A note on even Clifford algebras of skew quadric hypersurfaces

TL;DR

This paper investigates the structure of even Clifford algebras and Cohen-Macaulay modules over skew quadric hypersurfaces, revealing their algebraic properties and categorical equivalences in both odd and even dimensions.

Contribution

It provides explicit descriptions of the even Clifford algebra and stable categories for skew quadric hypersurfaces, extending classical results to a noncommutative setting.

Findings

Even Clifford algebra is isomorphic to matrix algebras or their squares depending on dimension parity.

Stable categories are equivalent to derived categories of modules over k or k^2.

The hypersurfaces have finite Cohen-Macaulay representation type.

Abstract

Let $S_\alpha = k\langle x_1,\dots,x_n\rangle /(x_i x_j - \alpha_{ij} x_j x_i)$ be a standard graded skew polynomial algebra over an algebraically closed field $k$ of characteristic not equal to $2$. We show the following results. When $n$ is odd and $f = x_1x_2 + \cdots + x_{n-2}x_{n-1} + x_n^2$ is a normal element of $S_\alpha$, the even Clifford algebra of the skew quadric hypersurface $S_\alpha/(f)$ is isomorphic to a full matrix algebra $M_{2^{(n-1)/2}}(k)$, and the stable category $\underline{\mathsf{CM}}^{\mathbb Z}(S_\alpha/(f))$ of graded maximal Cohen-Macaulay modules over $S_\alpha/(f)$ is triangle equivalent to the derived category $\mathsf{D}^b(\mathsf{mod}\,k)$. When $n$ is even and $f = x_1x_2 + \cdots + x_{n-1}x_n$ is a normal element of $S_\alpha$, the even Clifford algebra of $S_\alpha/(f)$ is isomorphic to $M_{2^{(n-2)/2}}(k)^2$, and the stable category $\underline{\mathsf{CM}}^{\mathbb Z}(S_\alpha/(f))$ of graded maximal Cohen-Macaulay modules over $S_\alpha/(f)$ is triangle equivalent to the derived category $\mathsf{D}^b(\mathsf{mod}\,k^2)$. As a consequence, $S_\alpha/(f)$ is of finite Cohen-Macaulay representation type in both cases. These results demonstrate that $S_\alpha/(f)$ is a natural noncommutative generalization of the homogeneous coordinate ring of a smooth quadric hypersurface.