论文标题
简要介绍$ q $形的派生类别
A brief introduction to the $Q$-shaped derived category
论文作者
论文摘要
链复合物可以看作是与关系的某个颤抖的表示，$ q^{\ permatatorName {cpx}} $。顶点是整数，每个整数$ q $都有一个箭头$ q \ xrightArrow {} q-1 $，并且关系连续箭头构成$ 0 $。因此，经典的派生类别$ \ mathscr {d} $可以视为$ q^{\ permatatorName {cpx}} $的表示类别。 Iyama和Minamoto的见解是，$ \ Mathscr {d} $表现良好的原因是，被视为一个小类别，$ q^{\ permatatorName {cpx}} $具有serre functor。将$ \ mathscr {d} $的构造概括到具有serre functor的其他关系的其他Quivers中，导致$ q $形的派生类别$ {\ Mathscr {d}} _ q $。 利用Hovey和Gillespie的方法，我们在最近三篇论文中开发了$ {\ Mathscr {d}} _ q $的理论。本文简要介绍了$ {\ mathscr {d}} _ q $，该Q $针对已经熟悉经典派生类别的读者。
A chain complex can be viewed as a representation of a certain quiver with relations, $Q^{\operatorname{cpx}}$. The vertices are the integers, there is an arrow $q \xrightarrow{} q-1$ for each integer $q$, and the relations are that consecutive arrows compose to $0$. Hence the classic derived category $\mathscr{D}$ can be viewed as a category of representations of $Q^{\operatorname{cpx}}$. It is an insight of Iyama and Minamoto that the reason $\mathscr{D}$ is well behaved is that, viewed as a small category, $Q^{\operatorname{cpx}}$ has a Serre functor. Generalising the construction of $\mathscr{D}$ to other quivers with relations which have a Serre functor results in the $Q$-shaped derived category ${\mathscr{D}}_Q$. Drawing on methods of Hovey and Gillespie, we developed the theory of ${\mathscr{D}}_Q$ in three recent papers. This paper offers a brief introduction to ${\mathscr{D}}_Q$, aimed at the reader already familiar with the classic derived category.
