On the generators of the polynomial algebra as a module over the Steenrod algebra

Võ Phúc, Đặng; Sum, Nguyễn

doi:10.1016/j.crma.2015.09.002

Topology

On the generators of the polynomial algebra as a module over the Steenrod algebra
[Sur les générateurs de l'algèbre polynomiale comme module sur l'algèbre de Steenrod]

Présenté par : the Editorial Board

Võ Phúc, Đặng ¹ ; Sum, Nguyễn ¹

¹ Department of Mathematics, Quy Nhơn University, 170 An Dương Vương, Quy Nhơn, Bình Định, Viet Nam

Comptes Rendus. Mathématique, Tome 353 (2015) no. 11, pp. 1035-1040.

Résumé
Abstract

Soient $A$ l'algèbre de Steenrod mod-2 et $P_{k} : = F_{2} [x_{1}, x_{2}, \dots, x_{k}]$ l'algèbre polynomiale graduée à k générateurs sur le corps à deux éléments $F_{2}$ , chaque générateur étant de degré 1. Nous étudions le problème suivant soulevé par F. Peterson : déterminer un système minimal de générateurs comme module sur l'algèbre de Steenrod pour $P_{k}$ , problème appelé hit problem en anglais. Dans ce but, nous étudions le hit problem en degré $(k - 1) (2^{d} - 1)$ , avec $d > 0$ . Cette solution implique un résultat de Mothebe [4,5].

Let $P_{k} : = F_{2} [x_{1}, x_{2}, \dots, x_{k}]$ be the polynomial algebra over the prime field of two elements, $F_{2}$ , in k variables $x_{1}, x_{2}, \dots, x_{k}$ , each of degree 1. We are interested in the Peterson hit problem of finding a minimal set of generators for $P_{k}$ as a module over the mod-2 Steenrod algebra, $A$ . In this paper, we study the hit problem in degree $(k - 1) (2^{d} - 1)$ , with d a positive integer. Our result implies the one of Mothebe [4,5].

Reçu le : 2015-03-15
Accepté le : 2015-09-03
Publié le : 2015-10-21

DOI : 10.1016/j.crma.2015.09.002

Mots clés : Steenrod squares, Peterson hit problem, Polynomial algebra

Affiliations des auteurs :

Võ Phúc, Đặng ¹ ; Sum, Nguyễn ¹

¹ Department of Mathematics, Quy Nhơn University, 170 An Dương Vương, Quy Nhơn, Bình Định, Viet Nam

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Võ Phúc, Đặng; Sum, Nguyễn. On the generators of the polynomial algebra as a module over the Steenrod algebra. Comptes Rendus. Mathématique, Tome 353 (2015) no. 11, pp. 1035-1040. doi : 10.1016/j.crma.2015.09.002. http://www.numdam.org/articles/10.1016/j.crma.2015.09.002/

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