Lifting low-dimensional local systems

Let k be a field of characteristic p > 0. Denote by W-r(k) the ring of truntacted Witt vectors of length r >= 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 -> GL(d)(k) lift to a representation G -> GL(d)(W-r(k))? We work in the class of cyclotomic pairs (Definition 4.3), first introduced inDeClercq and Florence (https://arxiv.org/abs/2009.11130, 2018) 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 Z[1/p], smooth curves over algebraically closed fields, and affine schemes over 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 = infinity, 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 = 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).