Let $X$ be a smooth and projective scheme over $\mathbf{Z}_p$.

We call $\mathfrak{X}$ the ringed space whose topological space is the topological space of the special fiber of $X$, and whose structure sheaf is the henselianization of the structure sheaf $\mathcal{O}^h$ of $X$ along $V(p\mathcal{O}_X)$.

(see for instance here)

$\mathcal{O}^h$ is a sheaf of Noetherian rings and so it’s coherent. We say a sheaf of $\mathcal{O}^h$-modules on $\mathfrak{X}$ is a **vector bundle** if it is locally free of finite local ranks and call $\text{Vect}_{\mathfrak{X}}$ the category of vector bundles on $\mathfrak{X}$.

Reduction modulo $p$ gives a functor

$$\text{Vect}_{\mathfrak{X}}\to\text{Vect}_X$$

Is this an equivalence of categories? Is it essentially surjective?

Note that if $X$ was affine the functor is only essentially surjective, but it does induce bijections on isomorphism classes. This is a property of henselian pairs.

closed fiber$X_0$ of $X$. This functor is not essentially surjective. Already for an invertible sheaf on $X_0$, there is an obstruction class in $H^2(X_0,\mathcal{O}_{X_0})$ that obstructs lifting the invertible sheaf mod $p^2$, much less to the Henselization. $\endgroup$ – Jason Starr Feb 2 at 10:47