The behavior of differential forms under purely inseparable extensions
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 \(p\).
scientific article; zbMATH DE number 6934126
scientific article