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Let R be a regular local ring, containing a finite field. Let G be areductive group scheme over R. We prove that a principal G-bundle over R istrivial, if it is trivial over the fraction field of R. In other words, if K isthe fraction field of R, then the map of pointed sets H^1_{et}(R,G) \toH^1_{et}(K,G), induced by the inclusion of R into K, has a trivial kernel.Certain arguments used in the present preprint do not work if the ring Rcontains a characteristic zero field. In that case and, more generally, in thecase when the regular local ring R contains an infinite field this result isproved in joint work due to R.Fedorov and I.Panin (see [FP]). Thus theGrothendieck--Serre conjecture holds for regular local rings containing afield.

Proof of Grothendieck--Serre conjecture on principal G-bundles over regular local rings containing a finite field