Derived categories of skew quadric hypersurfaces

Abstract

The existence of a full strong exceptional sequence in the derived category of a smooth quadric hypersurface was proved by Kapranov. In this paper, we present a skew generalization of this result. Namely, we show that if $S$ is a standard graded $(\pm 1)$-skew polynomial algebra in $n$ variables with $n \geq 3$ and $f = x_1^2+\cdots +x_n^2 \in S$, then the derived category $\operatorname{\mathsf{D^b}}(\operatorname{\mathsf{qgr}} S/(f))$ of the noncommutative scheme $\operatorname{\mathsf{qgr}} S/(f)$ has a full strong exceptional sequence. The length of this sequence is given by $n-2+2^r$ where $r$ is the nullity of a certain matrix over $\mathbb F_2$. As an application, by studying the endomorphism algebra of this sequence, we obtain the classification of $\operatorname{\mathsf{D^b}}(\operatorname{\mathsf{qgr}} S/(f))$ for $n=3, 4$.