Let $R$ be a ring and $I$ be an ideal of $R$. Let $\widehat{R}$ be a $I$-adic completion of $R$. It is known that $\widehat{R}$ is a flat $R$-module. Is there any geometric intuition for this fact?
For example, let $R = k[X_{1}, \dots, X_{n}]$ and $I = (X_{1}, \dots, X_{n})$, so that $\widehat{R} = k[[X_{1}, \dots, X_{n}]]$. In this case, $\mathrm{Spec}(R) = \mathbb{A}_{k}^{n}$. What is $\mathrm{Spec}(\widehat{R})$, and is there any explanation for why $\mathrm{Spec}(\widehat{R})\to \mathbb{A}_{k}^{n}$ is flat?
Here is my vague intuition : You can imagine this scheme as a formal open disk around zero, and open immersion are flat.
More generally I believe you can interpret the spectrum of the completion as a "formal neighborhood" of $Z(I)$ in $\text{Spec}(R)$. There is probably something about it in Hartshorne, in the section about formal schemes.