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).