Lifting mod $p$ representations of arithmetic fundamental groups of a non-affine scheme over a finite field of characteristic $p$

Question

Let $X$ be a geometrically irreducible scheme (not necessarily affine) over $\mathbb{F}_{p}$ and let $ \pi_{1}(X) $ be the arithmetic etale fundamental group of $ X $. Let $ \overline{\mathbb{F}}_{p} $ be an algebraic closure of $ \mathbb{F}_{p} $, $ \overline{\mathbb{Q}}_{p} $ be an algebraic closure of $ \mathbb{Q}_{p} $, and $ \overline{\mathbb{Z}}_{p} $ be the integral closure of $ \mathbb{Z}_{p} $ in $ \overline{\mathbb{Q}}_{p} $. If $\overline{\rho}:\pi_{1}(X)\to {\rm GL}_{d}(\overline{\mathbb{F}_{p}})$ is a continuous representation, then is it true that $ \overline{\rho}$ has a lift $\rho:\pi_{1}(X)\to {\rm GL}_{d}(\overline{\mathbb{Z}_{p}})$? It is true if $X$ is affine. Any references would be highly appreciated.

Answer 1

I think the answer is no. There is a construction, due to Godeaux--Serre, which shows that (*) if $\Gamma$ is any finite group, then over any field $k$ there exists a smooth geometrically connected complete intersection $X$ in $\mathbf{P}^N$ of any chosen dimension $d \geq 1$ which admits a free action of $\Gamma$. If $d \geq 2$, then a Lefschetz hyperplane theorem shows that $X_{\overline{k}}$ has trivial $\pi_1$. If $Y = X/\Gamma$, then by SGA1, Exposé X, Corollaire 2.2, we have a short exact sequence \begin{equation} 1 \to \pi_1(Y_{\overline{k}}) = \Gamma \to \pi_1(Y) \to \mathrm{Gal}(\overline{k}/k) \to 1 \end{equation} Note that (**) the $\Gamma$-cover $X \to Y$ (defined over $k$!) splits the first map in this sequence and shows $\pi_1(Y) \cong \Gamma \times \mathrm{Gal}(\overline{k}/k)$. Thus it suffices to show that there exist $\overline{\mathbf{F}}_p$-representations of finite groups which do not lift to $\overline{\mathbf{Z}}_p$-representations.

Here is an example due to Serre. Let $\Gamma = (\mathbf{Z}/p)^6$, and let $\overline{\rho}\colon \Gamma \to \mathrm{GL}_5(\overline{\mathbf{F}}_p)$ be the strictly upper-triangular representation mapping to the upper right $2 \times 3$-block. If $\overline{\rho}$ lifts to $\Gamma \to \mathrm{GL}_5(\overline{\mathbf{Z}}_p)$, then we obtain a faithful representation $\rho\colon \Gamma \to \mathrm{GL}_5(\overline{\mathbf{Q}}_p)$. Note that such a $\rho$ can be diagonalized since $\Gamma$ is abelian, but the order $p$ diagonal matrices in $\mathrm{GL}_5(\overline{\mathbf{Q}}_p)$ form a group of order $p^5$, so $\rho$ cannot be faithful.

EDIT: in response to the questions in the comments.

(*) For a complete argument, see my writeup here with Bogdan Zavyalov (this is based on section 20 here and page 51 here, neither of which applies verbatim in the present generality). Strictly speaking, the writeup there only mentions commutative $\Gamma$, but the general case is the same. The idea is as follows: first, find a projective space $\mathbf{P}^N$ with an action of $\Gamma$ which is free away from a closed subset of codimension $\geq d$. (Consider the tautological action of $\Gamma$ on $\mathbf{P}(\mathcal{O}(k[\Gamma])^n)$ for large $n$.) The naive quotient $\mathbf{P}^N/\Gamma$ is projective, so we can use Bertini's theorem to find a smooth geometrically connected complete intersection $Y$ in $\mathbf{P}^N/\Gamma$ of dimension $d$. The preimage $X$ of $Y$ in $\mathbf{P}^N$ is a complete intersection of dimension $d \geq 1$, hence geometrically connected, and $X \to Y$ is a $\Gamma$-torsor, so $X$ is smooth with free $\Gamma$-action.

(**) There is a map $\pi_1(Y) \to \pi_1(X/Y) = \Gamma$ given by restriction of automorphisms, and the restriction map $\Gamma = \pi_1(X_{\overline{k}}/Y_{\overline{k}}) = \pi_1(Y_{\overline{k}}) \to \pi_1(X/Y) = \Gamma$ is the identity.