Let k be a field of characteristic p>0\documentclass[12pt]{minimal}
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\begin{document}$$p>0$$\end{document}. Denote by Wr(k)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {W}_r(k)$$\end{document} the ring of truntacted Witt vectors of length r≥2\documentclass[12pt]{minimal}
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\begin{document}$$r \ge 2$$\end{document}, 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⟶GLd(k)\documentclass[12pt]{minimal}
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\begin{document}$$G\longrightarrow \mathrm {GL}_d(k)$$\end{document} lift to a representation G⟶GLd(Wr(k))\documentclass[12pt]{minimal}
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\begin{document}$$G\longrightarrow \mathrm {GL}_d(\mathbf {W}_r(k))$$\end{document}? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in De Clercq 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[1p]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}[\frac{1}{p}]$$\end{document}, smooth curves over algebraically closed fields, and affine schemes over Fp\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_p$$\end{document}. 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\documentclass[12pt]{minimal}
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\begin{document}$$d=2$$\end{document} and r=2\documentclass[12pt]{minimal}
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\begin{document}$$r=2$$\end{document}. When d=2\documentclass[12pt]{minimal}
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\begin{document}$$d=2$$\end{document} and r=∞\documentclass[12pt]{minimal}
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\begin{document}$$r=\infty $$\end{document}, we show that any 2-dimensional representation of Gstably lifts to a representation over W(k)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {W}(k)$$\end{document}: see Theorem 6.1. When p=2\documentclass[12pt]{minimal}
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\begin{document}$$p=2$$\end{document} and k=F2\documentclass[12pt]{minimal}
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\begin{document}$$k=\mathbb {F}_2$$\end{document}, we prove the same results, up to dimension d=4\documentclass[12pt]{minimal}
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\begin{document}$$d=4$$\end{document}. We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3).
