# ag.algebraic geometry – Locally free modules that spread near points in closed fibers

Suppose that $$R$$ is a discrete valuation ring, with a uniformizer $$pi$$, and $$A$$ is a smooth $$R$$-algebra. Fix $$M$$ an $$A$$-module.

I’m trying to understand whether “locally free” is a property of $$A$$-modules that “lifts from closed fibers and spreads to small neighborhoods”.

Question
Assume that $$M_0 := M/(pi)$$ is a locally free module over $$A_0 := A/(pi)$$ (of possibly infinite rank. Let’s assume, however, it is of countably infinite rank).

Let $$mathfrak{p}$$ be any point of $$text{Spec}(A)$$ containing $$pi$$.

Is there an open neighborhood $$U subset text{Spec}(A)$$ of $$mathfrak{p}$$ such that $$Mvert_U$$ is $$mathcal{O}_U$$-locally free?