Indecomposable A-module summands in H*V which are unstable algebras
type: Item
instance of: scholarly article
title: Indecomposable A-module summands in H*V which are unstable algebras
zbMATH Open document ID: 0686.55010
DOI: 10.1007/BF02571230
author: Hans-Werner Henn
author: Lionel Schwartz
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174164
review text: Let p be a prime and denote by A the mod p Steenrod algebra. We determine the indecomposable A-module summands of \(H*(({\mathbb{Z}}/p)^ d;{\mathbb{F}}_ p)\) which admit the structure of an unstable A-algebra. In fact, it turns out that this is equivalent to the problem of determining those indecomposable A-module summands which arise as the mod p cohomology of a space (or even a classifying space of a finite group). We reduce this problem to one in modular representation theory, namely for which d and p is the projective cover of the trivial one dimensional \(GL(d,{\mathbb{F}}_ p)\) representation \({\mathbb{F}}_ p\) a permutation module? Our solution to this latter problem makes use of the classification of subgroups of \(GL(d,{\mathbb{F}}_ p)\) acting transitively on \(({\mathbb{F}}_ p)^ d\setminus \{0\}\) and hence depends on the classification of finite simple groups (on Feit-Thompson's odd order theorem if \(p=2)\).
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 4124602
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: unstable A-algebra
MaRDI profile type: Publication
cites work: The Segal conjecture for elementary abelian \(p\)-groups
cites work: Homotopy limits, completions and localizations
cites work: G. B. Segal's Burnside ring conjecture for \((Z/2)^k\)
cites work: Q3684278
cites work: Q3922829
cites work: Solvable, flag-transitive, rank 3 collineation groups
cites work: A study of certain modular representations
cites work: Classifying spaces with injective mod p cohomology
cites work: The Categories of Unstable Modules and Unstable Algebras Over the Steenrod Algebra Modulo Nilpotent Objects
cites work: Finite permutation groups of rank 3
cites work: Stable decompositions of classifying spaces of finite abelianp-groups
cites work: Endliche Gruppen I
cites work: Q3659962
cites work: Q3952291
cites work: Sur la structure des A-modules instables injectifs
cites work: Sur les ${\scr U}$-injectifs
cites work: Sur les foncteurs dérivés de la déstabilisation
cites work: The Affine Permutation Groups of Rank Three
cites work: The Sullivan conjecture on maps from classifying spaces
cites work: Splitting \(B({\mathbb{Z}}/p)^ n\) and \(BT^ n\) via modular representation theory
Recommended article: On the classification of unstable \(H^{\ast}V\)-\(A\)-modules
Recommended article: On the indecomposable modules of algebras stably equivalent to hereditary algebras
Recommended article: Q4008601
Recommended article: Unstable modules over the Steenrod algebra revisited
Recommended article: Unstable modules over the Steenrod algebra
Recommended article: A note on indecomposable modules
Recommended article: On indecomposable modules over the Virasoro algebra
Recommended article: Q4519445
Recommended article: Q4784383
Recommended article: On the structure of indecomposable modules over Artin algebras
description: scientific article; zbMATH DE number 4124602