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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.

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  • $\begingroup$ Are you just looking for references about formal groups, or do you have a specific question in mind? $\endgroup$
    – S. Carnahan ♦
    Commented Jan 5, 2012 at 13:07
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    $\begingroup$ The same idea works for any group scheme. $\endgroup$
    – Keerthi Madapusi
    Commented Jan 5, 2012 at 14:24
  • $\begingroup$ At first, references would do... $\endgroup$
    – Veen
    Commented Jan 5, 2012 at 15:12
  • $\begingroup$ Have you done any explicit computations on your favorite nonaffine group? $\endgroup$
    – Lubin
    Commented Jan 6, 2012 at 2:45
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    $\begingroup$ You can find an account of the group law on an elliptic curve in any of the standard references, like Silverman. But I think you'll learn much more by working everything out on your own in explicit computations. Take your favorite elliptic curve... and if you don't have one, I recommend $Y^2+Y=X^3$ over the rationals, because it already has an inflection point at $(0,0)$. Take two points on the curve, get an equation for the line between'em, and continue... You'll get the group law of your nonaffine group. Now express all as series in terms of a parameter at the origin. There you are! $\endgroup$
    – Lubin
    Commented Jan 6, 2012 at 17:20

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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.

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    $\begingroup$ As an old geezer who likes doing hand computations for improving my understanding, I think OP would benefit greatly from a less abstract viewpoint. $\endgroup$
    – Lubin
    Commented Jan 6, 2012 at 17:28
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    $\begingroup$ I agree that it is extremely helpful to work out a few explicit examples by hand and I like your elliptic curve example. I think it is also handy to have an abstract viewpoint available. An even more elementary example is to work this out in an affine case such as $\mathbb{G}_m$ to obtain $\hat\mathbb{G}_m$. $\endgroup$
    – Justin Noel
    Commented Jan 6, 2012 at 22:20
  • $\begingroup$ @Justin Noel, absolutely! One must crawl before trying to run. $\endgroup$
    – Lubin
    Commented Jan 7, 2012 at 1:15

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