On Kn\"orrer periodicity for quadric hypersurfaces in skew projective spaces

We study the structure of the stable category $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ of graded maximal Cohen-Macaulay module over $S/(f)$ where $S$ is a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree 1, and $f =x_1^2 + \cdots +x_n^2$. If $S$ is commutative, then the structure of $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is well-known by Kn\"orrer's periodicity theorem. In this paper, we prove that if $n\leq 5$, then the structure of $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to ${\mathbb P}^1$.Comment: 12 pages, v2, v3: minor change

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 $d\geq 2$ 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

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

Hochschild cohomology related to graded down-up algebras with weights (1,n)(1,n)

Let $A=A(\alpha, \beta)$ be a graded down-up algebra with $({\rm deg}\,x, {\rm deg}\,y)=(1,n)$ and $\beta \neq 0$, and let $\nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $\nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $\nabla A$ for the case $n \geq 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether $\left(\begin{smallmatrix} 1&0 \end{smallmatrix}\right)\left(\begin{smallmatrix} \alpha &1 \\ \beta &0 \end{smallmatrix}\right)^n\left(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right)$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $n\geq 3$ in the context of Grothendieck groups.Comment: 13 page

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

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

Basis of Self-organized Proportion Regulation Resulting from Local Contacts

One of the fundamental problems in biology concerns the method by which a cluster of organisms can regulate the proportion of individuals that perform various roles or modes as if each individual knows a whole situation without a leader. A specific ratio exists in various species at multiple levels from the process of cell differentiation in multicellular organisms to the situation of social dilemma in a group of human beings. This study found a common basis of regulating a collective behavior which is realized by a series of local contacts between individuals. The most essential behavior of individuals in this theory is to change its internal mode through sharing information in contact with others. Our numerical simulations with cellular automata model realize to regulate the ratio of population of individuals who has either two kinds of modes. From the theoretical analysis and numerical calculations, we found that asymmetric properties in local contacts, are essential for adaptive regulation in response to the global information such a group size and whole density. Furthermore, a discrete system is crucial in no-leader groups to realize the flexible regulation, and the critical condition which eliminates overlap with one another (excluded volume effect) also affects the resulting proportion in high density. The foremost advantage of this strategy is that no global information is required for each individual, and only a couple of mode switching can achieve the whole proportion regulation. The simple mechanism say that proportion regulation in well-organized groups in nature can be realized through and limited to local contacts, and has a potential to solve various phenomena that microscopic individuals behaviors connect to macroscopic orderly behaviors.Comment: 18 pages, 8 figures, and supporting informatio

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

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