On Singer's invariant-theoretic description of the lambda algebra: A mod \(p\) analogue
type: Item
instance of: scholarly article
title: On Singer's invariant-theoretic description of the lambda algebra: A mod \(p\) analogue
zbMATH Open document ID: 0853.55019
DOI: 10.1016/0022-4049(94)00047-M
published in: Journal of Pure and Applied Algebra
publication date: 1997-01-09T00:00:00Z
review text: The lambda algebra \(\Lambda_p\) at the prime \(p\) is known as the \(E_1\)-term of the Adams spectral sequence for computing the homotopy groups of spheres at a prime \(p\). In his paper [Trans. Am. Math. Soc. 280, 673-693 (1983; Zbl 0533.55013)] \textit{W. M. Singer} describes the lambda algebra \(\Lambda_2\) in terms of modular invariant theory at the prime 2. As the authors write, this paper gives a \(\text{mod } p\) analogue of Singer's description of the lambda algebra for an odd prime \(p\). Actually, they construct a differential coalgebra \(\Gamma^+\) which is dual to the lambda algebra \(\Lambda_p\) using modular invariants of \(\text{GL} (n, \mathbb{Z}/p)\). Associating to this, they also construct a complex \(\Gamma^+ M\) for an \(A(p)\)-module \(M\) whose homology is \(\text{Tor}^{A(p)} (\mathbb{Z}/ p,M)\) as the image of the total power. This complex has advantages: simpler differentials and the diagonal \(A(p)\)-action.
reviewed by: Katsumi Shimomura
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55Q45
zbMATH DE Number: 838659
zbMATH Keywords: lambda algebra
zbMATH Keywords: Adams spectral sequence
zbMATH Keywords: homotopy groups of spheres
zbMATH Keywords: modular invariant
author: Nguyen Hu'u Viet Hu'ng
author: Nguyễn Sum
MaRDI profile type: Publication
cites work: The mod-\(p\) lower central series and the Adams spectral sequence
cites work: Q4102036
cites work: Cohomology operations derived from modular invariants
cites work: Equivariant stable homotopy theory. With contributions by J. E. McClure
cites work: Q5343374
cites work: On the action of the Dyer-Lashof algebra in \(H*(G)\)
cites work: The homology of iterated loop spaces
cites work: The Steenrod algebra and its dual
cites work: Invariant Theory and the Lambda Algebra
cites work: A new chain complex for the homology of the Steenrod algebra
cites work: Q3670170
cites work: Classifying spaces, Steenrod operations and algebraic closure
full work available at URL: 0022-4049(94)00047-m
OpenAlex ID: W2139909178
description: scientific article; zbMATH DE number 838659