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 $\hat{G}=Spf(\hat{\mathcal O\_{G,e}})$.


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.



Question about formal group schemes
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 $\widehat{G}$. 


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.


