The behavior of differential forms under purely inseparable extensions

摘要

Let F be a field of characteristic 2. In this paper we give a complete computation of the kernel of the homomorphism H-2(m+1) (F) H-2(m+1)(L) induced by scalar extension, where L/F is a purely inseparable extension (of any degree), H-2(m+1) (F) is the cokernel of the Artin-Schreier operator (sic) : Omega(m)(F) - Omega(m)(F)/d Omega(m-1)(F) given by: xdx(1)/x(1) Lambda ...Lambda dx(m)/x(m) - (x(2)- x) dx(1)/x(1) Lambda ...Lambda dx(m)/x(m) + d Omega(m-1)(F), where Omega(m)(F) is the space of absolute m-differential forms over F and d is the differential operator. Other related results are included. (C) 2018 Elsevier B.V. All rights reserved.