FORMAL SCHEMES AND FORMAL GROUPS
NEIL P. STRICKLAND
Contents
1. Introduction 2
1.1. Notation and conventions 3
1.2. Even periodic ring spectra 3
2. Schemes 3
2.1. Points and sections 6
2.2. Colimits of schemes 8
2.3. Subschemes 9
2.4. Zariski spectra and geometric points 11
2.5. Nilpotents, idempotents and connectivity 12
2.6. Sheaves, modules and vector bundles 13
2.7. Faithful flatness and descent 16
2.8. Schemes of maps 22
2.9. Gradings 24
3. Nonaffine schemes 25
4. Formal schemes 28
4.1. (Co)limits of formal schemes 29
4.2. Solid formal schemes 31
4.3. Formal schemes over a given base 33
4.4. Formal subschemes 35
4.5. Idempotents and formal schemes 38
4.6. Sheaves over formal schemes 39
4.7. Formal faithful flatness 40
4.8. Coalgebraic formal schemes 42
4.9. More mapping schemes 46
5. Formal curves 49
5.1. Divisors on formal curves 49
5.2. Weierstrass preparation 53
5.3. Formal differentials 56
5.4. Residues 57
6. Formal groups 59
6.1. Group objects in general categories 59
6.2. Free formal groups 63
6.3. Schemes of homomorphisms 65
6.4. Cartier duality 66
6.5. Torsors 67
7. Ordinary formal groups 69
____________
Date: November 17, 2000.
1
2 NEIL P. STRICKLAND
7.1. Heights 70
7.2. Logarithms 72
7.3. Divisors 72
8. Formal schemes in algebraic topology 73
8.1. Even periodic ring spectra 73
8.2. Schemes associated to spaces 74
8.3. Vector bundles and divisors 81
8.4. Cohomology of Abelian groups 84
8.5. Schemes associated to ring spectra 84
8.6. Homology of Thom spectra 85
8.7. Homology operations 87
References 89
1.Introduction
In this paper we set up a framework for using algebraic geometry to study the
generalised cohomology rings that occur in algebraic topology. This idea was pr*
*ob
ably first introduced by Quillen [21] and it implicitly or explicitly underlies*
* much
of our understanding of complex oriented cohomology theories, exemplified by the
work of Morava. Most of the results presented here have close and wellknown an*
*a
logues in the algebrogeometric literature, but with different definitions or t*
*echnical
assumptions that are often inconvenient for topological applications. Our aim h*
*ere
is merely to put everything together in a systematic way that naturally incorpo*
*rates
the phenomena that we see in topology while discarding complications that never
arise there. In more detail, in the classical situation one is often content to*
* deal
with finite dimensional, Noetherian schemes. Nilpotents are seen as a somewhat
peripheral phenomenon, and formal schemes are only introduced at a late stage in
the exposition. Schemes are defined as spaces with extra structure. The idea of*
* a
scheme as a functor occurs in advanced work (a nice example is [16]) but is usu*
*ally
absent from introductory treatments. For us, however, it is definitely most nat*
*ural
to think of schemes as functors. Our schemes are very often not Noetherian or f*
*inite
dimensional, and nilpotents are of crucial importance. We make heavy use of for
mal schemes, and we need to define these in a more general way than is traditio*
*nal.
On the other hand, we can get a long way using only affine schemes, whereas the
usual treatment devotes a great deal of attention to the nonaffine case.
Section 2 is an exposition of the basic facts of algebraic geometry that is w*
*ell
adapted to the viewpoint discussed above, together with a number of useful exam
ples.
In Section 3, we give a basic account of nonaffine schemes from our point of
view.
In Section 4, we give a very general definition of formal schemes which follo*
*ws
naturally from our description of ordinary (or "informal") schemes. We then work
out the basic properties of the category of formal schemes, such as the existen*
*ce
of limits and colimits and the behaviour of regular monomorphisms (or "closed
inclusions").
FORMAL SCHEMES AND FORMAL GROUPS 3
In Section 6, we discuss the Abelian monoid and group objects in the category*
* of
formal schemes. We then specialise in Section 7 to the case of smooth, commutat*
*ive,
onedimensional formal groups, which we call "ordinary formal groups".
Finally, in Section 8, we construct functors from the homotopy category of sp*
*aces
(or suitable subcategories) to the category of formal schemes. We use the work *
*of
Ravenel, Wilson and Yagita [24] to show that spaces whose Morava Ktheory is
concentrated in even degrees give formal schemes with good technical properties.
We also discuss what happens to a number of popular spaces under our functors.
Further applications of this point of view appear in [26, 27, 7, 11] and a numb*
*er of
other papers in preparation.
1.1. Notation and conventions. We write Rings for the category of rings (by
which we always mean commutative unital rings) and Setsfor the category of sets.
For any ring R, we write Mod R for the category of Rmodules, and AlgR for the
category of Ralgebras. Given a category C, we usually write C(X; Y ) for the
set of Cmorphisms from X to Y . We write CX for the category of objects of C
over X. More precisely, on object of CX is a pair (Y; u) where u: Y !Z, and
CX ((Y; u); (Z; v)) is the set of maps f :Y ! Z in C such that vf = u.
We write F for the category of all functors Rings! Sets.
1.2. Even periodic ring spectra. We now give a basic topological definition,
as background for some motivating remarks to be made in subsequent sections.
Details of topological applications will appear in Section 8. The definition be*
*low
will be slightly generalised there, to deal with unpleasantness at the prime 2.
Definition 1.1.An even periodic ring spectrum is a commutative and associative
ring spectrum E such that
1. ss1E = 0
2. ss2E contains a unit.
The example to bear in mind is the complex Ktheory spectrum KU. Suitable
versions of Morava Etheory and Ktheory are also examples, as are periodised
versions of MU and H; we write MP and HP for these. See Section 8 for more
details.
2. Schemes
In this section we set up the basic categorical apparatus of schemes. We then
discuss limits and colimits of schemes, and various kinds of subschemes. We com
pare our functorial approach with more classical accounts by discussing the Zar*
*iski
space of a scheme. We then discuss various issues about nilpotent and idempotent
functions. We define sheaves over functors, and show that our definition works
as expected for schemes. We then define flatness and faithful flatness for maps*
* of
schemes, and prove descent theorems for schemes and sheaves over faithfully flat
maps. Finally, we address the question of defining a "scheme of maps" Map (X; Y*
* )
between two given schemes X and Y .
Definition 2.1.An affine scheme is a covariant representable functor
X :Rings! Sets:
We make little use of nonaffine schemes, so we shall generally omit the word
"affine". A map of schemes is just a natural transformation. We write X for
4 NEIL P. STRICKLAND
the category of schemes, which is a full subcategory of F. We write spec(A) for*
* the
functor represented by A, so spec(A)(R) = Rings(A; R) and spec(A) is a scheme.
Remark 2.2. If E is an even periodic ring spectrum and Z is a finite spectrum
we define ZE = spec(E0Z). This gives a covariant functor Z 7! ZE from finite
complexes to schemes. We also write SE = spec(E0).
Definition 2.3.We write A1 for the forgetful functor Rings ! Sets. This is
isomorphic to spec(Z[t]) and thus is a scheme. Given any functor X 2 F, we write
OX for the set of natural maps X ! A1. (This can actually be a proper class for
general X, but it will always be a set in the cases that we consider.) Note tha*
*t OX
is a ring under pointwise operations.
Our category of schemes is equivalent to the algebraic geometer's category of
affine schemes, which in turn is equivalent (by Yoneda's lemma) to the opposite*
* of
the category of rings.
We now describe the duality between schemes and rings in more detail. The
Yoneda lemma tells us that Ospec(A)is naturally isomorphic to A. For any functor
X 2 F we have a tautological map : X ! spec(OX ). To define explicitly,
suppose we have a ring R and an element x 2 X(R); we need to produce a map
R (x): OX ! R. An element f 2 OX is a natural map f :X ! A1, so it has a
component fR :X(R) ! R, and we can define R (x)(f) = fR (x). If X = spec(A)
then is easily seen to be bijective. As schemes are by definition representab*
*le,
any scheme X is equivalent to spec(A) for some A, so we see that the map X !
spec(OX ) is always an isomorphism. Thus, the functor X ! OX is inverse to the
functor spec:Ringsop! X.
We next give some examples of schemes.
Example 2.4. A basic example is the "multiplicative group" Gm , which is defined
by
Gm (R) = Rx = the group of unitsRof:
This is a scheme because it is represented by Z[x1 ].
Example 2.5. The affine nspace An is defined by An(R) = Rn. This is a scheme
because it is represented by Z[x1; : :;:xn]. If f1; : :;:fm are polynomials in *
*n vari
ables over Z then there is an obvious natural map Rm ! Rn for all rings R, whi*
*ch
sends a_= (a1; : :;:am ) to (f1(a_); : :;:fn(a_)). Thus, this gives a map Am !*
* An of
schemes. These are in fact all the maps between these schemes. The key point is
of course that the set of ring maps Z[y1; : :;:ym ] Z[x1; : :;:xn] bijects na*
*turally
with the set of such tuples (f1; : :;:fm ). It is a good exercise to work out a*
*ll of the
identifications going on here.
We next define the scheme FGL of formal group laws, which will play a central
r^ole in the applications of schemes to algebraic topology.
Example 2.6. A formal group law over a ring R is a formal power series
X
F (x; y) = aklxkyl2 R[[x; y]]
k;l0
FORMAL SCHEMES AND FORMAL GROUPS 5
satisfying
F (x; 0)= x
F (x; y)= F (y; x)
F (F (x; y);=z)F (x; F (y; z)):
We can define a scheme FGL as follows:
FGL(R) = { formal group laws overR}:
To see that FGL is a scheme, we considerPthe ring L0 = Z[akl k; l > 0] and the
formal power series F0(x; y) = x + y + aklxkyl 2 L0[[x; y]]. We then let I be
the ideal in L0 generated by the coefficients of the power series F0(x; y)  F0*
*(y; x)
and F0(F0(x; y); z)  F0(x; F0(y; z)). Finally, set L = L0=I. It is easy to see*
* that
FGL = spec(L). The ring L is called the Lazard ring. It is a polynomial ring in
countably many variables; there is a nice exposition of the proof in [2, Part I*
*I].
Recall that MP denotes the 2periodic version of MU; a fundamental theorem of
Quillen [19, 20] (also proved in [2]) identifies the scheme SMP := spec(MP 0) *
*with
FGL .
Example 2.7. Given any diagram of schemes {Xi}, we claim that the functor
X = limXi (which is defined by (lim Xi)(R) = lim(Xi(R))) is also a scheme.
i  i oip
Indeed, suppose that Xi= spec(Ai). As spec:Rings ! X is an equivalence, we
get a diagram of rings Ai with arrows reversed. It is wellknown that the categ*
*ory
of rings has colimits, and it is clear that X = spec(lim!Ai).
i
In particular, if X and Y are schemes, we have a scheme XxY with (XxY )(R) =
X(R) x Y (R) and OXxY = OX OY (because coproducts of rings are tensor
products). Similarly, if we have maps X f!Z g Y then we can form the pullback
(X xZ Y )(R) = X(R) xZ(R)Y (R) = {(x; y) 2 X(R) x Y (R)  f(x) = g(y)}:
This is represented by the tensor product OX OZ OY .
We write 1 for any onepoint set, and also for the constant functor 1(R) = 1.
Thus 1 = spec(Z), and this is the terminal object in X or F.
Example 2.8. Let Z and W be finite CW complexes, and let E be an even periodic
ring spectrum. There is a natural map (Z x W )E ! ZE xSE WE . This will
be an isomorphism if E1Z = 0 = E1W and we have a K"unneth isomorphism
E*(Z x W ) = E*(Z) E* E*(W ). This holds in particular if H*Z is a free Abelian
group, concentrated in even degrees.
Example 2.9. An invertible power series over a ring R is a formal power series
f 2 R[[x]]such that f(x) = wx + O(x2) for some w 2 Rx . This implies, of course,
that f has a compositioninverse g = f1 , so that f(g(x)) = x = g(f(x)). We wr*
*ite
IPS(S) for the set of such f, which is easily seen to be a scheme. It is actual*
*ly a
group scheme, in that IPS(R) is a group (under composition), functorially in R.
The group IPS acts on FGL by
(f; F ) 7! Ff Ff(x; y) = f(F (f1 x; f1 y)):
An isomorphism between formal group laws F and G is an invertible series f
such that f(F (a; b)) = G(f(a); f(b)). Let FIbe the following scheme:
FI(R) = {(F; f; G)  F; G 2 FGL (R) and f :F ! G is an isomorphism}:
6 NEIL P. STRICKLAND
There is an evident composition map
FIxFGL FI! FI ((F; f; G); (G; g; H)) 7! (F; gf; H):
Moreover, there is an isomorphism
IPS x FGL ! FI (F; f) 7! (F; f; Ff):
One can describe these maps by giving implicit formulae in the representing rin*
*gs
OIPS, OFGL an OFI, but this should be avoided where possible. Note that for each
R we can regard FGL (R) as the set of objects of a groupoid, whose morphism
set is FI(R). In other words, the schemes FGL and FI define a groupoid scheme.
It is known that FI = spec(MP0MP ) (this follows easily from the description of
MU*MU in [2]).
Example 2.10. We now give an example for which representability is less obviou*
*s.
We say that an effective divisor of degree n on A1 over a scheme Y is a subsche*
*me
D Y x A1 = spec(OY [x]) such that OD is a quotient of OY [x] and is free of
rank n over OY . We let X(R) = Div+n(A1)(R) denote the set of such divisors over
spec(R), and we claim that X = Div+n(A1) is a scheme. Firstly, it is a functor
of R: given a ring map u: R ! R0 and a divisor D over R we get a divisor
uD = spec(R0R OD ) = spec(R0) xspec(R)D over R0. Next, given a divisor D as
above and an element y 2 R[x], we let (y) be the map u 7! uy, which is an Rlin*
*ear
endomorphism of the modulePOD ' Rn. The map (x) thus has a characteristic
polynomial fD (t) = ni=0ai(D)tni 2 R[t]. One checks that the map ai:X ! A1
is natural, so we get an element ai of OX . As fD (t) is monic, we have a0 = 1.*
* The
remaining ai's give us a map X ! An.
The CayleyHamilton theorem tells us that fD ((x)) = 0, but it is clear that
fD ((x)) = (fD (x)) and fD (x) = (fD (x))(1), so we find that fD (x) = 0 in OD
and thus that OD is a quotient of R[x]=fD (x). On the other hand, it is clear t*
*hat
R[x]=fD (x) is also free over R of rank n, and it follows that OD = R[x]=fD (x).
Given this, we see that D is freely and uniquely determined by the coefficients
a1; : :;:an, so that our map X ! An is an isomorphism. This shows in particular
that X is a scheme. (I learned this argument from [4].)
2.1. Points and sections. Let X be a scheme. A point of X means an element
x 2 X(R) for some ring R. We write Ox for R, which conveniently allows us to
mention x before giving R a name. Recall that points x 2 X(R) biject with maps
spec(R) !X. We say that x is defined over R, or over spec(R).
We can also think of an element of R as a point of the scheme A1 over R. If
f 2 OX then f is a natural map X(S) !S for all rings S, so in particular we ha*
*ve
a map X(R) !R. We thus have f(x) 2 Ox = R.
Example 2.11. Let F be a point of FGL , in other words a formal group law over
some ring R. We can write
[3](x) = F (x; F (x; x)) = 3x + u(F )x2 + v(F )x3 + O(x4)
for certain scalars u(F ) and v(F ). This construction associates to each point*
* F 2
FGL a point v(F ) 2 A1 in a natural way, thus giving an element v 2 OFGL.
Of course, we know that OFGL is the Lazard ring L, which is generated by the
coefficients aklof the universal formal group law
X
Funiv(x; y) = aklxkyl
k;l
FORMAL SCHEMES AND FORMAL GROUPS 7
Using this formal group law, we find that
[3](x) = 3x + 3a11x2 + (a211+ 8a12)x3 + O(x4)
This means that
v(Funiv) = a211+ 8a12
It follows that for any F over any ring R, the element v(F ) is the image of a2*
*11+8a12
under the map L !R classifying F .
Example 2.12. For any scalar a, we have a formal group law
Ha(x; y) = x + y + axy:
The construction a 7! Ha gives a natural transformation h: A1(R) ! FGL(R), in
other words a map of schemes h: A1 !FGL . This can be thought of as a family
of formal group laws, parametrised by a 2 A1. It can also be thought of as a si*
*ngle
formal group law over Z[a] = OA1.
Example 2.13. The point of view described above allows for some slightly schiz*
*o
phrenic constructions, such as regarding the two projections ss0; ss1: X x X !*
* X
as two points of X over X2. Indeed, this is the universal example of a scheme
Y equipped with two points of X defined over Y . Similarly, we can think of the
identity map X ! X as the universal example of a point of X. This is analogous*
* to
thinking of the identity map of K(Z; n) as a cohomology class u 2 HnK(Z; n); th*
*is
is of course the universal example of a space with a given ndimensional cohomo*
*logy
class.
Definition 2.14.For any functor X :Rings! Sets, we define a category Points(X),
whose objects are pairs (R; x) with x 2 X(R). The maps (R; x) ! (S; y) are ring
maps f :R !S such that X(f)(x) = y.
Remark 2.15. Let X be a scheme. The following categories are equivalent:
(a) The category XX of schemes Y equipped with a map u: Y ! X.
(b) The category of representable functors Y 0:Points(X) !Sets.
(c) The category of representable functors Y 00:XopX!Sets.
(d) The category AlgopOXof algebras R over OX .
(e) The category Points(X)op of pairs (R; x) with x 2 X(R).
By Yoneda, an element x 2 X(R) corresponds to a map x0:spec(R) ! X. Simi
larly, a map v :Z ! X gives a map v*: OX ! OZ, making OZ into an OX algebra.
This can also be regarded as an element of spec(OX )(OZ ) = X(OZ ). With this
notation, the equivalence is as follows.
a
Y (S)= Y 0(S; z)
z2X(S)
Y 0(S; z)= preimage of z 2 X(S) under u: Y (S) !X(S)
0
= Y 00(spec(S) z!X)
Y 00(Z v!X)= Y 0(OZ ; v*)
R = OY
Y = spec(R):
For us, the most important part of this will be the equivalence (a),(b).
8 NEIL P. STRICKLAND
Remark 2.16. If E is an even periodic ring spectrum and SE = spec(E0) then
we can regard the construction Z 7! ZE = spec(E0Z) as a functor from finite
complexes to XSE.
Definition 2.17.If X is a scheme over another scheme Y , and y 2 Y (R) is a
point of Y , we write Xy = spec(R) xY X, which is a scheme over spec(R). Here
we have used the map spec(R) !Y corresponding to the point y 2 Y (R) to form
the pullback spec(R) xY X. We call Xy the fibre of X over the point y.
2.2. Colimits of schemes. The category of rings has limits for small diagrams,
and the category of schemes is dual to that of rings, so it has colimits for sm*
*all
diagrams. However, it seems that these colimits only interact well with our geo
metric point of view if they have some additional properties (this is also the *
*reason
for Mumford's geometric invariant theory, which is much more subtle than any
thing that we consider here.) One good property that often occurs (with C = X or
C = XY ) is as follows.
Definition 2.18.Let C be a category with finite products, and let {Xi} be a
diagram in C. We say that an object X with a compatible system of maps Xi! X
is a strong colimit of the diagram if W x X is the colimit of {W x Xi} for each
W 2 C. We define strong coproducts and strong coequalisers as special cases of
this, in the obvious way.
ExampleQ2.19. The categories X and XY have strong finite coproducts, and O` iX*
*i=
op
iOXi. Indeed, byQthe usual duality Rings = X, we see that`the`coproduct exis*
*ts
and has O` iXi= iOXi. Thus,Qwe needQonly check that ZxY iXi= iZxY Xi,
or equivalently that OZ OY iOXi = iOZ OY OXi, which is clear because the
indexing set is finite. Note that when Y = 1 is the terminal object, we have XY*
* = X,
so we have covered that case as well.
As a special case of the above, we can make the following definition.
Definition 2.20.Given a finite set A, we can define an associated constant sche*
*me
A_by
a
A_= 1
a2A
`
(where 1 is the terminal object in X). This has the property that X xA_= a2AX
for all X. We also have OA_= F (A; Z), which denotes the ring of functions from
the set A to Z; this is a ring under pointwise operations.
Remark 2.21. It is not the case that (X qY )(R) = X(R)qY (R) (unlike the case
of products and pullbacks). Instead, we have
(X q Y )(R) = {(S; T; x; y)  S; T R ; R = S x T ; x 2 X(S) ; y 2 Y (T )}:
To explain this, note that an element of (XqY )(R) is (by Yoneda) a map spec(R)*
* !
X q Y . This will be given by a decomposition spec(R) = spec(S) q spec(T ) and
maps spec(S) !X and spec(T ) !Y . Clearly, if R does not split nontrivially a*
*s a
product of smaller rings then we have the naive rule (X q Y )(R) = X(R) q Y (R).
Similarly, the initial scheme ; = spec(0) has ;(R) = ; unless R = 0 in which
case ;(R) has a single element.
FORMAL SCHEMES AND FORMAL GROUPS 9
Example 2.22. Let f :X ! Y be a map of schemes. Let XnYdenote the fibre
product of n copies of X over Y , so that the symmetric group n acts on XnY,
covering the trivial action on Y . Suppose that the resulting map f* :OY ! OX
makes OX into a free module over OY . We then claim that there is a strong
colimit for the action of n on XnY. To see this, write A = OX and B = OY and
C = ABn , so that XnY= spec(C). Our claim reduces easily to the statement
that B0B (Cn ) = (B0B C)n for every algebra B0 over B. To see that this
holds, choose a basis for A over B. This gives an evident basis for C over B, w*
*hich
is permuted by the action of n. Clearly Cn is a free module over B, with one
generator for each norbit in our basis for C. There is a similar description f*
*or
(B0B C)n , which quickly implies our claim.
Some more circumstances in which colimits have unexpectedly good behaviour
are discussed in [7], which mostly follows ideas of Quillen [21].
2.3. Subschemes. Recall that an element of OX is a natural map X ! A1. Thus,
if x is a point of X then f(x) is a scalar (more precisely, if x 2 X(R) then f(*
*x) 2 R)
and we can ask whether f(x) = 0, or whether f(x) is invertible.
Definition 2.23.Given a scheme X and an ideal I OX , we define a scheme
V (I) by
V (I)(R) = {x 2 X(R)  f(x) = 0 for allf 2 I}:
One checks that V (I) = spec(OX =I), so this really is a scheme. Schemes of this
form are called closed subschemes of X.
Given an element f 2 OX , we define a scheme D(f) by
D(f)(R) = {x 2 X(R)  f(x) 2 Rx }:
One checks that D(f) = spec(OX [1=f]), so this really is a scheme. Schemes of t*
*his
form are called basic open subschemes of X.
A locally closed subscheme is a basic open subscheme of a closed subscheme.
Such a thing has the form D(f) \ V (I) = spec(OX [1=f]=I).
Remark 2.24. Recall that a map f :R ! S of rings is said to be a regular epi
morphism if and only if it is the coequaliser of some pair of maps T _____w_wR,*
* which
happens if and only if it is the coequaliser of the obvious maps R xS R _____w_*
*wR. It
is easy to check that this holds if and only if f is surjective. Given this, we*
* see that
the regular monomorphisms of schemes are precisely the closed inclusions, and t*
*hat
composites and pushouts of regular monomorphisms are regular monomorphisms.
Example 2.25. The map h in Example 2.12 gives an isomorphism between A1
and the closed subscheme V ((aij i + j > 2)) of FGL . The multiplicative group
Gm is an open subscheme of A1.
Example 2.26. If X is a scheme and e 2 OX satisfies e2 = e then it is easy to *
*check
that D(e) = V (1  e), so this subscheme is both open and closed. Moreover, we
have XP= D(e)qD(1e). More generally, if we`have idempotents e1; : :;:em 2 OX
with iei= 1 and eiej = ffiijei then X = iD(ei), and every splitting of X as*
* a
finite coproduct occurs in this way.
Example 2.27. Suppose X = spec(k[x]) is the affine line over a field k, and ; *
* 2
k. The closed subscheme V (x  ) = spec(k[x]=(x  )) ' spec(k) corresponds to
the point of the affine line; it is natural to refer to it as {}. The closed s*
*ubscheme
10 NEIL P. STRICKLAND
V ((x  )(x  )) corresponds to the pair of points {; }. If = , this is to be
thought of as the point with multiplicity two, or as an infinitesimal thickeni*
*ng of
the point .
We can easily form the intersection of locally closed subschemes:
D(a) \ V (I) \ D(b) \ V (J) = D(ab) \ V (I + J):
We cannot usually form the union of basic open subschemes and still have an
affine scheme. Again, it would be easy enough to consider nonaffine schemes, b*
*ut
it rarely seems to be necessary. Moreover, a closed subscheme V (a) determines
the complementary open subscheme D(a) but not conversely; D(a) = D(a2) but
V (a) 6= V (a2) in general.
We say that apscheme_X is reduced if OX has no nonzero nilpotents, and write
Xred= spec(OX = 0), which is the largest reduced closed subscheme of X. More
over, if Y X is closed then Yred= Xredif and only if X(k) = Y (k) for every fi*
*eld
k (we leave the proof as an exercise).
We define the union of closed subschemes by V (I) [ V (J) = V (I \ J). We
also define the schematic union by V (I) + V (J) = V (IJ). This is a sort of "u*
*nion
with multiplicity" _ in particular, V (I) + V (I) 6= V (I) in general. In the p*
*revious
example, we have
{} [ {} = V ((x  )2)
which is a thickening of {}. Note that V (IJ)red= V (I \ J)red, because (I \ J)2
IJ I \ J.
We shall say that X is connected if it cannot be split nontrivially as Y q Z,*
* if
and only if there are no idempotents in OX other than 0 and 1.
We shall say that a scheme X is integral if and only if OX is an integral dom*
*ain,
and that X is irreducible if and only if Xredis integral. We also say that X is
Noetherian if and only if the ringSOX is Noetherian. If so, then Xredcan be wri*
*tten
in a unique way as a finite union iYi with Yi an integral closed subscheme. T*
*he
schemes Yi are called the irreducible components of Xred; they are precisely the
schemes V (pi) for pi a minimal prime ideal of OX . See [18, section 6] for th*
*is
material. S
Suppose that X is Noetherian and reduced, say X = i2SSYias above for some
finiteTset S. Suppose that S = S0 q S00. Write X0 = S0Yi = V (I0), where
I0= S0pi, and similarly for X00and I00. If we then write
(I0) = {a 2 OX  a(I0)N = 0 forN 0};
we find that (I0) = I00and thus V ((I0)) = X00.
Example 2.28. Take Z = spec(k[x; y]=(xy2)) and set
X = V (y) = spec(k[x])
X0 = V (y2) = spec(k[x; y]=(y2))
Y = V (x) = spec(k[y])
Then X is the xaxis, Y is the yaxis and X0 is an infinitesimal thickening of *
*X.
The schemes X and Y are integral, and X0 is irreducible because X0red= X. The
scheme Z is reducible, and its irreducible components are X and Y .
FORMAL SCHEMES AND FORMAL GROUPS 11
2.4. Zariski spectra and geometric points. If R is a ring, we define the Zariski
space to be
zar(R) = { prime idealsp < R }:
If X is a scheme, we write Xzar= zar(OX ). Note that
V (I)zar= zar(OX =I) = {p 2 Xzar I p}
D(f)zar= zar(OX [1=f]) = {p 2 Xzar f 62 p}
(X q Y )zar= Xzarq Yzar
There is a map
(X x Y )zar! Xzarx Yzar;
but it is almost never a bijection.
Suppose that Y; Z X are locally closed; then
(Y \ Z)zar= Yzar\ Zzar:
If Y and Z are closed then
(Y [ Z)zar= (Y + Z)zar= Yzar[ Zzar:
We give Xzarthe topology with closed sets V (I)zar. A map of schemes X ! Y
then induces a continuous map Xzar! Yzar.
Suppose that R is an integral domain, and that x 2 X(R). Then x gives a map
x*: OX ! R, whose kernel px is prime. We thus have a map X(R) !Xzar, which
is natural for monomorphisms of R and arbitrary morphisms of X.
A geometric point of X is an element of X(k), for some algebraically closed f*
*ield
k. Suppose that either OX is a Qalgebra, or that some prime p is nilpotent in
OX . Let k be an algebraically closed field of the appropriate characteristic, *
*with
transcendence degree at least the cardinality of OX . Then it is easy to see t*
*hat
X(k) !Xzaris epi.
A useful feature of the Zariski space is that it behaves quite well under col*
*im
its [21, 7]. The following proposition is another example of this.
Proposition 2.29.Suppose that a finite group G acts on a scheme X. Then
(X=G)zar= Xzar=G.
Proof.Write S = OX and R = SG = OX=G . Given a prime p 2 zar(R) = (X=G)zar,
the fibre F over p in zar(S) = Xzaris just zar(Sp=pSp) (see [18, Section 7]). We
need to prove that F is nonempty, and that G acts transitively on F .
As localisation is exact, we have (Sp)G = Rp, so we can replace R by Rp and
thus assume that R is localQat p. With this assumption, we have F = zar(S=pS).
For a 2 S we write fa(t) = g2G(t  ga) 2 S[t]G = R[t], so that fa is a monic
polynomial with fa(a) = 0. This shows that S is an integral extension over R, so
F 6= ; and there are no inclusions between the elements of F [18, Theorem 9.3].
Let q and r be two points ofTF , so they are prime ideals in S with q\R = qG *
*= p
and r \ R = rG = p. Write I = g2Gg:q S. If a 2 I then g:a 2 q for all g so
fa(t) 2 tG+ q[t] but also fa(t) is Ginvariant so fa(t) 2 tG+ qG [t] tG+*
* r[t].
AsTfa(a) = 0 we conclude that a is nilpotent mod r but r is prime so a 2 r. Thus
g2G g:q r. As r is prime, we deduce that g:q r for some g 2 G. As there are
no inclusions between the elements of F , we conclude that g:q = r. Thus G acts_
transitively on F , which proves that (X=G)zar= Xzar=G. __
12 NEIL P. STRICKLAND
A number of interesting things can be detected by looking at Zariski spaces. *
*For
example, Xzarsplits as a disjoint union if and only if X does _ see Corollary 2*
*.40.
We also use the space Xzarto define the Krull dimension of X.
Definition 2.30.If there is a chain p0 < : :<:pn in Xzar, but no longer chain,
then we say that dim(X) = n. If there are arbitrarily long chains then dim(X) =*
* 1.
Example 2.31. The terminal object 1 has dimension one (because there are chains
(0) < (p) of prime ideals in Z). If OX is a field then dim(X) = 0. If OX is Noe*
*therian
then dim(Gm x X) = 1 + dim(X) and dim(An x X) = n + dim(X) [18, Section 15].
In particular, we have dim(An) = dim(1 x An) = n + 1.
Example 2.32. The schemes FGL , IPS and FIall have infinite dimension.
2.5. Nilpotents, idempotents and connectivity.
Proposition 2.33.Suppose that e 2 R is idempotent, and f = 1  e. Then
eR = R=f = R[e1] = {a 2 R  fa = 0}:
Moreover, this is a ring with unit e, and we have R = eR x fR as rings. *
*___
` n
Proposition 2.34.If X is a scheme, then splittingsPX = i=1Xi biject with *
* __
systems of idempotents {e1; : :;:en} with iei= 1 and eiej = ffiijej. *
* __
Example 2.35. Let Mult(n) be the scheme of polynomials OE(u) of degree at most
nPsuch that OE(1) = 1 and OE(uv) = OE(u)OE(v). Such a series can be written as *
*OE(u) =
n i P
i=0eiu , and the conditions on OE are equivalent to iei= 1 and eiej = ffiij*
*ej. In
other words, the`elements ei are orthogonal idempotents. Using this, we see eas*
*ily
that Mult(n) = ni=01.
Example 2.36. Now let E(n) be the scheme of n x n matrices A over R such that
A2 = A. Define ffA (u) = uA + (1  A) = (u  1)A + 1 2 Mn(R[u]) and OEA (u) =
det(ffA (u)) 2 R[u]. We find easily that ffA (1) = 1 and ffA (uv) = ffA`(u)ffA *
*(v), so
OEA (u) 2 Mult(n)(R). This construction`gives a map E(n) ! Mult(n) = ni=01,
which gives a splitting E(n) = ni=0E(n; i), where E(n; i) is the scheme of n *
*x n
matrices A such that A2 = A and OEA (u) = ui.
Note that the function A 7! trace(A) lies in OE(n)and that E(n; i) is contain*
*ed
in the closed subscheme E0(n; i) = {A  trace(A) = i}. However, if n > 0 but
n = 0 in R then E0(n; 0)(R) and E0(n; n)(R) are not disjoint, which shows that
E0(n; i) 6= E(n; i) in general.
For any ring R, we let Nil(R) denote the set of nilpotents in R.
Proposition 2.37.Nil(R) is the intersection of all prime ideals in R.
Proof.[18, Section 1] ___
Proposition 2.38 (Idempotent Lifting).Suppose that e 2 R= Nil(R) is idempo
tent. Then there is a unique idempotent "e2 R lifting e.
Proof.Choose a (not necessarily idempotent) lift of e to R, call it e, and write
f = 1  e. We know that ef is nilpotent, say enfn = 0. Define
c = en + fn  1 = en + fn  (e + f)n
This is visibly divisible by ef, hence nilpotent; thus en + fn = 1 + c is inver*
*tible.
Define
"e= en=(1 + c) "f= fn =(1 + c) = 1  "e
FORMAL SCHEMES AND FORMAL GROUPS 13
Then "eis an idempotent lifting e. If "e1is another such then "e1"fis idempoten*
*t. It
lifts ef = 0, so it is also nilpotent. It follows that "e1"f= 0 and "e1= "e"e1.*
*_Similarly,
"e= "e"e1, so "e= "e1. __
Theorem 2.39 (Chinese Remainder Theorem).Suppose that {Iff} is a finite fam
ily of ideals in R, which are pairwise coprime (i.e. Iff+ Ifi= R when ff 6= fi)*
*. Then
" Y
R= Iff= R=Iff
ff ff
Proof.[18, Theorems 1.3,1.4] ___
`
Corollary 2.40.Suppose that zar(R)p=_ ffzar(R=Iff)Q(a finite coproduct). Then
there are unique ideals Jff Iff Jffsuch that R ' ffR=Jff.
T
Proof.Proposition 2.37 implies that ffIffis nilpotent. If ff 6= fi then no p*
*rime
ideal contains Iff+ Ifi, so Iff+ Ifi= R. Now use the Chinese remainder theorem,_
followed by idempotent lifting. __
Remark 2.41. There are nice topological applications of these ideas in [15, 7]*
*, for
example.
2.6. Sheaves, modules and vector bundles. The simplest definition of a sheaf
over a scheme X is just as a module over the ring OX . (It would be more accura*
*te
to refer to this as a quasicoherent sheaf of Omodules over X, but we shall ju*
*st
call it a sheaf.) However, we shall give a different (but equivalent) definitio*
*n which
fits more neatly with our emphasis on schemes as functors, and which generalises
more easily to formal schemes.
Definition 2.42.A sheaf over a functor X 2 F consists of the following data:
(a) For each (R; x) 2 Points(X), a module Mx over R.
(b) For each map f :(R; x) ! (S; y) in Points(X), an isomorphism (f) =
(f; x): S R Mx !My of Smodules.
The maps (f; x) are required to satisfy the functorality conditions
(i)In the case f = 1: (R; x) !(R; x) we have (1; x) = 1: Mx !Mx.
(ii)Given maps (R; x) f!(S; y) g!(T; z), the map (gf; x) is just the composi*
*te
T R Mx = T S S R Mx 1(f;x)!T S My (g;y)!Mz:
We write SheavesXfor the category of sheaves over X. This has direct sums (with
(M N)x = Mx Nx) and tensor products (with (M N)x = Mx R Nx when
x 2 X(R)). The unit for the tensor product is the sheaf O, which is defined by
Ox = R for all x 2 X(R).
Remark 2.43. If M and N are sheaves over a sufficiently bad functor X, it can
happen that SheavesX(M; N) is a proper class. This will not be the case if X is*
* a
scheme or a formal scheme, however.
Example 2.44. Let x be a point of A1(R), or in other words an element of R.
Define Mx = R=x; this gives a sheaf over A1. Note that Mx = 0 if x is invertibl*
*e,
but Mx = R if x = 0. Thus, M is concentrated at the origin of A1.
Definition 2.45. 1.Let X be a functor in F. If N is a module over the ring
OX = F(X; A1), we define a sheaf N"over X by N"x= R OX N, where we
use x to make R into an algebra over OX .
14 NEIL P. STRICKLAND
`
2. If M is a sheaf over X and R is a ring, we write A(M)(R) = x2X(R)Mx. If
f :R !S is a homomorphism, we define a map A(M)(R) !A(M)(S), which
sends Mx to Mf(x)by m 7! (f; x)(1 m). This gives a functor A(M) 2 FX .
3. If M is a sheaf over X, we define (X; M) = FX (X; A(M)). Thus, an element
u 2 (X; M) is a system of elements ux 2 Mx for all rings R and points
x 2 X(R), which behave in the obvious way under maps of rings. If M = O
then A(O) = A1x X and (X; O) = OX . It follows that (X; M) is a module
over OX for all M.
4. If Y is a scheme over X, we also define (Y; M) = FX (Y; A(M)).
Proposition 2.46.For any functor X 2 F, the functor (X; ): SheavesX !
Mod OX is right adjoint to the functor N 7! "N.
Proof.For typographical convenience, we will write T N for "Nand GM for (X; M).
We define maps j :N ! GT N and ffl: T GM ! M as follows. Let n be an element
of N; for each point x 2 X(R), we define j(n)x = 1 n 2 R OX N = (T N)x,
giving a map j as required. Next, we define fflx: (T GM)x = R OX (X; M) !Mx
by fflx(a u) = aux. We leave it to the reader to check the triangular identit*
*ies __
(fflT)(T j) = 1T and (Gffl)(jG ) = 1G , which show that we have an adjunction. *
* __
Proposition 2.47.Let X be a scheme, and let x0 2 X(OX ) be the tautologi
cal point, which corresponds to the identity map of OX under the isomorphism
X = spec(OX ). Then there is a natural isomorphism (X; M) = Mx0, and
(X; ): SheavesX! Mod OX is an equivalence of categories.
Proof.First, we define a map ff: (X; M) !Mx0 by u 7! ux0. Next, suppose that
m 2 Mx0. If x 2 X(R) for some ring R then we have a corresponding ring map
^x:f 7! f(x) from (OX ; x0) to (R; x). We define fi(m)x = (^x; x0)(m) 2 Mx. One
can check that this gives an element fi(m) 2 (X; M), and that fi :Mx0 !(X; M)
is inverse to ff. It follows that (X; "N) = "Nx0, which is easily seen to be th*
*e same
as N. Also, if N = Mx0 then "Nx= R OX Mx0, and (^x; x0) gives an isomorphism
of this with Mx, so N" = M. It follows that the functor N 7! N" is inverse_to
(X; ). __
It follows that when X is a scheme, the category SheavesX is Abelian. Because
tensor products preserve colimits and finite products, we see that the functors
M 7! Mx preserve colimits and finite products.
We next need some recollections about finitely generated projective modules. *
*If
M is such a module over a ring R and p 2 zar(R) then Mp is a finitely generated
module over the local ring Rp and thus is free [18, Theorem 2.5], of rank rp(M)*
* say.
Note that rp(M) is also the dimension of (p) R M over the field (p) = Rp=pRp.
If this is independent of p then we call it r(M) and say that M has constant ra*
*nk.
Clearly, if any two of M, N and M N have constant rank then so does the third
and r(M N) = r(M) + r(N). Also, if r(M) = 0 then M = 0.
Definition 2.48.Let M be a sheaf over a functor X. If Mx is a finitely generated
projective module over Ox for all x 2 X, we say that M is a vector bundle or lo*
*cally
free sheaf over X. If in addition Mx has rank one for all x, we say that M is a*
* line
bundle or invertible sheaf .
If X is a scheme, a sheaf M is a vector bundle if and only if (X; M) is a fin*
*itely
generated projective module over OX . However, this does not generalise easily *
*to
FORMAL SCHEMES AND FORMAL GROUPS 15
formal schemes, so we do not take it as the definition. It is not hard to check*
* that
Mx has constant rank r for all R and all x 2 X(R) if and only if Mx has dimensi*
*on
r over K for all algebraically closed fields K and all x 2 X(K).
Remark 2.49. In algebraic topology, it is very common that the naturally occur
ring projective modules are free, and thus that the corresponding vector bundles
and line bundles are trivialisable. However, they are typically not equivarian*
*tly
trivial for important groups of automorphisms, so it is conceptually convenient*
* to
avoid choosing bases. The main example is that if Z is a finite complex and V is
a complex vector bundle over Z with Thom complex ZV then eE0ZV gives a line
bundle over ZE . A choice of complex orientation on E gives a Thom class and th*
*us
a trivialisation, but this is not invariant under automorphisms of E.
` n
Example 2.50. Recall the scheme E(n) = i=0E(n; i) of Example 2.36. A point
of E(n)(R) is an nxn matrix A over R with A2 = A. This means that MA = A:Rn
is a finitely generated projective Rmodule, so this construction defines a vec*
*tor
bundle M over E(n). If A is a point of E(n; i) (so that det((u1)A+1) = ui2 R[u*
*])
and R is an algebraically closed field, then elementary linear algebra shows th*
*at A
has rank i. It follows that the restriction of M to E(n; i) has rank i.
Let N be a vector bundle over an arbitrary scheme X. The associated projective
OX module is then a retract of a finitely generated free module, so there is a*
* matrix
A 2 E(n)(OX ) such that (X; N) = A:OnXfor some n. The point A 2 E(n)(OX )
corresponds to a map ff: X ! E(n), and we find`that ff*M = N. If Xi denotes
the preimage of E(n; i) under ff, then X = iXi and the restriction of N to Xi
has rank i.
Let X be a scheme. Using equivalence SheavesX ' Mod OX again, we see that
there are sheaves Hom (M; N) such that
SheavesX(L; Hom (M; N)) = SheavesX(L M; N):
In particular, we define M_ = Hom (M; O). If M is a vector bundle then we have
Hom (M; N)x = Hom R(Mx; Nx) and thus (M_ )x = Hom (Mx; R). In that case M_
is again a vector bundle and M__ = M. If M is a line bundle then we also have
M M_ = O.
Example 2.51. Let Y be a closed subscheme of X, with inclusion map j :Y ! X.
Then IY = {f 2 OX  f(y) = 0 for all pointsy 2 Y } is an ideal in OX and
OY = OX =IY . We define j*O to be the sheaf over X corresponding to the OX 
module OY . More explicitly, we have
(j*O)x = Ox=(f(x)  f 2 JY OX ):
We also let IY be the sheaf associated to the OX module IY , so that (IY )x =
Ox OX IY for all points x of X. Note that the sequence IY ae O i j*O is short
exact in SheavesX, even though the sequences (IY )x !OX i (j*O)x need only be
right exact.
Example 2.52. Given a sheaf N over a functor Y and a map f :X ! Y , we
can define a sheaf f*N over X by (f*N)x = Nf(x). The functor f* :SheavesY!
SheavesX clearly preserves colimits and tensor products. If N is a vector bundle
then so is f*N and we have f* Hom (N; M) = Hom (f*N; f*M) for all M. If X and
Y are schemes, we find that (X; f*N) = OX OY (Y; N).
16 NEIL P. STRICKLAND
Example 2.53. If the functor f* defined above has a right adjoint, we call it *
*f*. If
X and Y are schemes then we know from Proposition 2.47 that there is an essenti*
*ally
unique functor f*: SheavesX! SheavesYsuch that (Y; f*M) = (X; M) (where
the right hand side is regarded as an OY module using the map OX ! OY induced
by f). Using the fact that (X; f*N) = OX OY (Y; N) one checks that f* is
right adjoint to f* as required.
Proposition 2.54.If M is a vector bundle over a scheme X, then A(M) is a
scheme.
Proof.Write N = Mod OX((X; M); OX). Then for any map (x: OX ! R) 2 X(R)
we have Mx = Mod OX(N; R), where R is considered as an OX module via x. If
we let S be the symmetric algebra Sym`OX[N] then we have Mx`= AlgOX(S; R).
It follows easily that Rings(S; R) = xAlgOX ;x(S; R) = xMx = A(M)(R), so_
A(M) is representable as required. __
Definition 2.55.Given a line bundle L over a functor X, we define a functor
A(L)x over X by
a
A(L)x (R) = { isomorphisms u: R !Lx of Rmodules}:
x2X(R)
If X isLa scheme, an argument similar to the one for A(M) shows that A(L)x =
spec( n2Z Nn ), where N = Mod OX((X; L); OX) and N(n) means the dual
of Nn . In particular, A(L)x is a scheme in this case.
2.7. Faithful flatness and descent.
Definition 2.56.Let f :X ! Y be a map of schemes, and f* :XY ! XX the
associated pullback functor. We say that f is flat if f* preserves finite colim*
*its. By
Example 2.19, it is equivalent to say that f* preserves coequalisers. We say th*
*at f
is faithfully flat if f* preserves finite colimits and reflects isomorphisms.
Remark 2.57. Let f :X ! Y be faithfully flat. We claim that f* reflects finite
colimits, so that f*Z = lim!f*Zi if and only if Z = lim Zi. More precisely, if
i !i
{Zi} is a finite diagram in XY and {Zi !Z} is a cone under the diagram, then
{f*Zi! f*Z} is a colimit cone in XX if and only if {Zi! Z} is a colimit cone *
*in
XY . The "if" part is clear. For the "only if" part, write Z0 = lim!Zi, so we *
*have
i
a canonical map u: Z0 !Z. As f is flat we have f*Z0 = lim!f*Zi= f*Z. As f*
i
reflects isomorphisms, we see that u is an isomorphism if f*u is an isomorphism.
The claim follows.
Remark 2.58. Classically, a module M over a ring A is said to be flat if the
functor M A () is exact. It is said to be faithfully flat if in addition, when*
*ever
M A L = 0 we have L = 0. It turns out that f is (faithfully) flat if and only i*
*f the
associated ring map OY ! OX makes OX into a (faithfully) flat module over OY .
We leave this as an exercise (consider schemes of the form spec(OX L), where L
is an OX module and the ring structure is such that L:L = 0).
Remark 2.59. The idea of faithful flatness was probably first used in topology*
* by
Quillen [21]. He observed that if V is a complex vector bundle over a finite co*
*mplex
Z and F is the bundle of complete flags in V , then the projection map FE ! ZE
is faithfully flat. This idea was extended and used to great effect in [12].
FORMAL SCHEMES AND FORMAL GROUPS 17
We next define some other useful properties of maps, which do not seem to fit
anywhere else.
Definition 2.60.We say that a map f :X ! Y is very flat if it makes OX into a
free module over OY . A very flat map is flat, and even faithfully flat provide*
*d that
X 6= ;.
Definition 2.61.We say that a map f :X ! Y is finite if it makes OX into a
finitely generated module over OY .
Remark 2.62. A flat map f :X ! Y is faithfully flat if and only if the result*
*ing
map fzar:Xzar! Yzaris surjective [18, Theorem 7.3].
Example 2.63. An open inclusion D(a) ! X (where`a 2 OX ) is always flat. If
a1; : :;:am 2 OX generate the unit ideal then kD(ak) !X is faithfully flat.
Example 2.64. If D is a divisor on A1 over Y (as in Example 2.10) then D !Y
is very flat and thus faithfully flat.
Definition 2.65.Given a ring R and an Ralgebra S, we write I for the kernel of
the multiplication map S R S ! S, and 1S=R= I=I2, which is a module over S.
Given a map of schemes X ! Y , we define 1X=Y= 1OX=OY, which we think of
as a sheaf over X. We say that X is smooth over Y of relative dimension n if the
map X ! Y is flat and 1X=Yis a vector bundle of rank n over X (we allow the
case n = 1). In that case, we write kX=Yfor the k'th exterior power of 1X=Yover
OX , which is a vector bundle over X of rank nk .
Remark 2.66. If X and Y are reduced affine algebraic varieties over C, and X
is smooth over Y then the preimage of each point y 2 Y is a smooth variety of
dimension independent of y. The converse is probably not true but at least that*
* is
roughly the right idea. It has nothing to do with the question of whether the m*
*ap
X ! Y is a smooth map of manifolds. The latter only makes sense if X and Y
are both smooth varieties (in other words, smooth over spec(C)), and in that ca*
*se
it holds automatically for any algebraic map X ! Y .
The following two propositions summarise the basic properties of (faithfully)*
* flat
maps.
Proposition 2.67.Let X f!Y g!Z be maps of schemes. Then:
(a) If f and g are flat then gf is flat.
(b) If f and g are faithfully flat then gf is faithfully flat.
(c) If f is faithfully flat and gf is flat then g is flat.
(d) If f and gf are faithfully flat then g is faithfully flat.
Proof.All this follows easily from the definitions. *
*___
Proposition 2.68.Suppose we have a pullback diagram of schemes
W _____wXr
 
