上同调环有限生成的新方法
New approaches to finite generation of cohomology rings
摘要：
在支撑簇理论中，有限维（Hopf）代数$A$的表示可以通过使用$A$的上同调环将$A$的任何表示关联到代数簇来几何地研究。该理论的一个基本假设是$A$的上同调环及相应模的有限生成条件。本文介绍了研究有限生成条件的各种方法。首先，对任意有限维霍普夫代数$A$，在假设$R$在其不变子环$R^A$上是整数的条件下，证明了$A$-模上的有限生成条件可以被任意仿射交换$A$ -模代数$R$上的条件所取代.接下来，我们使用谱序列的论点，证明了有限生成条件成立某些过滤，粉碎和交叉积代数的正特征，如果相关的谱序列崩溃。最后，如果$A$定义在有理数数域上，我们构造了有限域上的另一个有限维霍普夫代数$A'$，其中$A$可以看作是$A'$的变形，并证明了如果有限生成条件对$A'$成立，那么同样的条件对$A$也成立.
英文摘要：
In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra $A$, we show that the finite generation condition on $A$-modules can be replaced by a condition on any affine commutative $A$-module algebra $R$ under the assumption that $R$ is integral over its invariant subring $R^A$. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if $A$ is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra $A'$ over a finite field, where $A$ can be viewed as a deformation of $A'$, and prove that if the finite generation condition holds for $A'$, then the same condition holds for $A$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
