Formal groups and formal group laws

Core idea

Formal group theory studies group multiplication in completed infinitesimal neighborhoods. A formal group is the intrinsic group object; a formal group law is its multiplication written in chosen power-series coordinates. This section uses general finite-dimensional, possibly noncommutative laws unless an entry explicitly invokes the classical one-dimensional commutative convention.

Formal power series and completion

• Formal power series ring
• Multivariable formal power series ring
• $I$-adic topology
• $I$-adic completion
• Substitution of formal power series
• Formal inverse function theorem

Laws, morphisms, and examples

• Formal group law
• One-dimensional commutative formal group law
• Morphism of formal group laws
• Additive and multiplicative formal group laws
• Formal group logarithm
• Height of a one-dimensional formal group law

Coordinate-free formal geometry

• Formal spectrum
• Formal scheme
• Formal affine space and the formal disc
• Formal group
• Formal group laws as coordinates on formal groups
• Formal completion at the identity
• Coordinate Hopf algebra of an affine formal group
• Affine formal groups and complete Hopf algebras

Formal Lie theory

• Lie algebra
• Lie algebra homomorphism
• Baker–Campbell–Hausdorff formula
• Tangent Lie algebra of a formal group
• Distribution algebra of a formal group
• Distribution algebra and universal enveloping algebra
• Equivalence between finite-dimensional Lie algebras and formal groups
• BCH group of a complete filtered Lie algebra
• Failure of tangent classification in positive characteristic

House convention

The central equivalence uses a characteristic-zero field $k$ and the category of finite-dimensional formally smooth formal groups whose pointed underlying formal scheme is a formal disc. It does not identify Lie algebras with global Lie groups, arbitrary formal schemes, or positive-characteristic formal groups.