Formal Groups (Lecture 11)

April 27, 2010
We begin by recalling our discussion of the Adams-Novikov spectral sequence:
Claim 1. Let X be any spectrum. Then MU∗ (X) is a module over the commutative ring L = π∗ MU, andcan therefore be understood as a quasi-coherent sheaf on the affine scheme Spec L which parametrizes formalgroup laws (here L denotes the Lazard ring). This quasi-coherent sheaf admits an action of the affine groupscheme G = Spec Z[b1 , b2 , . . .] which assigns to each commutative ring R the group {g ∈ R[[t]] : g(t) =t + b1 t2 + b2 t3 + · · · }, compatible with the action of G on Spec L by the construction(g ∈ G(R), f (x, y) ∈ FGL(R) ⊆ R[[x, y]]) 7→ gf (g −1 (x), g −1 (y)) ∈ FGL(R) ⊆ R[[x, y]]There is a spectral sequence {Erp,q , dr }, called the Adams-Novikov spectral sequence, with the following
properties. If X is connective, then {Erp,q , dr } converges to a finite filtration of πp−q X. Moreover, the groups
E2∗,q are given by the cohomology groups Hq (G; MU∗ X).
Equivalently, we can think of E2∗,q as the cohomology of the stack MsFG = Spec L/G with coefficients inthe sheaf FX determined by MU∗ (X) with its G-action.
To be more precise, we should observe that the ring L, and the ring Z[b1 , . . . , ] are all equipped a canonicalgrading. In geometric terms, this grading corresponds to an action of the multiplicative group Gm . Thisgroup acts on L by the formula(λ ∈ R× , f (x, y) ∈ FGL(R)) 7→ λf (λ−1 x, λ−1 y).
In fact, we can identify both Gm and G with subgroups of a larger group G+ , with G+ (R) = {g ∈ R[[x]] :g(t) = b0 t + b1 t2 + · · · , b0 ∈ R× }. This group can be identified with a semidirect product of the subgroup
Gm (consisting of those power series with bi = 0 for i > 0) and G (consisting of those power series withb0 = 1), and this semidirect product acts on Spec L by substitution.
For any spectrum X, MU∗ (X) is a graded L-module, and the action of G on MU∗ (X) is compatible
with the grading. In the language of algebraic geometry, this means that MUeven (X) = ⊕n MU2n (X) canbe regarded as a representation of the group G+ , compatible with the action of G+ on Spec L. In thelanguage of stacks, this means that MUeven (X) can be regarded as a quasi-coherent sheaf on the quotient
stack Spec L/G+ .
Definition 2. The quotient stack Spec L/G+ is called the moduli stack of formal groups and will be denotedby MFG .
To understand MFG , it will be useful to have a more conceptual way of thinking about formal grouplaws. Let R be a commutative ring and let f (x, y) ∈ R[[x, y]] be a formal group law over R. We let AlgR
denote the category of commutative R-algebras. We can associate to f a functor Gf : AlgR → Ab from R to
the category of abelian groups: namely, we let Gf (A) = {a ∈ A : (∃n)an = 0} ⊆ A, with the group structure
given by (a, b) 7→ f (a, b). Note that this expression makes sense: though f has infinitely many terms, if aand b are nilpotent then only finitely many terms are nonzero. We will call Gf the formal group associatedto f .
1Remark 3. The condition that f ∈ R[[x, y]] define a formal group law is equivalent to the requirement thatthe above formula defines a group structure on Gf (A) for every R-algebra A.
Suppose that we are given two formal group laws f, f 0 ∈ R[[x, y]] and an isomorphism α : Gf ' Gf 0 of
the corresponding formal groups. In particular, for every R-algebra A, α determines a bijection αA from theset {a ∈ A : a is nilpotent } with itself. To understand this bijection, let us treat the universal case where A
contains an element a such that an+1 = 0. This is the truncated polynomial ring A = R[t]/tn+1 . In this case,
α carries t to another nilpotent element, necessarily of the form b0 t+b1 t2 +. . .+bn−1 tn . Since α is functorial,
it follows that for any commutative R-algebra A containing an element a with an = 0, we have αA (a) =b0 a + b1 a2 + · · · + bn−1 an . In particular, if A = R[t]/tn , we deduce that αA (t) = b0 t + b1 t2 + . . . + bn−2 tn−1 .
In other words, the coefficients bi which appear are independent of n. We conclude that there exists a powerseries g(t) = b0 t + b1 t2 + · · · such that αA (a) = g(a) for every commutative ring a. Since αA is a bijection
for any A, we conclude that g is an invertible power series. Since αA is a group homomorphism, we deducethat g satisfies the formula f 0 (g(x), g(y)) = gf (x, y): that is, the formal group laws f and f 0 differ by thechange-of-variable g.
Definition 4. Let R be a commutative ring. An coordinatizable formal group over R is a functor G : AlgR →Ab which has the form Gf , for some formal group law f ∈ R[[x, y]].
We regard the coordinatizable formal group laws (and isomorphisms between them) as a subcategory of
the category Fun(AlgR , Ab) of functors from AlgR to abelian groups. We have just seen that this subcategory
admits a less invariant description: it is equivalent to a category whose objects are formal group lawsf ∈ R[[x, y]], and whose morphisms are maps g such that f 0 (g(x), g(y)) = gf (x, y).
The coordinatizable formal group laws over R do not satisfy descent in R. Consequently, it is convenient
to make the following more general definition:
Definition 5. Let R be a commutative ring. A formal group law over R is a functor G : AlgR → Absatisfying the following conditions:(1) The functor G is a sheaf with respect to the Zariski topology. In other words, if A is a commutative
R-algebra with a pair of elements x and y such that x + y = 1, then G(A) can be described as the1]).
subgroup of G(A[ x1 ]) × G(A[ y1 ]) consisting of pairs which have the same image in G(A[ xy(2) The functor G is a coordinatizable formal group law locally with respect to the Zariski topology. Thatis, we can choose elements r1 , r2 , . . . , rn ∈ R such that r1 +· · ·+rn = 1, such that each of the composite
functors
AlgR[ r1 ] → AlgR → Abi

