群上同调中的Dickson不变量、正则性和计算
Dickson invariants, regularity and computation in group cohomology
摘要：
本文研究域$k$上有限群$G$的上同调环$H^*(G,k)$的交换代数.本文将Benson和Carlson提出的拟正则序列的概念与上同调环的局部上同调联系起来。我们给出了一些稍微加强版本的准正则性，并将其中之一与Castelnuovo-芒福德正则性。证明了原意义或加强意义下的拟正则序列的存在性当且仅当Dickson不变量构成同一意义下的拟正则序列.证明涉及的概念，虚拟的射影，介绍了卡尔森，彭和惠勒。作为这一研究的副产品，我们给出了一个新的证明Bourguiba-Zarati定理的深度和Dickson不变量，在有限群上同调的上下文中，没有使用机器的Steenrod代数上的不稳定模。最后，我们描述了一个改进的Carlson算法计算有限群的上同调使用有限初始段的一个投射分解的平凡模。与Carlson的算法相反，我们的算法在计算过程中不依赖于验证任何假设，并且总是保证工作。
英文摘要：
In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo--Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler. As a by-product of this investigation, we give a new proof of the Bourguiba--Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra. Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.
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