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Comptes Rendus. Mathématique
Homological algebra
The mod p Margolis homology of the Dickson–Mùi algebra
Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 229-236.

Let E n =(/p) n be regarded as the translation group on itself. It is considered as a subgroup of the symmetric group 𝕊 p n on p n letters. We completely compute the mod p Margolis homology of the Dickson–Mùi algebra, i.e. the homology of the image of the restriction Res(𝕊 p n ,E n ):H * (𝕊 p n ;𝔽 p )H * (E n ;𝔽 p ) with the differential to be the Milnor operation Q j , for p an odd prime and for any n, j. The motivation for this problem is that, the Margolis homology of the Dickson–Mùi algebra plays a key role in study of the Morava K-theory K(j) * (B𝕊 m ) of the symmetric group 𝕊 m on m letters. The main tool of our work is the notion of “critical” elements. The mod p Margolis homology of the Dickson–Mùi algebra concentrates on even degrees. It is analogous to the mod 2 Margolis homology of the Dickson algebra.

Soit E n =(/p) n le groupe agissant sur lui même par les translations. On le considère comme sous-groupe du groupe symétrique 𝕊 p n en p n lettres. Dans cette note on calcule entièrement l’homologie de Margolis modulo p de l’algèbre de Dickson–Mùi, i.e. l’homologie de l’image de la restriction Res(𝕊 p n ,E n ):H * (𝕊 p n ;𝔽 p )H * (E n ;𝔽 p ) en choisissant pour différentielles les opérations de Milnor Q j , pour p un nombre premier impair et pour tout n, 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) * (B𝕊 m ) du groupe symétrique 𝕊 m en m lettres. L’outil principal de notre travail est la notion d’éléments « critiques ». L’homologie de Margolis mod p de l’algèbre de Dickson–Mùi concentre en degrés pairs. Elle est analogue à l’homologie de Margolis mod 2 de l’algèbre de Dickson.

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DOI: https://doi.org/10.5802/crmath.151
Classification: 55S05,  55S10,  55N99
Nguyễn H. V. Hưng 1

1. Department of Mathematics, HUS, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Vietnam
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Nguyễn H. V. Hưng. The mod $p$ Margolis homology of the Dickson–Mùi algebra. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 229-236. doi : 10.5802/crmath.151. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.151/

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