Self-maps of modules over the Steenrod algebra
type: Item
instance of: scholarly article
title: Self-maps of modules over the Steenrod algebra
zbMATH Open document ID: 0755.55009
DOI: 10.1016/0022-4049(92)90055-K
author: John H. Palmieri
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: If \(A\) is the \(\text{mod} p\) Steenrod algebra, then its dual \(A_ *\) is isomorphic (as algebra) to \(\mathbb{F}_ 2[\xi_ 1,\xi_ 2,\dots]\) when \(p=2\) and to \(\mathbb{F}_ p[\xi_ 1,\xi_ 2,\dots]\otimes E[\tau_ 0,\tau_ 1,\dots]\) when \(p\) is odd [\textit{J. W. Milnor}, Ann. Math., II. Ser. 67, 150-171 (1958; Zbl 0080.380)]. The Milnor basis of \(A\) is the dual of the monomial basis of \(A_ *\). Let \(P^ s_ t\) be the Milnor basis element dual to \(\xi^{p^ s}_ t\), and when \(p\) is odd let \(Q_ t\) be dual to \(\tau_ t\). When \(s<t\) then \((P^ s_ t)^ p=0\), and for \(p\) odd, \((Q_ t)^ 2=0\) for all \(t\). Therefore these elements are called differentials, and given any differential \(x\) and any \(A\)- module \(M\), one can define the homology of \(M\) with respect to \(x\) by: \[ H(M,P^ s_ t)={\text{ker }P^ s_ t: M\to M\over \text{im}(P^ s_ t)^{p-1}: M\to M}, \] and for \(p\) odd: \[ H(M,Q_ t)={\text{ker }Q_ t: M\to M\over\text{im }Q_ t: M\to M}. \] Given a differential \(x\), define its slope, \(\text{slope}(x)\), by \(\text{slope}(P^ s_ t)=p| P^ s_ t|/2\), and \(\text{slope}(Q_ t)=| Q_ t|\). A module \(M\) is called of type \(\langle m,n\rangle\) (written \(M\langle m,n\rangle\)) if and only if \(H(M,x)=0\) whenever \(\text{slope}(x)<m\) or \(\text{slope}(x)>n\). The main result of the paper is the following: Theorem A. Fix a differential \(x\) with \(\text{slope}(x)=m\). Given a finite module \(M=M\langle m,\infty\rangle\) with \(H(M,x)\neq 0\), there is an element \(v\in \text{Ext}^{k,km}_ A(M,M)\), for some \(k\), which is non-nilpotent under Yoneda composition. The author declares that this is a Steenrod algebra analogue of the theorem of Hopkins and Smith that any finite \(p\)-local spectrum \(X\) with \(K(n-1)(X)=0\) and \(K(n)(X)\neq 0\) has a \(v_ n\)-map, a non-nilpotent self-map that induces an isomorphism on \(K(n)\)-homology [\textit{M. J. Hopkins} and \textit{J. Smith}, Nilpotence and stable homotopy theory. II (to appear)]. The author uses…
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 61228
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: Milnor basis
zbMATH Keywords: \(A\)-module
reviewed by: Ioan Pop
MaRDI profile type: Publication
cites work: Modules over the Steenrod algebra
cites work: A vanishing theorem in homological algebra
cites work: Q3805461
cites work: Nilpotence and stable homotopy theory. II
cites work: Cohomology of sub-Hopf-algebras of the Steenrod algebra II
cites work: Q3345437
cites work: Vanishing lines for modules over the Steenrod algebra
cites work: The Steenrod algebra and its dual
cites work: Some quotient Hopf algebras of the dual Steenrod algebra
cites work: The Cohomology Algebras of Finite Dimensional Hopf Algebras
Recommended article: Q3339776
Recommended article: Commutators in the Steenrod algebra
Recommended article: Q3787139
Recommended article: Q3834724
Recommended article: On the double power operation
Recommended article: Finite complexes with A(n)-free cohomology
Recommended article: Certain properties of realizable modules over the Steenrod algebra
Recommended article: The Steenrod Algebra is a Prime Ring
Recommended article: A cochain complex associated to the Steenrod algebra
Recommended article: Q3772921
description: scientific article; zbMATH DE number 61228