f g
 
u u
Y ______Z:ws
Then:
18 NEIL P. STRICKLAND
(a) If s is flat then r is flat.
(b) If s is faithfully flat then r is faithfully flat.
(c) If g is faithfully flat and r is flat then s is flat.
(d) If g and r are faithfully flat so s is faithfully flat.
Proof.Consider the functor f*: XW ! XY , which sends a scheme U u!W over
W to the scheme U fu!Y over Y . Colimits in XW are constructed by forming
the colimit in X and equipping it with the obvious map to W . This means that
f* preserves and reflects colimits, as does g*. For any scheme V over X, we have
W xX V = (Y xZ X) xX V = Y xZ V , or in other words f*r*V = s*g*V in XY . It
follows that if s* preserves or reflects finite colimits then so does r*, which*
* gives (a)
and (b).
For part (c), suppose that g is faithfully flat and r is flat. This implies *
*that
sf = gr is flat. Also, part (b) says that any pullback of a faithfully flat ma*
*p is
faithfully flat, and f is a pullback of g so f is faithfully flat. As sf is fla*
*t, part (c)
of the previous proposition tells us that s is flat, as required. A similar_arg*
*ument
proves (d). __
Proposition 2.69.Let f :X ! Y be a faithfully flat map, and let {Vi} be a fini*
*te
diagram in XY . If {f*Vi} has a strong colimit in XX , then {Vi} has a strong c*
*olimit
in XX . In other words, f* reflects strong finite colimits.
Proof.Write V = lim!Vi. Given a map g :X0 !X, we need to show that g*V =
i
lim!g*Vi. To see this, form the pullback square
i
f0
Y 0_____wX0
 
