REALISING FORMAL GROUPS
N. P. STRICKLAND
Abstract.We show that a large class of formal groups can be realised fun*
*c-
torially by even periodic ring spectra.
1. Introduction
Let FG be the category of formal groups (of the sort usually considered in a*
*lge-
braic topology) over affine schemes. Thus, an object of FG consists of a pair (*
*G, S),
where S is an affine scheme, G is a formal group scheme over S, and a coordinate
x can be chosen such that OG ' OS[[x]] as OS-algebras. A morphism from (G0, S0)
to (G1, S1) is a commutative square
G0 ---~p-!G1
?? ?
y ?y
S0 ----!p S1
such that the induced map G0 -!p*G1 is an isomorphism of formal group schemes
over S0.
Next, recall that an even periodic ring spectrum is a commutative and associ*
*ative
ring spectrum E such that E1 = 0 and E2 contains a unit (which implies that E '
2E as spectra). Here we are using the usual notation Ek = Ek(point) = ß-kE.
We write EPR for the category of even periodic ring spectra.
Given an even periodic ring spectrum E, we can form the scheme SE := spec(E0)
and the formal group scheme GE = spf(E0CP 1) over SE . This construction gives
rise to a functor : EPR op-! FG.
It is a natural problem to try to define a realisation functor R: FG -! EPR *
*op
with R(G, S) ' (G, S), or at least to do this for suitable subcategories of FG*
* .
For example, if we let LFG denote the category of Landweber exact formal groups,
and put LEPR = {E 2 EPR | (E) 2 LFG }, one can show that the functor
: LEPR op -! LFG is an equivalence; this is essentially due to Landweber, but
details of this formulation are given in [4, Proposition 8.43]. Inverting this*
* gives
a realisation functor for LFG , and many well-known spectra are constructed usi*
*ng
this. In particular, this gives various different versions of elliptic cohomolo*
*gy, based
on various universal families of elliptic curves over rings such as Z[1_6, c4, *
*c6].
In Definition 3.13, we will introduce a full subcategory category GFG FG *
*of
ög od" formal groups. Among many other things, this contains all formal groups
over fields of characteristic not equal to 2, and all formal groups over rings *
*of the
____________
Date: November 5, 2002.
1991 Mathematics Subject Classification. 55N20,55N22.
Key words and phrases. generalized cohomology, formal group .
1
2 N. P. STRICKLAND
form Z[1=n] with n an even integer (this is proved as Theorem 3.14). Most formal
groups in GFG are not Landweber exact. Our main result is as follows.
Theorem 1.1. There is a realisation functor R: GFG -! EPR , with R '
1: GFG -!GFG .
The results of [5] give all the required objects in EPR ; the new content of*
* the
theorem is the analysis of morphisms. Here we explain the formal part of the
construction; in Section 4 we will give additional details and prove that we ha*
*ve
the required properties. The functor R actually arises as UV -1for a pair of fu*
*nctors
GFG -V E U-!EPR in which V isWan equivalence. The definition of E involves the
periodic bordism spectrum MP = n2Z 2nMU. We will verify in the appendix
that this can be constructed as a strictly commutative ring, so we have a topol*
*ogical
category Mod 0 of MP -modules. We write DMod 0 for the derived category, and
EPA 0 for the category of even periodic commutative ring objects in DMod 0. The
unit map j :S -! MP gives a functor j*: EPA 0-! EPR , and the objects of the
category E are the objects E 2 EPA 0for which the associated formal group law is
"stably realisable" in a sense to be explained later. The morphism set E(E0, E1)
is a subset of EPR (j*E0, j*E1), the functor V :E -! GFG is given by , and the
functor U :E -!EPR is given by j*. We say that a map f :j*E0 -!j*E1 in EPR is
good if there is a commutative ring object A in the derived category of MP ^ MP*
* -
modules together with maps f0:E0 -!(1 ^ j)*A and f00:(j ^ 1)*A -!E1 in EPA 0
such that f00is an equivalence and f is equal to the composite
*f0 j*f00
j*E0 j---!(j ^ j)*A ---! j*E1.
The morphisms in the category E are just the good maps. To prove Theorem 1.1,
we need to show that
(3)The composite of two good maps is good, so E really is a category.
(2)For any map (j*E0) -! (j*E1) of good formal groups, there is a unique
good map j*E0 -!j*E1 inducing it, so that V is full and faithful.
(1)For any good formal group (G, S) there is an object E 2 EPA 0such that
(j*E) ' (G, S), so V is essentially surjective.
To prove statement (k), we need to construct modules over the k-fold smash power
of MP . It will be most efficient to do this for all k, and use cosimplicial id*
*eas to
organize the functors between the various module categories.
Almost everywhere in this paper, we will assume that 2 has been inverted; we*
* do
not know whether there are any good formal groups in which this is not the case.
We found in [5] that when 2 is not inverted, there are nontrivial obstructions *
*to
various realisation problems, which can only be calculated with considerable la*
*bour.
