Cohomological kernels of purely inseparable field extensions
Let \(F\) be a field of characteristic \(p > 0\) and \(F^p = \{\lambda^p\colon \lambda \in F\}\) the subfield of \(F\) consisting of the \(p\)-th powers of its elements. We consider finite extensions \(L\) of \(F\) that are purely inseparable and modular; this means that \(L\) is a tensor product over \(F\) of simple purely inseparable extensions, that is, for some \(x_1, \ldots , x_s \in F\) which are \(p\)-independent over \(F^p\) (i.e. \([F^p(x_1, \ldots , x_s)\colon F^p] = p^s\)), \(L = F(\alpha_1, \ldots , \alpha_s)\), where \(\alpha_i^{p^{r_i}}\), for \(r_i \in \mathbb{N}\). The exponent of \(L/F\) is defined to be the maximal \(r_i\). The paper under review is devoted to the study of the kernel \(H_{p^m}^{n+1}(L/F)\) of the homorphism \(H_{p^m}^{n+1}(F) \to H_{p^m}^{n+1}(L)\) of the Kato-Milne cohomology groups, for \(m, n, s \ge 1\). When \(m = 1\), this has been done by Sobiech, and independently, in case \(p = 2\), by Laghribi and the first and third-named authors (see [\textit{M. Sobiech}, J. Algebra 499, 151--182 (2018; Zbl 1390.11077); \textit{R. Aravire} et al., J. Pure Appl. Algebra 223, No. 1, 439--457 (2019; Zbl 1454.12002)]. When \(L\) has exponent \(1\), i.e. \(L^p \subseteq F\), the kernel is the sum of the kernels of the simple subextensions, but when \(L\) has larger exponent it is significantly more complex. The paper under review determines \(H_{p^m}^{n+1}(L/F)\), for \(L = F(\alpha )\), \(\alpha^{p^e} = x\) and all \(m, n, e \ge 1\). Whereas the results in case \(m = 1\) have been obtained upon the basis of the theory of differential forms (for a reference to this background, see Chapter 9 of the book [\textit{P. Gille} and \textit{T. Szamuely}, Central simple algebras and Galois cohomology. Cambridge: Cambridge University Press (2006; Zbl 1137.12001)], the results for \(m > 1\) also require the de Rham-Witt complex. As a corollary, when \(L/F\) is modular and \(m\) exceeds the exponent of \(L/F\), it is proved that \(H_{p^m}^2(L/F)\) equals the sum of the kernels of the simple subextensions of \(F\) in \(L\).
differential modules de Rham Witt modules
scientific article; zbMATH DE number 7229703
scientific article