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 $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ of graded maximal Cohen--Macaulay module over $S/(f)$ can be completely computed using the four graphical operations. As a consequence, $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is equivalent to the derived category $\mathsf{D^b}(\operatorname{\mathsf{mod}} k^{2^r})$, 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 $\binom{r+1}{2}$.