A note on even Clifford algebras of skew quadric hypersurfaces

Abstract

Let $S_{\alpha }=k⟨x_1,\dots ,x_n⟩/(x_i{x}_j-{\alpha }_{ij}{x}_j{x}_i)$ 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 $f=x_1{x}_2+\cdots +x_{n-2}{x}_{n-1}+x_n^2$ 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 $f=x_1{x}_2+\cdots +x_{n-1}{x}_n$ 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.

18 pages