g0 g
 
u u
Y ______X:wf
We know from Proposition 2.68 that f0 is faithfully flat. Because f is flat, we*
* have
f*V = lim!f*Vi. By hypothesis, this colimit is strong, so (g0)*f*V = lim(g0)*f*
**Vi.
i !i
As gf0 = fg0, we have (f0)*g*V = lim!(f0)*g*Vi. As f0 is faithfully flat, the *
*functor
i *
* __
(f0)* reflects colimits, so g*V = lim!g*Vi as required. *
*__
i
Proposition 2.70.If f :X ! Y is faithfully flat and Y ! Z is arbitrary then
the diagram
X xY X _____w_wX f!Y
is a strong coequaliser in XZ.
Proof.As f* :XY ! XX reflects strong coequalisers, it is enough to show that t*
*he
above diagram becomes a strong coequaliser after applying f*. Explicitly, we ne*
*ed
to show that the following is a strong coequaliser:
X xY X xY X _____w_wd0d1X xY X d!X;
FORMAL SCHEMES AND FORMAL GROUPS 19
where
d0(a; b; c)= (b; c)
d1(a; b; c)= (a; c)
d(a; b)= b:
In fact, one can check that this is a split coequaliser, with splitting given b*
*y the
maps
X xY X xY X s X xY X t X;
where
s(a; b)= (a; b; b)
t(a)= (a; a):
As split coequalisers are preserved by all functors, they are certainly strong_*
*co
equalisers. __
Now suppose that f :X ! Y is faithfully flat, and that U is a scheme over X.
We will need to know when U descends to Y , in other words when there is a sche*
*me
V over Y such that U = V xY X. Given a point a 2 X(R), we regard a as a map
spec(R) !X and write Ua for the pullback of U along this map, which is a scheme
over spec(R).
Definition 2.71.Let f :X ! Y be a map of schemes, and let U be a scheme over
X. A system of descent data for U consists of a collection of maps a;b:Ua ! Ub
of schemes over spec(R), for any ring R and any pair of points a; b 2 X(R) with
f(a) = f(b). These maps are required to be natural in (a; b), and to satisfy t*
*he
cocycle conditions a;a= 1 and a;c= b;cO a;b.
We write Xf for the category of pairs (U; ), where U is a scheme over X and
is a system of descent data.
Remark 2.72. The naturality condition for the maps a;bjust means that they
give rise to a map ss*0U ! ss*1U of schemes over X xY X.
Remark 2.73. Note also that the cocycle conditions imply that a;bO b;a= 1, so
a;bis an isomorphism.
Definition 2.74.If V is a scheme over Y and f :X ! Y then there is an obvious
system of descent data for U = f*V , in which a;bis the identity map of Ua =
Vf(a)= Vf(b)= Ub. We can thus consider f* as a functor XY ! Xf. We say that
a system of descent data on U is effective if (U; ) is equivalent to an object*
* in
the image of f*. It is equivalent to say that there is a scheme V over Y and an
isomorphism OE: U ' f*V such that
1
a;b= (Ua OE!Vf(a)= Vf(b)OE!Ub)
for all (a; b).
Definition 2.75.Given a map f :X ! Y , a scheme U g! X over X, and a
system of descent data for U, we define U q!QU to be the coequaliser of the
maps d0; d1: U xY X ! U defined by
d0(u; a)= u
d1(u; a)= g(u);a(u):
20 NEIL P. STRICKLAND
We note that d0 and d1 have a common splitting s: u 7! (u; g(u)), so we have a
reflexive coequaliser. We also note that there is a unique map r :QU ! Y such
that rq = fg, so we can think of QU as a scheme over Y .
Proposition 2.76 (Faithfully flat descent).If f :X ! Y is faithfully flat, th*
*en
the functor f* :XY ! Xf is an equivalence, with inverse given by Q. Moreover,
the coequaliser in XY that defines QU is a strong coequaliser.
Proof.Firstly, it is entirely formal to check that Q is left adjoint to f*. Nex*
*t, we
claim that Qf* = 1, or in other words that the projection map f*V = V xY X ! V
is a coequaliser of the maps d0; d1: V xY X xY X ! V xY X. Explicitly, we need
to show that (v; a) 7! v is the coequaliser of (v; b; a) 7! (v; b) and (v; b; a*
*) 7! (v; a).
This is just the same as Proposition 2.70. Thus Qf* = 1 as claimed.
We now show that f*QU = U. As f* preserves coequalisers, it will be enough
to show thatfthe*projectiondf*U = U xY X ! U is the coequaliser of the fork
U xY XxY X _____w_wf*d1U xY 0X. More explicitly, we need to show that the map (*
*u; a) 7!
u is the coequaliser of the maps (u; a; b) 7! (u; b) and (u; a; b) 7! (g(u);a(u*
*); b). In
fact, it is a split coequaliser, with splitting given by the maps u 7! (u; g(u)*
*) and
(u; a) 7! (u; a; a). Thus, f*Q = 1 as claimed. We also see that the coequalis*
*er
defining QU becomes split and thus strong after applying f*. It follows from *
* __
Proposition 2.69 that it was a strong coequaliser in the first place. *
* __
Corollary 2.77.If f :X ! Y is faithfully flat, then the functor f* :XY !_XX is
faithful. __
We also have a similar result for sheaves.
Definition 2.78.Let f :X ! Y be a map of schemes, and let M be a sheaf over
X. A system of decent data for M consists of a collection of maps a;b:Ma !Mb of
Rmodules, for every ring R and every pair of points a; b 2 X(R) with f(a) = f(*
*b).
These are supposed to be natural in (a; b) and to satisfy the conditions a;a= 1*
* and
b;cO a;b= a;c. We write Sheavesffor the category of sheaves over X equipped
with descent data. The pullback functor f* can be regarded as a functor from
SheavesYto Sheavesf.
Proposition 2.79.If f is faithfully flat, then the functor f* :SheavesX! Sheav*
*esf
is an equivalence of categories.
The proof is similar to that of Proposition 2.76, and is omitted.
We shall say that a statement holds locally in the flat topology or fpqc loca*
*lly if
it is true after pulling back along a faithfully flat map. (fpqc stands for fid*
*element
plat et quasicompact; the compactness condition is automatic for affine scheme*
*s).
Suppose that a certain statement S is true whenever it holds fpqclocally. We t*
*hen
say that S is an fpqclocal statement.
Remark 2.80. Let X be a topological space. We say that aSstatement S holds
locally on X if and only`if there is an open covering X = iUi such that S holds
on each Ui. Write Y = iUi, so Y ! X is a coproduct of open inclusions and is
surjective. We could call such a map an "disjoint covering map". We would then
say that S holds locally if and only if it holds after pulling back along a dis*
*joint
covering map. One can get many analogous concepts varying the class of maps
in question. For example, we could use covering maps in the ordinary sense. In
the category of compact smooth manifolds, we could use submersions. This is the
conceptual framework in which the above definition is supposed to fit.
FORMAL SCHEMES AND FORMAL GROUPS 21
Example 2.81. Suppose that N is a sheaf on Y which vanishes fpqclocally. This
means that there is a faithfully flat map f :X ! Y such that (X; f*N) = OX OY
(Y; N) = 0. By the classical definition of faithful flatness, this implies that*
* N = 0.
In other words, the vanishing of N is an fpqclocal condition.
Example 2.82. Let N be a sheaf over Y , and let n be an element of (Y; N) that
vanishes fpqclocally. This means that there is a faithfully flat map f :X ! Y
such that the image of n in (X; f*N) = OX OY (Y; N) is zero. Let g be the
projection X xY X ! Y . One can show that the diagram
*
(Y; N) f!(X; f*N) _____w_w(X xY X; g*N)
is an equaliser. Indeed, it becomes split after tensoring with OX over OY , a*
*nd
that functor reflects equalisers by the classical definition of faithful flatne*
*ss. In
particular, the map marked f* is injective, so n = 0. Thus, the vanishing of n *
*is
an fpqclocal condition.
Example 2.83. Suppose that M is a vector bundle of rank r over a scheme X.
We claim that M is fpqclocally free of rank r, in other words that there is a
faithfully flat map f :W ! X such that f*M ' Or. To prove this, choose a
matrix A 2 Mn(OX ) such that (X; M) = A:OnX. If R is a ring and x 2 X(R) then
A(x) 2 Mn(R) and Mx = A(x):Rn. Let W (R) be the set of triples (x; P; Q) such
that x 2 X(R) and P and Q are matrices over R of shape r x n and n x r such that
det(P A(x)Q) is invertible. This is easily seen to be a scheme over X. In fact,*
* it
is an open subscheme of the scheme of all triples (x; P; Q), which can be ident*
*ified
with A2nrx X. It follows that W is flat over X. Moreover, if R is a field then
elementary linear algebra tells us that the map W (R) ! X(R) is surjective, so
that W is faithfully flat over R. If (x; P; Q) is a point of W then A(x)Q: Rr *
*!Mx
is a split monomorphism. By comparison of ranks, it is an isomorphism. It follo*
*ws
that M becomes free after pulling back to W .
Example 2.84. Proposition 2.68 tells us that flatness and faithful flatness are
themselves fpqclocal properties.
Example 2.85. Let M be a vector bundle of rank r over a scheme X, as in Ex
ample 2.83. Let Bases(M) be the functor of pairs (x; B) where x is a point of X
and B :Orx! Mx is an isomorphism. Note that Bases(M)(R) can be identified
with the set of tuples (x; b1; : :;:br; fi1; : :;:fir) such that bi 2 Mx and fi*
*j 2 M_x
and fij(bi) = ffiij, so Bases(M) is a closed subscheme of A(M)rXxX A(M_ )rX.
It is clear that M becomes free after pulling back along the projection
f :Bases(M) !X:
If M = Or is free, then Bases(M) is just the scheme GL rxX, where GL ris the
scheme of invertible r x r matrices. It's not hard to see that OGLr = Z[xi;j 0
i; j < r][det(xij)1] is torsionfree, and clearly GL r(k) 6= ; for all fields *
*k, and
one can conclude that the map GL r! 1 = spec(Z) is faithfully flat. It follows
that Bases(M) is faithfully flat over X when M is free. Even if M is not free, *
*we
see from Example 2.83 that it is fpqclocally free, so the map Bases(M) ! X is
fpqclocally faithfully flat. As remarked in Example 2.84, faithful flatness is*
* itself
a local condition, so Bases(M) !X is faithfully flat.
22 NEIL P. STRICKLAND
Example 2.86. Any monic polynomial f 2 R[x] can be factored as a product of
linear terms, locally in the flat topology. Indeed, suppose
mX
f = (1)mk amk xk
0
with a0 = 1. It is well known that S = Z[x1; : :x:m] is free of rank m! over
T = Sm = Z[oe1; : :o:em ], where oek is the k'thQelementary symmetric functio*
*n in
the x's. A basis is given by the monomials xff= xffkkfor which ffk < k. We can
map T to R by sending oek to ak, and then observeQthat U = S T R is free and
thus faithfully flat over R. Clearly f(x) = k(x  xk) in U[x], as required.
We conclude this section with some remarks about open mappings. We have
to make a slightly twisted definition, because in our affine context we do not *
*have
enough open subschemes. Suppose that f :X ! Y is a map of spaces, and that
W X is closed. We can then define W 0= {y 2 Y  f1 y W } = f(W c)c.
Clearly f is open iff (W closed implies W 0closed). We will define openness for
maps of schemes by analogy with this.
Definition 2.87.Let f :X ! Y be a map of schemes. For any closed subscheme
W X, we define a subfunctor W 0of Y by
W 0(R) = {y 2 Y (R)  Wy = Xy}:
We say that f is open if for every W , the corresponding subfunctor W 0 Y is
actually a closed subscheme.
Proposition 2.88.A very flat map is open.
Proof.Let f :X ! Y be very flat. Write A = OX and B = OY, and choose a basis
A = B{eff}. Suppose that W = V (I) is a closed subschemePof X. Let {gfi} be
a system of generators of I, so we can write gfi= ffgfiffefffor suitable elem*
*ents
gfffi2 A. Consider a point y 2 Y (R), corresponding to a map y*: B ! R. This
will lie in W 0(R) iff R B A = R B (A=I), iff the imagePof I in R B A = R{eff}
is zero. This image is generated by the elements hfi= ffy*(gfiff)eff. Thus,*
* it
vanishes iff y*(gfiff) = 0 for all ff and fi. This shows that W 0= V (I0), whe*
*re_
I0= (gfiff), so W 0is a closed subscheme as required. _*
*_
2.8. Schemes of maps.
Definition 2.89.Let Z be a functor Rings! Sets, and let X and Y be functors
over Z. For any ring R, we let Map Z(X; Y )(R) be the class of pairs (z; u), wh*
*ere
z 2 Z(R) and u: Xz ! Yz is a map of functors over spec(R). If this is a set
(rather than a proper class) for all R, then we get a functor Map Z(X; Y ) 2 F.*
* This
is clearly the case whenever X, Y and Z are all schemes. However, the functor
Map Z(X; Y ) need not itself be a scheme.
When Z = 1 is the terminal scheme we will usually write Map (X; Y ) rather th*
*an
Map 1(X; Y ).
Remark 2.90. It is formal to check that
FZ(W; MapZ (X; Y )) = FZ(W xZ X; Y ) = FW (W xZ X; W xZ Y ):
In particular, if X, Y , Z and Map Z(X; Y ) are all schemes then we have
XZ(W; MapZ (X; Y )) = XZ(W xZ X; Y ) = XW (W xZ X; W xZ Y ):
FORMAL SCHEMES AND FORMAL GROUPS 23
Example 2.91. It is not hard to see that maps Anxspec(R) !Am xspec(R) over
spec(R) biject with mtuples of polynomialsLover R in n variables, so Map(An; A*
*m )(R) =
R[x1; : :;:xn]m , which is isomorphic to n2N R (naturally in R). This functor
is not a representable (it does not preserve infinite products, for example) so
Map (An; Am ) is not a scheme. It is a formal scheme, however.
Example 2.92. Write D(n)(R) = {a 2 R  an+1 = 0}, so
D(n) = spec(Z[x]=xn+1)
Q n
is a scheme. We find that Map (D(n); A1)(R) = R[x]=xn+1 ' i=0R, so that
Map (D(n); A1) ' An+1 is a scheme.
Example 2.93. Let E be an even periodic ring spectrum. As U(n) is a com
mutative Hspace, we see that E0(U(n)) is a ring, so we can define a scheme
spec(E0(U(n))). We will see later that there is a canonical isomorphism
spec(E0(U(n))) ' Map SE((CP n1)E ; Gm ):
We now give a proposition which generalises the last two examples.
Proposition 2.94.Let Z be a scheme and let X and Y be schemes over Z, and
suppose that X is finite and very flat over Z. Then Map Z(X; Y ) is a scheme.
Proof.Let R be a ring, and z a point of Z(R), giving a map ^z:OZ ! R. We
need to produce an algebra B over OZ such that the maps B ! R of OZ algebras
biject with maps Xz ! Yz of schemes over spec(R), or equivalently with maps
ROZ OY ! ROZ OX of Ralgebras, or equivalently with maps OY ! ROZ OX
of OZalgebras.
Write O_X= Hom OZ(OX ; OZ) and A = SymOZ [O_X OZ OY ]. Then
AlgOZ(A; R) = Hom OZ(O_X OZ OY ; R) = Hom OZ(OY ; R OZ OX ):
A suitable quotient B of A will pick out the algebra maps from OY to OW OZ OXP.
To be more explicit,Plet {e1; : :;:en} be a basis for OX over OZ , with 1 = i*
*diei
and eiej = kcijkek. Let {ffli} be the dual basis for O_X. Then B is A mod the
relations
X
fflk ab= cijk(ffli a)(fflj b)
i;j
ffli 1= di:
More abstractly, if we write q for the projection A ! B and j for the inclusion
O_XOY !A, then B is the largest quotient of A such that the following diagrams
24 NEIL P. STRICKLAND
commute:
Y1
OY OY O_X _____wOY O_X

_  
1X   _
u  O_ _______wOZj
OY OY O_X O_X  X 
  
twist  1j  j
 qj  
u   
OY O_X OY O_X  _ u _____ u
 OX OY qj wB
qjqj 
 
u u
B B ____________wBB
We conclude that spec(B) has the defining property of Map Z(X; Y ). _*
*__
2.9. Gradings. In this section, we show that graded rings are essentially the s*
*ame
as schemes with an action of the multiplicative group Gm .
Definition 2.95.A gradingLof a ring R is a system of additive subgroups Rk R
for k 2 Z such that R = kRk and 1 2 R0 and RjRk Rj+k for all j; k. We say
that a map g :R ! S between graded rings is homogeneous if g(Rk) Sk for all
k.
Proposition 2.96.Let X be a scheme. Then gradings of OX biject with actions
of the group scheme Gm on X. Given such actions on X and Y , a map f :X ! Y
is Gm equivariant if and only if the corresponding map OY ! OX is homogeneous.
Proof.Given an action of Gm on X, we define (OX )k to be the set of maps f :X !
A1 such that f(u:x) = ukf(x) for all rings R and points u 2 Gm (R), x 2 X(R). It
is clearLthat 1 2 (OX )0 and that (OX )j(OX )k (OX )j+k. We need to check that
OX = k(OX )k.PFor this, we consider the map ff*: OX ! OGm xX = OX [u1 ].
If ff*(f) = kukfk (so fk = 0 for almost allPk), then we find that the fk are *
*the
unique functions X ! A1 such thatPf(u:x) = kukfk(x) for all u and x. By
taking u = 1, we see that f = kfk. We also find that
X X
ukvkfk(x) = f((uv):x) = f(u:(v:x)) = ujvkfkj(x):
k j;k
By working in the universal case R = OX [u1 ; v1 ] and comparing coefficients, *
*we
seeLthat fkj = ffijkfk so that fk 2 (OX )k. It follows easily that the addition*
* map
k(OX )k ! OX is an isomorphism, with inverse f 7! (fk)k2Z. Thus, we have a
grading of OX .
Conversely,Psuppose we have a grading (OX )*. We can then write any element
f 2 OX asP k fk with fk 2 (OX )k and fk = 0 for almost all k. We define
ff*(f) = kukfk, and check that this gives a ring map OX ! OX [u1 ]. One can
also check that ff = spec(ff*): Gm xX ! X is an action, and that this construc*
*tion_
is inverse to the previous one. __
Example 2.97. Recall the scheme FGL from Example 2.6. We can let Gm act on
FGLPby (u:F )(x; y) = uF (x=u; y=u); this gives a grading of OFGL. Write F (x; *
*y) =
i;jaij(F )xiyj, and recall that the elements aijgenerate OFGL. It is clear t*
*hat
FORMAL SCHEMES AND FORMAL GROUPS 25
P
(u:F )(x; y) = i;ju1ijaij(F )xiyj, so that aij(u:F ) = u1ijaij(F ), so ai*
*jis
homogeneous of degree 1  i  j. This is of course the same as the grading comi*
*ng
from the isomorphisms OFGL = ss0MP = ss*MU, except that all degrees are halved.
3.Nonaffine schemes
Let E be the category of (not necessarily affine) schemes in the classical se*
*nse,
as discussed in [9] for example. In this section we show that E can be embedded*
* as
a full subcategory of F, containing our category X of affine schemes. We show t*
*hat
our definition of sheaves over functors gives the right answer for functors com*
*ing
from nonaffine schemes, and we investigate the schemes Pn from this point of v*
*iew.
This theory is useful in topology when one wants to study elliptic cohomology, *
*for
example [11]. The results here are surely known to algebraic geometers, but I do
not know a reference.
Given a ring A, we write zar(A) for the Zariski spectrum of A, considered as *
*an
object of E in the usual way. The results of this section will allow us to iden*
*tify
zar(A) with spec(A). Of course, in most treatments, spec(A) is defined to be wh*
*at
we call zar(A).
Definition 3.1.Given a scheme X 2 E, we define a functor F X 2 F by
F X(R) = E(zar(R); X):
It is wellknown that
E(zar(R); zar(A)) = Rings(A; R);
so that F (zar(A)) = spec(A).
Proposition 3.2.The functor F :E !F is full and faithful.
Proof.Let X; Y 2 E be schemes; we need to show that the map F :E(X; Y ) !
F(F X; F Y ) is an isomorphism. First suppose that X is affine, say X = zar(A).
Then the Yoneda lemma tells us that
F(F X; F Y ) = F(spec(A); F Y ) = F Y (A) = E(zar(A); Y ) = E(X; Y )
as required.
Now let X be an arbitrary scheme. We can cover X by open affine subschemes
Xi, and for each i and j we can cover Xi\ Xj by open affine subschemes Xijk.
This gives rise to a diagram as follows.
Q _________Q
E(X; Y )v____________wiE(Xi; Y )________wwijkE(Xijk; Y )
  
  
F  F ' F '
  
u Q u _____ Q u
F(F X; F Y )________wJiF(F Xi; F Y_)___ww ijkF(F Xijk; F Y ):
Standard facts about the category E show that the top line is an equaliser. The
affine case of our proposition shows that the middle and righthand vertical ar*
*rows
are isomorphisms. If we can prove that the map J is injective, then a diagram c*
*hase
will show that the lefthand vertical map is an isomorphism, as required.
Suppose we have two maps f; g :F X ! F Y and that Jf = Jg, or in other words
fFXi = gFXi for all i. We need to show that f = g. Consider a ring R and a
point x 2 F X(R), or equivalently a map W = zar(R) x!X. We need to show that
26 NEIL P. STRICKLAND
f(x) = g(x) as maps from W to Y . We can cover W by open affine subschemes
Ws such that x: Ws !X factors through Xifor some i. As fFXi = gFXi, we see
that f(x) O js = g(x) O js, where js:Ws !W is the inclusion. As the schemes_Ws_
cover W , we see that f(x) = g(x) as required. __
Proposition 3.3.Let X 2 E be a scheme. Then the category of quasicoherent
sheaves of Omodules over X is equivalent to the category of sheaves over F X.
Proof.Let M be a quasicoherent sheaf of Omodules over X. Consider a ring R
and a point x 2 F X(R), corresponding to a map x: zar(R) ! X. We can pull
M back along this map to get a quasicoherent sheaf of Omodules over zar(R),
whose global sections form a module G(M)x = (zar(R); x*M) over R. It is not
hard to see that this construction gives a sheaf GM over the functor F X. If X
is affine then we know from Proposition 2.47 that sheaves over F X are the same
as modules over OX , and it is classical that these are the same as quasicohere*
*nt
sheaves of Omodules over X, so the functor G is an equivalence in this case.
Now let X 2 E be an arbitrary scheme, and let N be a sheaf over F X. We can
cover X by open affine subschemes Xi, and we can cover Xi\ Xj by open affine
subschemes Xijk. By the affine case of the proposition, we can identify Ni= NF*
*Xi
with a quasicoherent sheaf Mi of Omodules over Xi. The obvious isomorphism
NiFXijk= NjFXijkgives an isomorphism MiXijk= MjXijk(because our functor
G is an equivalence for the affine scheme Xijk). One checks that these isomorph*
*isms
satisfy the relevant cocycle condition, so we can glue together the sheaves Mi *
*to
get a quasicoherent sheaf M over X. One can also check that this construction i*
*s __
inverse to our previous one, which implies that G is an equivalence of categori*
*es. __
From now on we will not usually distinguish between X and F X.
We next examine how projective spaces fit into our framework. Let Pn be the
scheme obtained by gluing together n +Q1 copies of An in the usual way. In more
detail, we consider the scheme An+1 = ni=0A1, and let Uibe the closed subsche*
*me
where xi = 1, so Ui ' An. If j 6= i we let Vijbe the open subscheme of Ui where
xj is invertible. We define OEij:Vij! Vjiby
OEij(x0; : :;:xn) = (x0; : :;:xn)=xj:
We use these maps to glue the Ui's together to get a scheme Pn.
We define a sheaf Li over Ui by Li;a_= Ra_ Rn+1 for a_2 Ui(R). Note that if
ssi:Rn+1 !R is the i'th projection then ssi induces an isomorphism Li;a_!R, so
Li;a_is a line bundle over Ui. If a_2 Vij(R) then it is clear that Li;a_= Lj;OE*
*ij(a_). It
follows that the bundles Liglue together to give a line bundle L over Pn. From *
*the
construction, we see that there is a short exact sequence L ae On+1 i V , in wh*
*ich
V is a vector bundle of rank n. We also write O(k) for the (k)'th tensor power*
* of
L, which is again a line bundle over Pn.
Proposition 3.4.For any ring R, we can identify Pn(R) = E(zar(R); Pn) with the
set of submodules M Rn+1 such that M is a summand and has rank one.
This will be proved after a lemma.
Definition 3.5.Write Qn(R) for the set of submodules M Rn+1 such that L is a
rankone projective module and a summand, or equivalently Rn+1=M is a projective
module of rank n. Given a map R !R0we have a map Qn(R) !Qn(R0) sending
M to R0R M, which makes Qn into a functor.
FORMAL SCHEMES AND FORMAL GROUPS 27
We now define a map fl :Pn ! Qn, which will turn out to be an isomorphism.
Consider a ring R and a point x 2 Pn(R), corresponding to a map x: spec(R) !Pn.
By pulling back the sequence L ae On+1 i V and identifying sheaves over spec(R)
with Rmodules, we get a short exact sequence x*L ae Rn+1 i x*V . Here x*L
and x*V are projective, with ranks one and n respectively, so x*L 2 Qn(R). We
define fl(x) = x*L.
Lemma 3.6. Let W be an affine scheme, and let W1; : :;:Wm be a finite cover of
W by basic affine open subschemes Wi= D(ai). Then there is an equaliser diagram
Y Y
F(W; Qn) ! F(Wi; Qn) _____w_wF(Wi\ Wj; Qn):
i ij
` `
Proof.Write W 0= iWi and W 00= ijWi \ Wj, so that the evident map
f :W 0! W is faithfully flat and W 00= W 0xW W 0. We can thus use Propo
sition 2.79 to identify SheavesW with the category Sheavesf of sheaves on W 0
equipped with descent data. It follows that for any sheaf F on W , the subsheav*
*es of
F biject with subsheaves K f*F that are preserved by the descent data for f*F .
This condition is equivalent to the condition ss*0K = ss*1K (fss0)*F = (fss1)**
*F_.
Now take F = On+1, and the lemma follows easily. __
Proof of Proposition 3.4.Suppose we have two points x 2 Ui(R) Pn(R) and
y 2 Uj(R) Pn(R), and that fl(x) = fl(y). It then follows easily from the defin*
*itions
that x = y.
Now suppose we have two points x; y 2 Pn(R) such that fl(x) = fl(y). We write
W = spec(R), so x: W ! Pn. We can cover W by basic affine open subsets
W1; : :;:Wm with the property that each x(Wk) is contained in some Ui, and each
y(Wk) is contained in some Uj. This implies (by the previous paragraph) that
x = y as maps Wk !Pn. We can now deduce from Lemma 3.6 that x = y. Thus,
fl :Pn(R) !Qn(R) is always injective.
Now consider a point M 2 Qn(R), so M is a sheaf over W = spec(R). We claim
that we can cover W by basic open subschemes V such that MV lies in the image
of fl :F(V; Pn) ! F(V; Qn). Indeed, as M is projective, we can start by coveri*
*ng
W with basic open subschemes on which M is free. It is easy to see that over su*
*ch
a subscheme, there exist maps O u!On+1 v!O such that the image of u isPM and
vu = 1. If we write u and v in terms of bases in the obvious way then iuivi= *
*1,
so the elements uigenerate the unit ideal, so the basic open subschemes D(ui) f*
*orm
a covering. On D(ui) we can define x = (u0; : :;:un)=ui 2 Ui, and it is clear t*
*hat
fl(x) = M.
We can thus choose a basic open covering W = W1 [ : :[:Wm and maps
xk: Wk ! Pn such that fl(xk) = MWk . Let xjk be the restriction of xj to
Wjk = Wj \ Wk. We then have fl(xjk) = MWjk = fl(xkj) and fl is injective
so xjk= xkj. We also have a diagram
Q n ______Q n
F(W; Pn) v________wiF(Wi; P ) ______wwijF(Wij; P )
  
