论文标题
广义的Koszul代数和Koszul二元性
Generalized Koszul Algebra and Koszul Duality
论文作者
论文摘要
我们定义了广义的koszul模块和戒指，并为$ \ mathbb {n} $ - 分级环开发了广义的koszul理论，其零部分noetherian noeetherian semiperfect。该理论专门针对由Beilinson-Ginzburg-Soergel开发的Artinian Semimple的分级环的古典Koszul理论，以及由Green和Martin {é} Z-Villa开发的Noetherian Semipertect Rings的未分级Koszul理论。令$ a $为左有限$ \ mathbb {n} $ - 以度数$ 1 $生成的分级戒指，$ a_0 $ noyetherian semiperfect，$ j $是其分级的jacobson激进分子和$ s = a/j $。由$ a $的koszul dual，我们的意思是yoneda ext ring $ \ usewinline {\ text {ext}} _ a^\ bullet（s，s）$。如果$ a $是广义的koszul环，而$ m $是一个广义的koszul模块，那么证明$ a $ a $ a $ a $ a $ as $ \ text {gr} _j a $和koszul dual of koszul dual of koszul dual的koszul dual dual of koszul dual of koszul dual of $ m $ $ m $ $ $ us $ \文本{gr}如果$ a $是本地有限的代数，则证明以下声明是等效的：$ a $是广义的koszul； koszul dual $ \ usewandline {\ text {ext}} _ a^\ bullet（s，s）$的$ a $是（经典）koszul; $ \ text {gr} _j a $是（经典）koszul; $ a $的另一个环$ a^{op} $是广义的koszul。还可以证明，如果$ a $是有限全球尺寸的概括性的koszul，则$ a $在$ a $ a $的koszul dual dual是自注的情况下是常规的。
We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martin{é}z-Villa. Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect, $J$ be its graded Jacobson radical and $S=A/J$. By the Koszul dual of $A$ we mean the Yoneda Ext ring $\underline{\text{Ext}}_A^\bullet(S,S)$. If $A$ is a generalized Koszul ring and $M$ is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of $A$ is $\text{Gr}_J A$ and the Koszul dual of the Koszul dual of $M$ is $\text{Gr}_J M$. If $A$ is a locally finite algebra, then the following statements are proved to be equivalent: $A$ is generalized Koszul; the Koszul dual $\underline{\text{Ext}}_A^\bullet(S,S)$ of $A$ is (classically) Koszul; $\text{Gr}_J A$ is (classically) Koszul; the opposite ring $A^{op}$ of $A$ is generalized Koszul. It is also proved that if $A$ is generalized Koszul with finite global dimension then $A$ is generalized AS regular if and only if the Koszul dual of $A$ is self-injective.
