# Une description fonctorielle des K-théories de Morava des 2-groupes abéliens élémentaires

Abstract : The aim of this PhD thesis is to study, from a functorial point of view, the mod $2$ Morava $K$-theories of elementary abelian $2$-groups. Namely, we study the covariant functors $V \mapsto K(n)^*(BV^{\sharp})$ for the prime $p=2$ and $n$ a positive integer. The case $n=1$, which follows directly from the work of Atiyah on topological $K$-theory, gives us a coanalytic functor which contains no non-constant polynomial sub-functor. This is very different from the case $n>1$, where the above-mentioned functors are analytic. The theory of Henn-Lannes-Schwartz provides a correspondence between analytic functors and unstable modules over the Steenrod algebra. We determine the unstable module corresponding to the analytic functor $V \mapsto K(2)^*(BV^{\sharp})$, by studying the relation between this functor and the Hopf ring structure of the homology of the $\Omega$-spectrum associated to the theory $K(2)^*(-)$.
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https://hal.archives-ouvertes.fr/tel-01565626
Contributor : Quyet Nguyen L.C. <>
Submitted on : Thursday, July 20, 2017 - 8:30:18 AM
Last modification on : Friday, May 10, 2019 - 12:14:02 PM

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• HAL Id : tel-01565626, version 1

### Citation

Quyet Nguyen L.C.. Une description fonctorielle des K-théories de Morava des 2-groupes abéliens élémentaires. Mathématiques [math]. Université d'Angers, 2017. Français. ⟨tel-01565626v1⟩

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