Cohomological kernels of mixed extensions in characteristic 2 (Q2333353)

In this paper, the authors continue their study of computing kernels of restriction maps in Kato-Milne cohomology in characteristic \(2\). 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 called Artin-Schreier map given by \(\wp: \Omega^n_F\to \Omega^n_F/d\Omega^{n-1}_F\) 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 study the kernel of the natural restriction homomorphism \(H^{n+1}_2(F)\to H^{n+1}_2(E)\) for certain algebraic extensions \(E/F\). In the sequel, let \(K/F\) be a purely inseparable extension and let \(L/F\) be a separable (bi)quadratic extension. The kernels for \(E=L\) and \(E=K\) are known (in the case \(E=K\), see the authors' paper [J. Pure Appl. Algebra 223, No. 1, 439--457 (2019; Zbl 1454.12002)] or, more generally in arbitrary positive characteristic, \textit{M. Sobiech} [J. Algebra 499, 151--182 (2018; Zbl 1390.11077)]). In another paper by the first two authors [Proc. Am. Math. Soc. 141, No. 12, 4191--4197 (2013; Zbl 1283.11065)], it is shown that the kernel for the compositum \(E=K\cdot L\) with \(K\) multiquadratic and \(L\) quadratic is obtained by adding the kernels for \(K\) and for \(L\). In the present paper, the authors show that this is still true if \(K\) is multiquadratic and \(L\) is biquadratic. They also compute the kernels for \(E=K\cdot L\) for \(K\) arbitrary purely inseparable and \(L\) quadratic, and for \(K\) simple purely inseparable and \(L\) biquadratic. It turns out that in general, the sum of the kernels for \(K\) and for \(L\) in these latter two cases will be strictly contained in the kernel for \(K\cdot L\).