Milnor K-groups and Function Fields of Hypersurfaces in Positive Characteristic

Let X be an integral affine or projective hypersurface over a field F of characteristic p > 0, and let F (X) denote its function field. In a recent article, Dolphin and Hoffmann obtained an explicit description of the kernel of the natural restriction homomorphism between the rings of absolute Kähler differentials of F and F (X), respectively. In this note, we examine the possibility of deriving an analogous result for mod-p Milnor K-theory using the Bloch-Gabber-Kato theorem.