Nilpotence relations in the mod-\(p\) Steenrod algebra
Let \(p\) be a prime and let \(\mathcal A\) denote the mod-\(p\) Steenrod algebra. Thus \(\mathcal A\) is generated as an algebra by the operations \(Sq^i\) for \(p=2\), and \(P^i,\beta\) for \(p\) odd. \textit{J. Milnor} [J. Ann. Math. (2) 67, 150--171 (1958; Zbl 0080.38003)] showed that every positive dimensional element of \(A\) is nilpotent. We say that an element \(a\) of \(\mathcal A\) has nilpotency \(k\), written \(\text{Nil}(a)=k\), if \(a^k=0\) and \(a^{k-1} \neq 0\). \textit{K. G. Monks} [ Bol. Soc. Mat. Mex., II. Ser. 37, No.1-2, 401--416 (1992; Zbl 0848.55013)] investigated \(\text{Nil}(a)\) for Milnor basis elements \(a = P(r_1, \dots, r_m)\) of \(\mathcal A\) for \(p=2\). In this paper the case \(p\) odd is studied and upper and lower bounds are obtained for \(\text{Nil}(P(r_1,\dots,r_m))\). In some cases the upper and lower bounds determine \(\text{Nil}(P(n))\) exactly, for example Theorem 1.4 of the paper proves the following result. For \(n \geq 2\), \(\text{Nil}(P(n)) = 2p\) if \(n=p\), and \(\text{Nil}(P(n)) = [p/n] + 1\) if \(n <p\).
scientific article; zbMATH DE number 1787452