has the form Gf for some formal group law f ∈ R[ r1i ][[x, y]].
By definition, the moduli stack of the formal groups MFG is the functor which assigns to each commutative
ring R the category of formal group laws over R (the morphisms in this category are given by isomorphisms).
There is a canonical map of stacks MsFG = Spec L/G → Spec L/G+ = MFG . To understand this map(and the failure of general formal groups to be coordinatizable) it is useful to introduce a definition.
Definition 6. Let G be a formal group over R. The Lie algebra of G is the abelian group g = ker(G(R[t]/(t2 )) →G(R)).
Note that if G = Gf for some formal group law f , we get a group isomorphism g ' tR[t]/(t2 ) ' R(since f (x, y) = x + y to order 2). In fact, g is not just an abelian group: for each λ ∈ R, the equation
t 7→ λt determines a map from R[t]/(t2 ) to itself, which induces a group homomorphism g → g. When G iscoordinatizable, this is the usual action of R on itself by multiplication. It follows by descent that the above
formula always determines an action of R on g. Since g ' R locally for the Zariski topology, we deduce thatg is an invertible R-module: that is, it determines a line bundle on the affine scheme Spec R.

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Proposition 7.
to R.

(1) A formal group G over R is coordinatizable if and only if its Lie algebra g is isomorphic(2) The quotient stack MsFG parametrizes pairs (G, α), where G is a formal group and α : g ' R is atrivialization of its Lie algebra.
Proof. We have already established that g ' R when G is coordinatizable. Conversely, fix an isomorphism
g ' R. After localizing Spec R, the group G becomes coordinatizable: that is, we can write G ' Gf for somef ∈ R[[x, y]]. Modifying f by the action of Gm , we may assume that this isomorphism is compatible withour trivialization of g. The trouble is that these isomorphisms might not glue. The obstruction to gluing1them determines a cocycle representing a class in HZar(Spec R, G). We claim that this group vanishes. Thisis because the group G is an iterated extension of copies of the additive group (A ∈ AlgR ) 7→ (A, +), which
has no cohomology on affine schemes.
Assertion (2) is just a translation of the following observation: if f, f 0 ∈ R[[x, y]] are formal group laws,
then an isomorphism of formal groups Gf ' Gf 0 respects the trivializations of the Lie algebras of Gf andGf 0 if and only if it is given by a power series of the form g(t) = t + b1 t2 + · · · (a power series of the formg(t) = b0 t + · · · acts on the Lie algebras by multiplication by the scalar b0 ).
We can think of the assignment (R, G) 7→ g−1 as defining a line bundle ω on the moduli stack MFG .
In fact, MsFG is just the total space of ω with the zero section removed (equivalently, the moduli stack oftrivializations of ω).
We can now be a little bit more precise about the E2 -term of the Adams-Novikov spectral sequence.
Translating our gradings into algebraic geometry, we get the following result:
Claim 8. For any spectrum X, the bordism groups MUeven (X) form a module over the Lazard ring L 'π∗ MU which carries a compatible action of the group scheme G+ , and therefore determines a sheaf Feven on
MFG = Spec L/G+ . The E2 -term of the Adams-Novikov spectral sequence satisfies
E22a,b = Hb (MFG ; Feven ⊗ω a ).
Similarly, the odd homotopy groups MUodd (X) determine a sheaf Fodd on MFG satisfying
E22a+1,b = Hb (MFG ; Fodd ⊗ω a ).
In order to exploit Claim 8, we will need to understand the structure of the moduli stack MFG . This will
be our goal in the next lecture.

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