Vol. 306, No. 2, 2020

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Cohomological kernels of purely inseparable field extensions

Roberto Aravire, Bill Jacob and Manuel O’Ryan

Vol. 306 (2020), No. 2, 385–419
Abstract

Let F be a field of characteristic p and L a finite purely inseparable extension. The kernel Hpn+1(LF) = ker(Hpn+1F Hpn+1L) has been described by Sobiech, and independently when p = 2 by Aravire, Laghribi, and O’Ryan. When L has exponent 1, the kernel is the sum of the kernels of the simple subextensions, but when L has larger exponent it is significantly more complex. This paper determines Hpmn+1(LF) for L = F(xpe) and all m,n,e 1. Whereas the results when m = 1 used the theory of differential forms, the results for m > 1 require the de Rham Witt complex. The m > 1 case clarifies why the “messy” generators in the m = 1 case arose, as they emerge from relations in the de Rham Witt complex. As a corollary, when L is modular over F and m exceeds the exponent of L, the kernel Hpm2(LF) is the sum of the kernels of the simple subextensions.

Keywords
field theory, inseparable extensions, cohomological kernels
Mathematical Subject Classification 2010
Primary: 12F15, 12G05
Secondary: 16K50
Milestones
Received: 28 July 2019
Revised: 10 January 2020
Accepted: 14 January 2020
Published: 13 July 2020
Authors
Roberto Aravire
Instituto de Ciencias Exactas y Naturales, Facultad de Ciencias
Universidad Arturo Prat
Iquique
Chile
Bill Jacob
Mathematics Department
University of California, Santa Barbara
Santa Barbara, CA
United States
Manuel O’Ryan
Instituto de Matematica y Fisica
Universidad de Talca
Avenida Lircay S/N
Talca
Chile