We may return to this in future work.
2. Preliminaries
2.1. Differential forms. Let (G, S) be a formal group, and let I OG be the
augmentation ideal. Recall that the cotangent space of G at zero is the module
!G = I=I2. If x is a coordinate on G that vanishes at zero, then we write dx for
the image of x in I=I2, and note that !G is freely generated over OS by dx. We
REALISING FORMAL GROUPS 3
define a graded ring D(G, S)* by
(
0 if k is odd
D(G, S)k = (-k=2)
!G if k is even.
Here the tensor products are taken over OS, and !Gn means the dual of !G|n|
when n < 0. Where convenient, we will convert to homological gradings by the
usual rule: D(G, S)k = D(G, S)-k.
Now let E be an even periodic ring spectrum with (E) = (G, S). We then
have OG = E0CP 1 and I = Ee0CP 1 and one checks easily that the inclusion
S2 = CP 1-! CP 1 gives an isomorphism !G = I=I2 = eE0S2 = E-2. Using the
periodicity of E, we see that this extends to a canonical isomorphism D( (E))* '
E*.
It also follows from this analysis (or from more direct arguments) that a map
f :E0 -!E1 in EPR is a weak equivalence if and only if ß0f is an isomorphism.
2.2. Groups and laws. We define a category FGL as follows. The objects are
pairs (R, F ), where R is a commutative ring, and F is a formal group law over *
*R.
A morphism from (R0, F0) to (R1, F1) consists of a ring homomorphism OE: R0 -!
R1, together with a formal power series f(x) 2 R1[[x]] satisfying f(0) = 0 and
f0(0) 2 Rx1and
(OE*F0)(f(x), f(y)) = f(F1(x, y)) 2 R1[[x, y]].
Here OE*F0 means the FGL over R1 obtained by applying OE to the coefficients of
F0. The composition is given by (OE1, f1) O (OE0, f0) = (OE1OE0, f2), where f2*
*(x) =
(OE1*f0)(f1(x)).
Given (R, F ) 2 FGL we can make R[[x]] into a Hopf algebra over R (in a suit*
*ably
completed sense) by defining
_(x) = F (x 1, 1 x) 2 R[[x 1, 1 x]] = R[[x]]b R[[x]].
This makes the formal scheme spec(R) x bA1= spf(R[[x]]) into a formal group over
spec(R). This construction gives a functor Q: FGL op-! FG , which is easily seen
to be full and faithful.
If (G, S) 2 FG then we can choose a coordinate x on G that vanishes at zero,
and we find that there is a unique FGL F over OS such that
_(x) = F (x 1, 1 x) 2 OG2 = OS[[x 1, 1 x]].
We can regard x as a map G -!Ab1, and as such it induces an isomorphism G -!
Q(OS, F ). It follows that Q is essentially surjective and thus an equivalence.
2.3. Periodic bordism. Consider the homology theory MP*(X) = MU*(X)
Z[u, u-1], where u has homological degreeW2 (and thus cohomological degree -2).
This is represented by the spectrum MP = n2Z 2nMU, with an evident ring
structure. It is well-known that MU is an E1 ring spectrum; see for example [3,
Section IX]. It is also shown there that MU is an H21ring spectrum, which means
(as explained in [3, Remark VII.2.9]) that MP is an H1 ring spectrum; this is
weaker than E1 in theory, but usually equivalent in practise. As one would expe*
*ct,
MP is actually an E1 ring spectrum; a proof is given in the appendix. It follo*
*ws
from [2, Proposition II.4.3] that one can construct a model for MP that is a st*
*rictly
4 N. P. STRICKLAND
commutative ring spectrum (or "S-algebra"). We may also assume that it is a
cofibrant object in the category of all strictly commutative ring spectra.
For typographical convenience, we write MP (r) for the (r +1)-fold smash power
MP ^ . .^.MP , which is again a strictly commutative ring. The spectra MP (r)
fit together into a cosimplicial object in the usual way. In the category of st*
*rictly
commutative ring spectra, the coproduct is the smash product. It follows formal*
*ly
that the smash product of cofibrant objects is cofibrant, so in particular the *
*objects
MP (r) are all cofibrant.
The formal group (MP ) has a canonical coordinate ex, giving a canonical for*
*mal
group law eFand an isomorphism Q(ß0MP, eF) = (MP ). By a well-known theorem
of Quillen, this formal group law is universal: for any FGL F over any ring R, *
*there
is a unique map c[F ]: ß0MP -! R carrying eFto F . In this situation, we use the
notation (R, F ) to refer to R with the ß0MP -algebra structure obtained from c*
*[F ].
Next consider MP0MP = ß0MP (1). There are two maps 1 ^ j, j ^ 1: MP -!
