# Vector bundles on henselian schemes

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.

• Reduction modulo $p$ gives a functor from locally free sheaves on $\mathfrak{X}$ to the 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. – Jason Starr Feb 2 '19 at 10:47

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.