Cohomological kernels of mixed extensions in characteristic 2

Abstract

Let F be a field of characteristic 2, ΩFm the space of m-differential forms over F, d:ΩFm−1⟶ΩFm the differential operator, and H2m+1(F) the cokernel of the Artin-Schreier operator ℘:ΩFm⟶ΩFm/dΩFm−1 given on generators by: ℘(xdx1x1∧⋯∧dxmxm)=(x2−x)dx1x1∧⋯∧dxmxm+dΩFm−1. It was shown in [8] that given a multiquadratic purely inseparable extension L/F and a separable quadratic extension K/F, then the kernel of the natural homomorphism H2m+1(F)⟶H2m+1(K⋅L) is equal to the sum of the kernels of the extensions L/F and K/F. In this paper we extend this result to the compositum of a finite purely inseparable multiquadratic extension with a separable biquadratic extension, and we prove that this splitting property of kernels is no longer true by computing the kernel of the compositum of a separable quadratic (or biquadratic) extension with a simple purely inseparable extension of arbitrary degree.