules has a tilting object. In particular, the category ${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$ is triangle equivalent to the derived category of a finite dimensional algebra.Comment: 28 pages, an error in the previous version has been correcte

Ample Group Action on AS-regular Algebras and Noncommutative Graded Isolated Singularities

In this paper, we introduce a notion of ampleness of a group action $G$ on a right noetherian graded algebra $A$, and show that it is strongly related to the notion of $A^G$ to be a graded isolated singularity introduced by the second author of this paper. Moreover, if $S$ is a noetherian AS-regular algebra and $G$ is a finite ample group acting on $S$, then we will show that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\cal D}^b(\operatorname{mod} \nabla S*G)$ where $\nabla S$ is the Beilinson algebra of $S$. We will also explicitly calculate a quiver $Q_{S, G}$ such that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\mathcal D}^b(\operatorname{mod} kQ_{S, G})$ when $S$ is of dimension 2.Comment: 25 page

Noncommutative matrix factorizations with an application to skew exterior algebras

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras.Comment: 25 page

Noncommutative Kn\"orrer's periodicity theorem and noncommutative quadric hypersurfaces

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Kn\"orrer's periodicity theorem is a powerful tool to study Cohen-Macaulay representation theory since it reduces the number of variables in computing the stable category $\underline {\operatorname{CM}}(A)$ of maximal Cohen-Macaulay modules over a hypersurface $A$, so, in this paper, we show a noncommutative graded version of Kn\"orrer's periodicity theorem. Moreover, we prove another way to reduce the number of variables in computing the stable category $\underline {\operatorname{CM}}^{\mathbb Z}(A)$ of graded maximal Cohen-Macaulay modules if $A$ is a noncommutative quadric hypersurface. Under high rank property defined in this paper, we also show that computing $\underline {\operatorname{CM}}^{\mathbb Z}(A)$ over a noncommutative smooth quadric hypersurface $A$ up to 6 variables can be reduced to one or two variables cases. In addition, we give a complete classification of $\underline {\operatorname{CM}}^{\mathbb Z}(A)$ over a smooth quadric hypersurface $A$ in a skew $\mathbb P^{n-1}$ up to 6 variables without high rank property using graphical methods.Comment: 29 page

A categorical characterization of quantum projective spaces

Let $R$ be a finite dimensional algebra of finite global dimension over a field $k$. In this paper, we will characterize a $k$-linear abelian category $\mathscr C$ such that $\mathscr C\cong \operatorname {tails} A$ for some graded right coherent AS-regular algebra $A$ over $R$. As an application, we will prove that if $\mathscr C$ is a smooth quadric surface in a quantum $\mathbb P^3$ in the sense of Smith and Van den Bergh, then there exists a right noetherian AS-regular algebra $A$ over $kK_2$ of dimension 3 and of Gorenstein parameter 2 such that $\mathscr C\cong \operatorname {tails} A$ where $kK_2$ is the path algebra of the 2-Kronecker quiver.Comment: 31 pages, v2: The proof of Theorem 3.10 of the first version was not correct. Accordingly, the statement of the main result (Theorem 4.1) has been revised, v3: minor reviso

The classification of 3-dimensional noetherian cubic Calabi-Yau algebras

It is known that every 3-dimensional noetherian Calabi-Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S. P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi-Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi-Yau algebras. As an application, we show that if $S$ is a 3-dimensional noetherian cubic Calabi-Yau algebra and $\sigma$ is a graded algebra automorphism of $S$, then the homological determinant of $\sigma$ can be calculated by the formula $\operatorname{hdet} \sigma=(\operatorname{det} \sigma)^2$ with one exception.Comment: 19 page

