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arXiv · 2410.09483

Study of various categories gravitating around $(\varphi,\Gamma)$-modules

Nataniel Marquis
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Abstract

Functors involved in Fontaine equivalences decompose as extension of scalars and taking of invariants between full subcategories of modules over a topological ring equipped with semi-linear continuous action of a topological monoid. We give a general framework for these categories and the functors between them. We define the categories of \'etale projective $\mathcal{S}$-modules over $R$ to englobe categories that will correspond by Fontaine-type equivalences to finite free representations of a group. We study their preservation by base change, taking of invariants by a normal submonoid of $\mathcal{S}$ and coinduction to a bigger monoid. We define and study categories corresponding to finite type continuous representations over $\mathbb{Z}_p$ through the notions of finite projective $(r,\mu)$-d\'evissage and of topological \'etale $\mathcal{S}$-modules over $R$.

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Nataniel Marquis. 2024-10-12. Study of various categories gravitating around $(\varphi,\Gamma)$-modules. https://arxiv.org/abs/2410.09483

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