MP (1), giving rise to formal group laws eF0, eF1over ß0MP (1). These arise fro*
*m two
different coordinates on the same formal group, so there is a canonical isomorp*
*hism
eb:eF1-!eF0defined by a power series eb(x) = P k 0ebkxk+1 2 (ß0MP (1))[[x]]. It*
* is
well-known that this is universal in the following sense: given any ring R and *
*any
isomorphism b: F1 -!F0 of formal group laws over R, there is a unique ring map
c[F0- b F1]: ß0MP (1) -! R carrying eFito Fi and ebto b. (This can be deduced
from the corresponding result for MU, but note that in our periodic setting we *
*get
all isomorphisms, not just the strict ones.) We use the notation (R, F0- bF1) to
refer to R with the ß0MP (1)-algebra structure obtained from this map. It is cl*
*ear
by construction that
- bF ]
(ß0MP -ß0(1^j)----!ß0MP (1) c[F0---1---!R)= c[F0]
- bF ]
(ß0MP -ß0(j^1)----!ß0MP (1) c[F0---1---!R)= c[F1].
The same circle of ideas shows that ß0MP (1) = ß0MP [eb0,eb1, . .].[eb-10] (whe*
*re
ß0MP (1) is regarded as an algebra over ß0MP using the map j ^ 1).
More generally, we find that ß0MP (r) is the universal example of a ring equi*
*pped
with r + 1 different formal group laws (F0, . .,.Fr) and a chain of isomorphisms
F0- . .-. Fr between them, and that ß0MP (r) is obtained from a polynomial
algebra on countably many generators over ß0MP by inverting r of the generators.
One also checks that ß1MP (r) = 0 for all r. The module ß2MP (r) is free of
rank one over ß0MP (r), but there are a number of different natural choices of
generator. The most natural way around this is to use the formalism of Section *
*2.1,
and regard ß2MP (r) as !GMP(r).
2.4. Module categories. We write Mod rfor the category of MP (r)-modules (in
the strict sense, not the homotopical one). Note that a map f :A0 -!A1 of stric*
*tly
commutative ring spectra gives a functor f* :Mod A1 -!Mod A0, which is just the
identity on the underlying spectra (and thus preserves weak equivalences). It f*
*ollows
easily that for any two maps A0 f-!A1 g-!A2, the functor f*g* is actually equal
(not just naturally isomorphic or naturally homotopy equivalent) to (gf)*. Thus,
the categories Mod rfit together to give a simplicial category Mod *.
REALISING FORMAL GROUPS 5
Remark 2.1. For us, a simplicial category means a simplicial object in the cate*
*gory
of categories. Elsewhere in the literature, the same phrase is sometimes used *
*to
refer to categories enriched over the category of simplicial sets, which is a r*
*ather
different notion.
Next, we write DMod r the derived category of Mod r, as in [2, Chapter III]. *
*As
usual, there are two different models for a category such as DMod r:
(a) One can take the objects to be the cofibrant objects in Mod r, and mor-
phisms to be homotopy classes of maps; or
(b) One can use all objects in Mod r and take morphisms to be equivalence
classes of öf rmal fractions", in which one is allowed to invert weak eq*
*uiv-
alences.
We will use model (b). This preserves the strong functorality mentioned previou*
*sly,
and ensures that DMod * is again a simplicial category.
We also write EPA rfor the category of even periodic commutative ring objects
in DMod r, giving another simplicial category. (Note that periodicity is actua*
*lly
automatic, because MP (r) is itself periodic.) We reserve the letter j for the *
*unit
map S -! MP . We generally write i :MP -! E for the unit map of an object in
EPA 0, and , for unit maps in EPA 1.
If E 2 EPA 0then the canonical coordinate exon (MP ) gives a coordinate i*ex*
*on
(j*E) and thus a formal group law F over ß0E, with c[F ] = ß0i :ß0MP -! ß0E,
and a canonical isomorphism (j*F ) ' Q(ß0E, F ). A map f :j*E0 -! j*E1
in EPR gives a map (f): (j*E1) -! (j*E0) in FG , corresponding to a map
(ß0f, b): (ß0E0, F0) -! (ß0E1, F1) in FGL ; the power series b 2 (ß0E1)[[t]] is*
* char-
acterised by the fact that f*x0 = b(x1) in E01CP 1.
3. Basic realisation results
Let R be a strictly commutative ring spectrum that is even and periodic, such
that R0 is an integral domain. The main examples will be R = MP (r) for r 0. *
*Let
D be the derived category of R-modules, and let R be the category of commutative
ring objects A 2 D such that ß1A = 0 and ß0A has no 2-torsion. We also write
R0 for the category of commutative algebras over ß0R without 2-torsion. We say
that an object A0 2 R0 is strongly realisable if there exists an object A 2 R a*
*nd
an isomorphism A0 -!ß0A such that for any B 2 R, the induced map R(A, B) -!
R0(A0, ß0B) is an isomorphism.
