We develop the homological algebra of coefficient systems on a group, in particular from the point of view of calculating higher limits. We show how various sequences of modules associated to a class of subgroups of a given group can be analysed by methods from homological algebra. We are particularly interested in when these sequences are exact, or, if not, when their homology is equal to the higher limits of the coefficient system.
We record here one easily stated consequence.
Proposition 0.3. For any non-trivial finite group, the cohomology ring His generated by elements in degree at most |G| − 1 and the relations between them (as a graded commutative algebra) are generated in degrees at most 2(|G| − 1). This bound is weak, although it can be improved somewhat at the cost of a more complicated formulation, but previously no such bound was known.We go on to prove some other conjectures of Benson on the regularity of the cohomology of other classes of groups [3], specifically for compact Lie groups and virtual Poincaré duality groups.The proof uses standard techniques in equivariant cohomology based on work of Quillen [30] and a paper of Duflot [14].David Green and Simon King have verified Benson's Conjecture computationally for all groups of order less than 256 [20].There is a survey of the properties of group cohomology from the point of view of commutative algebra by Benson [3].
Castelnuovo-Mumford regularityWe will work in the category of Z-graded rings and modules, so maps will respect the grading, elements will be homogeneous, etc., without specific mention. For a module M = i∈Z M i , we will write M ≥d = i≥d M i and similarly for other
Consider a group acting on a polynomial ring over a finite field. We study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries. For example, any indecomposable module that appears as a summand must also appear in low degree, and so the number of isomorphism types of such summands is finite. There are also applications to invariant theory, giving a priori bounds on the degrees of the generators.
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