Lifting low-dimensional local systems

Abstract

Let k be a field of characteristic \(p>0\). Denote by \(\mathbf {W}_r(k)\) the ring of truntacted Witt vectors of length \(r \ge 2\), 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 \(G\longrightarrow \mathrm {GL}_d(k)\) lift to a representation \(G\longrightarrow \mathrm {GL}_d(\mathbf {W}_r(k))\)? 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 \(\mathbb {Z}[\frac{1}{p}]\), 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 \(\mathbf {W}(k)\): 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).