The results of [5] provide a good supply of strongly realisable rings, excep*
*t that
we need a little translation between the even periodic framework and the usual
graded framework. Suppose that A0 2 R0, and put T = spec(A0). We have a unit
map j :ß0R -!A0 and thus a map spec(j): T -! SR ; we can pull back the formal
group GR along this to get a formal group H := spec(j)*GR over T . From this
we get a map j*: R* = D(GR , SR )* -! D(H, T )*, which agrees with j in degree
zero. Indeed, if we choose a generator u of R2 over R0, then j* is just the map
R0[u, u-1] -! A0[u, u-1] obtained in the obvious way from j. It is easy to check
that A0 is strongly realisable (as defined in the previous paragraph) iff D(H, *
*T )* is
strongly realisable over R* (as defined in [5]).
Definition 3.1. A short ordinal is an ordinal ~ of the form n.! + m for some
n, m 2 N. A regular sequence in a ring R0 is a system of elements (xff)ff<~for *
*some
short ordinal ~ such that xffis not a zero-divisor in the ring (S-1R0)=(xfi| fi*
* < ff).
6 N. P. STRICKLAND
An object A0 2 R0 is a localised regular quotient (or LRQ) of R0 if A0 = (S-1R0*
*)=I
for some subset S R0 and some ideal I S-1R0 that can be generated by a
regular sequence.
Remark 3.2. We have made a small extension of the usual notion of a regular
sequence, to ensure that any LRQ of an LRQ of R0 is itself an LRQ of R0; see
Lemma 3.9.
Proposition 3.3. If A0 is an LRQ of R0 in which 2 is invertible, then it is str*
*ongly
realisable.
Proof.This is essentially [5, Theorem 2.6], translated into a periodic setting *
*as
explained above. Here we are using a slightly more general notion of a regular
sequence, but all the arguments can be adapted in a straightforward way. The
main point is that any countable limit ordinal has a cofinal sequence, so homot*
*opy
colimits can be constructed using telescopes in the usual way. Andrey Lazarev
has pointed out a lacuna in [5]: it is necessary to assume that the elements x*
*ff
are all regular in S-1R0 itself, which is not generally automatic. However, we *
*are
assuming that R0 is an integral domain so this issue does not arise.
Proposition 3.4. Suppose that
o A and B are strong realisations of A0 and B0
o The ring A0 R0 B0 has no 2-torsion
o The natural map A0 R0 B0 -!(A ^R B)0 is an isomorphism.
Then A ^R B is a strong realisation of A0 R0 B0.
Proof.This follows from [5, Corollary 4.5].
Proposition 3.5. If A0 2 R0 is strongly realisable, and B0 is an algebra over A0
that is free as a module over A0, then B0 is also strongly realisable.
Proof.This follows from [5, Proposition 4.13].
Proposition 3.6. Suppose that R0 is a polynomial ring in countably many variabl*
*es
over Z, that A0 2 R0, and that A0 = Z[1=n] as a ring (for some n). Then A0 is
an LRQ of R0, and thus is strongly realisable if n is even.
Proof.Choose a system of polynomial generators {xk | k 0} for R0 over Z. Put
ak = j(xk) 2 A0 = Z[1=n] and yk = xk - ak 2 R0[1=n]. It is clear that R0[1=n] =
Z[1=n][yk | k 0], that the elements yk form a regular sequence generating an *
*ideal
I say, and that A0 = R0[1=n]=I.
Proposition 3.7. Suppose that R0 is a polynomial ring in countably many variabl*
*es
over Z, that A0 2 R0, and that A0 is a field (necessarily of characteristic dif*
*fer-
ent from 2). Then A0 is a free module over an LRQ of R0, and thus is strongly
realisable.
Proof.For notational simplicity, we assume that A0 has characteristic p > 2; the
case of characteristic 0 is essentially the same.
Choose a set X of polynomial generators for R0 over Z. Let K be the subfield
of A0 generated by the image of j, or equivalently by j(X). We can choose a
subset Y X such that j(Y ) is a transcendence basis for K over Fp. This means
that the subfield L0 of K generated by j(Y ) is isomorphic to the rational func*
*tion
field Fp(Y ), and that K is algebraic over L0. Put S = Z[Y ] \ (pZ[Y ]), so L0*
* =
REALISING FORMAL GROUPS 7
(S-1Z[Y ])=p. Next, list the elements of X \ Y as {x1, x2, . .}., and let Lk be*
* the
subfield of K generated by {xi| i k}. (We will assume that X \ Y is infinite;*
* if
not, the notation changes slightly.) As xk is algebraic over Lk-1, there is a m*
*onic
polynomial fk(t) 2 Lk-1[t] with Lk = Lk-1[xk]=fk(xk). As Lk-1 is a quotient of
the ring Pk-1 := S-1Z[Y, x1, . .,.xk-1], we can choose a monic polynomial gk(t)*
* 2
Pk-1[t] lifting fk, and put zk := gk(xk) 2 Pk S-1R0. It is not hard to check *
*that
the sequence (p, z1, z2, . .).is regular in S-1R0, and that (S-1R0)=(zi| i > 0)*
* = K,
so K is an LRQ of R0. It is clear that A0 is free over the subfield K.
