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 formal power series algebras.

Specifically, fixing a formal coordinate chart, then the product operation of a formal group is entirely expressed as a formal power series in two variables, satisfying conditions. This is called a formal group law, a concept that goes back to Bochner and Lazard.

Commutative formal group laws of dimension 1 notably appear in algebraic topology (originating in work by Novikov, Buchstaber and Quillen, see Adams 74, part II), where they express the behaviour of complex oriented cohomology theories evaluated on infinite complex projective space (i.e. on the classifying space $B U(1) \simeq \mathbb{C}P^\infty$). In particular complex cobordism cohomology theory in this context gives the universal formal group law represented by the Lazard ring. The height of formal groups induces a filtering on complex oriented cohomology theories called the chromatic filtration.

More recently Morel and Marc Levine consider the algebraic cobordism of smooth schemes 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 laws

Definition

An (commutative) adic ring is a (commutative) topological ring $A$ and an ideal $I \subset A$ such that

1. the topology on $A$ is the $I$-adic topology;

2. the canonical morphism

$A \longrightarrow \underset{\longleftarrow}{\lim}_n (A/I^n)$

to the limit over quotient rings by powers of the ideal is an isomorphism.

A homomorphism of adic rings is a ring homomorphism that is also a continuous function (hence a function that preserves the filtering $A \supset \cdots \supset A/I^2 \supset A/I$). This gives a category $AdicRing$ and a subcategory $AdicCRing$ of commutative adic rings.

The opposite category of $AdicRing$ (on Noetherian rings) is that of affine formal schemes.

Similarly, for $R$ any fixed commutative ring, then adic rings under $R$ are adic $R$-algebras. We write $Adic A Alg$ and $Adic A CAlg$ for the corresponding categories.

Example

For $R$ a commutative ring and $n \in \mathbb{N}$ then the formal power series ring

$R[ [ x_1, x_2, \cdots, x_n ] ]$

in $n$ variables with coefficients in $R$ and equipped with the ideal

$I = (x_1, \cdots , x_n)$

is an adic ring (def. 1).

Proposition

There is a fully faithful functor

$AdicRing \hookrightarrow ProRing$

from adic rings (def. 1) to pro-rings, given by

$(A,I) \mapsto ( (A/I^{\bullet})) \,,$

i.e. for $A,B \in AdicRing$ two adic rings, then there is a natural isomorphism

$Hom_{AdicRing}(A,B) \simeq \underset{\longleftarrow}{\lim}_{n_2} \underset{\longrightarrow}{\lim}_{n_1} Hom_{Ring}(A/I^{n_1},B/I^{n_2}) \,.$
Definition

For $R \in CRing$ a commutative ring and for $n \in \mathbb{N}$, a formal group law of dimension $n$ over $R$ is the structure of a group object in the category $Adic R CAlg^{op}$ from def. 1 on the object $R [ [x_1, \cdots ,x_n] ]$ from example 1.

Hence this is a morphism

$\mu \;\colon\; R[ [ x_1, \cdots, x_n ] ] \longrightarrow R [ [ x_1, \cdots, x_n, \, y_1, \cdots, y_n ] ]$

in $Adic R CAlg$ satisfying unitality, associativity.

This is a commutative formal group law if it is an abelian group object, hence if it in addition satisfies the corresponding commutativity condition.

This is equivalently a set of $n$ power series $F_i$ of $2n$ variables $x_1,\ldots,x_n,y_1,\ldots,y_n$ such that (in notation $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)$, $F(x,y) = (F_1(x,y),\ldots,F_n(x,y))$)

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

A 1-dimensional commutative formal group law according to def. 2 is equivalently a formal power series

$\mu(x,y) = \underset{i,j \geq 0}{\sum} a_{i,j} x^i y^j$

(the image under $\mu$ in $R[ [ x,y ] ]$ of the element $t \in R [ [ t ] ]$) such that

1. (unitality)

$\mu(x,0) = x$
2. (associativity)

$\mu(x,\mu(y,z)) = \mu(\mu(x,y),z) \,;$
3. (commutativity)

$\mu(x,y) = \mu(y,x) \,.$

The first condition means equivalently that

$a_{i,0} = \left\{ \array{ 1 & if i = 0 \\ 0 & otherwise } \right. \;\;\;\;\,, \;\;\;\;\; a_{0,i} = \left\{ \array{ 1 & if i = 0 \\ 0 & otherwise } \right. \,.$

Hence $\mu$ is necessarily of the form

$\mu(x,y) \;=\; x + y + \underset{i,j \geq 1}{\sum} a_{i,j} x^i y^j \,.$

The existence of inverses is no extra condition: by induction on the index $i$ one finds that there exists a unique

$\iota(x) = \underset{i \geq 1}{\sum} \iota(x)_i x^i$

such that

$\mu(x,\iota(x)) = x \;\;\;\,, \;\;\; \mu(\iota(x),x) = x \,.$

Hence 1-dimensional formal group laws over $R$ are equivalently monoids in $Adic R CAlg^{op}$ on $R[ [ x ] ]$.

