斜二次超曲面的导出范畴
Derived categories of skew quadric hypersurfaces
摘要：
Kapranov证明了光滑二次超曲面的导出范畴中存在完全强例外序列。在本文中，我们提出了一个斜推广的结果。也就是说，如果$S$是一个标准的分次$(\pm 1)$ -斜多项式代数，且$n \geq 3$和$f = x_1^2+\cdots +x_n^2 \in S$都是$n$变量，则非交换方案$\operatorname{\mathsf{qgr}} S/(f)$的导出范畴$\operatorname{\mathsf{D^b}}(\operatorname{\mathsf{qgr}} S/(f))$有一个完全强例外序列.这个序列的长度由$n-2+2^r$给出，其中$r$是$\mathbb F_2$上某个矩阵的零值。作为应用，通过研究该序列的自同态代数，得到了$\operatorname{\mathsf{D^b}}(\operatorname{\mathsf{qgr}} S/(f))$对$n=3, 4$的分类.
英文摘要：
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 $n=3, 4$.
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