Bases in the \(\operatorname{mod}\, p\) Steenrod algebra, \(\beta\) included
type: Item
instance of: scholarly article
title: Bases in the \(\operatorname{mod}\, p\) Steenrod algebra, \(\beta\) included
zbMATH Open document ID: 1412.55016
published in: Journal of Pure and Applied Algebra
publication date: 2019-04-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55S05
zbMATH DE Number: 7047900
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: reduced power
zbMATH Keywords: Bockstein element
zbMATH Keywords: monomial basis
zbMATH Keywords: Adem relation
author: D. Yu. Emelyanov
author: Th. Yu. Popelenskii
reviewed by: Geoffrey Powell
MaRDI profile type: Publication
full work available at URL: j.jpaa.2018.11.007
OpenAlex ID: W2900491746
cites work: Monomial bases in the Steenrod algebra
cites work: On monomial bases in the mod \(p\) Steenrod algebra
cites work: Monomial Bases in the Mod-p Steenrod Algebra
cites work: The Steenrod algebra and its dual
cites work: ON CARTAN'S THEOREM
cites work: Cohomologie modulo 2 des complexes d'Eilenberg-MacLane
cites work: Cohomology Operations (AM-50)
cites work: Generators and relations for the Steenrod algebra
DOI: 10.1016/J.JPAA.2018.11.007
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description: scientific article; zbMATH DE number 7047900