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 $n\ge 3$ and $f=x_1^2+\cdots +x_n^2\in S$, then the derived category ${\mathsf{D}}^{\mathsf{b}}(\mathrm{qgr}S/(f))$ of the noncommutative scheme $\mathrm{qgr}S/(f)$ 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 ${\mathsf{D}}^{\mathsf{b}}(\mathrm{qgr}S/(f))$ for $n=3,4$.

23 pages

Cited by in corpus (2)

• Generalized Knörrer's Periodicity Theorem
• Noncommutative conics in Calabi-Yau quantum projective planes