  In this series of three articles, we study structural properties of smooth
profinite groups, a class designed to extend classical Kummer theory for
fields, with coefficients in $p$-primary roots of unity. Enhancing coefficients
to arbitrary $G$-linearized line bundles in Witt vectors, smooth profinite
groups provide a powerful formalism which sheds light on several conjectures in
Galois cohomology, Galois representations and local systems.
  In this first article, we introduce our main protagonists: cyclotomic pairs,
smooth profinite groups, Witt modules and $(G,S)$-cohomology. With this robust
axiomatic, we prove a first lifting theorem for $G$-linearized torsors under
line bundles (Theorem A). It leads, in the second article, to the proof of the
existence of mod $p^2$ liftings of mod $p$ Galois representations, of all
fields and of all dimensions (Theorem B). With this in hand, we prove in the
third article Theorem D, the smoothness theorem, stating that mod $p$
cohomology of a smooth profinite group lifts mod $p^2$, in all cohomological
degrees. In the particular case of Galois cohomology, we obtain a new proof of
the Norm Residue Isomorphism Theorem.
