Varieties and local cohomology for chromatic group cohomology rings
type: Item
instance of: scholarly article
title: Varieties and local cohomology for chromatic group cohomology rings
zbMATH Open document ID: 0959.55005
DOI: 10.1016/S0040-9383(98)00048-2
published in: Topology
publication date: 2001-03-21T00:00:00Z
review text: This important paper lies at the confluence of two major tributaries to the main stream of algebraic topology. The central importance of complex oriented multiplicative cohomology theories, in particular the \(2\)-periodic ones, has been evident for almost three decades. Their systematic study and applications have made significant bodies of algebra, number theory and algebraic geometry seemingly an inescapable part of the standard topologist's toolkits, particularly in the structural theory of the stable homotopy category. This may have something to do with the non-geometric nature of many of the central examples such Morava \(K\)-theories (only rational cohomology, \(K\)-theory and perhaps elliptic cohomology are known to have truly geometric constructions). However, even \(K\)-theory leads via its stable operations to many subtle number theoretic phenomena which are manifestations of this kind of structure. Equivariant topology (for finite and compact Lie groups) is now a well developed domain which casts its shadows into non-equivariant stable homotopy theory (see for example the relationships between Tate cohomology and periodicity). In this paper an interesting topic is considered which naturally uses ideas from both of these areas, namely periodic cohomology of classifying spaces of finite groups. It is worth remarking that the influential work of \textit{M. J. Hopkins, N. J. Kuhn}, and \textit{D. C. Ravenel} [Generalized group characters and complex oriented cohomology theories, J. Am. Math. Soc. 13, No. 3, 553-594 (2000)] has already considered similar questions about the \(E\)-cohomology of the classifying space a finite group, \(E^*(BG)\), but their results tend to be most useful when \(E^*\) contains an inverse for the order of \(G\), \(|G|\). The present paper addresses the more delicate issues occurring when for example \(E^*\) is \(p\)-local for a prime \(p\) dividing \(|G|\). Thus these two bodies of work are in a certain sense complementary although…
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 14B15
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 1317252
zbMATH Keywords: periodic cohomology theory
zbMATH Keywords: scheme
zbMATH Keywords: variety
author: John Greenlees
author: Neil Strickland
reviewed by: Andrew Baker
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1317252