This is a question that I suspect is simply a matter of technical issues written down or clarified somewhere in the literature, but which I can't find.

Suppose we're working over an arbitrary base scheme $S$, maybe with some unspecified basic niceness assumptions. Following, e.g., the terminology used on p. 493 of Hazewinkel's tome *Formal Groups and Applications*, we can define the category of formal groups over $S$ simply as the group objects in formal schemes. Some of these formal groups' underlying spaces can be written as a relative formal spectrum, for example in the one-dimensional case $\underline{\operatorname{Spf}}_S\mathcal{O}_S[[t]]$, in which case they are called smooth by Hazewinkel. I've also variously seen the smooth ones called formal Lie groups, or even the term "formal groups" used exclusively to refer to these special ones, as in the notes of Weinstein on The geometry of Lubin–Tate spaces.

In any case, these formal groups that possess such a framing give rise to formal group laws, which is great. My question is, **what are the most natural/general conditions on a formal group that imply that it has a framing**? For example, I guess certainly at minimum it should be connected and formally smooth, and I believe this is enough if $S$ is the spectrum of a DVR, or even a local ring.

But in general it seems like the sheaf on $S$ given by the relative tangent space also needs to be free. For example, if we take a non-stacky open modular curve over $\mathbb{Z}[1/N]$ or even $\mathbb{Q}$, and look at the formal completion of the universal elliptic curve over it, I believe it can't have a framing since the corresponding vector bundle is in general nontrivial. (This point is what motivated the question, because on the bottom of p. 19 of Artin and Mazur - Formal groups arising from algebraic varieties, it seems to be claimed that a pretty general class formal groups of including the universal elliptic curve example is of Lie type, which seems false.)

Is this triviality of the relative tangent bundle sufficient? It seems like you could probably prove something like this using formal smoothness by lifting jets over increasing thickenings, but I'd love a place this is all written down.