Nice triples in the DVR context

A theory of standard triples was invented by V. Voevodsky in [14] to construct the triangulated category of motives. Being inspired by that theory a theory of nice triples was invented in [7] in order to attack the Grothendieck–Serre conjecture and related problem. However both the mentioned theories were developed for smooth varieties over a field. In the present paper a theory of nice triples is developed for smooth schemes over a DVR. Theorem 1.4 is used in [12] as one of a major step in the proof of the Grothendieck–Serre conjecture in the constant mixed characteristic case.