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](https://arxiv.org/pdf/1702.00679.pdf))


$\mathcal{O}^h$ 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


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



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



Vector bundles on henselian schemes
I am just posting my comment as an answer. Reduction modulo $p$ defines a functor from the category of locally free $\mathcal{O}\_{\mathfrak{X}}$-modules to the category of locally free sheaves on the closed fiber $X\_0$. This functor is not essentially surjective. In fact, given an invertible sheaf $\mathcal{L}\_0$ on $X\_0$, already the obstruction to lift the invertible sheaf modulo $p^2$ is an element in $H^2(X\_0,\mathcal{O}\_{X\_0})$. 


For instance, given two smooth, proper, relative curves over $\mathbb{Z}\_p$, say $C$ and $D$, such that the closed fibers $C\_0$ and $D\_0$ happen to be isomorphic, for the fiber product $X=C\times\_{\text{Spec}\ \mathbb{Z}\_p} D$, for the graph $\Gamma\_0$ of the given isomorphism, for $\mathcal{L}\_0:= \mathcal{O}\_{X\_0}(\Gamma\_0)$, the obstruction class to lifting $\mathcal{L}\_0$ modulo $p^2$ is essentially the "difference of the Kodaira-Spencer classes" of the two families. There are many similar examples.


