The behavior of differential forms under purely inseparable extensions

Let F be a field of characteristic 2. In this paper we give a complete computation of the kernel of the homomorphism H2m+1(F)⟶H2m+1(L) induced by scalar extension, where L/F is a purely inseparable extension (of any degree), H2m+1(F) is the cokernel of the Artin-Schreier operator ℘:ΩFm⟶ΩFm/dΩFm−1 given by: xdx1x1∧⋯∧dxmxm↦(x2−x)dx1x1∧⋯∧dxmxm+dΩFm−1, where ΩFm is the space of absolute m-differential forms over F and d is the differential operator. Other related results are included.