DERIVED CATEGORIES OF SKEW QUADRIC HYPERSURFACES

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(1)(2) + . . . + x(n)(2) is an element of S, then the derived category D-b(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 + 2(r) where r is the nullity of a certain matrix over F-2. As an application, by studying the endomorphism algebra of this sequence, we obtain the classification of D-b(qgr S/(f)) for n = 3, 4.