fl fl fl
  
u Q u n _____ Q u n
F(W; Qn) ________w iF(Wi; Q ) _____wwijF(Wij; Q ):
28 NEIL P. STRICKLAND
The top row is unchanged if we replace F by E, and this makes it clear that it *
*is an
equaliser diagram. The bottom row is an equaliser diagram by Lemma 3.6. We have
alreadyQseen that the verticalQmaps are injective. The elements xigive an eleme*
*nt of
iF(Wi; Pn), whose image in iF(Wi; Qn) is the same as that of M 2 F(W; Qn).
We conclude by diagram chasing that there is an element x 2 F(W; Pn) such that *
*__
fl(x) = M. Thus fl is also surjective, as required. *
*__
Definition 3.7.SupposePthat we have elements a0; : :;:an 2 R, which generate
the unit ideal, say ibiai = 1. Let M be the submodule of Rn+1 generated
by a_= (a0; : :;:an). The elements bj define a map Rn+1 ! R which carries L
isomorphically to R. It follows that M 2 Qn(R); the submodules M that occur in
this way are precisely those that are free over R. We write [a0 : : :::an] for *
*the
corresponding point of Pn(R). Most of the time, when working with points of Pn,
we can assume that they have this form, and handle the general case by localisi*
*ng.
We finish this section with a useful lemma.
Lemma 3.8. We have [a0 : : :::an] = [a00: : :::a0n] if and only if there is a *
*unit
u 2 Rx such that ua0j= aj for all j, if and only if aia0j= aja0ifor all i and j.
Proof.The first equivalence is clear if we think in terms of Qn(R). For the sec*
*ond,
suppose that ua0j= aj for all j. Then aia0j= u1aiaj = a0iaj as required. Con
versely, suppose that aia0j=Paja0ifor all iPand j. We can choose sequencesPb0; *
*: :;:bn
and b00;P: :;:b0nsuch that iaibi= 1 and ia0ib0i= 1. Now define u = iaib0i*
*and
v = ja0jb0j. Then
X X
ua0j= b0iaia0j= b0ia0iaj = aj:
i i
Moreover, we have
X X
u bja0j= bjaj = 1;
j j
so u is a unit as required. ___
4. Formal schemes
In this section we define formal schemes, and set up an extensive categorical
apparatus for dealing with them, and generalise our results for schemes to form*
*al
schemes as far as possible. We define the subcategory of solid formal schemes, *
*which
is convenient for some purposes. We also define functors from various categori*
*es
of coalgebras to the category of formal schemes, which are useful technical too*
*ls.
Finally, we study the question of when Map Z(X; Y ) is a formal scheme.
Definition 4.1.A formal scheme is a functor X :Rings! Sets that is a small
filtered colimit of schemes. More precisely, there must be a small filtered cat*
*egory
I and a functor i 7! Xi from I to X F = [Rings; Sets] such that X = lim!Xi in
i
F, or equivalently X(R) = lim!Xi(R) for all R. We call such a diagram {Xi} a
i
presentation of X. We write bXfor the category of formal schemes.
Example 4.2. The most basic example is the functor bA1 defined by bA(R) =
Nil(R). This is clearly the colimit over N of the functors D(N) = spec(Z[x]=xN+*
*1 ).
We also define bAn(R) = Nil(R)n.
FORMAL SCHEMES AND FORMAL GROUPS 29
Example 4.3. More generally, given a scheme X and a closed subscheme Y =
V (I), we define a formal scheme X^Y= lim!V (IN ).
N
Example 4.4. For a common example not of the above type, consider the functor
bA(1)(R) = L n2NNil(R), so X = lim bAn, which is again a formal scheme.
!n
Example 4.5. If Z is an infinite CW complex and {Zff} is the collection of fini*
*te
subcomplexes and E is an even periodic ring spectrum, we define ZE = lim!(Zff)*
*E .
ff
This is clearly a formal scheme.
We can connect this with the framework of [8, Section 8] by taking C to be the
category Ringsop. From this point of view, a formal scheme is an indrepresenta*
*ble
contravariant functor from Ringsopto Sets. We shall omit any mention of uni
verses here, leaving the settheoretically cautious reader to lift the necessar*
*y details
from [8, Appendice], or to avoid the problem in some other way.
Given two filtered diagrams X :I !X and Y :J !X we know from [8, 8.2.5.1]
that
bX(limXi; limYj) = limlimX(Xi; Yj):
! !  !
i j i j
It follows that bXis equivalent to the category whose objects are pairs (I; X) *
*and
whose morphisms are given by the above formula. We will feel free to use either
model for bXwhere convenient.
Proposition 4.6.A functor X :Rings! Setsis a formal scheme if and only if
(a) X preserves finite limits, and
(b) There`is a set of schemes Xiand natural maps Xi! X such that the resulting
map iXi(R) !X(R) is surjective for all R.
Proof.This is essentially [8, Theoreme 8.3.3]. To see this, let D be the catego*
*ry of
schemes over X. A map spec(R) ! X is the same (by Yoneda) as an element of
X(R), so Dop is equivalent to the category Points(X). This category corresponds
to the category C=F of the cited theorem. Thus, by the equivalence (i),(iii) of*
* that
theorem, we see that X is a formal scheme if and only if X preserves finite lim*
*its,
and D has a small cofinal subcategory. (Grothendieck actually talks about finite
colimits, but in our case that implicitly refers to colimits in Ringsopand thus*
* limits
in Rings.) It is shown in the proof of the theorem that if X preserves finite l*
*imits,
then D is a filtered category, so we can use [8, Proposition 8.1.3(c)] to recog*
*nise
cofinal subcategories. This means that a small collection {Xi} of schemes over X
gives a cofinal subcategory if and only if each map from a scheme Y to X factors
through some Xi. By writing Y `= spec(R) and using the Yoneda lemma, it is *
*__
equivalent to say that the map iXi(R) !X(R) is surjective for all R. *
*__
4.1. (Co)limits of formal schemes.
Proposition 4.7.The category bXhas all small colimits. The inclusion X ! bX
preserves finite colimits, and the inclusion bX!F = [Rings; Sets] preserves fi*
*ltered
colimits. Moreover, if X 2 X then the functor bX(X; ): bX! Setsalso preserves
colimits.
30 NEIL P. STRICKLAND
Proof.Apart from the last sentence, the proof is the same as that of [14, Theor*
*em
VI.1.6]. Johnstone assumes that C (which is Ringsopin our case) is small, but h*
*e__
does not really use this. The last sentence is [14, Lemma VI.1.8]. *
*__
Example 4.8. It is not hard to see that the functor Z 7! ZE of example 4.5
converts filtered homotopy colimits to colimits of formal schemes.
Suppose we have a diagram of formal schemes X :I ! Xb. For each i 2 I
we then have a filtered category J(i) and a functor X(i; ): J(i) ! X such that
X(i) = lim! X(i; j). For many purposes, it is convenient if we can take all the
J(i)
categories J(i) to be the same. This motivates the following definition.
Definition 4.9.A category I is rectifiable if for every functor X :I ! bXthere*
* is
a filtered category J and a functor Y :I x J ! X such that X(i) = lim!Y (i; j*
*) as
J
functors of i.
Proposition 4.10.If I is a finite category such that I(i; i) = {1} for all i 2 *
*I, then
I is rectifiable.
Proof.See [8, Proposition 8.8.5]. ___
Proposition 4.11.If I is a discrete small category (in other words, a set), the*
*n I
is rectifiable.
Proof.As X(i) is a formal scheme, there is a filtered categoryQJ(i) and a funct*
*or
Z(i; ): J(i) ! X such that X(i) = lim! Z(i; j). Write J = iJ(i), let ssi:J*
* !
J(i)
J(i) be the projection, and let Y (i; ) be the composite functor J ssi!J(i) Z*
*(i;)!X.
It is easy to check that J is filtered and that ssiis cofinal, so X(i) = lim!Y*
* (i; j), as
__ J
required. __
Proposition 4.12.The category bXhas finite limits, and the inclusions X !Xb!
F preserve all limits that exist. Moreover, finite limits in bXcommute with fil*
*tered
colimits.
Proof.First consider a diagram X :I !Xb indexed by a finite rectifiable catego*
*ry.
We define U(R) = limX(i)(R), which gives a functor Rings !Sets. It is well
I
known that this is the inverse limit of the diagram X in the functor category
F, so it will suffice to show that U is a formal scheme. As I is rectifiable, *
*we
can choose a diagram Y :I x J ! X as in Definition 4.9. As X has limits, we
can define Z(j) = limY (i; j) 2 X, and then define W = lim Z(j) 2 bX. Then
i !j
W (R) = lim!limY (i; j)(R). As filtered colimits commute with finite limits in*
* the
j  i
category of sets, this is the same as limlimY (i; j)(R) = lim X(i)(R) = V (R).
i!j  i
Thus V = W is a formal scheme, as required.
Both finite products and equalisers can be considered as limits indexed by re*
*c
tifiable categories, and we can write any finite limit as the equaliser of two *
*maps
between finite products. This shows that bXhas finite limits.
Now let {Xi} be a diagram of formal schemes, let X be a formal scheme, and
let {fi:X ! Xi} be a cone. If this is a limit cone in bXthen we must have
X(R) = bX(spec(R); X) = limbX(spec(R); Xi) = lim Xi(R), which means that it
i  i
FORMAL SCHEMES AND FORMAL GROUPS 31
is a limit cone in F (because limits in functor categories are computed pointwi*
*se).
The converse is equally easy, so the inclusion bX! F preserves and reflects li*
*mits.
Similarly, the inclusion X !F preserves and reflects limits, and it follows th*
*at the
same is true of the inclusion X !Xb. ___
4.2. Solid formal schemes.
Definition 4.13.A linear topology on a ring R is a topology such that the cosets
of open ideals are open and form a basis of open sets. One can check that such
a topology makes R into a topological ring. We write LRings for the category
of rings with a given linear topology, and continuous homomorphisms. For any
ring S, the discrete topology is a linear topology on S, so we can think of Rin*
*gs
as a full subcategory of LRings. Given a linearly topologised ring R, we define
spf(R): Rings! Setsby
spf(R)(S) = LRings(R; S) = lim!Rings(R=J; S);
J
where J runs over the directed set of open ideals. Clearly this defines a func*
*tor
spf:LRingsop! bX.
Definition 4.14.Let R be a linearly topologised ring. The completion of R is the
ring bR= limR=I, where I runs over the open ideals in R. There is an evident m*
*ap
I _
R !Rb, and_the composite R !Rb! R=I is surjective so we have R=I = bR=I for
some ideal I bR. These ideals form a filtered system, so we can give bRthe line*
*ar
topology for which they are a base of neighbourhoods of zero. It is easy to che*
*ck
that bbR= bRand that spf(Rb) = spf(R). We say that R is complete, or that it is*
* a
formal ring, if R = bR. Thus bRis always a formal ring. We write FRings for the
category of formal rings.
Definition 4.15.Given a formal scheme X, we recall that OX = bX(X; A1). This
is again a ring under pointwise operations. If {Xi} is a presentation of X then
OX = limOXi.
i
For any point x of X we define Ix = {f 2 OX  f(x) = 0 2 Ox}. From a slightly
different point of view, we can think of x as a map Y = spec(Ox) ! X and Ix
as the kernel of the resulting map OX ! OY. As the informal schemes over X
form a filtered category, we see that the ideals Ix form a directed system. Thu*
*s,
there is a unique linear topology on OX , such that the ideals Ix form a base of
neighbourhoods of zero. With this topology, if {Xi} is a presentation of X, then
OX = limOXi as topological rings.
i
Note that
bX(X; spf(R)) = limbX(Xi; spf(R)) = limLRings(R; OX ) = LRings(R; OX);
  i
i i
so that O: bX!LRingsop is left adjoint to spf:LRingsop! bX. In particular, we
have a unit map X ! spf(OX ) in bX, and a counit map R ! Ospf(R)in LRings.
The latter is just the completion map R !Rb.
Definition 4.16.We say that a formal scheme X is solid if it is isomorphic to
spf(R) for some linearly topologised ring R. We write bXsolfor the category of *
*solid
formal schemes.
32 NEIL P. STRICKLAND
In the earlier incarnation of this paper [25] we defined formal schemes to be
what we now call solid formal schemes. While only solid formal schemes seem to
occur in the cases of interest, the category of all formal schemes has rather b*
*etter
categorical properties, so we use it instead.
Example 4.17. Any informal scheme X is a solid formal scheme (because the zero
ideal is open).
Example 4.18. The formal scheme bAnis solid. To see this, consider the formal
power series ring R = Z[[x1; : :;:xn]], with the usual linear topology defined *
*by the
ideals Ik, where I = (x1; : :;:xk). This is clearly a formal ring, and bAn= spf*
*(R).
Example 4.19. If R is a complete Noetherian semilocal ring with Jacobson radic*
*al
I (for example, a complete Noetherian local ring with maximal ideal I) then it *
*is
natural to give R the linear topology defined by the ideals Ik, and to define s*
*pf(R)
using this. With this convention, the set bX(spf(R); spf(S)) (where S is anoth*
*er
ring of the same type) is just the set of local homomorphisms S ! R. Thus, the
categories of formal schemes used in [26] and [7] embed as full subcategories o*
*f our
category bX.
Example 4.20. Let Z be an infinite CW complex with finite subcomplexes {Zff},
and let E be an even periodic ring spectrum. Let Jffbe the kernel of the map
E0Z ! E0Zff. These ideals define a linear topology on E0Z. In good cases E0Z
will be complete and we will have ZE = spf(E0Z), so this is a solid formal sche*
*me.
See Section 8 for technical results that guarantee this.
Proposition 4.21.
(a) If X is a solid formal scheme then OX is a formal ring.
(b) A formal scheme X is solid if and only if it is isomorphic to spf(R) for s*
*ome
formal ring R, if and only if the natural map X ! spf(OX ) is an isomorph*
*ism.
(c) The functor X 7! Xsol= spf(OX ) is left adjoint to the inclusion of bXsoli*
*n bX.
(d) The functor R 7! bRis left adjoint to the inclusion of FRings in LRings.
(e) The functors R 7! spf(R) and X 7! OX give an equivalence between bXsoland
FRingsop.
Proof.(a): If X is solid then X = spf(R) for some linearly topologised ring R, *
*so
OX = Ospf(R)= bRwhich is a formal ring.
(b): If X is solid then X = spf(R) as above, but spf(R) = spf(Rb) so we may
assume that R is formal. We find as in (a) that OX = R and thus that the map
X ! spf(OX ) = spf(R) is an isomorphism. The converse is easy.
(c): Let T denote the functor X 7! Xsol. This arises from an adjunction,
so it is a monad. On the other hand, if R = OX then R is formal by (a), so
R = Ospf(R)= OXsol. By applying spf(), we see that (Xsol)sol= Xsol, so T 2= T
and T is an idempotent monad. Moreover, bXsolis the subcategory of formal schem*
*es
for which the unit map jX :X ! T X is an isomorphism. It is wellknown that
this is automatically a reflective subcategory. In outline, if Y is solid and*
* X is
arbitrary and f :X ! Y , then f0 = j1YO T f :Xsol! Y is the unique map such
that f0O jX = f.
(d): The proof is similar.
(e): If R is formal then spf(R) is solid and Ospf(R)= bR= R. If X is solid_th*
*en_
OX is formal (by (a)) and X = spf(OX ) (by (b)). __
FORMAL SCHEMES AND FORMAL GROUPS 33
Definition 4.22.Let R, S and T be linearly topologised rings, and let R !S and
R ! T be continuous homomorphisms. We then give S R T the linear topology
defined by the ideals I T + S J, where I runs over open ideals in S and J runs
over open ideals in T . This is easily seen to be the pushout of S and T under *
*R in
LRings. We also define S bRT to be the completion of S R T . If R, S and T are
formal then S bRT is the pushout in FRings (because completion is left adjoint *
*to
the inclusion FRings! LRings).
Proposition 4.23.The subcategory bXsol bXis closed under finite products and
arbitrary coproducts. It also has its own colimits for arbitrary diagrams, whi*
*ch
need not be preserved by the inclusion bXsol!bX.
Proof.One can check that spf(R S) = spf(Rb S) = spf(R) x spf(S), which givesQ
finite products. Let {Ri i 2 I} be a family of formal rings, and write R = i*
*Ri.
We give this ring the product topology,Qwhich is the same as the linear topology
defined by the ideals of the form iJi,`where Ji is open in Ri and Ji = Ri for
almost all i. We claim that spf(R) = ispf(Ri).Q
To see this, let J denote the set ideals J = Q iJias above. This is easily se*
*en to
be a directed set. For J 2 J we see that`R=J = iRi=Ji, where almost all terms*
* in
the product are zero. Thus spec(R=J) = i2Ispec(Ri=Ji), where almost all terms
in the coproduct`are empty. As colimits commute with coproducts, we see that
spf(R) = Ilim!spec(Ri=Ji). As the projection from J to the set of open ideal*
*s in
J `
Ri is cofinal, we see that lim!spec(Ri=Ji) = spf(Ri), so that spf(R) = Ispf(*
*Ri)
J
as claimed.
Now let {Xi} be an arbitrary diagram of solid formal schemes, and let X be its
colimit in bX. As the functor Y 7! Ysolis left adjoint to the inclusion bXsol!*
*bX, we
see that Xsolis the colimit of our diagram in bXsol. *
*___
Remark 4.24. We will see in Corollary 4.40 that bXsolis actually closed under
finite limits.
Example 4.25. As a special case of the preceeding proposition, consider an inf*
*inite
set A. Let R be the`ring of functions u: A ! Z with the product topology, so
that A_= spf(R) = a2A1. We call formal schemes of this type constant`formal
schemes. More generally, given a formal scheme X we write A_X= a2AX. If X is
solid then A_X= spf(C(A; OX)), where C(A; OX) is the ring of functions A !OX ,
under the evident product topology. Clearly, if E is an even periodic ring spec*
*trum
and we regard A as a discrete space then AE = A_x SE .
4.3. Formal schemes over a given base. Let X be a formal scheme. Write
bXXfor the category of formal schemes over X, and XX for the full subcategory
of informal schemes over X. We also write Points(X) for the category of pairs
(R; x), where R is a ring and x 2 X(R); the maps are as in Definition 2.14. Aga*
*in,
the Yoneda isomorphism X(R) = bX(spec(R); X) gives an equivalence Points(X) =
XopX. Moreover, formal schemes Y over X biject with indrepresentable functors
Y 0:Points(X) !Sets by the rules
Y 0(R; x)= preimage of x under the map Y (R) !X(R)
a
Y (R)= Y 0(R; x):
x2X(R)
34 NEIL P. STRICKLAND
Now consider a formal scheme X with presentation {Xi}, indexed by a filtered
category I. We next investigate the relationship between the categories bXX and
bXXi, which we now define.
Definition 4.26.Given a diagram {Xi} as above, we write D{Xi}for the category
of diagrams {Yi}: I !Xbequipped with a map of diagrams {Yi} !{Xi}. For any
such diagram {Yi} and any map u: i !j in I, we have a commutative square
Yi ______YjwYu
 
 
 
u u
Xi _____wXj:Xu
We write bX{Xi}for the full subcategory of D{Xi}consisting of diagrams {Yi} for
which all such squares are pullbacks.
We define functors F :D{Xi}! bXXand G: bXX! D{Xi}by
F {Yi}= lim!Yi
i
GY = {Y xX Xi}:
Proposition 4.27.The functor F is left adjoint to G, and it preserves finite li*
*mits.
The functor G is full and faithful, and its image is bX{Xi}. The functors F and*
* G
give an equivalence between bXXand bX{Xi}.
Moreover, if W is an informal scheme over X and {Yi} 2 bX{Xi}, then any
factorisation W ! Xi ! X of the given map W ! X gives an isomorphism
W xX F {Yi} = W xXi Yi.
Proof.A map F {Yi} ! Z is the same as a compatible system of maps Yi ! Z
over X. As the map Yi! X has a given factorisation through Xi, this is the same
as a compatible system of maps Yi! Z xX Xi= G(Z)iover Xi, or in other words
a map {Yi} !G(Z). Thus F is left adjoint to G.
As filtered colimits commute with finite limits, we see that F G(Y ) = lim!(*
*Y xX
i
Xi) = Y xX lim!Xi= Y . This means that
i
D{Xi}(GY; GZ) = bXX(Y; F GZ) = bXX(Y; Z);
so G is full and faithful. This means that G is an equivalence of bXXwith its i*
*mage,
and it is clear that the image is contained in bX{Xi}. The commutation of fini*
*te
limits and filtered colimits also implies that F preserves finite limits.
We now prove the last part of the proposition; afterwards we will deduce that
the image of G is precisely bX{xi}. Consider an informal scheme W and a map
f :W ! X, and an object {Yi} of bX{Xi}. Let J be the category of pairs (i; g),
where i 2 I and g :W ! Xi and the composite W g!Xi !X is the same as f.
It is not hard to check that J is filtered and that the projection functor J !*
* I is
FORMAL SCHEMES AND FORMAL GROUPS 35
cofinal. For each (i; g) 2 J we have a pullback diagram
W xXi Yi _____wYi
 
 
 
 
 
u u
W ________Xi:w
By taking the colimit over J we get a pullback diagram
lim!W xXi Yi____wF {Yi}

 
 
 
u u
W ___________X:w
On the other hand, for each map u: (i; g) ! (j; h) in J we have Yi = XixXj Yj
(by the definition of bX{Xi}) and thus W xXi Yi= W xXj Yj. It follows easily th*
*at
for each (i; g) the map W xXi Yi! lim!W xXj Yj is an isomorphism, and thus (by
the diagram) that W xX F {Yi} = W xXi Yi.
Now take W = Xi and g = 1 in the above. We find that XixX F {Yi} = Yi,
and thus that F G{Yi} = {Yi}, and thus that {Yi} is in the image of G. This sho*
*ws
that the image of G is precisely bX{Xi}, as required. *
*___
Definition 4.28.Let Y be a formal scheme over a formal scheme X. We say that
Y is relatively informal over X if for all informal schemes X0 over X, the pull*
*back
Y xX X0 is informal.
Proposition 4.29.The category of relatively informal schemes over X has limits,
which are preserved by the inclusion into bXX.
Proof.We can write X as the colimit of a filtered diagram of informal schemes
Xi. It is clear that the category of relatively informal schemes is equivalent*
* to
the subcategory C of bX{Xi}consisting of systems {Yi} of informal schemes. As
the category of informal schemes has limits, we see that the category of inform*
*al
schemes over Xi has limits. Moreover, for each map Xi! Xj, the functor XixXj
(): XXj ! XXi preserves limits. Given this, it is easy to check that C has li*
*mits,
as required. As the inclusion X !Xbpreserves limits, one can check that the sa*
*me
is true of the inclusions XXi !XbXiand C !Xb{Xi}= bXX. ___
4.4. Formal subschemes.
Definition 4.30.We say that a map f :X ! Y of formal schemes is a closed
inclusion if it is a regular monomorphism in bX. (This means that it is the equ*
*aliser
of some pair of arrows Y _____w_wZ, or equivalently that it is the equaliser of*
* the pair
Y _____w_wY qX Y .) A closed formal subscheme of a formal scheme Y is a subfunc*
*tor
X of Y such that X is a formal scheme and the inclusion X ! Y is a closed
inclusion.
Remark 4.31. The functor Z 7! Z(R) is representable (by spec(R)). It follows
that if f :V ! W is a monomorphism in bXthen V (R) !W (R) is injective for all
36 NEIL P. STRICKLAND
R, so V is isomorphic to a subfunctor of W . If f is a regular monomorphism, th*
*en
the corresponding subfunctor is a closed subscheme.
Example 4.32. Let J be an ideal in OX , generated by elements {fi  i 2 I} say.
We define
V (J)(R) = {x 2 X(R)  f(x) = 0 for allf 2 J} = {x  fi(x) = 0 for alli}:
Q
Define a scheme AI by AI(R) = i2IR (this is representedQby the polynomial
algebra Z[xi  i 2 I]). This is just the product i2IA1; by Proposition 4.12,*
* it
does not matter whether we interpret this in X or bX. It follows that there is *
*a map
f :X ! AI with components fi, and another map g :X ! AI with components 0.
Clearly V (J) is the equaliser of f and g, and thus it is a closed formal subsc*
*heme of
X. There is a natural map OX =J ! OV (J)which is an isomorphism in most cases
of interest, but I suspect that this is not true in general (compare Remark 4.3*
*9).
Example 4.33. If X is an informal scheme and Y is a closed informal subscheme
of X then the evident map X^Y! X is a closed inclusion.
Proposition 4.34.A map f :X ! Y of informal schemes is a closed inclusion in
bXif and only if it is a closed inclusion in X.
Proof.It follows from Proposition 4.7 that the pushout Y qX Y is the same wheth*
*er
constructed in X or bX. It follows in turn from Proposition 4.12 that the equal*
*iser
of the two maps Y _____w_wY qX Y is the same whether constructed in X or bX. The
map f is a closed inclusion if and only if X maps isomorphically to this equali*
*ser,_
so the proposition follows. __
Proposition 4.35.If X 2 bXand Y 2 X, then a map f :X ! Y is a closed
inclusion if and only if there is a directed set of closed informal subschemes *
*Yi of
Y such that X = lim!Yi.
i
Proof.First suppose that f is a closed inclusion. We can write X as a colimit of
informal schemes, say X = lim!Xi. Write Zi= Y qXi Y . One checks that these
i2I
schemes give a functor I !X, and that lim!Zi= Y qX Y . Let Yibe the equaliser*
* of
i
the two maps X _____w_wZi, so that Yiis a closed informal subscheme of X, and a*
*gain
the schemes Yigive a functor I !X. As finite limits commute with filtered coli*
*mits
in bX, we see that lim!Yi is the equaliser of the maps Y _____w_wlimZi = Y qX *
*Y .
i !i
This is just X, because f is assumed to be a regular monomorphism.
Conversely, suppose that {Yi} is a directed family of closed subschemes of an
informal scheme Y . Write Zi = Y qYiY and Z = lim!Zi. By much the same
i
logic as above, we see that there is a pair of maps Y _____w_wZ whose equaliser*
*_in
X = lim!Yi, so that X is a closed formal subscheme of Y . _*
*_
i
Proposition 4.36.A map f :X ! Y in bXis a closed inclusion if and only if for
all informal schemes Y 0and all maps Y 0! Y , the pulledback map f0:X0 !Y 0
is a closed inclusion.
Proof.It is clear that the condition is necessary, because in any category a pu*
*llback
of a regular monomorphism is a regular monomorphism. For sufficiency, suppose
that f :X ! Y is such that all maps of the form f0:X0! Y 0are closed inclusio*
*ns.
Write Y as a colimit of informal schemes Yiin the usual way, and let fi:Xi! Yi*
*be
FORMAL SCHEMES AND FORMAL GROUPS 37
the pullback of f along the map Yi! Y . As finite limits in bXcommute with fil*
*tered
colimits, we see that X = lim!Xi. By assumption, fi is a closed inclusion. Wri*
*te
i
Zi = YiqXi Yi, so Xi is the equaliser of the fork Yi_____w_wZi. Write Z = lim!*
*Zi.
i
As finite limits in bXcommute with filtered colimits, we see that X is the equa*
*liser_
of the maps Y _____w_wZ, and thus that f is a closed inclusion. *
* __
Proposition 4.37.Let X f!Y g!Z be maps of formal schemes. If f and g are
closed inclusions, then so is gf. Conversely, if gf is a closed inclusion and g*
* is a
monomorphism then f is a closed inclusion.
Proof.The second part is a formal statement which holds in any category: if we
have maps X f!Y g!Z such that gf is the equaliser of a pair Z _____w_wpqW , t*
*hen
a diagram chase shows that X f!Y is the equaliser of pg and qg and thus is a
regular monomorphism.
For the first part, we can assume by Proposition 4.36 that Z is an informal
scheme. We then know from Proposition 4.35 that there is a filtered system of
closed subschemes Zi of Z such that Y is the colimit of the Zi. The maps Y ! Z
and Zi !Y ! Z are closed inclusions, so the second part tells us that Yi !Y
is a closed monomorphism. Let Xi be the preimage of Zi Y under the map
f :X ! Y . The maps Xi ! Zi and Zi ! Z are closed inclusions of informal
schemes, so the composite Xi! Z is easily seen to be a closed inclusion (becau*
*se
closed inclusions in the informal category are just dual to surjections of ring*
*s). As
filtered colimits commute with pullbacks, we see that X = lim!Xi. It follows f*
*rom
i __
Proposition 4.35 that X ! Z is a closed inclusion. __
Proposition 4.38.Any closed formal subscheme of a solid formal scheme is again
solid.
Proof.Let W f!X _____w_wghY be an equaliser diagram, and suppose that X is sol*
*id.
We need to show that W is solid. Choose a presentation Y = lim!Yi for Y . Let
i2I
J be the set of tuples j = (J; i; g0; h0), where J is an open ideal in OX and i*
* 2 I and
g0; h0:V (J) !Yi and the following diagram commutes.
0
V (J)_____wYi_wgh0
v 
 