Definition 3.8. Let A0 be a commutative algebra over ß0MP with no 2-torsion.
Regard ß0MP as a subring of ß0MP (r) via any one of the r + 1 obvious ring maps
j0, . .,.jr: MP -! MP (r). We say that A0 is stably realisable if, given any map
ß0MP (r) -! A0 extending the given map ß0MP -! A0, the resulting ß0MP (r)-
algebra is strongly realisable. (Using the symmetric group action on MP (r), we
see that this does not depend on which ji we use.)
Lemma 3.9. An LRQ of an LRQ is an LRQ.
Proof.Suppose that B = (S-1A)=(xff| ff < ~) and C = (T -1B)=(yfi| fi < ~),
where ~ and ~ are short ordinals, and the x and y sequences are regular in S-1A
and T -1B respectively. Let T 0be the set of elements of A that become invertib*
*le
in T -1B; clearly S T 0and T -1B = ((T 0)-1A)=(xff| ff < ~). As (T 0)-1A is a
localisation of S-1A and localisation is exact, we see that x is a regular sequ*
*ence
in (T 0)-1A as well. After multiplying by suitable elements of T 0if necessary,*
* we
may assume that yfilies in the image of A (this does not affect regularity, as *
*the
elements of T 0are invertible). We then put zff= xfffor ff < ~, and let z~+fibe
any preimage of yfiin A for 0 fi < ~. This gives a regular sequence in (T 0)-*
*1A
indexed by ~ + ~, such that C = ((T 0)-1A)=(zfl| fl < ~ + ~) as required.
Proposition 3.10. If A0 is an LRQ of ß0MP in which 2 is invertible, then A0 is
stably realisable.
Proof.We know from Section 2.3 that ß0MP (r) is a polynomial ring in countably
many variables over ß0MP , in which r of the variables have been inverted, so we
can write
ß0MP (r) = ß0MP [x1, x2, . .].[x-11, . .,.x-1r].
Suppose that A0 = (S-1ß0MP )=I. Put
B0 = A0[x1, x2, . .].[x-11, . .,.x-1r],
which is evidently an LRQ of ß0MP (r). Let f :ß0MP (r) -! A0 be any map
extending the given map ß0MP -!A0, and put ak = f(xk) 2 A0, and yk =
xk - ak 2 B0. Clearly B0 is a localisation of A0[yk | k > 0], the sequence of y*
*'s
is regular in B0, and B0=(yk | k > 0) = A0 as ß0MP (r)-algebras. It follows that
A0 is an LRQ of an LRQ, and thus an LRQ, over ß0MP (r). It is thus strongly
realisable as required.
Corollary 3.11. If A0 is an algebra over ß0MP that is just Z[1=n] as a ring for
some even integer n, then A0 is stably realisable.
Proposition 3.12. Let A0 be an algebra over ß0MP that is a field of characteris*
*tic
not equal to 2. Then A0 is stably realisable.
Proof.This is immediate from Proposition 3.7.
8 N. P. STRICKLAND
It will be convenient to restate the above results in slightly different lang*
*uage.
Definition 3.13. Let F be a formal group law over a ring R. We say that F is
strongly (resp. stably) realisable if R, regarded as an algebra over ß0MP by the
classifying map of F , is strongly (resp. stably) realisable.
Let (G, S) be a formal group. We say that a coordinate x on G is strongly
(resp. stably) realisable if the associated formal group law over OS is strong*
*ly
(resp. stably) realisable. We say that G is good if it admits a stably realis*
*able
coordinate. We write GFG for the category of good formal groups.
In these terms, our results give:
Theorem 3.14. Suppose that R is a field of characteristic not equal to 2, or th*
*at
R = Z[1=n] for some even integer n. Then every formal group law over R is stably
realisable, so every formal group over spec(R) is good.
4. Proof of the main theorem
Let E denote the class of objects E 2 EPA 0for which the formal group (j*E)
is good.
Proposition 4.1. For any good formal group (G, S), there exists E 2 E with
(j*E) ' (G, S).
Proof.Choose a stably realisable coordinate x on G, and let F be the resulting
formal group law over OS. As F is stably realisable, there exists E 2 EPA 0
together with an isomorphism OE: ß0E -!OS of rings such that the composite
ß0MP -ß0i-!ß0E OE-!OS
is just c[F ]. If we let F 0be the formal group law over ß0E classified by ß0i *
*then
(j*E) = Q(ß0E, F 0) ' Q(OS, F ) ' (G, S) as required.
Proposition 4.2. Suppose we have objects E0, E1 2 E, together with a map
p: (j*E1) -! (j*E0) in GFG . Then there is a unique good map f :j*E0 -!j*E1
such that (f) = p.
Proof.For i = 0, 1 we let xi be the coordinate on (j*Ei) supplied by the unit
map ii:MP -! Ei and let Fi be the corresponding formal group law over ß0Ei.
