Question about formal group schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:08:50Z http://mathoverflow.net/feeds/question/84944 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84944/question-about-formal-group-schemes Question about formal group schemes Veen 2012-01-05T10:32:53Z 2012-01-06T12:27:15Z <p>If $G$ is a group scheme of finite type over a field $k$, then one can study it's Hopf Algebra if it is affine. This is clear, but now if $G$ is not affine, one seems to do the following: complete the local ring $\mathcal O_{G,e}$ of the zero point with respect to it's maximal ideal and then one gets a comultiplication on this completion. Furthermore, associate to it the formal group $\hat{G}=Spf(\hat{\mathcal O_{G,e}})$.</p> <p>As these things (I mean the non-affine case) are not clear to me, I would like to know if there is some more detailed treatment of this anywhere in the literature. I couldn't find anything very satisfying in the standard books.</p> http://mathoverflow.net/questions/84944/question-about-formal-group-schemes/85050#85050 Answer by Justin Noel for Question about formal group schemes Justin Noel 2012-01-06T12:27:15Z 2012-01-06T12:27:15Z <p>Show your product on $G$ restricts to a product on formal neighborhoods of the identity (via the Hopf algebra correspondence you mentioned these are coalgebra structures on the quotients of the powers of the maximal ideal of $\mathcal O_{G,e}$), these small group schemes form a directed system, then take the associated Ind-scheme. This is $\widehat G$ and the directed system corresponds to the inverse system defining $\widehat{\mathcal{O}}_{G,e}$. Since products commute with Ind systems, the multiplications on the formal neighborhoods define a multiplication on $\widehat{G}$. </p> <p>Dually we obtain comultiplications on each of our quotient rings, which defines a comultiplication on the pro/topological-ring $\widehat{\mathcal{O}}_{G,e}$. I don't know a particular reference for this. I learned about formal groups through Hazewinkel's book. Formal schemes you can learn about in EGA.I.10.</p>