Example

Any power series of the form $f(x) = x + a_2 x^2 + a_3 x^3 + \ldots$ in $R[ [x] ]$ has a functional or compositional inverse $f^{-1}(x)$ in the monoid $x R[ [x] ]$ under composition. Thus we may define a 1-dimensional formal group law by the formula $\mu(x, y) = f^{-1}(f(x) + f(y))$. That this is in some sense the typical way that 1-dimensional formal group laws arise is the content of Lazard's theorem.

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 $k$-algebras is considered; this category is equivalent to the category of complete cocommutative $k$-Hopf algebras.

For a generalization over operads see (Fresse).

Properties

In characteristic 0

Proposition

The quotient moduli stack $\mathcal{M}_{FG} \times Spec \mathbb{Q}$ of formal group over the rational numbers is isomorphic to $\mathbf{B}\mathbb{G}_m$, the delooping of the multiplicative group (over $Spec \mathbb{Q}$). 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 1d commutative formal group law

It is immediate that there exists a ring carrying a universal formal group law. For observe that for $\underset{i,j}{\sum} a_{i,j} x_1^i x_1^j$ an element in a formal power series algebra, then the condition that it defines a formal group law is equivalently a sequence of polynomial equations on the coefficients $a_k$. For instance the commutativity condition means that

$a_{i,j} = a_{j,i}$

and the unitality constraint means that

$a_{i 0} = \left\{ \array{ 1 & if \; i = 1 \\ 0 & otherwise } \right. \,.$

Similarly associativity is equivalently a condition on combinations of triple products of the coefficients. It is not necessary to even write this out, the important point is only that it is some polynomial equation.

This allows to make the following definition

Definition

The Lazard ring is the graded commutative ring generated by elements $a_{i j}$ in degree $2(i+j-1)$ with $i,j \in \mathbb{N}$

$L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below)$

quotiented by the relations

1. $a_{i j} = a_{j i}$

2. $a_{10} = a_{01} = 1$; $\forall i \neq 1: a_{i 0} = 0$

3. the obvious associativity relation

for all $i,j,k$.

The universal 1-dimensional commutative formal group law is the formal power series with coefficients in the Lazard ring given by

$\ell(x,y) \coloneqq \sum_{i,j} a_{i j} x^i y^j \in L[ [ x , y ] ] \,.$
Remark

The grading is chosen with regards to the formal group laws arising from complex oriented cohomology theories (prop.) where the variable $x$ naturally has degree -2. This way

$deg(a_{i j} x^i y^j) = deg(a_i,j) + i deg(x) + j deg(y) = -2 \,.$

The following is immediate from the definition:

Proposition

For every ring $R$ and 1-dimensional commutative formal group law $\mu$ over $R$ (example 2), there exists a unique ring homomorphism

$f \;\colon\; L \longrightarrow R$

from the Lazard ring (def. 3) to $R$, such that it takes the universal formal group law $\ell$ to $\mu$

$f_\ast \ell = \mu \,.$
Proof

If the formal group law $\mu$ has coefficients $\{c_{i,j}\}$, then in order that $f_\ast \ell = \mu$, i.e. that

$\underset{i,j}{\sum} f(a_{i,j}) x^i y^j = \underset{i,j}{\sum} c_{i,j} x^i y^j$

it must be that $f$ is given by

$f(a_{i,j}) = c_{i,j}$

where $a_{i,j}$ are the generators of the Lazard ring. Hence it only remains to see that this indeed constitutes a ring homomorphism. But this is guaranteed by the vary choice of relations imposed in the definition of the Lazard ring.

What is however highly nontrivial is this statement:

Theorem

(Lazard's theorem)

The Lazard ring $L$ (def. 3) is isomorphic to a polynomial ring

$L \simeq \mathbb{Z}[ t_1, t_2, \cdots ]$

in countably many generators $t_i$ in degree $2 i$.

Remark

The Lazard theorem 1 first of all implies, via prop. 3, that there exists an abundance of 1-dimensional formal group laws: given any ring $R$ then every choice of elements $\{t_i \in R\}$ defines a formal group law. (On the other hand, it is nontrivial to say which formal group law that is.)

Deeper is the fact expressed by the Milnor-Quillen theorem on MU: the Lazard ring in its polynomial incarnation of prop. 1 is canonically identieif with the graded commutative ring $\pi_\bullet(M U)$ of stable homotopy groups of the universal complex Thom spectrum MU. Moreover:

1. MU carries a universal complex orientation in that for $E$ any homotopy commutative ring spectrum then homotopy classes of homotopy ring homomorphisms $M U \to E$ are in bijection to complex orientations on $E$;

2. every complex orientation on $E$ induced a 1-dimensional commutative formal group law (prop.)

3. under forming stable homtopy groups every ring spectrum homomorphism $M U \to E$ induces a ring homomorphism

$L \simeq \pi_\bullet(M U) \longrightarrow \pi_\bullet(E)$

and hence, by the universality of $L$, a formal group law over $\pi_\bullet(E)$.

This is the formal group law given by the above complex orientation.

Hence the universal group law over the Lazard ring is a kind of decategorification of the universal complex orientation on MU.

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$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ 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 geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ 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

Quillen's theorem on MU is due to

1-Dimensional formal groups

A basic introduction is in

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

An elementary discussion of associativity of the formal group law of elliptic curves is in

• Stefan Friedl, An elementary proof of the group law for elliptic curves Stefan Friedl (arXiv:1710.00214)

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