The map p gives us a map OE: ß0E0 -!ß0E1 of rings together with an isomorphism
b: F1 -!OE*F0 of formal group laws over ß0E1. We know that maps MP -! j*E1 in
EPR biject with coordinates on (j*E1), so there is a unique map oe :MP -! j*E1
carrying the canonical coordinate exon (MP ) to b(x1). We let fi0 be the compo*
*site
MP (1) oe^i1---!(j*E1) ^ (j*E1) mult---!j*E1
(so fi0 2 EPR (MP (1), j*E1)) and we put
fi = ß0fi0: ß0MP (1) -!ß0E1.
It is clear that fi = c[OE*F0 b-F1].
We next introduce a category B = B(E0, p, E1) as follows. The objects are tr*
*iples
(A, f0, f00) where
(a)A is an object of EPA 1, with unit map , :MP (1) -!A.
(b) f0 is a morphism E0 -!(1 ^ j)*A in EPA 0.
(c)f00is an isomorphism (j ^ 1)*A -!E1 in EPA 0.
REALISING FORMAL GROUPS 9
(d) The composite
*f0 j*f00
f = `(A, f0, f00) := (j*E0 j---!(j ^ j)*A ---! j*E1)
satisfies ß0f = OE: ß0E0 -!ß0E1 and f*x0 = b(x1) 2 E01CP 1.
The morphisms from (A, f0, f00) to (B, g0, g00) in B are the isomorphisms u: A *
*-!B
in EPA 1for which ((1 ^ j)*u)f0 = g0 and g00((j ^ 1)*u) = f00.
The maps of the form `(A, f0, f00) are precisely the good maps that induce p,
and isomorphic objects of B have the same image under `. It will thus suffice to
show that all objects of B are isomorphic.
We first claim that for any object (A, f0, f00) in B, the composite
(MP -i0!j*E0 f-!j*E1) 2 EPR (MP, j*E1)
is equal to oe. Indeed, we certainly have i0*(ex) = x0 and f*(x0) = b(x1) by co*
*ndi-
tion (d), so (fi0)*(ex) = b(x1) as required.
Next, we claim that the composite
*f00
fl0 = (MP (1) ,-!(j ^ j)*A j---!j*E1) 2 EPR (MP (1), j*E1)
is equal to fi0. Indeed, MP (1) is just the coproduct in EPR of two copies of *
*MP ,
so it will suffice to check that fl0O (j ^ 1) = fi0O (j ^ 1) and fl0O (1 ^ j) =*
* fi0O (1 ^ j).
By construction, we have fi0O (1 ^ j) = oe = fi0 and fi0O (j ^ 1) = i1. Now con*
*sider
the following diagram, in which we have implicitly applied forgetful functors d*
*own
to EPR .
MP -1^j---!MP (1)-j^1---MP
? ? ?
i0?y ,?y ?yi1
E0 ----!f0 A ----!f00E1
As f0 comes from a ring map E0 -! (1 ^ j)*A in EPA 0, it is certainly a map of
MP -algebras in the naive, homotopical sense. In particular, it preserves unit*
*s,
which means that the left hand square commutes. Similarly, so does the right ha*
*nd
square. From the right hand square, we see that fl0 O (j ^ 1) = i1. From the le*
*ft
hand square, we see that fl0O (1 ^ j) = fi0 = oe. It follows that fl0 = fi0 as *
*claimed,
and so ß0fl0 = fi = c[OE*F0 b-F1]. We deduce that ß0A is isomorphic as an algeb*
*ra
over ß0MP (1) to (ß0E1, OE*F0 b-F1). As F1 is a stably realisable formal group *
*law,
we see that this algebra is strongly realisable, so maps A -!A0in EPA 1biject w*
*ith
maps ß0A -!ß0A0of ß0MP (1)-algebras.
Now suppose we have another object (B, g0, g00) in CB. We then have an iso-
morphism v = (g00)-1f00:(j ^ 1)*A -!(j ^ 1)*B in EPA 1. If we let Ø be the unit
map for B, the previous paragraph tells us that following diagram commutes (with
both composites equal to fi):
ß0MP (1) -ß0,---!ß0A
? ?
ß0Ø?y ?yß0f00
ß0B ----!ßß0E1
0g00
It follows easily that ß0v is a map of ß0MP (1)-algebras, so by strong realisab*
*lility
it agrees with ß0u for a unique map u: A -!B in EPA 1. As ß0u is an isomorphism,
10 N. P. STRICKLAND
the same is true of u. We also claim that (j ^ 1)*u = v as a map from (j ^ 1)*A
to (j ^ 1)*B in EPA 0. Indeed, the ß0MP -algebra ß0((j ^ 1)*A) is isomorphic
to ß0E1 and so is strongly realisable, so maps out of (j ^ 1)*A are determined
by their effect on ß0, and ß0((j ^ 1)*u) = ß0v by construction. This shows that
g00O ((j ^ 1)*u) = f00.
Finally, we claim that ((1 ^ j)*u) O f0 = g0 as maps from E0 to (1 ^ j)*B in
EPA 0. As ß0E0 is strongly realisable, it again suffices to check the equation *
*on ß0.
