Combinatorial study of stable categories of graded Cohen--Macaulay modules over skew quadric hypersurfaces

Abstract

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let $S$ be a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree $1$ and $f=x_1^2+\cdots +x_n^2\in S$. We prove that the stable category ${\underline{\mathsf{CM}}}^{\mathbb{Z}}(S/(f))$ of graded maximal Cohen--Macaulay module over $S/(f)$ can be completely computed using the four graphical operations. As a consequence, ${\underline{\mathsf{CM}}}^{\mathbb{Z}}(S/(f))$ is equivalent to the derived category ${\mathsf{D}}^{\mathsf{b}}(\mathrm{mod}{k}^{2^r})$, and this $r$ is obtained as the nullity of a certain matrix over ${\mathbb{F}}_2$. Using the properties of Stanley--Reisner ideals, we also show that the number of irreducible components of the point scheme of $S$ that are isomorphic to ${\mathbb{P}}^1$ is less than or equal to $\left(\genfrac{}{}{0ex}{}{r+1}{2}\right)$.

10 pages, v2: minor changes