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Let R be a discrete valuation ring with infinite residue field and X a smooth projective curve over R. Let G be a simple simply-connected group scheme over R and E a principal G-bundle over X. It is proved that E is trivial locally for the Zariski topology on X providing E is trivial over the generic point of X. The main aim of the present paper is to develop a method rather than to get a very strong concrete result.

A very short proof of an unpublished result due to O. Gabber is given. More exactly, let R be a regular local ring containing a finite field k. Let G be a simply-connected reductive group scheme over k. It is proved that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. This is the mentioned unpublished result due to O. Gabber. In this paper, this result is derived from a purely geometric one, proved in another paper of the author and stated in the Introduction.

Let R be a semilocal Dedekind domain, and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus $$ {\mathbb{G}}_{m,R} $$ . It is proved that the kernel of the map $$ {H}_{\overset{\prime }{e}t}^1\left(R,\kern0.5em G\right)\to {H}_{\overset{\prime }{e}t}^1\left(K,\kern0.5em G\right) $$ induced by the inclusion of R into K is trivial. This result partially extends the Nisnevich theorem.

A version of the lemma proved by M. Levine and F. Morel in their book “Algebraic cobordisms,” is reformulated in the Chow group context. The obtained statement turns out to be valid in any characteristic and its proof is substantially shortened.

A smooth projective morphism p : T → S to a smooth variety S is considered. In particular, the following result is proved. The total direct image Rp*(ℤ/nℤ) of the constant etale sheaf ℤ/nℤ is locally (in Zariski topology) quasiisomorphic to a bounded complex \(\mathcal{L}\) on S that consists of locally constant, constructible etale sheaves of ℤ/nℤ-modules. Bibliography: 2 titles.