Affine formal groups and complete Hopf algebras

Statement

Fix a commutative base ring $k$ and an adic category in which affine formal schemes are formal spectra of complete separated $k$ -algebras with specified ideals of definition. Formal spectrum restricts to a contravariant equivalence

\left\{\text{affine adic formal groups over \(k\)}\right\} \simeq \left\{\text{admissible complete commutative Hopf \(k\)-algebras}\right\}^{\mathrm{op}}.

On the algebraic side, the comultiplication takes values in the completed tensor product, and all structure maps are continuous.

Correspondence on objects and morphisms

For an affine formal group $G=\operatorname{Spf}(A)$ , pullback of multiplication, identity, and inverse gives the complete coordinate Hopf algebra $\mathcal O(G)=A$ . Conversely, the Hopf structure maps on an admissible complete algebra $A$ dualize to the group-object diagrams on $\operatorname{Spf}(A)$ .

A formal-group homomorphism $f:G\to H$ corresponds to the continuous Hopf-algebra homomorphism

f^*:\mathcal O(H)\longrightarrow\mathcal O(G).

The reversal of arrows is the ordinary pullback-of-functions variance.

Scope

The topology and the admissible class of adic rings are part of the theorem. Forgetting them loses the completed tensor product and can make formal spectrum unavailable. The statement is affine; a non-affine formal group is encoded by a sheaf of complete coordinate algebras rather than by one global Hopf algebra.

References

Michiel Hazewinkel, Formal Groups and Applications, Pure and Applied Mathematics 78, Academic Press, 1978; AMS reprint, 2012. Publisher record. Relevant: Chapters I–II.
The Stacks Project Authors, “Formal schemes à la EGA.” Section 87.2, Tag 0AHY.