Bases in the \(\operatorname{mod}\, p\) Steenrod algebra, \(\beta\) included
The authors give monomial bases of the mod $p$ Steenrod algebra $A_p$ for $p$ odd, generalizing results of [\textit{D. Arnon}, J. Pure Appl. Algebra 96, No. 3, 215--223 (1994; Zbl 0840.55012)] for $p=2$ and extending their earlier work [J. Fixed Point Theory Appl. 17, No. 2, 341--353 (2015; Zbl 1326.55017)] to take into account the Bockstein, $\beta$. \par Recall that $A_p$ is generated by $\beta$ and $P^i$, $i \in \mathbb{N}$, subject to the Adem relations; the subalgebra $\overline{A}_p \subset A_p$ is generated by the $P^i$. This leads to the obvious notion of monomial bases of $A_p$ (respectively $\overline{A}_p $), indexed by multi-indices. \par The classical basis of admissible monomials is maximal with respect to a certain order $\epsilon L$ of the multi-indices. Here the authors use left (respectively right) lexicographic orders, $IL$ (resp. $IR$) that extend the orders $L$ (resp. $R$) used in their earlier work for $\overline{A}_p$. \par They identify explicitly the following four monomial bases, together with their determining properties: $\beta Z$, monomials minimal with respect to the $IR$-ordering; $\beta X$, minimal with respect to the $IL$-ordering; $ZA$, maximal with respect to the $IL$-ordering; $XC$, maximal with respect to the $IR$-ordering. \par They also construct a monomial basis $\beta C$ that extends the $C$-basis (due to Arnon for $p=2$ and \textit{I. Karaca} [Czech. Math. J. 55, No. 3, 699--707 (2005; Zbl 1081.55016)] for $\overline{A}_p$ with $p$ odd). They conclude by exhibiting generators of the mod $p$ cohomology of the Eilenberg-Mac Lane space $K (\mathbb{Z}/p, n)$ in terms of the basis $\beta C$, using the appropriate instability condition.
scientific article; zbMATH DE number 7047900
scientific article