 
 
u_______gu
X _______Ywwh
One can make J into a filtered category so that j 7! J is a cofinal functor to*
* the
directed set of open ideals of OX , and j 7! i is a cofinal functor to I (see t*
*he proof
of [8, Proposition 8.8.5]). The equaliser of g0and h0is a closed subscheme of V*
* (J),
so it has the form V (Ij) for some ideal Ij J. As equalisers commute with filt*
*ered
colimits, we see that W = lim!V (Ij). Let K be the set of ideals of the form I*
*j for
J
some j. The functor j 7! Ij from J to K is cofinal, so we have W = lim! V (I).
I2K
We can define a new linear topology on R = OX by letting the ideals I 2 K be a
base of neighbourhoods of zero, and we conclude that W = spf(R). Thus,_W_is
solid. __
38 NEIL P. STRICKLAND
Remark 4.39. In the above proof, suppose that Y is also solid, and let K be the
ideal in OX generated by elements of the form g*u  h*u with u 2 OY . One can
then check that OW = limOX =(K + J), where J runs over the open ideals in OX .
J T
The kernel of the map ss :OX ! OW is J (J + K), which is just the closure of *
*K.
One would like to say that ss was surjective, but in fact its cokernel is lim1*
*(J + K),
J
which can presumably be nonzero.
Corollary 4.40.The subcategory bXsol bXof solid formal schemes is closed under
finite limits.
Proof.We know from Proposition 4.23 that a finite product of solid schemes is *
* __
solid, and a finite limit is a closed formal subscheme of a finite product. *
* __
4.5. Idempotents and formal schemes.
Proposition 4.41.Let`X be a formal scheme. Then systems of formal subschemes
Xi such that X = PiXi biject with systems of idempotents ei 2 OX such that
eiej = ffiijeiand ieiconverges to 1 in the natural topology in OX . More expl*
*icitly,
wePrequire that for every open ideal J OX the set S = {i  ei62 J} is finite, *
*and
Sei= 1 (mod J).
` Q
Proof.SupposeQthat X = i2IXi. Then OX = bX(X; A1) = ibX(Xi;`A1) =
iOXi as rings. If K is a finite subset of I, we write XK = i2KXi. We then
have X = lim!XK , and this is a filtered colimit, so X(R) = lim XK (R) for all
K Q ! K
R. Using this, it is not hard to check that OX = iOXi as topological rings, w*
*here
the right hand side is given the productQtopology. Note that the product topolo*
*gy
is defined by the ideals of the form iJi, where Ji is an open ideal in OXi and
Ji= OXi for almost all i. Q
For each i there is an evident idempotent ei in OX = iOXi, whose j'th com
ponent is ffiij. This gives a system of idempotents as described in the proposi*
*tion.
Conversely, suppose we start with such a system of idempotents. For any
idempotent e 2 OX it is easy to check that D(e) = V`(1  e), so we can define
Xi = D(ei) = V (1  ei). We need to check that X = iXi. We can write
X = lim!Yj for some filtered system of informal schemes Yj. Let eijbe the image
J
of ei in OYj and write Zij= D(eij) = V (1  eij) Yj. As Yj is informal we know
that the kernel of the map OX ! OYj`is open and thus that eij= 0 for almost al*
*l i.
We thus have a decomposition Yj = iZij, in which only finitely many factors a*
*re
nonempty. If we fix i, it is easy to check that the schemes Zijare functors`of *
*j, and
that lim!Zij= Xi. As colimits commute with coproducts, we find that X = iXi
j __
as claimed. __
Corollary 4.42.Coproducts in bXor bXXare strong.
`
Proof.Let {Yi} be a family of schemes over X, and write Y = iYi. Let Z
be another`scheme over X, and write Zi = Z xX Yi. We need to show that
Z xX Y = iZi. To see this, take idempotents ei 2 OY as in the proposition,
so that Yi = D(ei) = V (1  ei). Let e0ibe the image of ei under the evident map
OZ ! OY; it is easy to check that Zi= D(e0i). As the idempotents eiare orthogo*
*nal
and sum to 1 and the map OZxXY !OY is a continuous map of topological rings,
we see that`the e0iare also orthogonal idempotents whose sum is 1. This_shows_t*
*hat
Z xX Y = iZi as claimed. __
FORMAL SCHEMES AND FORMAL GROUPS 39
4.6. Sheaves over formal schemes. In Section 2.6, we defined sheaves and vector
bundles over all functors, and in particular over formal schemes.
Remark 4.43. If M is a vector bundle and L is a line bundle over a formal sche*
*me
X, we can define functors A(M)(R) and A(L)x (R) just as in Definitions 2.45
and 2.55. We claim that these are formal schemes. Given a map f :W ! X, it is
easy to check that f*A(M) = A(f*M) (where the pullback on the left hand side is
computed in the functor category F). In particular, if W is informal then Propo*
*si
tion 2.54 shows that f*A(M) is a scheme. Now write X = lim!Xiin the usual way,
i
and let Mi be the pullback of M over Xi. We find easily that A(M) = lim!A(Mi),
i
so A(M) is a formal scheme. Similarly, A(L)x is a formal scheme.
Remark 4.44. If M is a sheaf such that Mx is an infinitely generated free modu*
*le
for all x, we find that A(M) is a formal scheme over X. Unlike the case of a ve*
*ctor
bundle, it is not relatively informal over X. We leave the proof as an exercise.
Remark 4.45. Let {Xi} be a presentation of a formal scheme X. If M is a sheaf
over X then one can check that (X; M) = lim(Xi; M). In particular, if X is sol*
*id
i
and MJ = (V (J); M) for all open ideals J OX we find that (X; M) = limMJ.
J
Moreover, if J K we find that MK = MJ=KMJ.
In particular, if N is an OX module we find that (X; "N) = limN=JN. We say
J
that N is complete if N = limN=JN. It follows that the functor N 7! "Nembeds
J
the category of complete modules as a full subcategory of SheavesX. Warning: it
seems that the functor N 7! limN=JN need not be idempotent in bad cases, so
J
limN=JN need not be complete.
J
We next consider the problem of constructing sheaves over filtered colimits.
Definition 4.46.Let {Xi} be a filtered diagram of functors, with colimit X. Let
Sheaves{Xi}denote the category of systems ({Mi}; OE) of the following type:
(a) For each i we have a sheaf Mi over Xi.
(b) For each u: i ! j (with associated map Xu: Xi !Xj) we have an isomor
phism OE(u): Mi' X*uMj.
(c) In the case u = 1: i !i we have OE(1) = 1.
(d) Given i u!j v!k we have OE(vu) = (X*uOE(v)) O OE(u).
Proposition 4.47.Let {Xi i 2 I} be a filtered diagram of functors, with colimit
X. The category Sheaves{Xi}is equivalent to SheavesX.
Proof.Given a sheaf M over X, we define a system of sheaves Mi = v*iM, where
vi:Xi ! X is the given map. If u: i ! j then vj O Xu = vi so we have a
canonical identification Mi= X*uMj, which we take as OE(u). This gives an object
of Sheaves{Xi}.
On the other hand, suppose we start with an object {Mi} of Sheaves{Xi}, and
we want to construct a sheaf M over X. Given a ring R and a point x 2 X(R),
we need to define a module Mx over R. As X = lim!Xi(R), we can choose i 2 I
i
and y 2 Xi(R) such that vi(y) = x. We would like to define Mx = Mi;y, but we
need to check that this is canonically independent of the choices made. We thus
let J be the category of all such pairs (i; y). Because X(R) = lim!Xi(R), we s*
*ee
i
40 NEIL P. STRICKLAND
that J is filtered. For each (i; y) 2 J we have an Rmodule Mi;y, and the maps
OE(u) make this a functor J ! Mod R. We define Mx = lim!Mi;y. Because this is
J
a filtered diagram of isomorphisms, each of the canonical maps Mi;y! Mx is an
isomorphism. We leave it to the reader to check that this construction produces*
* a_
sheaf, and that it is inverse to our previous construction. *
*__
Corollary 4.48.Let X ! Y be a map of formal schemes. To construct a sheaf
over X, it suffices to construct sheaves over W xY X in a sufficiently natural *
*way,
for all informal schemes W over Y . It also suffices to construct sheaves over *
*Xy in
a sufficiently natural way, for all points y of Y .
Proof.The two claims are really the same, as points of Y biject with informal
schemes over Y by sending a point y 2 Y (R) to the usual map spec(R) y!Y .
For the first claim, we choose a presentation Y = lim!Yi and write Xi= YixY
i
X, and note that X = lim!Xi. By assumption, we have sheaves Mi over Xi.
i
"Sufficiently natural" means that we have maps OE(u) making {Mi} into an object*
*__
of Sheaves{Xi}, so the proposition gives us a sheaf over X. *
*__
4.7. Formal faithful flatness.
Definition 4.49.Let f :X ! Y be a map of formal schemes. We say that f is
flat if the pullback functor f* :bXY! bXXpreserves finite colimits. We say tha*
*t f
is faithfully flat if f* preserves and reflects finite colimits.
Remark 4.50. For any map f :X ! Y of formal schemes, we know that f*
preserves all small coproducts. Thus f is flat if and only if f* preserves coeq*
*ualisers,
if and only if f* preserves all small colimits.
Definition 4.49 could in principle conflict with Definition 2.56; the followi*
*ng
proposition shows that this is not the case.
Proposition 4.51.A map f :X ! Y of informal schemes is flat (resp. faithfully
flat) as a map of informal schemes if and only if it is flat (resp. faithfully *
*flat) as a
map of formal schemes.
Proof.Recall that the inclusion X !Xbpreserves finite colimits. Given this, we*
* see
easily that a map that is formally flat (resp. faithfully flat) flat is also in*
*formally
flat (resp. faithfully flat).
Now suppose that f is informally flat. Let U _____w_wV ! W be a coequaliser
in bXY. By Proposition 4.10, we can find a filtered system of diagrams Ui_____w*
*_wVi
(with Uiand Viin X) whose colimit is the diagram U _____w_wV . We define Wito be
the coequaliser of Ui_____w_wVi. As colimits commute, we have W = lim!Wi. Clea*
*rly
i
all this can be thought of as happening over W and thus over Y . By assumption,*
* the
diagram f*Ui_____w_wf*Vi !f*Wi is a coequaliser. We now take the colimit over
i, noting that f* commutes with filtered colimits and that colimits of coequali*
*sers
are coequalisers. This shows that f*U _____w_wf*V ! f*W is a coequaliser. Thus,
f is flat.
Now suppose that f is informally faithfully flat, and let u: U ! V be a map
of formal schemes over Y such that f*u is an isomorphism. Choose a presentation
V = lim!Viand write Ui= U xV Vi, so that U = limUi. As f* preserves pullbacks,
i !i
we see that the map f*Ui! f*Vi is the pullback of the isomorphism f*U ! f*V
FORMAL SCHEMES AND FORMAL GROUPS 41
along the map f*Vi ! f*V , and thus that the map f*Ui ! f*Vi is itself an
isomorphism. As f is informally faithfully flat, we conclude that Ui ' Vi. By_
passing to colimits, we see that U ' V as claimed. __
Remark 4.52. Propositions 2.67, 2.68, 2.70 and 2.76 are general nonsense, valid
in any category with finite limits and colimits. They therefore carry over dire*
*ctly
to formal schemes.
Lemma 4.53. Let f :X ! Y be a map of formal schemes. Let XY be the category
of informal schemes with a map to Y , and let f*0:XY ! bXXbe the restriction of
f* to XY . If f*0preserves coequalisers, then f is flat.
Proof.Suppose that f*0preserves coequalisers. Let U _____w_wV ! W be a co
equaliser in bXY. By Proposition 4.10, we can find a filtered system of diagra*
*ms
Ui_____w_wVi (with Ui and Vi in X) whose colimit is the diagram U _____w_wV . *
*We
define Wi to be the coequaliser of Ui_____w_wVi. As colimits commute, we have
W = lim!Wi. Clearly all this can be thought of as happening over W and thus
i
over Y . By assumption, the diagram f*Ui_____w_wf*Vi! f*Wiis a coequaliser. We
now take the colimit over i, noting that f* commutes with filtered colimits and*
* that
colimits of coequalisers are coequalisers. This shows that f*U _____w_wf*V !_f*
**W_
is a coequaliser. Thus, f is flat. __
Proposition 4.54.Let f :X ! Y be a map of formal schemes. Suppose that
Y has a presentation Y = lim!Yi for which the maps fi:Xi = f*Yi ! Yi are
i
(faithfully) flat. Then f is (faithfully) flat.
Proof.First suppose that each fiis flat. Let U _____w_wV ! W be a coequaliser *
*of in
formal schemes over Y . By Lemma 4.53, it is enough to check that f*U _____w_wf*
**V !
f*W is a coequaliser. We know from Proposition 4.7 that bX(W; Y ) = lim!bX(W; *
*Yi),
i
so we can choose a factorisation W ! Yi! Y of the given map W ! Y , for some*
* i.
We then have f*W = W xY X = W xYiYixY X = W xYiXi= f*iW . Similarly, we
have f*V = f*iV and f*U = f*iU. As fiis flat, we see that f*U _____w_wf*V ! f*W
is a coequaliser, as required.
Now suppose that each fiis faithfully flat. Let s: U ! V be a morphism in bXY
such that f*s is an isomorphism. We need to show that s is an isomorphism. We
have a pullback square of the following form.
fi
Xi _____wYi
 
