The behavior of differential forms under purely inseparable extensions

Abstract 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 ℘ : Ω F m ⟶ Ω F m / d Ω F m − 1 given by: x d x 1 x 1 ∧ ⋯ ∧ d x m x m ↦ ( x 2 − x ) d x 1 x 1 ∧ ⋯ ∧ d x m x m + d Ω F m − 1 , where Ω F m is the space of absolute m -differential forms over F and d is the differential operator. Other related results are included.