# Formal group law is a group object in ...?

A formal group law over a commutative ring $R$, (by nLab) is a sequence of power srires

$$f_1,...,f_n\in R[[x_1,...,x_n,y_1,...,y_n]]$$

such that, using the notation

$$x=(x_1,...,x_n),y=(y_1,...,y_n),f=(f_1,...,f_n)$$

we have $$f(x,f(y,z))=f(f(x,y),z)$$ $$f(x,y)=x+y+\text{higher order terms}$$

I understand vaguely the idea of a formal group law as a power series expansion of the group law of a lie group or an algebraic group (actual or hypothetical) in the neighborhood of the identity. But I would be happy to know, if only for psychological reasons, if this definition can be recovered as simply a group object in some category.

In the nLab entry about formal groups, it is written that formal group laws are one approach to formal groups, and the later is a group object in 'infinitesimal spaces', but I was unable to understand what is an infinitesimal space from the linked entry. I would appreciate if someone could explain this circle of ideas or point to the relevant literature.

• A formal group law is a formal group with chosen coordinate. A formal group is a group object in the category of formal schemes, but (for most authors) not every group object in formal schemes is a formal group. It is a group object in formal Lie varieties. This is presented, e.g., in math.bu.edu/people/jsweinst/FRGLecture.pdf (p.5) or more categorically in math.rochester.edu/people/faculty/doug/otherpapers/st-fsfg.pdf by Neil Strickland (although he calls every group object in formal schemes a formal group). Jul 13, 2014 at 8:34
• The category whose objects are natural numbers and such that the morphisms from $n$ to $m$ are the $m$-tuples of formal power series in $n$ variables, with composition given by composition of power series. Jul 13, 2014 at 18:14
• @Will: that doesn't sound right; the problem is that you can't formally compose formal power series without some condition on constant terms vanishing. But maybe that's the only issue. Jul 14, 2014 at 17:55
• @QiaochuYuan Yes, I forgot to say that the constant terms of the power series all vanish. Jul 14, 2014 at 23:06
• @LennartMeier: thank you for the explanation and references. Jul 16, 2014 at 6:01

A formal group law over a scheme $S$ is a group object in the category of framed formal schemes over $S$. Objects in this category are formal schemes $X$ over $S$ equipped with an $S$-isomorphism $X \to \operatorname{Spf} \mathscr{O}_S[[t_1,\ldots,t_n]]$ for some $n$.

There is a functor from framed formal schemes over $S$ to formal schemes over $S$, given by forgetting the $S$-isomorphism. The essential image is the category of formal Lie varieties over $S$. This functor takes formal group laws to the class of formal groups that admit a framing.

• This is a concise and precise answer to my question. Thank you! Jul 16, 2014 at 6:03

Given a group scheme one can complete at the identity to get a group object in the category of formal schemes. Every such group object in the category of formal schemes obtained in this way is equivalent to a formal group law. The power series operations for formal group laws are induced hopf-algebra-like structure on the structure sheaf of the group scheme.

If we fix the base ring then we can classify formal group laws and state that every formal group law is isomorphic to one obtained in this way. In this sense we can say that every formal group law is a group object in the category of formal schemes. The converse of this is of course not true. There are group objects in the category of formal schemes which are not just formal group laws (take a group scheme and complete along something that isn't the identity).

This is talked about in "Baby Silverman".

• Could you tell me what is ´Baby Silverman´ ？ Dec 5, 2021 at 4:37