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

Đặng Võ Phúc; Nguyễn Sum

doi:10.1016/j.crma.2015.09.002

Topology

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

Presented by: the Editorial Board

Đặng Võ Phúc ¹; Nguyễn Sum ¹

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

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

Abstract
Résumé

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].

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].

Received: 2015-03-15
Accepted: 2015-09-03
Published online: 2015-10-21

DOI: 10.1016/j.crma.2015.09.002

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

Author's affiliations:

Đặng Võ Phúc ¹; Nguyễn Sum ¹

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

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     title = {On the generators of the polynomial algebra as a module over the {Steenrod} algebra},
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     pages = {1035--1040},
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Đặng Võ Phúc; Nguyễn Sum. On the generators of the polynomial algebra as a module over the Steenrod algebra. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1035-1040. doi : 10.1016/j.crma.2015.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.002/

References
Cited by

[1] D.P. Carlisle; R.M.W. Wood The boundedness conjecture for the action of the Steenrod algebra on polynomials, Manchester, 1990 (N. Ray; G. Walker, eds.) (Lond. Math. Soc. Lect. Note Ser.), Volume vol. 176, Cambridge University Press, Cambridge (1992), pp. 203-216 (MR1232207)

[2] M.C. Crabb; J.R. Hubbuck Representations of the homology of BV and the Steenrod algebra II, Sant Feliu de Guíxols, 1994 (Prog. Math.), Volume vol. 136, Birkhäuser Verlag, Basel, Switzerland (1996), pp. 143-154 (MR1397726)

[3] M. Kameko Products of projective spaces as Steenrod modules, The Johns Hopkins University, ProQuest LLC, Ann Arbor, MI, 1990 (PhD thesis 29 p., MR2638633)

[4] M.F. Mothebe Generators of the polynomial algebra $F_{2} [x_{1}, \dots, x_{n}]$ as a module over the Steenrod algebra, The University of Manchester, UK, 1997 (PhD thesis)

[5] M.F. Mothebe Dimension result for the polynomial algebra $F_{2} [x_{1}, \dots, x_{n}]$ as a module over the Steenrod algebra, Int. J. Math. Math. Sci. (2013) (MR3144989)

[6] T.N. Nam $A$ -générateurs génériques pour l'algèbre polynomiale, Adv. Math., Volume 186 (2004), pp. 334-362 (MR2073910)

[7] F.P. Peterson Generators of $H^{⁎} (R P^{\infty} \times R P^{\infty})$ as a module over the Steenrod algebra, Abstr. Amer. Math. Soc., Volume 833 (April 1987), pp. 55-89

[8] S. Priddy On characterizing summands in the classifying space of a group, I, Amer. J. Math., Volume 112 (1990), pp. 737-748 (MR1073007)

[9] J. Repka; P. Selick On the subalgebra of $H_{⁎} ({(R P^{\infty})}^{n}; F_{2})$ annihilated by Steenrod operations, J. Pure Appl. Algebra, Volume 127 (1998), pp. 273-288 (MR1617199)

[10] W.M. Singer The transfer in homological algebra, Math. Z., Volume 202 (1989), pp. 493-523 (MR1022818)

[11] W.M. Singer On the action of the Steenrod squares on polynomial algebras, Proc. Amer. Math. Soc., Volume 111 (1991), pp. 577-583 (MR1045150)

[12] N.E. Steenrod; D.B.A. Epstein Cohomology Operations, Ann. Math. Stud., vol. 50, Princeton University Press, Princeton, NJ, 1962 (MR0145525)

[13] N. Sum On the hit problem for the polynomial algebra, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013), pp. 565-568 (MR3095107)

[14] N. Sum On the Peterson hit problem, Adv. Math., Volume 274 (2015), pp. 432-489 (MR3318156)

[15] G. Walker; R.M.W. Wood Weyl modules and the mod 2 Steenrod algebra, J. Algebra, Volume 311 (2007), pp. 840-858 (MR2314738)

[16] R.M.W. Wood Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Camb. Philos. Soc., Volume 105 (1989), pp. 307-309 (MR0974986)

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