From the definition of B we have
ß0(f00) O ß0(f0) = OE = ß0(g00) O ß0(g0): ß0E0 -!ß0E1,
and by construction we have ß0u = ß0(g00)-1ß0(f00); it follows that ß0(u)Oß0(f0*
*) =
ß0(g0) as required.
This shows that there is actually a unique isomorphism between any two objects
of B, so in particular the function ` is constant on B, so there is a unique go*
*od map
inducing p, as claimed.
Lemma 4.3. For any E 2 E, the identity map 1: j*E -!j*E is good.
Proof.Note that the multiplication map MP (1) = MP ^ MP -! MP is a map
of ring spectra (in the strict sense) and so induces a functor ~*: EPA 0-! EPA 1
with (1 ^ j)*~*E = (j ^ 1)*~*E = E on the nose. We can thus take A = ~*E and
f0 = f00= 1E to show that 1E is good.
Proposition 4.4. Suppose we have objects E0, E1, E2 2 E and good morphisms
j*E0 f-!j*E1 g-!j*E2. Then the composite gf is also good.
Proof.Let Fi be the formal group law for Ei, and let (OE, b) and (_, c) be the
morphisms of formal group laws coming from f and g (so in particular OE = ß0f
and _ = ß0g). Choose objects A, B 2 EPA 1and maps
f0:E0 -!(1 ^ j)*A
f00:(j ^ 1)*A '-!E1
g0:E1 -!(1 ^ j)*B
g00:(j ^ 1)*B '-!E2
exhibiting the goodness of f and g. Next, observe that we have a chain
_*OE*F0 _*b--_*F1 c-F2
of formal group laws over ß0E2. This makes ß0E2 into an algebra over ß0MP (2),
and it is strongly realisable as such, because F2 is stably realisable by assum*
*ption.
We can thus choose an object P 2 EPA 2 and an isomorphism w :ß0P -! ß0E2
of ß0MP (2)-algebras. We now include ß0MP (1) in ß0MP (2) by the map j ^
1 ^ 1: MP (1) -! MP (2). The resulting algebra structure on ß0E2 classifies the
chain _*F10c-F2.0From the proof of Proposition 4.2, we see that the isomorphism
ß0B -ß0g--!ß0E2 is also a ß0MP (1)-algebra map with respect to this structure.
This means that we have an isomorphism (ß0g00)-1w :ß0((j ^ 1 ^ 1)*P ) -!ß0B of
ß0MP (1)-algebras, and by strong realisability, it comes from a unique isomorph*
*ism
v :(j ^ 1 ^ 1)*P -! B in EPA 1.
REALISING FORMAL GROUPS 11
Next, we have maps of algebras over ß0MP (1) as follows
00 b
ß0A -ß0f--!(ß0E1, OE*F0- F1)
-_!(ß _*b
0E2, _*OE*F0 -- _*F1)
-w-1--!ß *
0((1 ^ 1 ^ j) P ).
As the first map is an isomorphism, we know that ß0A is strongly realisable, so
there is a unique map u: A -!(1 ^ 1 ^ j)*P in EPA 1inducing the composite map
on ß0.
We now put
C = (1 ^ j ^ 1)*P 2 EPA 1
0 (1^j)*u
h0= (E0 f-!(1 ^ j)*A -----! (1 ^ j ^ j)*P = (1 ^ j)*C)
*v g00
h00= ((j ^ 1)*C = (j ^ j ^ 1)*P -(j^1)----!(j ^ 1)*B -! E2).
As v and g00are isomorphisms, the same is true of h00. We claim that after forg*
*etting
down to EPR , we have h00h0= gf; this will prove that gf is good as claimed. We
certainly have h00h0 = g00vuf0 and gf = g00g0f00f0 so it will suffice to show t*
*hat
vu = g0f00:A -!B in EPR . For this, it will be enough to prove that the followi*
*ng
diagram in EPA 0commutes.
*u
(j ^ 1)*A(j^1)-----!(j ^ 1 ^ j)*P
? ?
f00?y' '?y(1^j)*v
E1 ----!g0 (1 ^ j)*B.
As this is a diagram in EPA 0 and ß0((j ^ 1)*A) ' ß0E1 is strongly realisable,
it will be enough to check that the diagram commutes after applying ß0. By
construction we have ß0(u) = w-1 O _ O ß0(f00) and _ = ß0(g) = ß0(g00) O ß0(g0)
and ß0(v) = ß0(g00)-1 O w. It follows directly that the above diagram commutes *
*on
homotopy, groups, so it commutes in EPA 0, so it commutes in EPR , so gf = h00h0
in EPR as explained previously. Thus, the map gf is good, as claimed.
