Lifting low-dimensional local systems

Let k be a field of characteristic $$p>0$$ . Denote by $$\mathbf {W}_r(k)$$ the ring truntacted Witt vectors length $$r \ge 2$$ , built out k. In this text, we consider following question, depending on given profinite group G. Q(G): Does every (continuous) representation $$G\longrightarrow \mathrm {GL}_d(k)$$ lift to {GL}_d(\mathbf {W}_r(k))$$ ? We work in class cyclotomic pairs (Definition 4.3), first introduced De Clercq and Florence ( https://arxiv.org/abs/2009.11130 2018) under name “smooth groups”. Using Grothendieck-Hilbert’ theorem 90, show that algebraic fundamental groups schemes are cyclotomic: spectra semilocal rings over $$\mathbb {Z}[\frac{1}{p}]$$ smooth curves algebraically closed fields, affine {F}_p$$ particular, absolute Galois fields fit into class. then give positive partial answer Q(G), for G: is positive, when $$d=2$$ $$r=2$$ When $$r=\infty $$ any 2-dimensional G stably lifts {W}(k)$$ : see Theorem 6.1. $$p=2$$ $$k=\mathbb {F}_2$$ prove same results, up dimension $$d=4$$ concrete application geometry: local systems low Zariski-locally (Corollary 6.3).