ui vi
 
u u
X ______Y:wf
As f*s is an isomorphism, we see that f*iv*is = u*if*s is an isomorphism. As fi*
* is
faithfully flat, we conclude that v*is: v*iU ! v*iV is an isomorphism for all *
*i. We
also know that U = lim!v*iU and V = lim v*iV , and it follows easily that s is*
* an
i ! i __
isomorphism. __
Proposition 4.55.Let M be a vector bundle of rank r over a formal scheme X.
Then there is a faithfully flat map f :Bases(M) !X such that f*M ' Or.
42 NEIL P. STRICKLAND
Proof.Let Bases(M)(R) be the set of pairs (x; B), where x 2 X(R) and B :Rr !
Mx is an isomorphism. Define f :Bases(M) ! X by f(x; B) = x. As in the
informal case (Example 2.85) we see that Bases(M) is a formal scheme over X, and
that f*M ' Or. If Xi is an informal scheme and u: Xi! X then one checks that
u*Bases(M) = Bases(u*M), which is faithfully flat over Xi by Example 2.85. It *
* __
follows from Proposition 4.54 that Bases(M) is faithfully flat over X. *
* __
Definition 4.56.A map f :X ! Y of formal schemes is very flat if for all infor*
*mal
schemes Y 0over Y , the scheme X0 = f*Y 0is informal and the map X0 ! Y 0is
very flat (in other words, OX0 is a free module over OY 0). Similarly, we say t*
*hat f
is finite if for all such Y 0, the scheme X0 is informal and the map X0! Y 0is*
* finite.
4.8. Coalgebraic formal schemes. Fix a scheme Z, and write R = OZ . We
next study the category CZ of coalgebras over R, and a certain full subcategory
C0Z. It turns out that there is a full and faithful embedding C0Z ! bXZ, and
that the categorical properties of CZ are in some respects superior to those of*
* bXZ.
Because of this, the categories CZ and C0Zare often useful tools for constructi*
*ng
objects of bXZwith specified properties. Our use of coalgebras was inspired by *
*their
appearance in [3], although it is assumed there that R is a field, which removes
many technicalities.
We will use R and Z as interchangeable subscripts, so
bXR= bXZ= {formal schemes overZ};
for example. Write MR = MZ and CR = CZ for the categories of modules and coal
gebras over R. (All coalgebras will be assumed to be cocommutative and counital*
*.)
It is natural to think of CZ as a "geometric" category, and we choose our nota
tion to reflect this point of view. In particular, we shall see shortly that C*
*Z has
finite products; we shall write them as U x V , although they are actually give*
*n by
the tensor product over R. We also write 1 for the terminal object, which is the
coalgebra R with R = fflR = 1R .
The following result is wellknown when R is a field, but we outline a proof *
*to
show that nothing goes wrong for more general rings.
Proposition 4.57.The category CZ has finite products, and strong colimits for
all small diagrams. The forgetful functor to MZ creates colimits.
Proof.Given two coalgebras U; V , we make U V into a coalgebra with counit
fflU fflV :U V ! R and coproduct
U V UV!U U V V 1o1!U V U V:
This is evidently functorial in U and V . There are two projections ssU = 1
fflV :U V ! U and ssV = fflU 1: U V ! V , and one checks that these are
coalgebra maps. One also checks that a pair of maps f :W ! U and W ! V yield
a coalgebra map h = (f; g) = (f g)O W :W ! U V , and that this is the unique
map such that ssU O h = f and ssV O h = g. Thus, U V is the categorical product
of U and V . Similarly, we can make R into a coalgebra with R = fflR = 1R , and
this makes it a terminal object in CZ.
Now suppose we have a diagram of coalgebras Ui, and let U = lim!Ui denote
i
the colimit in MZ. Because tensor products are right exact, we see that U U =
lim!Ui Uj, so there is an obvious map Ui Ui !U U. By composing with
i;j
FORMAL SCHEMES AND FORMAL GROUPS 43
the coproduct on Ui, we get a map Ui! U U. These maps are compatible with
the maps of the diagram, so we get a map U = lim!Ui !U U. We use this as
i
the coproduct on U. The counit maps Ui !R also fit together to give a counit
map U ! R, and this makes U into a coalgebra. One can check that this gives a
colimit in the category CZ. Thus, CZ has colimits and they are created in MZ. It
is clear from the construction that V x lim!Ui = lim(V x Ui), because tensoring
i ! i __
with V is right exact. __
Let f :R ! S = OY be a map of rings, and let Tf: MZ ! MY be the functor
M 7! SR M. This clearly gives a functor CZ ! CY which preserves finite products
and all colimits.
We now introduce a class of coalgebras with better than usual behaviour under
duality.
Definition 4.58.Let U be a coalgebra over R, and suppose that U is free as an
Rmodule, say U = R{ei  i 2 I}. For any finite set J of indices, we write UJ =
R{ei i 2 J}; if this is a subcoalgebra of U, we call it a standard subcoalgebr*
*a. We
say that {ei} is a good basis if each finitely generated submodule of U is cont*
*ained
in a standard subcoalgebra. We write C0Zfor the category of those coalgebras th*
*at
admit a good basis. It is easy to see that C0Zis closed under finite products.
Proposition 4.59.There is a full and faithful functor sch = schZ:C0Z! XbZ,
which preserves finite products and commutes with base change. Moreover, sch(U)
is always solid and we have Osch(U)= U_ := Hom R(U; R).
Proof.Let U be a coalgebra in C0Z. For each subcoalgebra V U such that V is a
finitely generated free module over R, we define V _= Hom R(V; R). We can clear*
*ly
make this into an Ralgebra using the duals of the coproduct and counit maps, so
we have a scheme spec(V _) over Z. We define sch(U) = lim!spec(V _) 2 bXZ. If *
*we
V
choose a good basis {ei i 2 I}for U then it is clear that the standard subcoal*
*gebras
form a cofinal family of V 's, so we have sch(U) = lim!spec(U_J), where J runs*
* over
J
the finite subsets of I for which UJ is a subcoalgebra. This is clearly a direc*
*ted,
and thus filtered, colimit. It follows that Osch(U)= limU_J= U_ . The resulting
J
topology on U_ = Hom R(U; R) is just the topology of pointwiseQconvergence, whe*
*re
we give R the discrete topology. We can also think of this as IR, and the top*
*ology
is just the product topology. It is clear from this that sch(U) is solid.
If V is another coalgebra with good basis, then the obvious basis for U R V is
also good. Moreover, if UJ and VK are standard subcoalgebras of U and V , then
UJ R VJ is a standard subcoalgebra of U R V , and the subcoalgebras of this
form are cofinal among all standard subcoalgebras of U R V . It follows easily
that sch(U x V ) = sch(U R V ) = lim! spec(U_J) xZ spec(VK_). As finite limits
J;K
commute with filtered colimits in bX, this is the same as sch(U) xZ sch(V ).
Now consider a map Y = spec(S) !Z of schemes. The claim is that the functors
schY and schZ commute with base change, in other words that schY(S R U) =
Y xZ schZ(U). As pullbacks commute with filtered colimits, the right hand side *
*is_
just lim!spec(S R UJ), which is the same as the left hand side. *
*__
J
Definition 4.60.Let Z be an informal scheme. We write bX0Zfor the image of schZ,
which is a full subcategory of bXZ. We say that a formal scheme Y is coalgebraic
44 NEIL P. STRICKLAND
over Z if it lies in bX0Z. We say that Y is finitely coalgebraic over Z if OY i*
*s a finitely
generated free module over OZ, or equivalently Y is finite and very flat over Z*
*; this
easily implies that Y is coalgebraic over Z.
More generally, let Z be a formal scheme, and Y a formal scheme over Z. We
say that Y is (finitely) coalgebraic over Z if for all informal schemes Z0 over*
* Z,
the pullback Z0xZ Y is (finitely) coalgebraic over Z0. We again write bX0Zfor t*
*he
category of coalgebraic formal schemes over Z.
Example 4.61. Let Z be a space such that H*(Z; Z) is a free Abelian group,
concentrated in even degrees. It is not hard to check that E0Z is a coalgebra o*
*ver
E0 which admits a good basis, and that ZE = schE0(E0Z). Details are given in
Section 8.
Remark 4.62. The functor schX:C0X! bX0Xis an equivalence of categories, with
inverse Y 7! cY = Hom ctsOX(OY ; OX).
Remark 4.63. For any coalgebra U, we say that an element u 2 U is grouplike
if ffl(u) = 1 and (u) = u u, or equivalently if the map R ! U defined by
r 7! ru is a coalgebra map. We write GL (U) = CR (R; U) for the set of groupli*
*ke
elements. If U is a finitely generated free module over R, then it is easy to c*
*heck
that GL (U) = AlgR(U_ ; R). From this one can deduce that
XbZ(Y; schZ(U)) = GL (OY R U);
where we regard OY R U as a coalgebra over OY . This gives another useful
characterisation of schZ(U).
Proposition 4.64.Let {Ui} be a diagram in CZ with colimit U, and suppose that
U and Ui actually lie in C0Z. Then sch(U) is the strong colimit in bXZof the fo*
*rmal
schemes sch(Ui).
Proof.Note that U = lim!Ui as Rmodules (because colimits in CZ are created in
i
MZ), and it follows immediately that U_ = limU_ias rings. There are apparently
i
two possible topologies on U_ . The first is as in the definition of schR(U), w*
*here
the basic neighbourhoods of zero are the submodules ann(M), where M runs over
finitely generated submodules of U. The second is the inverse limit topology: f*
*or
each index i and each finitely generated submodule N of Ui, the preimage of the
annihilator of N under the evident map U_ ! U_iis a neighbourhood of zero. This
is just the same as the annihilator of the image of N in U, and neighbourhoods *
*of
this form give a basis for the inverse limit topology. Given this, it is easy t*
*o see
that the two topologies in question are the same. We thus have an inverse limit*
* of
topological rings. As the category of formal schemes is just dual to the catego*
*ry of
formal rings, we have a colimit diagram of formal schemes, so sch(U) = lim!sch*
*(Ui).
i
We need to show that the colimit is strong, in other words that for any formal
scheme T over Z we have T xZ schZ(U) = lim!(T xZ schZ(Ui)). First suppose that
i
T = spec(B) is an informal scheme. We then have T xZ schZ(U) = schT(B R U)
and similarly for each Ui, and B R U = lim!B R Ui because tensor products
i
are right exact. By the first part of the proof (with R replaced by B) we see t*
*hat
T xZ schZ(U) = lim!(T xZ schZ(Ui)) as required.
i
FORMAL SCHEMES AND FORMAL GROUPS 45
If T is a formal scheme, we write it as a strong filtered colimit of informal*
* schemes
Tk. The colimit of the isomorphisms Tk xZ schZ(U) = lim!(Tk xZ schZ(Ui)) is the
i __
required isomorphism T xZ schZ(U) = lim!(T xZ schZ(Ui)). __
i
Example 4.65. If X is coalgebraic over Y we claim that XnY=n is a strong colim*
*it
for the action of n on XnY. To see this, we first suppose that Y is informal and
X = schY(U) for some coalgebra U that is free over X with good basis {ei i 2 I}
say. Then XnY= schY(Un ), and the set of terms ei_= ei1 : : :ein for i_=
(i1; : :;:in) 2 In is a good basis for In. For each orbit j 2 In=n, we choose an
element i_of the orbit and let fj be the image of ei_in Un =n. We find that the
terms fj form a good basis for Un =n, so this coalgebra lies in C0Y. It follow*
*s from
Proposition 4.64 that XnY= schY(Un =n), and that this is a strong colimit. For
a general base Y , we choose a presentation Y = lim!Yi and write Xi = X xY Yi
i
and Zi= (Xi)nYi=n. By what we have just proved, this is an object of bX{Yi}, wi*
*th
lim!Zi= XnY=n. It is now easy to see that this is a strong colimit, using the *
*ideas
i
of Proposition 4.27.
We conclude this section with a result about gradings.
Proposition 4.66.Let Y be a coalgebraic formal scheme over an informal scheme
X, and suppose that X and Y have compatible actions of Gm . Then cY has a
natural structure as a graded coalgebra over OX .
Proof.Write R = OX and U = cY . Proposition 2.96 makes R into a graded
ring. Next, observe that OY = U_Pand OGm xY = U_ bZ[t1 ], which is the ring
of doubly infinite Laurent series k2Zaktk such that ak 2 U_ and ak ! 0 as
k ! 1. Thus, the action ff: Gm x YP! Y gives a continuous homomorphism
ff*: U_ ! U_ bZ[t1 ], say ff*(a) = kaktk. The basic neighbourhoods of zero in
U_ are the kernels of the maps U_ ! W _, where W is a standard subcoalgebra
of U. Similarly, the basic neighbourhoods of zero in U_ bZ[t1 ] are the kernels*
* of
the maps to V _[t1 ], where V is a standard subcoalgebra. Thus, continuity means
that for every standard subcoalgebra V U, there is a standard subcoalgebra W
such that whenever a(W ) = 0 we have ak(V ) = 0 for all k. In particular, it fo*
*llows
that the mapPssk: a ! ak is continuous. Just as in the proof of Proposition 2.*
*96,
we see that kak = a and that ssjssk = ffijkssk. It follows that U_ is a kind*
* of
completed direct sum of the subgroups image(ssk). We would like to dualise this
and thus split U as an honest direct sum.
First, we need to show that the maps ssi have a kind of Rlinearity.PLet r be*
* an
element of R, and let ri be the part in degree i, so that r = iri and riP= 0 *
*for
almost all i. Using the compatibility of the actions, we find that (ra)i= jrj*
*aij
(which is really a finite sum).
Suppose that u 2 U. Choose a standard subcoalgebra V containing u, and let
W be a standard subcoalgebra such that whenever a(W ) = 0 we have ai(V ) = 0
for all i.
Suppose that a 2 U_ . It follows from our asymptotic conditionPon Laurent ser*
*ies
that ai(u) = 0 when i is large, so we can define Ok(u)(a) = iai(u)i+k2 R. We
46 NEIL P. STRICKLAND
then have
X
Ok(u)(ra)= ((ra)i(u))i+k
X i
= (rjaij(u))i+k
i;j
X
= rj(aij(u))i+kj
i;j
X
= rjam (u)m+k
m;j
= rOk(u)(a):
Thus, the map Oj(u): U_ ! R is Rlinear. Clearly, if a(W ) = 0 then Oj(u)(a) =*
* 0,
so Oj(u) can be regarded as an element of (U_ =ann(W ))_ = W __= W (because
W is a finitely generated free module). More precisely, there is a unique eleme*
*nt
uj 2 U such that Oj(u)(a) = a(uj) for all a, and in fact uj 2 W .
Next, we choose a finite set of elements in U_ which project to a basis for W*
* _.
We can then choose a number N such that bi(u) = 0 whenever b lies in that set a*
*nd
i > N. Because aj(u) = 0 for all j whenever a(W ) = 0, we conclude that ai(u)*
* = 0
for all a 2 U_ and all i such that i > N. ItPfollows thatPui= 0 when iP> N.*
* This
justifies the followingPmanipulation: a(u) = i;jai(u)j = ja(uj) = a( juj).
We conclude that u = juj. We define a map OEi:U ! U by OEi(u) = ui, and
we defineLUi= image(OEi). We leave it to the reader to check that OEiOEj = ffii*
*jOEj, so
that U = iUi, and that this grading is compatible with the Rmodule structure_
and the coalgebra structure. __
4.9. More mapping schemes. Recall the functor Map Z(X; Y ) , given in Defini
tion 2.89. We now prove some more results which tell us when Map Z(X; Y ) is a
scheme or a formal scheme.
First, note that for any functor W over Z, we have
FZ(W; MapZ (X; Y )) = FZ(W xZ X; Y ) = FW (W xZ X; W xZ Y ):
Indeed, if W is informal then this follows from the definitions and the Yoneda
lemma, by writing W in the form spec(R). The general case follows from this
by taking limits, because every functor is the colimit of a (not necessarily sm*
*all
or filtered) diagram of representable functors. It is also not hard to give a d*
*irect
proof.
Conversely, suppose we have a functor M over Z and a natural isomorphism
FZ(W; M) ' FZ(W xZ X; Y ) for all informal schemes W over Z. It is then easy
to identify M with Map Z(X; Y ).
Lemma 4.67. Let X and Y be functors over Z, and suppose that X and Z are
formal schemes. Then Map Z(X; Y )(R) is a set for all R, so the functor Map Z(X*
*; Y )
exists.
Proof.We have only a set of elements z 2 Z(R), so it suffices to check that for*
* any
such z there is only a set of maps Xz !Yz of functors over spec(R). Here Xz is*
* a
formal scheme, with presentation {Wi}Qsay. Clearly F(Wi; Yz) = Yz(OWi) is a set*
*,_
and Fspec(R)(Xz; Yz) is a subset of iF(Wi; Yz). __
FORMAL SCHEMES AND FORMAL GROUPS 47
Recall also from Proposition 2.94 that Map Z(X; Y ) is a scheme when X, Y and
Z are all informal schemes, and X is finite and very flat over Z.
Definition 4.68.We say that a formal scheme Y over Z is of finite presentation
if there is an equaliser diagram in bXZof the form
Y ! An x Z _____w_wAm x Z:
Theorem 4.69. Let X and Y be formal schemes over Z. Then Map Z(X; Y ) is a
formal scheme if
(a) X is coalgebraic over Z and Y is relatively informal over Z, or
(b) X is finite and very flat over Z, or
(c) X is very flat over Z and Y is of finite presentation over Z.
This will be proved at the end of the section, after some auxiliary results.
Lemma 4.70. If Z0is a functor over Z then MapZ0(XxZZ0; Y xZZ0) = Map Z(X; Y )xZ
Z0.
Proof.If W is a scheme over Z0 then
FZ0(W; MapZ (X; Y ) xZ Z0)= FZ(W; MapZ (X; Y ))
= FZ(W xZ X; Y )
= FZ0(W xZ X; Y xZ Z0)
= FZ0(W xZ0(X xZ Z0); Y xZ Z0):
Thus, Map Z(X; Y ) xZ Z0 has the required universal property. _*
*__
Lemma 4.71. If X is a strong colimit of formal schemes Xi and Map Z(Xi; Y )
is a formal scheme and is relatively informal over Z for all i then Map Z(X; Y )
is a formal scheme and is equal to limMap Z(Xi; Y ) (where the inverse limit is
i
computed in bXZ).
Note that coproducts and filtered colimits are always strong, so the lemma ap
plies in those cases.
Proof.Because Map Z(Xi; Y ) is relatively informal, Proposition 4.29 allows us *
*to
form the limit limMapZ (Xi; Y ) in bXZ. If W is a formal scheme over Z then we
i
have
bXZ(W; limMap (Xi; Y ))= limbXZ(W; Map (Xi; Y ))
 Z  Z
i i
= limbXZ(W xZ Xi; Y )
i
= bXZ(lim!W xZ Xi; Y )
i
= bXZ(W xZ X; Y ):
This proves that limMapZ (Xi; Y ) = Map Z(X; Y ) as required. _*
*__
i
We leave the next lemma to the reader.
48 NEIL P. STRICKLAND
Lemma 4.72. Suppose that Y is an inverse limit of a finite diagram of formal
schemes {Yi} over Z. Then Map Z(X; Y ) = limMap Z(X; Yi), where the limit
i
is computed in FZ. Thus, if Map Z(X; Yi) is a formal scheme for all i, then_
Map Z(X; Y ) is a formal scheme. __
Lemma 4.73. Let {Zi} be a filtered system of informal schemes with colimit Z.
Let X and Y be formal schemes over Z, with Xi = X xZ Zi and Yi = Y xZ Zi.
If Map Zi(Xi; Yi) is a formal scheme for all i then Map Z(X; Y ) is a formal sc*
*heme
and is equal to lim!MapZi(Xi; Yi).
i
Proof.Lemma 4.70 tells us that the system of formal schemes
Mi= Map Zi(Xi; Yi)
defines an object of the category bX{Zi}of Proposition 4.27. Thus, if we define
M = lim!Miwe find that bXZ(W; M) is the set of maps of diagrams {W xZ Zi} !
i
{Mi} over {Zi}. This is the same as the set of maps of diagrams {W xZ Xi} =
{W xZ ZixZiXi} !{Yi} over {Zi}. By the adjunction in Proposition 4.27, this
is the same as the set of maps W xZ X = lim!W xZ Xi !Y over Z. Thus, M
i __
has the defining property of Map Z(X; Y ). __
Lemma 4.74. Let X be relatively informal over Z, and let {Yi} be a filtered sy*
*stem
of formal schemes over Z with colimit Y . If Map Z(X; Yi) is a formal scheme fo*
*r all
i, then Map Z(X; Y ) is a formal scheme and is equal to lim!MapZ(X; Yi).
i
Proof.Write M = lim!MapZ(X; Yi). Let W be an informal scheme over Z. As X
i
is relatively informal, we see that W xZ X is informal. It follows that the fun*
*ctors
bXZ(W; ) and bXZ(W xZ X; ) preserve filtered colimits. We thus have
bXZ(W; M)= limbXZ(W; Map (X; Yi))
! Z
i
= lim!bXZ(W xZ X; Yi)
i
= bXZ(W xZ X; lim!Yi)
i
= bXZ(W xZ X; Y );
as required. ___
Lemma 4.75. If X and Z are informal and X is very flat over Z then the functor
Map Z(X; A1 x Z) is a formal scheme.
Proof.We can choose a basis for OX over OZ and thus write OX as a filtered
colimit of finitely generated free modules Mi over OZ . From the definitions we
see that Map Z(X; A1)(R) is the set of pairs (x; u), where x 2 X(R) (making R
into an OX algebra) and u is a map R[t] ! R OZ OX of Ralgebras. This is of
course equivalent to an element of R OZ OX = lim!R OZ Mi. Thus, we see that
i __
Map Z(X; A1 x Z) = lim!A(Mi), which is a formal scheme. __
i
FORMAL SCHEMES AND FORMAL GROUPS 49
Proof of Theorem 4.69.We shall prove successively that Map Z(X; Y ) is a formal
scheme under any of the following hypotheses. Cases (3), (5) and (7) give the
results claimed in the theorem.
(1) X, Y and Z are informal, and X is finite and very flat. In this case MapZ *
*(X; Y )
is informal.
(2) Y is informal, and X is finite and very flat. In this case Map Z(X; Y ) *
*is
relatively informal.
(3) X is finite and very flat.
(4) Y and Z are informal, and X is coalgebraic. In this case, Map Z(X; Y ) is
informal.
(5) Y is relatively informal, and X is coalgebraic. In this case, Map Z(X; Y )*
* is
relatively informal.
(6) X and Z are informal, X is very flat, and Y is of finite presentation.
(7) X is very flat and Y is of finite presentation.
Proposition 2.94 gives case (1). For case (2), write Z = lim!Zi in the usual w*
*ay.
i
Then case (1) tells us that Map Zi(X xZ Zi; Y xZ Zi) is an informal scheme. Usi*
*ng
this and Lemma 4.73, we see that Map Z(X; Y ) is a formal scheme. Using case (1)
and Lemma 4.70 we see that MapZ (X; Y ) is relatively informal. In case (3), we*
* write
Y as a filtered colimit of informal schemes Yj. Case (2) tells us that Map Z(X;*
* Yj)
is a relatively informal scheme, so Lemma 4.74 tells us that Map Z(X; Y ) is a *
*formal
scheme. In case (4), it follows easily from the definitions that X can be writt*
*en as
the filtered colimit of a system of finite, very flat schemes Xi. It then follo*
*ws from
case (1) that Map Z(Xi; Y ) is an informal scheme. Using Lemma 4.71 we see that
Map Z(X; Y ) = limMap Z(Xi; Y ). This is an inverse limit of informal schemes,
i
and thus is an informal scheme. We deduce (5) from (4) in the same way that we
deduced (2) from (1). Case (6) follows easily from Lemmas 4.75 and 4.72. Again,_
the argument for (1))(2) also gives (6))(7). __
5. Formal curves
In this section, we define formal curves. We also study divisors, differentia*
*ls, and
meromorphic functions on such curves.
Let X be a formal scheme, and let C be a formal scheme over X. We say that
C is a formal curve over X if it is isomorphic in bXX to bA1x X. (In some sense,
it would be better to allow formal schemes that are only isomorphic to bA1x X
fpqclocally on X, but this seems unnecessary for the topological applications *
*so
we omit it.) A coordinate on C is a map x: C ! bA1giving rise to an isomorphism
C ' bA1x X.
Example 5.1. If E is an even periodic ring spectrum then (CP 1)E and (HP 1)E
are formal curves over SE .
5.1. Divisors on formal curves. Let C be a formal curve over X, and let D be
a closed subscheme of X. If X is informal, we say that D is a effective divisor*
* of
degree n on C if D is informal, and OD is a free module of rank n over OX . If X
is a general formal scheme, we say that D is a divisor if D xX X0 is a divisor *
*on
C xX X0, for all informal schemes X0 over X. If Y is a formal scheme over X, we
refer to divisors on C xX Y as divisors on C over Y .
50 NEIL P. STRICKLAND
Proposition 5.2.There is a formal scheme Div+n(C) over X such that maps Y !
Div+n(C) over X biject with effective divisors of degree n on C over Y . Moreov*
*er,
a choice of coordinate on C gives rise to an isomorphism Div+n(C) ' bAnx X.
Proof.This is much the same as Example 2.10. We define
Div+n(C)(R) =
{(a; D)  a 2 X(R) and D is an effective divisor of degree n}on:Ca
We make this a functor by pullback, just as in Example 2.10. To see that Div+n(*
*C)
is a formal scheme, choose a coordinate x on C. Given a point (a; D) as above,
we find that Ca = C xX spec(R) = spf(R[[x]]), where the topology on R[[x]]is
defined by the ideals (xk). We know that D is a closed subscheme of Ca, and
that D is informal. It follows that D = spec(R[[x]]=J) for some ideal J such th*
*at
xk 2 J for some k. Let (x)Pbe the endomorphism of OD given by multiplication
by x, and let fD (t) = ni=0ai(D)tni be the characteristic polynomial of (x).
As xk 2 J, we see that (x)k = 0. If R is a field, then we deduce by elementary
linear algebra that fD (t) = tn. If p is a prime ideal in R then by considering
the divisor spec((p)) xspec(R)D, we conclude that fD (t) = tn (mod p[t]). Using
Proposition 2.37, we deduce that ai(D) 2 Nil(R) for i > 0. Thus, the ai's give a
map Div+n(C) !Abnx X. As in Example 2.10, the CayleyHamilton theorem tells
us that fD (x) 2 J and thus that OD = R[x]=fD (x) = R[[x]]=fD (x).
Conversely, suppose we have elementsPb0; : :;:bn with b0 = 1 and bi 2 Nil(R)
for i >P0 and we define g(t) = ibitni and D = spf(R[[x]]=g(x)). In OD we have
xn =  i>0bixni, which is nilpotent, so x is nilpotent, so (g(x)) is open in*
* R[[x]].
This means that D is informal and that OD = R[x]=g(x), which is easily seen to *
*be
a free module of rank n over R. Thus, D is an effective divisor of rank n on Ca*
*. We
conclude that Div+n(C) is isomorphic to bAn, and in particular is a formal sche*
*me.
If Y is an arbitrary formal scheme over X, we can choose a presentation Y =
lim!Yi, so Yi is an informal scheme over X. The above tells us that maps Yi !
i
Div+n(C) over X biject with effective divisors of degree n on C over Yi. Thus, *
*maps
Y ! Div+n(C) over X biject with systems of divisors Diover Yi, such that for e*
*ach
map Yi !Yj we have Di = Dj xYjYi. Using Proposition 4.27, we see that these *
* __
biject with effective divisors of degree n on C over Y . *
*__
Example 5.3. It is essentially wellknown that BU(n)E = Div+n(GE ), where GE =
(CP 1)E . A proof will be given in Section 8.
Remark 5.4. It is not hard to check that for any map Y ! X of formal schemes
and any formal curve C over X we have Div+n(C xX Y ) = Div+n(C) xX Y (because
both sides represent the same functor bXY! Sets).
Definition 5.5.Let D be an effective divisor on a curve C over X. We shall defi*
*ne
an associated line bundle J(D) over C. By Corollary 4.48, it is enough to do th*
*is
in a sufficiently natural way when X is an informal scheme. In that case we have
OD = OC=J(D) for some ideal J(D) in OC. In terms of a coordinate x, we see from
the proof of Proposition 5.2 that J(D) is generated by a monic polynomial f(x)
whose lower coefficients are nilpotent. Thus f(x) = xn  g(x) where g(x)k = 0 s*
*ay.
If fh = 0 then xnkh = gkh = 0 so h = 0, so f is not a zerodivisor and J(D) is
free of rank one over OC . Thus, J(D) can be regarded as a line bundle over C as
required (using Remark 4.45).
FORMAL SCHEMES AND FORMAL GROUPS 51
Proposition 5.6.There is a natural commutative and associative addition oe :Div*
*+j(C)xX
Div+k(C) !Div+j+k(C), such that J(D + E) = J(D) J(E).
Proof.Let a: spec(R) !X be an element of X(R), and let D and E be effective di
visors of degrees j and k on Ca over spec(R). We then have D = V (J(D)) and E =
V (J(E)) where J(D) and J(E) are ideals in OCa. We define F = V (J(D)J(E)).
If we choose a coordinate x on C we find (as in the proof of Proposition 5.2) t*
*hat
J(D) = (fD (x)) and J(E) = (fE (x)), where fD and fE are monic polynomials
whose lower coefficients are nilpotent. This means that g = fD fE is a polynomi*
*al
of the same type, and it follows that F = V (g) is a divisor of degree j + k as
required. We define oe(D; E) = D + E = F . It is clear from the construction_th*
*at
J(D + E) = J(D) J(E). __
Proposition 5.7.Let C be a formal curve over a formal scheme X. Then Div+n(C) =
CnX=n, and this is a strong colimit. Moreover, the quotient map CnX! CnX=n
is faithfully flat.
Proof.First consider the case n = 1. Fix a ring R and a point a 2 X(R), and wri*
*te
Ca = C xX spec(R), which is a formal curve over Y = spec(R). A point c 2 C lying
over a is the same as a section of the projection Ca !Y . Such a section is a *
*split
monomorphism, and thus a closed inclusion; we write [c] for its image, which is*
* a
closed formal subscheme of Ca. The projection Ca ! Y carries [c] isomorphically
to Y , which shows that [c] is an effective divisor of degree 1 on C over Y . T*
*hus,
this construction gives a map C ! Div+1(C). If x is a coordinate on C then it *
*is
easy to see that x(c) 2 Nil(R) and [c] = spf(R[[x]]=(x  x(c))). Using this, we*
* see
easily that our map is an isomorphism, giving the case n = 1 of the Proposition.
We now use the iterated addition map CnX= Div+1(C)nX! Div+n(C) to get a
map CnX=n !Div +n(C).
Next, because C ' bA1x X, it is easy to see that C is coalgebraic over X and
thus (by Example 4.65) that CnX=n is a strong colimit. Given this, we can reduce
easily to the case where X is informal, say X = spec(R). Choose a coordinate
x on C. This gives isomorphisms ODiv+n(C)= R[[a1; : :;:an]]= S and OCnX =
R[[x1; : :;:xn]]= T and OCnX=n = T n . The claim is thus that the map S ! T n
is an isomorphism, and that T isQfaithfully flat over T n . The map S ! T n se*
*nds
aito the coefficient of xni in j(x  xj), which is (up to sign) the i'th ele*
*mentary
symmetric function of the variables xj. It is thus a wellknown theoremQof Newt*
*on
that S = T n . It is also wellknown that the elements of the form nj=1xdjjwi*
*th
0 dj < j form a basis for T over T n , so that T is indeed faithfully_flat_over
T n . __
We next consider pointed curves, in other words curves C equipped with a spec
ified "zerosection" 0: X ! C such that the composite X 0! C ! X is the
identity. If C is such a curve and x is a coordinate on C, we say that x is no*
*r
malised if x(0) = 0. If y is an unnormalised coordinate then x = y  y(0) is a
normalised one, so normalised coordinates always exist.
Definition 5.8.Let C be a pointed formal curve over X. Define
f :Div+n(C) !Div+n+1(C)
52 NEIL P. STRICKLAND
by f(D) = D + [0]. For n 2 Z with n < 0 we write Div+n(C) = ;. Define
a
Div+(C) = Div+n(C)
n0
Divn(C) = lim!(Div+n(C) f!Div+n+1(C) f!: :):
a
Div(C) = Divn(C)
n2Z
= lim!(Div+ (C) f!Div+(C) f!: :)::
It is not hard to see that fk`induces an isomorphism Divn(C) ' Divn+k(C), so
Div(C) can be identified with nDiv0(C) = Z_x Div0(C).
A choice of normalised coordinate on C gives an isomorphism Div+n(C) ' XxbAn.
Under this identification, f becomes the map
(x; a1; : :;:an) 7! (x; a1; : :;:an; 0):
We thus have an isomorphism Div0(C) = bA(1)(using the notation of Example 4.4)
and thus Div= Z_x bA(1).
Definition 5.9.Given a divisor D on a pointed curve C over X, we define the
Thom sheaf of D to be the line bundle L(D) = 0*J(D) over X. It is clear that
L(D + E) = L(D) L(E). Note that a coordinate on C gives a generator fD (x)
for J(D) and thus a generator uD for L(D), which we call the Thom class. This is
natural for maps of X, and satisfies uD+E = uD uE .
Definition 5.10.If C is a pointed formal curve over X, we define a functor
Coord(C) 2 FX by
Coord(C)(R) = {(a; x)  a 2 X(R) and x is a normalised coordinate on}Ca:
Proposition 5.11.The functor Coord(C) is a formal scheme over X, and is un
naturally isomorphic to Gm x A1 x X.
Proof.Choose a normalised coordinate x on C, and suppose that a 2 X(R). Then
any normalised function y :Ca !Ab1has the form
X
y(c) = f(x(c)) = ukx(c)k
k>0
for a uniquely determined sequence of coefficients uk. Moreover, y is a coordin*
*ate
if and only if f :bA1x spec(R) ! bA1x spec(R) is an isomorphism, if and only if
there is a power series g with g(f(t)) = t = f(g(t)). It is wellknown that th*
*is
happens if and only if u1 is invertible. Thus, the set of coordinates on Ca bij*
*ects
naturally with (Gm x A1 )(R), and Coord(C) ' Gm x A1 x X is a formal scheme,_
as required. __
Remark 5.12. We will see later that when E is an even periodic ring spectrum
and GE = (CP 1)E we have Coord(GE ) = spec(E0MP ).
FORMAL SCHEMES AND FORMAL GROUPS 53
5.2. Weierstrass preparation.
DefinitionP5.13.A Weierstrass series over a ring R is a formal power series g(x*
*) =
kakxk 2 R[[x]]such that there exists an integer n such that ak is nilpotent f*
*or
k < n, and an is a unit. The integer n is called the Weierstrass degree of g(x)*
*. (It
is clearly welldefined unlessPR = 0). A Weierstrass polynomial over a ring R i*
*s a
monic polynomial h(x) = nk=0bkxk such that bk is nilpotent for k < n.
The following result is a version of the Weierstrass Preparation Theorem; see*
* [6,
Theorem 3] (for example) for a more classical version.
Lemma 5.14. Let R be a ring, and let g(x) be a Weierstrass series over R, of
Weierstrass degree n. Then there is a unique ring map ff: R[[y]]!R[[x]]sending*
* y to
g(x), and this makes R[[x]]into a free module over R[[y]]with basis {1; x; : :;*
*:xn1}.
Proof.We can easily reduce to the case where an = 1. It is also easy to checkPt*
*hat
there isPa unique map ff sending y to g(x), and that it sends any series jbjy*
*j to
the sum jbjg(x)j, which is xadically convergent.
For any j 0 and 0 k < n we define zjk= g(x)jxk. Given any m 0 we can
write m = nj + k for some j 0 and 0 k < n, and we put wm = zjk. For any
Rmodule M, we define a map
Y
fiM : M ! M[[x]]
m
P
by fiM (b) = m bm wm . It is easy to check that this sum is again xadically
convergent. The claim in the lemma is equivalent to the statement that fiR is an
isomorphism.
Write I = (a0; : :;:an1). This isQfinitelyQgenerated, so the same is true of*
* Ir
for all r, and it follows that Ir m M = m IrM and so on. We also see that
wm = xm (mod I; xm+1 ).
Now consider a module M with IMP= 0, so that bwm = bxm (mod xm+1 ) for
b 2 M. Given any series c(x) = m cm xm 2 M[[x]],Pwe see by induction on m
that there is a unique sequence (bj) such that j K. It follows that Ij = 0 and Jj = OX when j > K.
Next, let p be a prime ideal in OX . As OX =p is an integral domain, it is clea*
*r that
modulo p we must have f(x) = akxk + : :a:nd g(x) = bkxk + : :f:or some k.
This implies that aibj 2 p whenever i + j < 0. As the intersection of all prime
ideals is the set of nilpotents, the elements aibj must be nilpotent when i + j*
* < 0.
If i 6= j then either i  j or j  i is negative, soTaibiajbj is nilpotent. I*
*t follows
that IiIj is nilpotent when i 6= j, and thus that jJj is nilpotent. It follow*
*spfrom_
the resultsQof Section 2.5 that there are unique ideals J0jsuch that Jj J0j *
*Jj
and OX = jOX =J0j. We take Xj = spec(OX =J0j); one can check that this has the
claimed properties.
Conversely, suppose that X has a decomposition of the type discussed. We
reduce easily to the case where X = Xk for somePk. After replacing f by xkf, we
may assume that k = 0. This means that f(x) = j2Zajxj, where a0 is invertible
and aj is nilpotent for j < 0 and aj = 0 for j 0. The invertibility or otherwi*
*se of
f is unaffected if we subtract off a nilpotent term, so we may assume that aj =*
* 0 __
for j < 0. The resulting series is invertible in OC and thus certainly in MC=X *
*. __
Definition 5.24.Let x be a coordinate on C,`and let f be an invertible element
of MC=X , so we have a decomposition X = kXk as above. If X = Xk then we
say that f has constant degree k. More generally, we let deg(f) be the map from*
* X
to the constant scheme Z_that takes the value k on Xk. One can check that these
definitions are independent of the choice of coordinate.
Lemma 5.25. Let x be a coordinate on C, and let f be an invertible element
of MC=X , with constant degree k. Then therePis a unique factorisation f(x) =
xku(x)g(x), where u(x) 2 OxC, and g(x) = j0 bjxj where b0 = 1 and bj is
nilpotent for j > 0 and bj = 0 for j 0.
Proof.Clearly we have h(x) = xN f(x) 2 OC for some N > 0. We see from
Lemma 5.23 that h(x) is a Weierstrass power series of Weierstrass degree N + k.
It follows from Corollary 5.16 that h(x) has a unique factorisation of the form
h(x) = k(x)u(x), where k(x) is a Weierstrass polynomial of degree N + k, and
u(x) 2 OxC. We write g(x) = xNk h(x); this clearly gives a factorisation of t*
*he_
required type, and one can check that it is unique. _*
*_
Proposition 5.26.Let C be a formal curve over a formal scheme X. For any ring
R, we define
Mer(C; Gm )(R) = {(u; f)  u 2 X(R) ; f 2 MxCu=spec(R)}:
Then Mer(C; Gm ) is a formal scheme over X, and there is a short exact sequence
of formal groups
Map (C; Gm ) ae Mer(C; Gm ) i Div(C);
which admits a noncanonical splitting.
Proof.As Map (C; Gm )(R) = {(u; f)  u 2 X(R) ; f 2 OxCu}, there is an obvious
inclusion Map (C; Gm ) ! Mer(C;PGm ) of groupvalued functors. Next, let Y (R)
be the set of series g(x) = j0 bjxj such that b0 = 1 and bj is nilpotent for
56 NEIL P. STRICKLAND
Q
j > 0 and bj = 0 for j 0. Then Y = lim! 0 0 and bj = 0 for j 0. We then have f0=f = d=x+u0=u+g0=*
*g.
It is clear that u0=u 2 R[[x]]so ae(u0=u) = 0. Similarly, we find that g0only i*
*nvolves
powers xk with k < 1. Moreover,Pif h(x) = 1  g(x) then h is a polynomial in 1*
*=x
and is nilpotent, and 1=g = Nk=0hk for some N so 1=g is a polynomial in 1=x.
It follows that g0=g only involves powers xk with k < 1, so ae(g0=g) = 0. Thus
ae(f0=f) = d as claimed.
Finally, suppose that n 6= 1. Note that (n + 1)ae(fn :f0) = ae((fn+1)0) = 0.*
* If
R is torsionfree we concludePthat ae(fn :f0) = 0. If R is not torsionfree, we*
* recall
that f(x) has the form 1i=maixi for some m, where ai is nilpotent for i < d a*
*nd
ad is invertible. Thus there is some N > 0 such thatPaNi= 0 for all i < d. Defi*
*ne
R0= Z[bi i m][1=bd]=(bNi m i < d) and g(x) = ibixi2 R0[[x]][1=x]x . It is
clear that R0 is torsionfree and thus that ae(gn:g0) = 0. There is an evident *
*map_
R0! R carying g to f, so we deduce that ae(fn :f0) = 0 as claimed. *
*__
Corollary 5.36.If g(x) 2 R[[x]]is a Weierstrass series of degree d > 0 and f(x)*
* 2
R[[x]][1=x] then ae(f(g(x))g0(x)) = dae(f(x)).
P
Proof.Suppose that f(x) = km akxk. We first observe that the claim makes
sense: as g is a Weierstrass series of degree d > 0 we know that g(0) is nilpot*
*ent,
so gN 2 R[[x]]x for some N, so gNk 2 R[[x]]xk for k 0. Moreover, Lemma 5.23
impliesPthat g is invertible in R[[x]][1=x]. Thus, the terms in the sum f(g(x)*
*) =
km akg(x)k are all defined, and the sum is convergent. We thus have
X
ae(f(g(x))) = akae(gk:g0) = d a1 = dae(f)
k
as required. ___
Definition 5.37.Let C be a formal curve over an affine scheme X. We write
M1C=Xfor MC=X OC 1C=X, which is a free module of rank one over MC=X . It is
easy to check that there is a unique map d: MC=X ! M1C=Xextending the usual
map d: OC !1C=Xand satisfying d(fg) = f d(g) + g d(f).
Corollary 5.38.Let C be a formal curve over an affine scheme X. Then there is
a natural residue map res= resC=X:M1C=X! OX such that
(a) res(df) = 0 for all f 2 MC=X .
(b) res((df)=f) = deg(f) for all f 2 MxC=X.
(c) If q :C ! C0 is an isogeny then res(q*ff) = deg(q)res(ff) for all ff.
Proof.Choose a coordinate x on C, so that any ff 2 MC=X OC 1C=Xhas a unique
expression ff = f(x)dx for some f 2 OX [[t]][1=t]. Define res(ff) = ae(f). If*
* y is
a different coordinate then x = g(y) for some Weirstrass series g of degree 1 a*
*nd
dx = g0(y)dy so ff = f(g(y))g0(y)dy and we know that ae(f(g(y))g0(y)) = ae(f) s*
*o our
definition is independent of the choice of the coordinate. The rest of the coro*
*llary_
is just a translation of the properties of ae. *
*__
See Remark 8.34 for a topological application of this.
FORMAL SCHEMES AND FORMAL GROUPS 59
6.Formal groups
A formal group over a formal scheme X is just a group object in the category
bXX. In this section, we will study formal groups in general. In the next, we w*
*ill
specialise to the case of commutative formal groups G over X with the property
that the underlying scheme is a formal curve; we shall call these ordinary form*
*al
groups. For technical reasons, it is convenient to compare our formal groups wi*
*th
group objects in suitable categories of coalgebras. To combine these cases, we
start with a discussion of Abelian group objects in an arbitrary category with *
*finite
products. We then discuss the existence of free Abelian formal groups, or of sc*
*hemes
of homomorphisms between formal groups. As a special case, we discuss the Carti*
*er
duality functor G 7! Hom (G; Gm ). Finally, we define torsors over a commutative
formal group, and show that they form a strict Picard category.
6.1. Group objects in general categories. Let D be a category with finite
products (including an empty product, in other words a terminal object). There *
*is
an evident notion of an Abelian group object in D; we write Ab D for the catego*
*ry
of such objects. We also consider the category Mon D of Abelian monoids in D.
A basepoint for an object U of D is a map from the terminal object to U. We
write BasedD for the category of objects of D equipped with a specified basepoi*
*nt.
There are evident forgetful functors
Ab D !Mon D !Based D !D:
If U 2 D and G 2 Ab D then the set D(U; G) has a natural Abelian group
structure. In fact, to give such a group structure is equivalent to giving maps
1 0!G oe G x G making it an Abelian group object, as one sees easily from
Yoneda's lemma.
Let {Gi} be a diagram in Ab D, and suppose that the underlying diagram in D
has a limit G. Then D(U; G) = limD(U; Gi) has a natural Abelian group structur*
*e.
i
It follows that there is a unique way to make G into an Abelian group object su*
*ch
that the maps G !Gi become homomorphisms, and with this structure G is also
the limit in Ab D. In other words, the forgetful functor Ab D ! D creates limi*
*ts.
Similarly, we see that all the functors Ab D ! Mon D ! BasedD ! D and their
composites create limits.
Suppose that G, H and K are Abelian group objects in D and that f :G !K
and g :H ! K are homomorphisms. One checks that the composite G x H fxg!
K x K oe!K is also a homomorphism, and that it is the unique homomorphism
whose composites with the inclusions G ! G x H and H ! G x H are f and g.
This means that G x H is the coproduct of G and H in Ab D, as well as being the*
*ir
product.
We next investigate another type of colimit in Ab D.
Definition 6.1.A reflexive fork in any category D is a pair of objects U; V , t*
*o
gether with maps d0; d1: U ! V and s: V ! U such that d0s = 1 = d1s. The
coequaliser of such a fork means the coequaliser of the maps d0 and d1.
Proposition 6.2.Let D be a category with finite products. Let
V _____wsU _____w_wd0d1V
60 NEIL P. STRICKLAND
be a reflexive fork in Mon D, and let U _____w_wd0d1V _____weW be a strong coeq*
*ualiser in
D. Then there is a monoid structure on W such that e is a homomorphism, and
this makes the above diagram into a coequaliser in Mon D.
Proof.Let oeU :U x U ! U and oeV :V x V ! V be the addition maps. We have
a commutative diagram as follows:
oeU
V x U _____wUsxxU1_______wU
  
