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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.

Let Pk:=F2[x1,x2,,xk] be the polynomial algebra over the prime field of two elements, F2, in k variables x1,x2,,xk, each of degree 1. We are interested in the Peterson hit problem of finding a minimal set of generators for Pk as a module over the mod-2 Steenrod algebra, A. In this paper, we study the hit problem in degree (k1)(2d1), with d a positive integer. Our result implies the one of Mothebe [4,5].

Soient A l'algèbre de Steenrod mod-2 et Pk:=F2[x1,x2,,xk] l'algèbre polynomiale graduée à k générateurs sur le corps à deux éléments F2, 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 Pk, problème appelé hit problem en anglais. Dans ce but, nous étudions le hit problem en degré (k1)(2d1), avec d>0. Cette solution implique un résultat de Mothebe [4,5].

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.002
Keywords: Steenrod squares, Peterson hit problem, Polynomial algebra
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, Volume 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/

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

[2] Crabb, M.C.; Hubbuck, J.R. 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] Kameko, M. Products of projective spaces as Steenrod modules, The Johns Hopkins University, ProQuest LLC, Ann Arbor, MI, 1990 (PhD thesis 29 p., MR2638633)

[4] Mothebe, M.F. Generators of the polynomial algebra F2[x1,,xn] as a module over the Steenrod algebra, The University of Manchester, UK, 1997 (PhD thesis)

[5] Mothebe, M.F. Dimension result for the polynomial algebra F2[x1,,xn] as a module over the Steenrod algebra, Int. J. Math. Math. Sci. (2013) (MR3144989)

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

[7] Peterson, F.P. Generators of H(RP×RP) as a module over the Steenrod algebra, Abstr. Amer. Math. Soc., Volume 833 (April 1987), pp. 55-89

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

[9] Repka, J.; Selick, P. On the subalgebra of H((RP)n;F2) annihilated by Steenrod operations, J. Pure Appl. Algebra, Volume 127 (1998), pp. 273-288 (MR1617199)

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

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

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

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

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

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

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

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