# the definition of pro-infinitesimal thickenings

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?

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Why not just read ahead in Expose II and see how Fontaine uses this notion in order to figure out what the right definition should be (or if it turns out to be an irrelevant concept)? –  user30180 Dec 26 '12 at 1:06
Well, that's how I arrived at my guess for the definition, but it's nice to have confirmation, especially since I've never studied French and am not quite sure I understand everything he says. In any case, I was hoping it wouldn't be so much trouble for some expert to say it...and I'm interested in the other parts of the question too. –  Tony Dec 26 '12 at 3:11