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 (±1)-skew polynomial algebra in n variables with n ≥ 3 and f = x 21 + ⋯ + x 2n ∈ S, then the derived category Db(qgr S/(f)) of the noncommutative scheme qgr S/(f) has a full strong exceptional sequence. The length of this sequence is given by n−2+2r 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 Db(qgr S/(f)) for n = 3, 4.