Coordinate Hopf algebra of an affine formal group

Definition

Let $k$ be a commutative ring and let $G=\operatorname{Spf}(A)$ be an affine formal group over $k$ , where $A$ is a complete separated commutative topological $k$ -algebra. Pullback along the multiplication, identity, and inverse of $G$ gives continuous maps

\Delta:A\longrightarrow A\widehat\otimes_k A,\qquad \varepsilon:A\longrightarrow k,\qquad S:A\longrightarrow A.

Together with the multiplication and unit of $A$ , these maps make $A$ a complete commutative Hopf algebra: the ordinary tensor product in the Hopf-algebra axioms is replaced by the completed tensor product. The resulting topological Hopf algebra $A=\mathcal O(G)$ is the coordinate Hopf algebra of $G$ .

The structure identities

The group axioms pull back, in the opposite direction, to

(\Delta\widehat\otimes\operatorname{id})\Delta = (\operatorname{id}\widehat\otimes\Delta)\Delta,

(\varepsilon\widehat\otimes\operatorname{id})\Delta =\operatorname{id}_A = (\operatorname{id}\widehat\otimes\varepsilon)\Delta,

and

m_A(S\widehat\otimes\operatorname{id})\Delta =u_A\varepsilon =m_A(\operatorname{id}\widehat\otimes S)\Delta.

Thus $\Delta$ , $\varepsilon$ , and $S$ encode multiplication, identity, and inversion on $G$ , respectively. Although $A$ is commutative as an algebra, $\Delta$ need not be cocommutative; cocommutativity of $\Delta$ is equivalent to commutativity of the formal group.

Scope

This construction is affine. Non-affine formal groups require sheaves of complete coordinate algebras rather than one global Hopf algebra. Over more general bases, completed tensor products and formal spectra also require explicit adic hypotheses. The precise anti-equivalence with complete Hopf algebras and the coordinate formula for formal group laws are separate results. Finite-order duals at the augmentation form the distribution algebra.

References

Michiel Hazewinkel, Formal Groups and Applications, Pure and Applied Mathematics 78, Academic Press, 1978; AMS reprint, 2012. Publisher record. Relevant: Chapters I and II on formal groups, formal group laws, and Hopf-algebra coordinates.
Michel Demazure and Pierre Gabriel, Groupes algébriques, tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson, 1970. Relevant: the formal-group and formal-Lie-group constructions.
The Stacks Project Authors, Formal Algebraic Spaces, Section 87.2: Formal schemes à la EGA and Section 87.5: Completed tensor product.