1xd01xd1 d0xd0d1xd1 d0d1
uu uu uu
V x V _____ V x V _______wVoeV
The right hand square commutes because d0 and d1 are homomorphisms, and
the left hand one because d0s = d1s = 1. Using this, we see that eoeV (1 x d0) =
eoeV (1 x d1), and a similar proof shows that eoeV (d0 x 1) = eoeV (d1 x 1). In*
* terms
of elements, this just says that e(d0(u) + v) = e(d1(u) + v). As our coequalis*
*er
diagram was assumed to be strong, we see that the diagram
V x U _____w_w1xd01xd1V x V _____w1xeV x W
is a coequaliser. This implies that there is a unique map o :V x W ! W with
o(1 x e) = eoeV :V x V ! W . Now consider the diagram
d0x1 oeV
U x V ______wV_xwVd1x1_____Vw
  
1xe  1xe e
  
u d0x1 u u
U x W _____Vwx_Wwd1x1____wW:o
We have already seen that eoeV (d0 x 1) = eoeV (d1 x 1), and it follows that o(*
*d0 x
1)(1 x e) = o(d1 x 1)(1 x e). As the relevant coequaliser is preserved by the
functor U x (), we see that 1U x e is an epimorphism, so we can conclude that
o(d0x1) = o(d1x1). As the functor ()xW preserves our coequaliser, this gives us
a unique map oeW :W xW ! W such that oeW (ex1) = o :V xW ! W . One checks
that this makes W into an Abelian group object, and that e is a homomorphism. __
One can also check that this makes W into a coequaliser in Ab D. _*
*_
Remark 6.3. The same result holds, with essentially the same proof, with Mon D
replaced by Ab D or BasedD. The same methods also show that a reflexive fork in
the category of Ralgebras (for any ring R) has the same coequaliser when compu*
*ted
in the category of Ralgebras, or of Rmodules, or of sets.
We next try to construct free Abelian groups or monoids on objects of D or
BasedD. If U 2 D and V 2 BasedD, we "define" objects M+ (U); N+ (V ) 2 Mon D
and M(U); N(V ) 2 Ab(D) by the equations
Mon D(M+ (U); M) = D(U; M)
Mon D(N+ (V ); M)= BasedD(V; M)
Ab D(M(U); G) = D(U; G)
Ab D(N(V ); G)= BasedD(V; G):
More precisely, if there is an object H 2 Ab D with a natural isomorphism
Ab D(H; G) = BasedD(V; G)
FORMAL SCHEMES AND FORMAL GROUPS 61
for all G 2 Ab D, then H is unique up to canonical isomorphism, and we write
N(V ) for H. Similar remarks apply to the other three cases. Given a monoid
object M, we also "define" its group completion G(M) 2 Ab D by the equation
Ab D(G(M); H) = Mon D(M; H).
There are fairly obvious ways to try to construct free group and monoid objec*
*ts,
using a mixture of products and colimits. However, there are two technical poin*
*ts
to address. Firstly, it turns out that we need our colimits to be strong colim*
*its
in the sense of Definition 2.18. Secondly, in some places we can arrange to use
reflexive coequalisers, which is technically convenient.
Proposition 6.4.Let U be an object of D. For each k 0, the symmetric group
k acts on Uk. Suppose`that the quotient Uk=k exists as a strong colimit and
also that L = k0 Uk=k exists as a strong coproduct. Then L = M+ (U).
Proof.Let I be the category with object set N, and with morphisms
(
I(j; k) = ; ifj 6= k
k ifj = k:
It is easy to see that there is a functor k 7! Uk from I to D, and that L is a
strong colimit of this functor. It follows that L x Um is the colimit of the f*
*unctor
k 7! Uk x Um , and thus (using the "Fubini theorem" for colimits) that
L x L = lim! Uk x Um :
(k;m)2IxI
Similarly, L x L x L is the colimit of the functor (k; m; n) 7! Uk x Um x Un fr*
*om
I x I x I to D.
Let jk: Uk ! L be the evident map. We then have maps UkxUm ' Uk+m jk+m!
L, and these fit together to give a map oe :L x L ! L. We also have a zero
map 0 = j0: 1 = U0 ! L. We claim that this makes L into a commutative
monoid object in D. To check associativity, for example, we need to show that
oe O (oe x 1) = oe O (1 x oe): L3 ! L. By the above colimit description of L3,*
* it is
enough to check this after composing with the map jk x jm x jn :Uk+m+n !L3,
and it is easy to check that both the resulting composites are just jk+m+n . We
leave the rest to the reader.
Now suppose we have a monoid M 2 Mon D and a map f :U ! M in D. We
k oe
then have maps fk = (Uk f! Mk ! M), which are easily seen to be invariant
under the action of k, so we get an induced map f0:L !M in D. We claim that
this is a homomorphism. It is clear that f0 O 0 = 0, so we need only check that
f O oe = oe O (f x f): L2 !M. Again, we need only check this after composing w*
*ith
the map jk x jm :Uk+m ! L2, and it then becomes easy. We also claim that f0 is
the unique homomorphism g :L ! M such that g O j1 = f. Indeed, we have jk =
k oe fk oe
(Uk j1!Lk ! L), so for any such g we have g O jk = (Uk ! Mk ! M) = f0O jk.
By our colimit description of L, we see that g = f0 as claimed.
This shows that monoid maps g :L !M biject naturally with maps f :U ! M, __
by the correspondence g 7! g O j1. This means that L = M+ (U) as claimed. _*
*_
Proposition 6.5.Let V be an object of BasedD, and suppose that Vk = V k=k
exists as a strong colimit for all k 0. The basepoint of U then induces maps
62 NEIL P. STRICKLAND
Vk ! Vk+1. Suppose also that the sequence of Vk's has a strong colimit L. Then
L = N+ (V ).
Proof.This is essentially the same as the proof of Proposition 6.4, and_is_left*
* to
the reader. __
We next try to construct group completions of monoid objects. We digress
briefly to introduce some convenient language. Let M be a monoid object, so that
D(U; M) is naturally a monoid for all U. We thus have a map fU :D(U; M3) =
D(U; M)3 !D(U; M2) defined by f(a; b; c) = (c + 2a; 3b + c) (for example). This
is natural in U, so Yoneda's lemma gives us a map f :M3 ! M2. From now on,
we will allow ourselves to abbreviate this definition by saying "let f :M3 ! M2
be the map (a; b; c) 7! (c + 2a; 3b + c)". This is essentially the same as thin*
*king of
D as a subcategory of [Dop; Sets], by the Yoneda embedding.
Given a monoid object M, we define maps d0; d1: M3 ! M2 and s: M2 ! M
by
d0(a; b;=x)(a; b)
d1(a; b;=x)(a + x; b + x)
s(a; b)= (a; b; 0):
This is clearly a reflexive fork in Mon D.
Proposition 6.6.If the above fork has a strong coequaliser q :M2 ! H in D,
then H has a unique group structure making q into a homomorphism of monoids,
and with that group structure we have H = G(M).
Proof.Firstly, Proposition 6.2 tells us that there is a unique monoid structure*
* on H
making q into a monoid map, and that this makes H into the coequaliser in Mon D.
We define a monoid map 0:M2 ! H by 0(a; b) = q(b; a). Clearly 0d0(a; b; x) =
qd0(b; a; x) and 0d1(a; b; x) = qd1(b; a; x) but qd0 = qd1 so 0d0 = 0d1, so the*
*re is
a unique map :H ! H with 0= q. We then have
q(a; b) + q(a;=b)q(a; b) + q(b; a)
= q(a + b; a + b)
= qd1(0; 0; a + b)
= qd0(0; 0; a + b)
= q(0; 0) = 0:
This shows that (1 + )q = 0, but q is an epimorphism so 1 + = 0. This means
that is a negation map for H, making it into a group object. We let j :M ! H
be the map a 7! q(a; 0), which is clearly a homomorphism of monoids. Clearly
q(a; b) = q(a; 0) + q(0; b) = j(a) + j(b) = j(a)  j(b).
Now let K be another Abelian group object, and let f :M ! K be a homo
morphism of monoids. We define f0:M2 ! K by f0(a; b) = f(a)  f(b). It
is clear that f0d0 = f0d1, so we get a unique monoid map f00:H ! K with
f00q = f0. In particular, we have f00j(a) = f00q(a; 0) = f0(a; 0) = f(a), so
that f00j = f. If g :H ! K is another homomorphism with gj = f then
gq(a; b) = g(j(a)  j(b)) = f(a)  f(b) = f00q(a; b), and q is an epimorphism
so g = f00.
This shows that group maps H ! K biject with monoid maps M ! K by the __
correspondence g 7! gj, which means that H = G(M) as claimed. __
FORMAL SCHEMES AND FORMAL GROUPS 63
6.2. Free formal groups. We next discuss the existence of free Abelian formal
groups.
Proposition 6.7.Let Y be a formal scheme over a formal scheme X. Write X
as a filtered colimit of informal schemes Xi, and put Yi = Y xX Xi. If M+ (Yi)
exists in Mon bXXifor all i, then M+ (Y ) exists and is equal to lim!M+ (Yi). *
*Similar
i
remarks apply to M(Y ) and (if Y has a given section 0: X ! Y ) to N+ (Y ) and
N(Y ).
Proof.We use the notation of Definition 4.26 and Proposition 4.27. It is clear *
*that
{M+ (Yi)} is the free Abelian monoid object on {Yi} in the category D{Xi}. As t*
*he
functor F :D{Xi} !XbX preserves finite limits, we see that L = lim!M+ (Yi) =
i
F {M+ (Yi)} is an Abelian monoid object in bXX. Using the fact that F preserves
finite products and is left adjoint to G, we see that
bXX(Lm ; Z) = D{Xi}({M+ (Yi)mXi}; {Z xX Xi})
for all Z 2 bXZ. Using this, one can check that
Mon bX(L; M)= Mon D{Xi}({M+ (Yi)}; {M xX Xi})
= D{Xi}({Yi}; {M xX Xi}) = bXX(Y; M);
as required. We leave the case of M(Y ) and so on to the reader. *
*___
Proposition 6.8.If Y is a coalgebraic formal scheme over X, then the free Abeli*
*an
monoid scheme M+ (Y ) exists. If Y also has a specified section 0: X ! Y (maki*
*ng
it an object of BasedbXX) then N+ (Y ) exists.
Proof.By the previous proposition, we may assume that X is informal, and that
Y = schX(U) for some coalgebra U over R = OX with a good basis {ei i 2 I}. We
know from Example`4.65 that YXk=k is a strong colimit for the action of k on YX*
*k.
Moreover, kY k=k exists as a strong`coproduct by Corollary 4.42. We conclude
from Proposition 6.4 that M+ (Y ) = kY k=k. In the based case, we observe
that the diagram {Y k=k} is just indexed by N and thus is filtered, and filtered
colimits exists and are strong in bXXby Proposition 4.12. Given this, Propositi*
*on_6.5
completes the proof. __
Remark 6.9. If X is informal we see that the coalgebra cM+ (Y ) is just the sy*
*m
metric algebra generated by cY over OX . In the based case, if e0 2 cY is the
basepoint then cN+ (Y ) = cM+ (Y )=(e0  1).
We next show that in certain cases of interest, the free Abelian monoid N+ (Y*
* )
constructed above is actually a group.
Definition 6.10.A good filtration of a coalgebra U over a ring R is a sequence *
*of
submodules FsU for s 0 such that
(a) ffl: F0U ! A is an isomorphism.
(b) For s > 0 the quotient GsU = FsU=Fs1U is a finitely generated free module
overSR.
(c) sFsU = UP
(d) (FsU) s=t+uFsU FtU.
64 NEIL P. STRICKLAND
We write C00= C00R= C00Zfor the category of coalgebras that admit a good filtra*
*tion.
1
Given a good filtration, we write j for the composite A ffl!F0U ae U. One can
check that this is a coalgebra map, so it makes U into a based coalgebra. A good
basepoint for U is a basepoint which arises in this way. We say that a very good
basis for U is a basis {e0; e1; : :}:for U over R such that
(i)e0 = j(1)
(ii)ffl(ei) = 0 for i > 0
(iii)There exist integers Ns such that {ei i < Ns} is a basis for FsU.
One can check that very good bases exist, and that a very good basis is a good
basis.
Proposition 6.11.If U and V lie in C00Zthen so do U x V and Uk=k. If we
choose a good basepoint for U then we can define N+ (U), and it again lies in C*
*00Z.
Proof.Choose good filtrationsPon U and V . Define a filtration on U x V = U V
by setting Fs(U V ) = s=t+uFtU FuV . It is not hard to check that this is
good. Essentially the same procedure gives a filtration of Um . This is invari*
*ant
under the action of the symmetric group m , so we get an induced filtration of *
*the
group of coinvariants Umm . Our filtrations on these groups are compatible as m
varies, so we get an induced filtration of N(U) = lim!Um . Using a very good
m m
basis for U and the associated monomial basis for N(U), we can check that the_
filtration of N(U) is good. __
Proposition 6.12.Let U be an Abelian monoid object in CZ, with addition map
oe :U x U = U U ! U. If U admits a good filtration such that the basepoint is
good and oe(FsU FtU) Fs+tU for all s; t 0, then U is actually an Abelian
group object.
Proof.First note that we can use oe to make U into a ring. We need to construct
a negation map (otherwise known as an antipode) O: U ! U, which must be a
coalgebraPmap satisfying oe(1 O) = jffl. In terms ofPelements, if (a) = 1 a*
* +
a0 a00then the requirement is that O(a) = jffl(a)  a0O(a00). The idea is to
use this_formula_to define O on FsU by recursion_onPs.
Write =  j__1: U ! U U. Note that (FsU) st=0FstU FtU, __
andPthat (ffl 1) = 0. Choose a very good basis {ei} for U, and write (ei) =
j;kaijkej ek. Suppose that Ns1 i < Ns, so that ei2 FsU \ Fs1U. If j > 0
and k Ns1 then ej_ ek 62 Fs(U U)Pso aijk= 0. On the other hand, the
equation (ffl 1) (ei) = 0 gives m ai0mem = 0 for all m, so ai0k= 0, so aijk*
*= 0
for all j. This applies for all k Ns1, and thus in particular for k i.
We now define O(ei) recursively by O(e0) = e0 and
X
O(ei) =  aijkejO(ek)
0k 0. By the previous paragraph, we actually have O(ei) =  k0 aijkejO(e*
*k),
and it follows that oe(1 O) = jffl as required. We still have to check that O*
* is
a coalgebra map. For the counit, it is clear that fflO(e0) = ffl(e0). If we a*
*ssume
inductively that ffl(O(ek)) = ffl(ek) = 0 for 0 < k < i then we find that
X
fflO(ei) =  aijkffl(ej)fflO(ek) = ai00= (ffl ffl) (ei) = ffl(ei) =*
* 0:
0k 2, but only quasicommutative when p = 2. The derivation Q in Defini
tion 8.1 is just the Bockstein map fi :KU=2 !KU=2.
Example 8.5. Let MP be the Thom spectrum associated to the tautological vir
tual bundle over Z x BU. It is more usual to consider the connected component
BU = 0Wx BU of Z x BU, giving the Thom spectrum MU. It turns out that
MP = k2Z 2kMU, and that this is an even periodic ring spectrum. Moreover,
a fundamental theorem of Quillen tells us that MP0 = L = OFGL.
Example 8.6. It turns out [5, 28] that given any ring E0 that can be obtained
from MP0[1_2] by inverting some elements and killing a regular sequence, there *
*is
a canonical even periodic ring spectrum E with ss0E = E0. If we work over MP0
rather than MP0[1_2] then things are more complicated, but typically not too di*
*ffer
ent in cases of interest, except that we only have quasicommutativity rather t*
*han
commutativity. Because MP0 = OFGL, the theory of formal group laws provides
us with many naturally defined rings E0 to which we can apply this result.
8.2. Schemes associated to spaces. Let E be an even periodic ring spectrum.
We write SE = spec(E0).
Example 8.7. As mentioned above, Quillen's theorem tells us that SMP = FGL .
Less interestingly, we have SHP = SK = 1 = spec(Z), the terminal scheme.
If Z is a finite complex, we write ZE = spec(E0Z) 2 XSE. This is a covariant
functor of Z. If Z is an arbitrary space, we write (Z) for the category of pairs
(W; w), where W is a finite complex and w is a homotopy class of maps W ! Z.
Lemma 8.8. The category (Z) is filtered and essentially small.
Proof.It is wellknown that every finite CW complex is homotopy equivalent to
a finite simplicial complex, and that there are only countably many isomorphism
types of finite simplicial complexes. It follows easily that (Z) is essentially*
* small.
If (W; w) and (V; v) are objects of (Z) then there is an evident map u: U =
V q W ! Z whose restrictions to V and W are v and w. Thus (U; u) 2 (Z), and
there are maps (V; v) !(U; u) (W; w) in (Z).
On the other hand, suppose we have a parallel pair of maps f0; f1: (V; v) !
(W; w) in (Z). Let U be the space (W q V x I)= ~, where (x; t) ~ ft(x) whenever
FORMAL SCHEMES AND FORMAL GROUPS 75
x 2 V and t 2 {0; 1}. Let g :W ! U be the evident inclusion, so clearly gf0 ' *
*gf1.
We are given that wf0 and wf1 are homotopic to v. A choice of homotopy between
wf0 and wf1 gives a map u: U ! X with ug = w. Thus g is a map (W; w) !(U; u)_
in (Z) with gf0 = gf1. This proves that (Z) is filtered. __
Remark 8.9. Let Z be a space with a given CW structure, and let CW (Z) be
the directed set of finite subcomplexes of Z. Then there is an evident functor
CW (Z) !(Z), which is easily seen to be cofinal. We can also define stable(Z)
to be the filtered category of finite spectra W equipped with a map w :W ! 1 Z*
*+ .
There is an evident stabilisation functor (Z) ! stable(Z), and one checks that
this is also cofinal.
Remark 8.10. Given two spaces Y and Z, there is a functor (Y ) x (Z) !
(Y x Z) given by ((V; v); (W; w)) 7! (V x W; v x w). This is always cofinal, as*
* one
can see easily from the previous remark (for example).
Definition 8.11.For any space Z, we write
ZE = lim! spec(E0W ) 2 bXSE:
(W;w)2(Z)
*
We also give E0Z the linear topology defined by the ideals I(W;w)= ker(E0Z w!
E0W ). Thus
spf(E0Z) = lim!spec(image(E0Z ! E0W )):
(Z)
We write bE0Z for the completion of E0Z. There is an evident map ZE ! spf(E0Z).
Also, if Y is another space then the projection maps Y Y x Z ! Z give rise *
*to
a map (Y x Z)E ! YE xSE ZE .
Remark 8.12. We know from [1] that the map E0Z ! lim E0W is surjective;
(Z)
the kernel is the ideal of phantom maps. It is clear that the map E0(Z)=I(W;w)!
E0W is injective, so the same is true of the map
limE0(Z)=I(W;w)! limE0W:
It follows by diagram chasing that bE0Z = limE0(Z)=I(W;w)= limE0W , and that
this is a quotient of E0Z. From this we see that E0Z is complete if and only if
there are no phantom maps Z ! E.
Definition 8.13.We say that Z is tolerable (relative to E) if ZE = spf(E0Z) and
(Y x Z)E = YE xSE ZE for all finite complexes Y .
Proposition 8.14.If Z is tolerable and Y is arbitrary then
(Y x Z)E = YE xSE ZE :
If Y is also tolerable then so is Y x Z, and bE0(Y x Z) = bE0(Y )b E0bE0(Z). Of
course if E0Y , E0Z and E0(Y x Z) are complete this means that E0(Y x Z) =
E0Y bE0E0Z.
Proof.If we fix V 2 (Y ) then the functor from (Z) to (V x Z) given by
W 7! V x W is clearly cofinal, so lim!(V x W )E = (V x Z)E , and this is the s*
*ame
W
as VE xSE ZE because Z is tolerable and V is finite. If we now take the colimit
76 NEIL P. STRICKLAND
over V and use the fact that filtered colimits of formal schemes commute with f*
*inite
limits, we find that lim! (V x W )E = YE xSE ZE . It follows from Remark 8.10
V;W
that (Y x Z)E = lim! (V x W )E , so the first claim follows.
V;W
Now suppose that Y is tolerable. Then
(Y x Z)E = YE xSE ZE
= spf(E0Y ) xSE spf(E0Z)
= spf(Eb0Y ) xSE spf(Eb0Z)
= spf(Eb0Y bE0Eb0Z):
It follows that bE0(Y x Z) = O(Y xZ)E= bE0Y bE0Eb0Z as claimed. It also follows
that (Y x Z)E is solid, and thus that (Y x Z)E = spf(E0(Y x Z)).
Now let X be a finite complex. We need to show that (X x Y x Z)E = XE xSE
(Y x Z)E = XE xSE YE xSE ZE . In fact, we have (X x Y )E = XE xSE YE because
Y is tolerable, and ((X x Y ) x Z)E = (X x Y )E xSE ZE because Z is tolerable,_
and the claim follows. __
Definition 8.15.A space Z is decent if H*Z is a free Abelian group, concentrated
in even degrees.
Example 8.16. The spaces CP 1 , BU(n), Z x BU, BSU and S2n+1 are all
decent.
Proposition 8.17.Let Z be a decent space. Then Z is tolerable for any E, and
ZE is coalgebraic over SE . Moreover, for any map E ! E0 of even periodic ring
spectra, the resulting diagram
ZE0 _____wZE
 
 
 