m-Koszul Artin-Schelter regular algebras

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian $m$-Koszul twisted Calabi-Yau or, equivalently, $m$-Koszul Artin-Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w,i) for a unique-up-to-scalar-multiples twisted superpotential w in a tensor power of some vector space V. By definition, D(w,i) is the quotient of the tensor algebra TV by the ideal generated by all i-th order left partial derivatives of w. We identify the group of graded algebra automorphisms of D(w,i) with a subgroup of GL(V). We show that the homological determinant of a graded algebra automorphism $\sigma$ of an $m$-Koszul Artin-Schelter regular algebra D(w,i) is the scalar hdet($\sigma$) given by the formula hdet($\sigma$) w =$\sigma^{\otimes m+i}$(w). It follows from this that the homological determinant of the Nakayama automorphism of an $m$-Koszul Artin-Schelter regular algebra is 1. As an application, we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian Artin-Schelter regular algebras of dimension 3.Comment: 21 page

The Classification of 3-Calabi-Yau algebras with 3 generators and 3 quadratic relations

Let $k$ be an algebraically closed field of characteristic not 2 or 3, $V$ a 3-dimensional vector space over $k$, $R$ a 3-dimensional subspace of $V \otimes V$, and $TV/(R)$ the quotient of the tensor algebra on $V$ by the ideal generated by $R$. Raf Bocklandt proved that if $TV/(R)$ is 3-Calabi-Yau, then it is isomorphic to $J({\sf{w}})$, the "Jacobian algebra" of some ${\sf{w}} \in V^{\otimes 3}$. This paper classifies the ${\sf{w}}\in V^{\otimes 3}$ such that $J({\sf{w}})$ is 3-Calabi-Yau. The classification depends on how ${\sf{w}}$ transforms under the action of the symmetric group $S_3$ on $V^{\otimes 3}$ and on the nature of the subscheme $\{\overline{{\sf{w}}}=0\} \subseteq \mathbb{P}^2$ where $\overline{{\sf{w}}}$ denotes the image of ${\sf{w}}$ in the symmetric algebra $SV$. Surprisingly, as ${\sf{w}}$ ranges over $V^{\otimes 3}-\{0\}$, only nine isomorphism classes of algebras appear as non-3-Calabi-Yau $J({\sf{w}})$'s.Comment: 26 pages v2. Minor changes after the referee's repor

Moduli of noncommutative Hirzebruch surfaces

We introduce three non-compact moduli stacks parametrizing noncommutative deformations of Hirzebruch surfaces; the first is the moduli stack of locally free sheaf bimodules of rank 2, which appears in the definition of noncommutative $\mathbb{P}^1$-bundle in the sense of Van den Bergh arXiv:math/0102005, the second is the moduli stack of relations of a quiver in the sense of arXiv:1411.7770, and the third is the moduli stack of quadruples consisting of an elliptic curve and three line bundles on it. The main result of this paper shows that they are naturally birational to each other. We also give an Orlov-type semiorthogonal decomposition for noncommutative $\mathbb{P}^1$-bundles, an explicit classification of locally free sheaf bimodules of rank 2, and a noncommutative generalization of the (special) McKay correspondence as a derived equivalence for the cyclic group $\left\langle \frac{1}{d}(1,1) \right\rangle$.Comment: 34 page

Quantum Projective Planes Finite over their Centers

For a $3$-dimensional quantum polynomial algebra $A=\mathcal{A}(E,\sigma)$, Artin-Tate-Van den Bergh showed that $A$ is finite over its center if and only if $|\sigma|<\infty$. Moreover, Artin showed that if $A$ is finite over its center and $E\neq \mathbb{P}^2$, then $A$ has a fat point module, which plays an important role in noncommutative algebraic geometry, however the converse is not true in general. In this paper, we will show that, if $E\neq \mathbb{P}^2$, then $A$ has a fat point module if and only if the quantum projective plane $\mathsf{Proj}_{{\rm nc}} A$ is finite over its center in the sense of this paper if and only if $|\nu^*\sigma^3|<\infty$ where $\nu$ is the Nakayama automorphism of $A$.In particular, we will show that if the second Hessian of $E$ is zero, then $A$ has no fat point module.Comment: 13 page