The mod 2 Margolis homology of the Dickson algebra

Nguyễn H. V. Hưng

doi:10.5802/crmath.68

Topologie algébrique

The mod 2 Margolis homology of the Dickson algebra

Nguyễn H. V. Hưng ¹

¹ Department of Mathematics, HUS, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Vietnam

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 505-510.

Résumé
Abstract

Dans cette note on calcule entièrement l’homologie de Margolis modulo 2 de l’algèbre de Dickson $D_{n}$ , i.e. l’homologie de $D_{n}$ en choisissant pour différentielles les opérations de Milnor $Q_{j}$ , pour tous $n$ et $j$ . La motivation pour cette étude est le rôle clé joué par cette homologie dans l’étude de la K-théorie de Morava $K {(j)}^{*} (B S_{m})$ du groupe symétrique $S_{m}$ en $m$ lettres.

Nous montrons que la conjecture de Pengelley–Sinha sur $H_{*} (D_{n}; Q_{j})$ pour $n \leq j$ est vraie si et seulement si $n = 1, 2$ . Pour $3 \leq n \leq j$ notre résultat montre que la conjecture est fausse à cause de l’occurence d’éléments « critiques » $h_{s_{1}, \dots, s_{k}}$ de degré $(2^{j + 1} - 2^{n}) + \sum_{i = 1}^{k} (2^{n} - 2^{s_{i}})$ dans cette homologie pour $0 < s_{1} < \dots < s_{k} < n$ et $k > 1$ .

We completely compute the mod 2 Margolis homology of the Dickson algebra $D_{n}$ , i.e. the homology of $D_{n}$ with the differential to be the Milnor operation $Q_{j}$ , for every $n$ and $j$ . The motivation for this problem is that, the Margolis homology of the Dickson algebra plays a key role in study of the Morava K-theory $K {(j)}^{*} (B S_{m})$ of the symmetric group on $m$ letters $S_{m}$ .

We show that Pengelley–Sinha’s conjecture on $H_{*} (D_{n}; Q_{j})$ for $n \leq j$ is true if and only if $n = 1$ or $2$ . For $3 \leq n \leq j$ , our result proves that this conjecture turns out to be false since the occurrence of some “critical elements” $h_{s_{1}, \dots, s_{k}}$ ’s of degree $(2^{j + 1} - 2^{n}) + \sum_{i = 1}^{k} (2^{n} - 2^{s_{i}})$ in this homology for $0 < s_{1} < \dots < s_{k} < n$ and $k > 1$ .

Reçu le : 2020-02-24
Révisé le : 2020-05-02
Accepté le : 2020-05-04
Publié le : 2020-07-28

DOI : 10.5802/crmath.68

Classification : 55S05, 55S10, 55N99

Affiliations des auteurs :

Nguyễn H. V. Hưng ¹

¹ Department of Mathematics, HUS, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Vietnam

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The mod~2 {Margolis} homology of the {Dickson} algebra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {505--510},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {4},
     year = {2020},
     doi = {10.5802/crmath.68},
     language = {en},
}

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Nguyễn H. V. Hưng. The mod 2 Margolis homology of the Dickson algebra. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 505-510. doi : 10.5802/crmath.68. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.68/

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