Soit l'algèbre polynomiale graduée à k générateurs sur le corps à deux éléments , chaque générateur étant de degré 1. En tant que cohomologie mod-2 du classifant , l'algèbre est dotée d'une structure naturelle de module sur l'algèbre de Steenrod . Dans cette Note, nous généralisons un résultat de Hưng pour le morphisme de Kameko . En appliquant ce résultat, nous montrons que la conjecture de Singer pour le transfert algébrique est vraie pour et le degré avec .
Let be the graded polynomial algebra over the prime field of two elements , in k generators , each of degree 1. Being the mod-2 cohomology of the classifying space , the algebra is a module over the mod-2 Steenrod algebra . In this Note, we extend a result of Hưng on Kameko's homomorphism . Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case and the degree with s an arbitrary positive integer.
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@article{CRMATH_2016__354_9_940_0, author = {T{\'\i}n, Nguyễn Khắc and Sum, Nguyễn}, title = {Kameko's homomorphism and the algebraic transfer}, journal = {Comptes Rendus. Math\'ematique}, pages = {940--943}, publisher = {Elsevier}, volume = {354}, number = {9}, year = {2016}, doi = {10.1016/j.crma.2016.06.005}, language = {en}, url = {http://www.numdam.org./articles/10.1016/j.crma.2016.06.005/} }
TY - JOUR AU - Tín, Nguyễn Khắc AU - Sum, Nguyễn TI - Kameko's homomorphism and the algebraic transfer JO - Comptes Rendus. Mathématique PY - 2016 SP - 940 EP - 943 VL - 354 IS - 9 PB - Elsevier UR - http://www.numdam.org./articles/10.1016/j.crma.2016.06.005/ DO - 10.1016/j.crma.2016.06.005 LA - en ID - CRMATH_2016__354_9_940_0 ER -
%0 Journal Article %A Tín, Nguyễn Khắc %A Sum, Nguyễn %T Kameko's homomorphism and the algebraic transfer %J Comptes Rendus. Mathématique %D 2016 %P 940-943 %V 354 %N 9 %I Elsevier %U http://www.numdam.org./articles/10.1016/j.crma.2016.06.005/ %R 10.1016/j.crma.2016.06.005 %G en %F CRMATH_2016__354_9_940_0
Tín, Nguyễn Khắc; Sum, Nguyễn. Kameko's homomorphism and the algebraic transfer. Comptes Rendus. Mathématique, Tome 354 (2016) no. 9, pp. 940-943. doi : 10.1016/j.crma.2016.06.005. http://www.numdam.org./articles/10.1016/j.crma.2016.06.005/
[1] A periodicity theorem in homological algebra, Math. Proc. Camb. Philos. Soc., Volume 62 (1966), pp. 365-377 (MR0194486)
[2] Modular representations on the homology of power of real projective space, Oaxtepec, 1991 (Tangora, M.C., ed.) (Contemp. Math.), Volume vol. 146 (1993), pp. 49-70 (MR1224907)
[3] Determination of , Topol. Appl., Volume 158 (2011), pp. 660-689 (MR2774051)
[4] Lambda algebra and the Singer transfer, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011), pp. 21-23 (MR2755689)
[5] The cohomology of the Steenrod algebra and representations of the general linear groups, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 4065-4089 (MR2159700)
[6] The image of Singer's fourth transfer, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 1415-1418 (MR2588792)
[7] Products of projective spaces as Steenrod modules, The Johns Hopkins University, ProQuest LLC, Ann Arbor, MI, USA, 1990 (PhD thesis 29 p., MR2638633)
[8] and , Topol. Appl., Volume 155 (2008), pp. 459-496 (MR2380930)
[9] Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2, Ann. Inst. Fourier (Grenoble), Volume 58 (2008), pp. 1785-1837 (MR2445834)
[10] The transfer in homological algebra, Math. Z., Volume 202 (1989), pp. 493-523 (MR1022818)
[11] On the action of the Steenrod squares on polynomial algebras, Proc. Amer. Math. Soc., Volume 111 (1991), pp. 577-583 (MR1045150)
[12] Cohomology Operations, Annals of Mathematics Studies, vol. 50, Princeton University Press, Princeton, NJ, USA, 1962 (MR0145525)
[13] On the hit problem for the polynomial algebra, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013), pp. 565-568 (MR3095107)
[14] On the Peterson hit problem, Adv. Math., Volume 274 (2015), pp. 432-489 (MR3318156)
[15] On the cohomology of the Steenrod algebra, Math. Z., Volume 116 (1970), pp. 18-64 (MR0266205)
[16] Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Camb. Philos. Soc., Volume 105 (1989), pp. 307-309 (MR0974986)
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