nLab
formal group

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Formal geometry

Group Theory

Contents

Idea

A formal group is a group object internal to infinitesimal spaces. More general than Lie algebras, which are group objects in first order infinitesimal spaces, formal groups may be of arbitrary infinitesimal order. They sit between Lie algebras and finite Lie groups or algebraic groups.

Since infinitesimal spaces are typically modeled as formal duals to algebras, formal groups are typically conceived as group objects in formal duals to power series algebras.

Formal group laws

One of the oldest formalisms is the formalism of formal group laws (early study by Bochner and Lazard), which are a version of representing a group operation in terms of coefficients of the formal power series rings. A formal group law of dimension nn is given by a set of nn power series F iF_i of 2n2n variables x 1,,x n,y 1,,y nx_1,\ldots,x_n,y_1,\ldots,y_n such that (in notation x=(x 1,,x n)x=(x_1,\ldots,x_n), y=(y 1,,y n)y=(y_1,\ldots,y_n), F(x,y)=(F 1(x,y),,F n(x,y))F(x,y) = (F_1(x,y),\ldots,F_n(x,y)))

F(x,F(y,z))=F(F(x,y),z) F(x,F(y,z))=F(F(x,y),z)
F i(x,y)=x i+y i+higherorderterms F_i(x,y) = x_i+y_i+\,\,higher\,\,order\,\,terms

Formal group laws of dimension 11 proved to be important in algebraic topology, especially in the study of cobordism, starting with the works of Novikov, Buchstaber and Quillen; among the generalized cohomology theories the complex cobordism is characterized by the so-called universal group law; moreover the usage is recently paralleled in the theory of algebraic cobordism of Morel and Levine in algebraic geometry. Formal groups are also useful in local class field theory; they can be used to explicitly construct the local Artin map according to Lubin and Tate.

Formal group schemes

Much more general are formal group schemes from (Grothendieck)

Formal group schemes are simply the group objects in a category of formal schemes; however usually only the case of the formal spectra of complete kk-algebras is considered; this category is equivalent to the category of complete cocommutative kk-Hopf algebras.

Formal groups over an operad

For a generalization over operads see (Fresse).

Properties

In characteristic 0

Proposition

The quotient moduli stack FG×Spec\mathcal{M}_{FG} \times Spec \mathbb{Q} of formal group over the rational numbers is isomorphic to B𝔾 m\mathbf{B}\mathbb{G}_m, the delooping of the multiplicative group (over SpecSpec \mathbb{Q}). This means that in characteristic 0 every formal group is determined, up to unique isomorphism, by its Lie algebra.

For instance (Lurie 10, lecture 12, corollary 3).

Universal formal group

Relation to complex oriented cohomology theories

Examples

Formal geometry is closely related also to the rigid analytic geometry.

(nlab remark: we should explain connections to the Witt rings, Cartier/Dieudonné modules).

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

General

  • Alexander Grothendieck et al. SGA III, vol. 1, Expose VIIB (P. Gabriel) ETUDE INFINITESIMALE DES SCHEMAS EN GROUPES (part B) 474-560
  • Benoit Fresse, Lie theory of formal groups over an operad, J. Alg. 202, 455–511, 1998, doi

Quillen's theorem on MU is due to

1-Dimensional formal groups

A basuc introduction is in

  • Carl Erickson, One-dimensional formal groups (pdf)

See also

  • Takeshi Torii, One dimensional formal group laws of height NN and N1N-1, PhD thesis 2001 (pdf)

  • Takeshi Torii, On Degeneration of One-Dimensional Formal Group Laws and Applications to Stable Homotopy Theory, American Journal of Mathematics Vol. 125, No. 5 (Oct., 2003), pp. 1037-1077 (JSTOR)

Revised on May 18, 2014 08:19:28 by Urs Schreiber (89.204.135.40)