A note on even Clifford algebras of skew quadric hypersurfaces

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.