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arXiv · 2111.02095

Henselianity in NIP $\mathbb{F}_p$-algebras

Will Johnson
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Abstract

We prove an assortment of results on (commutative and unital) NIP rings, especially $\mathbb{F}_p$-algebras. Let $R$ be a NIP ring. Then every prime ideal or radical ideal of $R$ is externally definable, and every localization $S^{-1}R$ is NIP. Suppose $R$ is additionally an $\mathbb{F}_p$-algebra. Then $R$ is a finite product of Henselian local rings. Suppose in addition that $R$ is integral. Then $R$ is a Henselian local domain, whose prime ideals are linearly ordered by inclusion. Suppose in addition that the residue field $R/\mathfrak{m}$ is infinite. Then the Artin-Schreier map $R \to R$ is surjective (generalizing the theorem of Kaplan, Scanlon, and Wagner for fields).

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Will Johnson. 2021-11-03. Henselianity in NIP $\mathbb{F}_p$-algebras. https://doi.org/10.2140/mt.2022.1.115

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