Lifting low-dimensional local systems

Abstract

Let $k$ be a field of characteristic $p>0$. Denote by $W_r(k)$ 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 $G⟶G{L}_d(k)$ lift to a representation $G⟶G{L}_d(W_r(k))$? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in [DCF] 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 $W(k)$: 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).

to appear in Math. Z