论文标题
偏斜四边形的派生类别
Derived categories of skew quadric hypersurfaces
论文作者
论文摘要
卡普拉诺夫证明了在平滑四边形超表面的派生类别中存在一个完整的强例序列。在本文中，我们提出了该结果的偏斜概括。即，我们表明，如果$ s $是标准分级$（\ pm 1）$ - 偏斜多项式代数，$ n $变量，带有$ n \ geq 3 $和$ f = x_1^2 +\ cdots +cdots +x_n^2 \ in S $，则是派生类别$ \ operatorName {\ Mathsf {d^b}}（\ operatorAtorName {\ MathSf {qgr}} s/（f））$的$ \ operatotorname {\ operatorname {\ m athssf {qgr}} s/（f）s/（f）s/（f）$具有完整的强度序列。此序列的长度由$ n-2+2^r $给出，其中$ r $是$ \ m athbb f_2 $的某个矩阵的无效。作为一个应用程序，通过研究该序列的内态代数，我们获得了$ \ peripatorName {\ Mathsf {\ MathSf {d^b}}（\ operatorname {\ m athsf {qgr}} s/（f））$ n = 3，4 $。
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 $n=3, 4$.
