Let $R$ be a ring and $A$ and $R$-algebra. A pro-infinitesimal thickening of $R$ is a pair $(D, \theta)$ such that $\theta: D \rightarrow R$ is surjective and $D$ is separated and complete for the $I:= ker \theta$-adic topology.
In Asterisque 223, Expose II, "Le corps des periodes p-adiques," Fontaine goes to define, for for $\mathfrak{p} \subset R$ an ideal and $V$ a complete, separated $R$-algebra for the $\mathfrak{p}$-adic topology, the pro-infinitesimal formal $\mathfrak{p}$-adic thickening "de maniere evidente."
Unfortunately, it's not evident to me what the definition should be. I suspect that it means $D$ is separated and complete with respect to the $(\mathfrak{p}, I)$-adic topology. But I don't understand why this is the natural definition to make.
So my first question is: what is the definition of pro-infinitesimal formal $\mathfrak{p}$-adic thickening...and why?
On that note, is the definition of thickenings is reminiscent of deformation theory. Is there a geometric motivation or context for it?
