Reduction modulo p of Noether’sproblem

Let R be a complete valuation ring of mixed characteristic (0, p) with algebraically closed fraction field K and residue field k.Let X/R be a smooth projective morphism.We show that if X k is stably rational, then H 3 K et (X K , ) is torsion-free.The proof uses the integral p-adic Hodge theory of Bhatt, Morrow and Scholze and the study of differential forms in positive characteristic.We then apply this result to study the Noether problem for finite p-groups.