the definition of pro-infinitesimal thickenings - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:01:19Z http://mathoverflow.net/feeds/question/117218 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117218/the-definition-of-pro-infinitesimal-thickenings the definition of pro-infinitesimal thickenings Tony 2012-12-26T00:49:23Z 2012-12-26T00:49:23Z <p>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. </p> <p>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."</p> <p>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. </p> <p>So my first question is: what is the definition of pro-infinitesimal formal $\mathfrak{p}$-adic thickening...and why?</p> <p>On that note, is the definition of thickenings is reminiscent of deformation theory. Is there a geometric motivation or context for it? </p>