  Let $k$ be a field of characteristic $p>0$. Denote by $W_r(k)$ the ring of
truntacted Witt vectors of length $r \geq 2$, built out of $k$. In this text,
we consider the following question, depending on a given profinite group $G$.
  $Q(G)$: Does every (continuous) representation $G\longrightarrow GL_d(k)$
lift to a representation $G\longrightarrow GL_d(W_r(k))$?
  We work in the class of cyclotomic pairs (Definition 4.3), first introduced
in [DCF] under the name "smooth profinite groups". Using Grothendieck-Hilbert'
theorem 90, we show that the algebraic fundamental groups of the following
schemes are cyclotomic: spectra of semilocal rings over
$\mathbb{Z}[\frac{1}{p}]$, smooth curves over algebraically closed fields, and
affine schemes over $\mathbb{F}_p$. In particular, absolute Galois groups of
fields fit into this class. We then give a positive partial answer to $Q(G)$,
for a cyclotomic profinite group $G$: the answer is positive, when $d=2$ and
$r=2$. When $d=2$ and $r=\infty$, we show that any $2$-dimensional
representation of $G$ stably lifts to a representation over $W(k)$: see Theorem
6.1. \\When $p=2$ and $k=\mathbb{F}_2$, we prove the same results, up to
dimension $d=4$.
  We then give a concrete application to algebraic geometry: we prove that
local systems of low dimension lift Zariski-locally (Corollary 6.3).

Algebraic Geometry