Commutative Algebra in the Cohomology of Groups  pp. 1-50

Commutative Algebra in the Cohomology of Groups

By Dave Benson

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Abstract. Commutative algebra is used extensively in the cohomology of groups. In this series of lectures, I concentrate on finite groups, but I also discuss the cohomology of finite group schemes, compact Lie groups, p-compact groups, infinite discrete groups and profinite groups. I describe the role of various concepts from commutative algebra, including finite generation, Krull dimension, depth, associated primes, the Cohen–Macaulay and Gorenstein conditions, local cohomology, Grothendieck's local duality, and Castelnuovo–Mumford regularity.

Introduction

The purpose of these lectures is to explain how commutative algebra is used in the cohomology of groups. My interpretation of the word “group” is catholic: the kinds of groups in which I shall be interested include finite groups, finite group schemes, compact Lie groups, p-compact groups, infinite discrete groups, and profinite groups, although later in the lectures I shall concentrate more on the case of finite groups, where representation theoretic methods are most effective. In each case, there are finite generation theorems which state that under suitable conditions, the cohomology ring is a graded commutative Noetherian ring; over a field k, this means that it is a finitely generated graded commutative k-algebra.

Although graded commutative is not quite the same as commutative, the usual concepts from commutative algebra apply. These include the maximal/prime ideal spectrum, Krull dimension, depth, associated primes, the Cohen–Macaulay and Gorenstein conditions, local cohomology, Grothendieck's local duality, and so on. One of the themes of these lectures is that the rings appearing in group cohomology theory are quite special.