The behavior of differential forms under purely inseparable extensions
type: Item
instance of: scholarly article
title: The behavior of differential forms under purely inseparable extensions
zbMATH Open document ID: 1454.12002
author: Roberto Aravire
author: Ahmed Laghribi
author: Manuel O'Ryan
published in: Journal of Pure and Applied Algebra
publication date: 2018-09-07T00:00:00Z
review text: Let \(F\) be a field of characteristic \(2\) and let \(\Omega^n_F\) be the \(F\)-vector space of absolute Kähler \(n\)-differentials. Then there is a well-defined homomorphism \[ \wp: \Omega^n_F\to \Omega^n_F/d\Omega^{n-1}_F\] with \[ \wp (a\frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n})=\overline{(a^2-a) \frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n}}, \] whose kernel resp. cokernel is denoted by \(\nu_F(n)\) resp. \(H^{n+1}_2(F)\). In this paper, the authors determine the kernel of the natural restriction homomorphism \(H^{n+1}_2(F)\to H^{n+1}_2(L)\) for an arbitrary purely inseparable algebraic extension \(L/F\). The paper also contains the computation of the kernels of the restriction maps \(\Omega^n_F\to \Omega^n_L\) in the case of so-called modular purley inseparable extensions and some slight generalizations. It should be noted that independently, \textit{M. Sobiech} [J. Algebra 499, 151--182 (2018; Zbl 1390.11077)] computed the kernels of the maps \(H^{n+1}_p(F)\to H^{n+1}_p(L)\) for arbitrary purely inseparable algebraic extensions \(L/F\) in any positive characteristic \(p\).
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12F15
Mathematics Subject Classification ID: 12G99
Mathematics Subject Classification ID: 13N05
zbMATH DE Number: 6934126
zbMATH Keywords: differential form
zbMATH Keywords: Kato-Milne cohomology
zbMATH Keywords: purely inseparable extension
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: j.jpaa.2018.03.019
OpenAlex ID: W2796163462
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Wikidata QID: Q130034717
DOI: 10.1016/J.JPAA.2018.03.019
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description: scientific article; zbMATH DE number 6934126