Proof of Theorem 1.1.We merely need to collect results together and explain the
argument in the introduction in more detail. Lemma 4.3 and Proposition 4.4 show
that we can make E into a category by taking the good maps from j*E0 to j*E1
as the morphisms from E0 to E1. Tautologically, we can define a faithful functor
U :E -! EPR by U(E) = j*E and U(f) = f. We then define V = U :E -! FG;
by the definition of E, this actually lands in GFG . Proposition 4.1 says that*
* V
is essentially surjective, and Proposition 4.2 says that V is full and faithful*
*. This
means that V is an equivalence, so we can invert it and define R = UV -1:GFG -!
EPR . As V = U we have R = 1, so R is the required realisation functor.
Appendix A. The product on MP
In this appendix we verify that MP can be constructed as an E1 ring spectrum.
Let U be a complex universe. For any finite-dimensional subspace U of U, we
write UL = U 0 < U U and UR = 0 U < U U. We let Grass(U U)
denote the Grassmannian of all subspaces of U U (of all possible dimensions),
12 N. P. STRICKLAND
and we write flU for the tautological bundle over this space, and Thom (U U)
for the associated Thom space. If U U0 < U then we define i: Grass(U2) -!
Grass((U0)2)0by i(A) = A (U0 U)R . On passing to Thom spaces we get a map
oe : U U Thom (U2) -! Thom ((U0)2). These maps can be used to assemble the
spaces Thom (U2) into a -inclusion prespectrum indexed by the complex subspaces
of U. We write TU for this prespectrum, and MPU for its spectrification.
Now let V be another complex universe, so we have a prespectrum TV over V,
and thus an external smash product TU ^extTV indexed on the complex subspaces
of U V of the form U V . The direct sum gives a map Grass(U2) x Grass(V 2) *
*-!
Grass((U V )2) which induces a map Thom (U2) ^ Thom (V 2) -!Thom ((U V )2).
These maps fit together to give a map TU^extTV -!TU V , and thus a map MPU^ext
MPV -! MPU V of spectra over U V. Essentially the same construction gives
maps
MPU1^ext. .^.extMPUr -!MPU1 ... Ur.
If U1 = . .U.r= U, then this map is r-equivariant.
Now suppose instead that we have a complex linear isometry f :U -! V. This
gives evident homeomorphisms Thom (U2) -!Thom ((fU)2), which fit together to
induce a map MPU -! f*MPV, which is adjoint to a map f*MPU -! MPV. We
next observe that this construction is continuous in all possible variables, in*
*cluding
f. (This statement requires some interpretation, but there are no new issues be*
*yond
those that are well-understood for MU; the cleanest technical framework is prov*
*ided
by [1].) It follows that they fit together to give a map LC(U, V) n MPU -!MPV of
spectra over V.
We now combine this with the product structure mentioned earlier to get a map
LC (Ur, U) n r (MPU ^ext. .^.extMPU) -!MPU.
This means that MPU has an action of the E1 operad of complex linear isometries,
as required.
All that is left is to check that the spectrum MP = MPC1 constructed above
has the right homotopy type. As T is a -inclusion prespectrum, we know that
spectrification works in the simplest possible way and that MP is the homotopy
colimit of the spectra
n`
-2n Thom (Cn Cn) = -2n Grassk+n(Cn Cn)fl,
k=-n
where Grassd(V ) is the space of d-dimensional subspaces of V . It is not hard *
*to
see that the maps of the colimit system preserve this splitting, so that MP is *
*the
wedge over all k 2 Z of the spectra
Xk := holim-! -2n Grassk+n(Cn Cn)fl.
n
This can be rewritten as
Xk = 2kholim-! -2(k+n)Grassk+n(Cm Cn)fl.
n,m
We can reindex by putting n = i - k and m = j + k, and then pass to the limit in
j. We find that
Xk = 2kholim-! -2iGrassi(C1 Ci)fl.
i
REALISING FORMAL GROUPS 13
It is well-known thatWGrassi(C1 Ci) is a model for BU(i), and it follows that
Xk = 2kMU, so MP = k 2kMU as claimed. We leave it to the reader to check
that the product structure is the obvious one.
References
[1]A. D. Elmendorf. The Grassmannian geometry of spectra. Journal of Pure and A*
*pplied Alge-
bra, 54:37-94, 1988.
[2]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and A*
*lgebras in
Stable Homotopy Theory, volume 47 of Amer. Math. Soc. Surveys and Monographs*
*. American
Mathematical Society, 1996.
[3]L. G. Lewis, J. P. May, and M. S. (with contributions by Jim E. McClure). Eq*
*uivariant Stable
Homotopy Theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verla*
*g, New York,
1986.
[4]N. P. Strickland. Formal schemes and formal groups. In J. Meyer, J. Morava, *
*and W. Wilson,
editors, Homotopy-invariant algebraic structures: in honor of J.M. Boardman,*
* volume 239 of
Contemporary Mathematics. American Mathematical Society, 1999.
[5]N. P. Strickland. Products on MU-modules. Transactions of the American Mathe*
*matical So-
ciety, 351:2569-2606, 1999.
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK
E-mail address: N.P.Strickland@sheffield.ac.uk