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


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.

  • 1
    $\begingroup$ 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. $\endgroup$ – Jason Starr Feb 2 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.


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