Self-maps of modules over the Steenrod algebra
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 Theorem A to give a proof of Margolis' construction to kill \(P^ s_ t\)-homology groups, at all primes [\textit{H. R. Margolis}, Spectra and the Steenrod algebra (1983; Zbl 0552.55002)].
scientific article; zbMATH DE number 61228
scientific article