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?