u u
SE0 _____wSE
is a pullback.
Proof.We may assume that Z is connected (otherwise treat each component sep
arately). As H1Z = 0 we see that ss1Z is perfect, so we can use Quillen's plus
construction to get a homology equivalence Z ! Z+ such that ss1(Z+ ) = 0. By
the stable Whitehead theorem, this map is a stable equivalence, so E0(Y x Z+ ) =
E0(Y xZ) for all Y . We may thus replace Z by Z+ and assume that ss1Z = 0. This
step is not strictly necessary, but it seems the cleanest way to avoid trouble *
*from
the fundamental group. Given this, it is wellknown that Z has a CW structure in
which all the cells have even dimension. It follows that the AtiyahHirzebruch *
*spec
tral sequence collapses and that E*Z is a free module over E*, with one generat*
*or
ei for each cell. As E* is twoperiodic, we can choose these generators in degr*
*ee
zero. Similarly, E*(Z x Z) is free on generators ei ej and thus is isomorphic to
E*(Z) E* E*(Z), so we can use the diagonal map to make E*Z into a coalgebra
over E*. By periodicity, E0(Z x Z) = E0(Z) E0 E0(Z) and E0Z is a coalgebra
over E0, and is freely generated as an E0module by the ei.
If W is a finite subcomplex of Z, it is easy to see that E0W is a standard
subcoalgebra of E0Z (in the language of Definition 4.58). Moreover, any finite
FORMAL SCHEMES AND FORMAL GROUPS 77
collection of cells lies in a finite subcomplex, so it follows that any finitel*
*y generated
submodule of E0Z lies in a standard subcoalgebra. It follows that {ei} is a good
basis for E0Z, so that E0Z 2 C0SE.
It follows from the above in the usual way that E ^ Z+ is equivalent as an E
module spectrum to a wedge of copies of E (one for each cell), and thus that E**
*Z =
Hom E*(E*Z; E*). Using the periodicity we conclude that E0Z = Hom E0(E0Z; E0).
It follows that spf(E0Z) = schSE(E0Z) is a solid formal scheme, which is coalge
braic over SE . It is also easy to check that spf(E0Z) is the colimit of the sc*
*hemes
spec(E0W ) as W runs over the finite subcomplexes. It follows from Remark 8.9
that spf(E0Z) = ZE .
Now let Y be another space. Let W be a finite subcomplex of Z, and let (V; v)
be an object of (Y ). The usual K"unneth arguments show that E0(W x V ) =
E0W E0 E0V , and thus that (W x V )E = WE xSE VE . Using Remark 8.10 we
conclude that
(Z x Y )E = lim!WE xSE VE = (lim!WE ) xSE (lim!VE ) = ZE xSE YE :
W;V W V
This proves that Z is tolerable. We leave it to the reader to check that_a_map
E ! E0 gives an isomorphism ZE0 = ZE xSE SE0. __
Example 8.18. It follows from the proposition that the spaces CP 1 , BU(n),
Z x BU, BSU and S2n+1 are all tolerable, and the corresponding schemes are
coalgebraic over SE . The case of CP 1 is particularly important. We note that
CP 1 = BS1 = K(Z; 2) is an Abelian group object in the homotopy category, so
GE = CP 1E is an Abelian formal group over SE . Because H*CP 1 = Z[[x]], the
AtiyahHirzebruch spectral sequence tells us that E0CP 1 = E0[[x]](although this
does not give a canonical choice of generator x). This means that GE ' bA1x SE
in BasedbXSE, so that GE is an ordinary formal group.
We next recall that for n > 0 there is a quasicommutative rings spectrum P (n*
*) =
BP=In with P (n)* = Fp[vk  k 0], where vk has degree 2(pk  1). The cleanest
construction now available is given in [5, 28], although of course there are mu*
*ch
older constructions using BaasSullivan theory. We also have P (0) = BP , with
P (0)* = Z(p)[vk  k > 0].
Definition 8.19.Let E be an even periodic ring spectrum. We say that E is an
exact P (n)module (for some n 0) if it is a modulespectrum over P (n), and t*
*he
sequence (vn; vn+1; : :):is regular on E*.
Proposition 8.20.Let E be an exact P (n)module. Let Z be a CW complex of
finite type such that K(m)*Z is concentrated in even degrees for infinitely many
m. If n = 0, assume that Hs(Z; Q) = 0 for s 0. Then Z is tolerable for E.
Remark 8.21. When combined with Proposition 8.14 this gives a useful K"unneth
theorem.
The proof will follow after Corollary 8.27. Many spaces are known to which th*
*is
applies: simply connected finite Postnikov towers of finite type, classifying s*
*paces
of many finite groups and compact Lie groups, the spaces QS2m , BO, ImJ and
BU<2m> for example. See [24] for more details. The proof of our proposition will
also rely heavily on the results of that paper.
78 NEIL P. STRICKLAND
We next need some results involving the procompletion of the category of gra*
*ded
Abelian groups, which we denote by Pro(Ab *). It is necessary to distinguish th*
*is
carefully from the category Pro(Ab )* of graded systems of progroups. A tower *
*of
graded groups can be regarded as an object in either category, but the morphisms
are different. A tower {A0* A1* . .}.in Pro(Ab *) is protrivial if for *
*all
j, there exists k > j such that the map Ak* ! Aj* is zero. It is protrivial *
*in
Pro(Ab )* if for all j and d there exists k such that the map Akd ! Ajd is zer*
*o.
Because k is allowed to depend on d, this is a much weaker condition than trivi*
*ality
in Pro(Ab *). Note also that if R* ! R0*is a map of graded rings, and {Mff*} is
a prosystem of R*modules that is trivial in Pro(Ab *), then the same is true *
*of
R0*R* M*. However, the corresponding statement for Pro(Ab )* is false.
Remark 8.22. If E is an exact P (n)module, we know from work [29] of Yagita
that the functor M 7! E*P(n)*M is an exact functor on the category of P (n)*P (*
*n)
modules that are finitely presented as modules over P (n)*. (This category is
Abelian, because the ring P (n)* is coherent.) It follows that E*Z = E* P(n)*
P (n)*Z for all finite complexes Z.
The following lemma is largely a paraphrase of results in [24].
Lemma 8.23. Fix n 0. Suppose that Z is a CW complex of finite type, and
write Zr for the rskeleton of Z. If n = 0 we also assume that Hs(Z; Q) = 0
for s 0. Let F r+1= ker(P (n)*Z ! P (n)*Zr) denote the (r + 1)'st Atiyah
Hirzebruch filtration in P (n)*Z. Then the tower {P (n)*Zr}r0 is isomorphic to
{P (n)*(Z)=F r+1}r0 in Pro(Ab *), and thus is MittagLeffler. Moreover, the gr*
*oups
P (n)*(Z)=F r+1are finitely presented modules over P (n)*, and their inverse li*
*mit
is P (n)*Z.
Proof.Write P = P (n) for brevity. Write Ar = P *Zr and
Br = P *(Z)=F r+1= image(P *Z ! Ar):
We then have an inclusion of towers {Br} ! {Ar}, for which we need to provide
an inverse in the Procategory. We claim that for each r, there exists m(r) > r
such that the image of the map Am(r)! Ar is precisely Br. We will deduce the
lemma from this before proving it. Define m0 = 0 and mk+1 = m(mk) > mk. By
construction, the map Amk+1 ! Amk factors through Bmk Amk . One checks
that the resulting maps Amk+1 ! Bmk are compatible as k varies, and that they
provide the required inverse. We also know that P *is a coherent ring, so the
category of finitely presented modules is Abelian and closed under extensions. *
*It
follows in the usual way that Ar is finitely presented for all r, and thus that*
* Br =
image(Am(r)! Ar) is finitely presented.
We now need to show that m(r) exists. By the basic setup of the Atiyah
Hirzebruch spectral sequence, it suffices to show that for large m, the first r*
* + 1
columns in the spectral sequence for P *Zm are the same as in the spectral sequ*
*ence
for P *Z. This is Lemma 4.4 of [24]. (When n = 0, we need to check that we are *
*in
the case P (0) = BP of their Definition 1.5. This follows from our assumption t*
*hat
Hs(Z; Q) = 0 for s 0.)
Finally, we need to show that P *Z = limP *(Z)=F r+1. This is essentially [2*
*4,
r __
Corollary 4.8]. __
FORMAL SCHEMES AND FORMAL GROUPS 79
Corollary 8.24.Let Z and n be as in the Lemma, and let E be an exact P (n)
module. Then E0Z is complete, and ZE = spf(E0Z), and E*Z = E*b P(n)*P (n)*Z.
Moreover we have isomorphisms
{E*(Zr)} ' {E*(Z)=F r+1}' {E* P(n)*P (n)*Zr}
' {E* P(n)*(P (n)*(Z)=F r+1)}
in Pro(Ab *).
Proof.We reuse the notation of the previous proof. We also define A0r= E*P* Ar
and B0r= E* P* Br. As Zr is finite we see that A0r= E*Zr. Next recall
that for any r we can choose m > r such that Br = image(Am ! Ar). As the
functor E* P* () is exact on finitely presented comodules, we see that B0ris
the image of the map A0m ! A0rand in particular that the map B0r! A0ris
injective. Next, the map E* P* P *Z ! E* P* P *Zm = E*Zm = A0mclearly
factors through E*Z, so our epimorphism P *Z ! Am ! Br gives an epimorphism
E* P* P *Z ! A0m! B0rwhich factors through E*Z, so the map E*Z ! B0ris
surjective. Thus B0r= image(E*Z ! E*Zr) = E*(Z)=F r+1. We can now apply
the functor E* P* () to the proisomorphisms in the Lemma to get the pro
isomorphisms in the present corollary. This makes it clear that the tower {E*Zr}
is MittagLeffler so the Milnor sequence tells us that
E*Z = limE*Zr = limE*(Z)=F r+1:
r r
This means in particular that E0Z is complete with respect to the linear topolo*
*gy
generated by the ideals F r+1, which is easily seen to be the same as the topol*
*ogy
in Definition 8.11. Moreover, we have an isomorphism {A0r} ' {B0r} in the Pro
category of groups, and it is easy to see from the construction that this is ac*
*tually
an isomorphism in the Pro category of rings as well, so by applying spec() we
get an isomorphism in the Ind category of schemes, which is just the category of
formal schemes. From the definitions we have ZE = lim!spec(A0r) and spf(E0Z) =
r __
lim!spec(B0r), so we conclude that ZE = spf(E0Z). __
r
Lemma 8.25. Let E and Z be as in Corollary 8.24, and suppose that K(m)*Z is
concentrated in even degrees for infinitely many m. Then the ring E*Z is Landwe
ber exact over P (n)*, so the function spectrum F (Z+ ; E) is an exact P (n)mo*
*dule.
Proof.We know from [24, Lemma 5.3] that P (m)*Z is concentrated in even de
grees for all m, and from [24, Corollary 4.6] that the tower {P (m)*Zr} has the
MittagLeffler property. It follows that the tower {P (m)oddZr} is protrivial.*
* Next,
consider the cofibration 2(pm 1)P (m) vm!P (m) ! P (m + 1) ! 2pm 1P (m).
This gives a proexact sequence of towers
0 !{P (m)evZr} vm!{P (m)evZr} !{P (m + 1)evZr} !0:
It follows that the sequence (vn; vn+1; : :):acts regularly on the tower {P (n)*
**Zr}.
Next, for any spectrum X we have a map P (m) ^ X ! P (m) ^ BP ^ X which
makes P (m)*X a comodule over P (m)*BP = BP*BP=In. Moreover, we have
P (m)*XP(m)*P (m)*BP = P (m)*XBP*BP*BP so this actually makes P (m)*X
intoma comodule over BP*BP . One can checkmfrom this construction that the maps
2(p 1)P (m) vm!P (m) !P (m+1) !2p 1P (m) give rise to maps of comod
ules, so our whole diagram of towers is a diagram of finitelypresented comodul*
*es
80 NEIL P. STRICKLAND
over P (n)*BP . The functor E* P(n)*() is exact on this category. It is easy to
conclude by induction that {E* P(n)*P (m)*Zr} ' {E*(Zr)=Im }, that the odd
dimensional part of these towers is protrivial, the towers are MittagLeffler,*
* and
the sequence (vn; vn+1; : :):is regular on the tower {E*(Zr)}. We can now pass
to the inverse limit (using the MittagLeffler property to show that the lim1t*
*erms
vanish) to see that the sequence (vn; vn+1; : :):is regular on E*(Z). *
* ___
Our next few results are closely related to those of [24, Section 9], althoug*
*h a
precise statement of the relationship would be technical.
Lemma 8.26. Let Z be a CW complex of finite type such that K(m)*Z is con
centrated in even degrees for infinitely many m. If n = 0 we also assume that
Hs(Z; Q) = 0 for s 0. Then for any finite spectrum W we have proisomorphisms
{P (n)*(Zr x W )}' {P (n)*Zr P(n)*P (n)*W }
' {P (n)*(Z)=F r+1P(n)*P (n)*W };
and these towers are MittagLeffler. Moreover, we have isomorphisms
P (n)*(Z x W ) = P (n)*Z P(n)*P (n)*W = P (n)*Z bP(n)*P (n)*W:
Proof.Write P = P (n) for brevity. The usual Landweber exactness argument
shows that W 7! P *Z P* P *W is a cohomology theory and thus that it coincides
with P *(Z x W ). We can also do the same argument with progroups. We saw
in the proof of the previous lemma that the sequence*(vn; vn+1; : :):acts regul*
*arly
on the progroup {P *Zr}, so the progroup {TorP1(P (m)*; P *Zr)} is trivial for
all m n. Any finitely presented comodule M* has a finite Landweber filtration
whose quotients have the form P (m)**for m n, and we see by induction on the
length of the filtration that {TorP1(M*; P *Zr)} is trivial. This implies that*
* the
construction M* 7! {M* P* P *Zr} gives an exact functor from finitely presented
comodules to Pro(Ab *), so that W 7! {P *W P* P *Zr} is a Pro(Ab *)valued
cohomology theory on finite complexes. The construction W 7! {P *(W x Zr)}
gives another such cohomology theory, and we have a natural transformation from
the first to the second that is an isomorphism when W is a sphere, so it is an
isomorphism in general. Thus {P *(Zr x W )} = {P *Zr P* P *W }, as claimed.
We have seen that the tower {P *Zr} is proisomorphic to {P *(Z)=F r+1}, so it
follows that {P *Zr P* P *W } ' {P *(Z)=F r+1P* P *W }. The second of these
is a tower of isomorphisms, so all three of our towers are MittagLeffler as cl*
*aimed.
As Z x W is the homotopy colimit of the spaces Zr x W , the Milnor sequence giv*
*es
an isomorphism P *(Z x W ) = limP *(Z)=F r+1P* P *W , and the right hand side
r *
*__
is by definition P *(Z)b P*P *W , which completes the proof. *
*__
Corollary 8.27.Let E be an exact P (n)module. Let Z be a CW complex of
finite type such that K(m)*Z is concentrated in even degrees for infinitely many
m. If n = 0 we also assume that Hs(Z; Q) = 0 for s 0. Then for any finite
spectrum W we have proisomorphisms
{E0(Zr x W )} ' {E0Zr E0 E0W } ' {E0(Z)=F r+1E0 E0W };
and these towers are MittagLeffler. Moreover, we have isomorphisms
E0(Z x W ) = E0Z E0 E0W = E0Z bE0E0W:
FORMAL SCHEMES AND FORMAL GROUPS 81
Proof.If we apply the functor E*P(n)*() to the proisomorphisms in the lemma,
we get the proisomorphisms in the corollary. We deduce in the same way as in
the lemma that E0(Z x W ) = E0Z bE0E0W . On the other hand, we see from
Lemma 8.25 that
E*(Z x W ) = [W+ ; F (Z+ ; E)]* = E*Z P(n)*P (n)*W =
E*Z E* (E* P(n)*P (n)*W ) = E*Z E* E*W:
Thus E0(Z x W ) = E0Z E0 E0W as claimed. ___
Proof of Proposition 8.20.Corollary 8.24 shows that ZE = spf(E0Z). Write
F r+1= ker(E0Z ! E0Zr);
so ZE = lim!V (F r). Let W be a finite complex. We then have
r
ZE xSE WE = lim!V (F r) xSE WE
r
= lim!spec(E0(Z)=F rE0 E0(W ))
r
= spf(E0(Z)b E0E0(W ))
= spf(E0(Z x W ));
where we have used Corollary 8.27. We can apply Lemma 8.23 to Y xZ and conclude
that spf(E0(Y x Z)) = (Y x Z)E , giving the required isomorphism (Y x Z)E_=
YE xSE ZE . __
8.3. Vector bundles and divisors. Let V be a complex vector bundle of rank
n over a tolerable space Z. We write P (V ) for the space of pairs (z; W ), wh*
*ere
z 2 Z and W is a line (i.e. a onedimensional subspace) in Vz. This is clearly *
*a fibre
bundle over Z with fibres CP n1. We write D(V ) = P (V )E . There is a tautolo*
*gical
line bundle L over P (V ), whose fibre over a pair (z; W ) is W . This is class*
*ified by
a map P (V ) ! CP 1. By combining this with the projection to Z, we get a map
P (V ) ! CP 1 x Z and thus a map D(V ) ! G xS ZE . The wellknown theorem
on projective bundles translates into our language as follows.
Proposition 8.28.The above map is a closed inclusion, making D(V ) into an
effective divisor of degree n on G.
Proof.Choose an orientation x of E, so x 2 E"0CP 1. We also write x for the
image of x under the map P (V ) ! CP 1, which is just the Euler class of L. We
claim that E*P (V ) is freely generated over E*Z by {1; x; : :;:xn1}, which wi*
*ll
prove the claim. This is clear when V is trivialisable. For the general case*
*, we
may assume that Z is a regular CW complex. The claim holds when Z is a finite
union of subcomplexes on which V is trivialisable, by a wellknown MayerVietor*
*is
argument. It thus holds when Z is a finite complex, and the general case follow*
*s_
by passing to colimits. __
Proposition 8.29.If V and W are two vector bundles over a tolerable space Z
then D(V W ) = D(V ) + D(W ).
Proof.Choose an orientation, and let x be the Euler class of the usual line bun*
*dle
over P (V W ). The polynomial fD(V W) (t) is the unique one of degree dim(V W )
of which x is a root, so it suffices to check that fD(V )(x)fD(W)(x) = 0. There*
* are
82 NEIL P. STRICKLAND
evident inclusions P (V ) ! P (V W ) P (W ) with P (V ) \ P (W ) = ;. Write
A = P (V W ) \ P (V ) and B = P (V W ) \ P (W ), so that A [ B = P (V W ).
By a wellknown argument, if a; b 2 E0P (V [ W ) and aA = 0 and bB = 0 then
ab = 0, so it suffices to check that fD(V )(x)B = 0 and fD(W)(x)A = 0. It is *
*not
hard to see that the inclusions P (V ) !B is a homotopy equivalence and thus t*
*hat_
fD(V )(x)B = 0, and the other equation is proved similarly. *
*__
Proposition 8.30.If M is a complex line bundle over a tolerable space Z, which *
*is
classified by a map u: Z ! CP 1, then D(M) is the image of the map (u; 1)E :ZE*
* !
(CP 1 x Z)E = G xS ZE .
Proof.This follows from the definitions, using the obvious fact that P (M) = Z.*
* ___
Proposition 8.31.There is a natural isomorphism BU(n)E = Div+n(G).
Proof.This is essentially wellknown, but we give some details to illustrate how
everything fits together. Let T (n) be the maximal torus in U(n), so that BT (n*
*) '
(CP 1 )n and BT (n)E = GnS. Thus, the inclusion i: T (n) ! U(n) gives a map
GnS! BU(n)E . If oe 2 n is a permutation, then the evident action of oe on T (*
*n)
is compatible with the action on U(n) given by conjugating with the associated
permutation matrix. This matrix can be joined to the identity matrix by a path
in U(n), so the conjugation is homotopic to the identity. Thus, our map GnS!
BU(n)E factors through a map Div+n(G) = GnS=n ! BU(n)E . On the other
hand, the tautological bundle Vn over BU(n) gives rise to a divisor D(Vn) over
BU(n)E and thus a map BU(n)E ! Div+n(G). The composite GnS= BT (n)E !
BU(n)E ! Div+n(G) = GnS=n classifies the divisor D(i*Vn). Let M1; : :;:Mn be
the evident line bundles over BT (n), so that i*Vn = M1: :M:n. One checks from
this and Propositions 8.29 and 8.30 that the composite is just the usual quotie*
*nt
map GnS! GnS=n, and thus the composite Div+n(G) ! BU(n)E ! Div+n(G) is
the identity.
Next, we take the space of nframes in C1 as our model for EU(n). There is
then a homeomorphism EU(n)=(S1 x U(n  1)) ! P (Vn) (sending (w1; : :;:wn)
to the pair (L; W ), where W is the span of {w1; : :;:wn} and L is the span of *
*w1).
The left hand side is a model for CP 1 x BU(n  1). By induction on n, we may
assume that BU(n  1)E = Gn1=n1. This gives a commutative diagram as
follows.
G x Gn1=n1 _____Pw(Vn)E'
 
 
 
 
u uu
Gn=n v________BU(n)E_w
The top horizontal is an isomorphism by induction and the right hand vertical *
*is
faithfully flat, and thus a categorical epimorphism. It follows that the bottom*
* map
is an epimorphism, but we have already seen that it is a split monomorphism,_so
it is an isomorphism as required. __
Definition 8.32.Let x be a coordinate on G. If V is a vector bundle of rank
n over a tolerable space Z, then we have D(V ) = spf(E0Z[[x]]=f(x)) for a unique
FORMAL SCHEMES AND FORMAL GROUPS 83
P n
monic polynomial f(x) = i=0ci(V )xni, with ci(V ) 2 E0Z. We call ci(V ) the
i'th Chern class of V .
Definition 8.33.We write L(V ) for L(D(V )), the Thom sheaf of D(V ), which
is a line bundle over ZE . It is easy to see that L(V ) = E"0ZV , where ZV =
P (C V )=P (V ) is the Thom space of V .
Remark 8.34. Let E be an even periodic ring spectrum and put G = GE =
(CP 1 )E and S = SE = spec(E0) as usual. Then the Thom spectra CP1nform a
tower, and there is a natural identification MG=S = lim!E0(CP1n). We also have
n
!G=S = Ee0CP 1= Ee0S2 = ss2E. The theory of invariant differentials identifies
M1G=Swith MG=S E0 !G=S = lim!E0(2CP1n). The S1equivariant Segal con
n
jecture gives an equivalence between holim2CP1nand the profinite completion
n
of S0, and one can show that the resulting map M1G=S= lim!E0(2CP1n) !E0
n
is just resG=S.
Proposition 8.35.There are natural isomorphisms
a
( BU(n))E = M+ (G) = Div+(G)
n
BUE = N+ (G) = N(G) = Div0(G)
(Z x BU)E = M(G) = Div(G)
(Z x BU)E = Map S(G; Gm ):
Proof.This is wellknown, and follows easily from Proposition 8.31 and the rema*
*rks
following Definition 5.8. The fourth statement follows from the third one_by Ca*
*rtier
duality. __
Next, recall that there is a "complex reflection map" r :S1 x CP+n1! U(n),
where r(z; L) has eigenvalue z on the line L < Cn and eigenvalue 1 on L? . This
gives an unbased map CP n1! U(n). We can also fix a line L0 < Cn and
define _r(z; L) = r(z; L)r(z; L0)1, giving a map _r:CP n1! SU(n). Moreover,
the Bott periodicity isomorphisms U = Z x BU and SU = BU give us maps
U(n) ! Z x BU and SU(n) ! BU. It is easy to see that (CP n1)E is the
divisor Dn = n[0] = spec(E0[[x]]=xn) on GE over SE .
Proposition 8.36.There are natural isomorphisms
(U(n))E = M(Dn)
(SU(n))E = N(Dn)
(U(n))E = Map S(Dn; Gm )
(SU(n))E = BasedMap S(Dn; Gm ):
Under these identifications, the map U(n) ! Z x BU gives the obvious map
M(Dn) !M(GE ) and so on.
Proof.For the second statement, it is enough (by Remark 6.9) to check that
E*(SU(n)) is the symmetric algebra generated by the reduced Ehomology of
CP n1. This is wellknown for ordinary homology, and it follows for all E by a
84 NEIL P. STRICKLAND
collapsing AtiyahHirzebruch spectral sequence. See [22, 23] for more details. *
*The
inclusion S1 = U(1) ! U(n) and the determinant map det:U(n) ! S1 give a
splitting U(n) = S1 x SU(n) of spaces and thus U(n) = Z x SU(n) of Hspaces
and the first claim follows in turn using this. The last two statements_follow*
*_by
Cartier duality. __
8.4. Cohomology of Abelian groups. Let A be a compact Abelian Lie group,
and write A* for the character group Hom (A; S1), which is a finitely generated
discrete Abelian group. Let G be an ordinary formal group over a base S. For any
point s 2 S(R) we write (Gs) = bXspec(R)(spec(R); Gs) for the associated group *
*of
sections. A coordinate gives a bijection between (Gs) and Nil(R), which becomes
a homomorphism if we use an appropriate formal group law to make Nil(R) a group.
We define a formal scheme Hom (A*; G) by
Hom (A*; G)(R) = {(s; OE)  s 2 S(R) and OE: A* !(Gs)}:
(If A* is a direct sum of r cyclic groups then this can be identified with a cl*
*osed
formal subscheme of GrSin an evident way, which shows that it really is a schem*
*e.)
Proposition 8.37.For any finite Abelian group A, there is a natural map BAE !
Hom (A*; G). This is an isomorphism if E is an exact P (n)module for some n.
Proof.An element ff 2 A* = Hom (A; S1) gives a map BA ! BS1 of spaces and
thus a map BAE ! (BS1)E = G of formal groups over S. One checks that the
resulting map A* !Ab bXS(BAE ; G) is a homomorphism, so by adjointing things
around we get a map BAE ! Hom (A*; G). If A is a torus then A* ' Zr and
BAE = Gr = Hom (A*; G), so our map is an isomorphism. Moreover, in this case
BA ' (CP 1 )r which is decent and thus tolerable for any E. If A = Z=m then
there is a wellknown way to identify BA with the circle bundle in the line bun*
*dle
Lm , where L is the tautological bundle over CP 1. This gives a long exact Gysin
sequence
E*BA E*CP 1 [m](x)E*CP 1 :
The second map here is multiplication by [m](x), which is the image of x under
the map G xm!G. If this map is injective then the Gysin sequence is a short
exact sequence and we have E0BA = E0CP 1 =[m](x), and we conclude easily that
spf(E0BA) = ker(G m!G) = Hom (A*; G). One can apply similar arguments to
the skeleta S2k+1=(Z=m) of BA and find that spf(E0BA) = BAE .
In the case of twoperiodic Morava Ktheory we recover the wellknown calcula
tion showing that K(n)*BA is concentrated in even degrees for all n. We also ha*
*ve
Hs(BA; Q) = 0 for s > 0 so Proposition 8.20 tells us that BAE is tolerable for *
*any
E that is an exact P (n)module for any n. Moreover, it is easy to see that [m]*
*(x)
is not a zerodivisor in this case so BAE = Hom (A*; G). We have just shown thi*
*s __
when A* is cyclic, but it follows easily for all A by Proposition 8.14. *
* __
8.5. Schemes associated to ring spectra. If R is a commutative ring spectrum
with a ring map E ! R, we have a scheme spec(ss0R) over SE . If Z is a finite
complex we can take R = F (Z+ ; E) and we recover the case ZE = spec(E0Z) =
spec(ss0R). If M is an arbitrary commutative ring spectrum, we can take R =
E ^ M. In this case we write ME = spec(E0M) for the resulting scheme. If Y
is a commutative Hspace we can take M = 1 Y+ , and we write Y E for ME =
FORMAL SCHEMES AND FORMAL GROUPS 85
spec(E0Y ) in this case. If we have a K"unneth isomorphism E0Y k = (E0Y )k
then E0Y is a Hopf algebra, so Y E is a group scheme over S. If Y is decent then
E0Y is a coalgebra with good basis. In this case Proposition 6.19 applies, and
we have a Cartier duality Y E = D(YE ) = Hom S(YE ; Gm ) and YE = D(Y E) =
Hom S(Y E; Gm ).
If {Rff} is an inverse system of ring spectra as above, we have a formal sche*
*me
lim!spec(ss0Rff). If Zffruns over the finite subcomplexes of a CW complex Z, t*
*hen
ff
the rings F (Zff+; E) give an example of this, and the associated formal scheme
is just ZE . Another good example is to take the tower of spectra E=pk, where
E is an even periodic ring spectrum such that E0 is torsionfree. More generall*
*y,
if E has suitable Landweber exactness properties then we can smash E with a
generalised Moore spectrum S=I (see [13, Section 4], for example) and get a new
even periodic ring spectrum E=I, and then we can consider a tower of these. The*
*re
are technicalities about the existence of products on the spectra E=I, which we
omit here.
8.6. Homology of Thom spectra. Let Z be a space equipped with a map Z z!
Z x BU, and let T (Z; z) be the associated Thom spectrum. It is wellknown that
T is a functor from spaces over Z x BU to spectra, which preserves homotopy
pushouts. Moreover, if (Y; y) is another space over Z x BU then we can use the
addition on Z x BU to make (Y x Z; (y; z)) into a space over Z x BU and we find
that T (Y x Z; (y; z)) = T (Y; y) ^ T (Z; z).
The above construction really needs an actual map Z z!Z x BU and not just a
homotopy class. However, we do have the following result.
Lemma 8.38. If Z is a decent space then the spectrum T (Z; z) depends only on
the homotopy class of z, up to canonical homotopy equivalence. Thus T can be
regarded as a functor from the homotopy category of decent spaces over Z x BU to
spectra. In particular, we can define T (Z; V ) when V is a virtual bundle over*
* Z.
Proof.Suppose we have two homotopic maps z0; z1: Z ! Z x BU. We can then
choose a map w :Z x I ! Z x BU such that wj0 = z0 and wj1 = z1, where jt(a) =
(a; t). The maps jt induce maps of spectra T (Z; zt) ft!T (Z x I; w), and the *
*Thom
isomorphism theorem implies that these give equivalences in homology so they are
weak equivalences. We thus have a weak equivalence f11f0: T (Z; z0) ! T (Z; z*
*1).
This much is true even when Z is not decent.
To see that our map is canonical when Z is decent, note that KU*Z is con
centrated in even degrees, so the space F of unpointed maps from Z to Z x BU
has trivial odddimensional homotopy groups with respect to any basepoint. We
can think of z0 and z1 as points of F , and w as a path between them. If w0 is
another path then then we can glue w and w0to get a map of S1 to F , which can
be extended to give a map u: D2 !F because ss1F = 0. It follows that we have a
86 NEIL P. STRICKLAND
commutative diagram as follows:
T (Z; z0)____________________Tw(Zfx0I;fw)l
 flfl A u
 flflffl AA 
 AAD 
f00 T (Z x D2; u) f1
 AAC fflflli 
 A A flfl 
u A 
T (Z x I; w0)u__________________T_(Z;fz1)0
1
It follows easily that f11O f0 = (f01)1 O f00, as required. *
* ___
A coordinate on GE is the same as a degree zero complex orientation of E, whi*
*ch
gives a multiplicative system of Thom classes for all virtual complex bundles. *
*In
particular, this gives isomorphisms E*T (Y; y) ' E*Y , which are compatible in *
*the
evident way with the isomorphisms T (Y x Z; (y; z)) = T (Y; y) ^ T (Z; z).
If Z z!{n}xBU(n) classifies an honest ndimensional bundle V over Z then we
have T (Z; z) = 1 ZV . In particular, the inclusion CP 1 = BU(1) ! {1} x BU
just gives the Thom spectrum 1 (CP 1 )L, which is wellknown to be the same as
1 CP 1 (without a disjoint basepoint).
Now let Z be a decent commutative Hspace. Let z :Z ! ZxBU be an Hmap,
and write M = T (Z; z). We note that addition gives a map (Z x Z; (z; z)) !(Z;*
* z)
of spaces over Z x BU and thus a map of spectra M ^ M ! M, which makes M
into a commutative ring spectrum. Similarly, the diagonal gives a map (Z; z) !
(Z x Z; (0; z)) and thus a map M ffi!1 Z+ ^ M. Finally, we consider the sheari*
*ng
map (a; b) 7! (a; a + b). This is an isomorphism (Z x Z; (z; z)) ! (Z x Z; (0;*
* z))
over Z x BU, which gives an isomorphism M ^ M ! 1 Z+ ^ M of spectra.
A choice of coordinate gives a Thom isomorphism E*M ' E*Z, which shows
that E*M is free and in even degrees. For the moment we just use this to show
that we have K"unneth isomorphisms, from which we will recover a more natural
statement about the relationship between E*Z and E*M.
Recall that we defined define ZE = spec(E0Z) = spec(E01 Z+ ) (which is a
commutative group scheme over S = SE ) and ME = spec(E0M). Our diagonal
map ffi gives an action of ZE on ME . The shearing isomorphism M ^M = 1 Z+ ^
M shows that the action and projection maps give an isomorphism ZE xS ME !
ME xS ME .
A choice of coordinate on G gives an isomorphism E0M ' E0Y . One can check
(using the multiplicative properties of Thom classes) that this is an isomorphi*
*sm
of E0Y comodule algebras, so it gives an isomorphism Y E ' ME of schemes,
compatible with the action of Y E. This means that ME is a trivialisable torsor*
* for
Y E.
In the universal case Y = Z x BU, this works out as follows. As mentioned
previously, we have a map CP 1 = {1} x BU(1) !Z x BU, and the Thom functor
gives a map 1 CP 1 ! MP . In particular, the bottom cell gives a map S2 =
CP 1! MP , or an element u 2 ss2MP . The inclusion {1} !ZxBU also gives an
element of ss2MP , which one checks is inverse to u. Thus, a ring map E0MP ! R
gives an E0algebra structure on R, and an E0module map "E0CP 1! R, which
sends E"0S2 into Rx . In other words, it gives a point s 2 SE (R) together with
an element y 2 Rb E0"E0CP 1. We can identify Rb E0"E0CP 1 with the ideal of
FORMAL SCHEMES AND FORMAL GROUPS 87
functions on Gs that vanish at zero, and the extra condition on the restriction*
* to
S2 says that y is a coordinate. This gives a natural map MP E! Coord(G). Well
known calculations show that E0MP is the symmetric algebra over E0 on "E0CP 1,
with the bottom class inverted. This implies easily that the map MP E! Coord(G)
is an isomorphism. Recall also that (Z x BU)E = Map (G; Gm ). Clearly, if u: G *
*!
Gm and x is a coordinate on G, then the product ux is again a coordinate. This
gives an action of Map (G; Gm ) on Coord(G), which makes Coord(G) into a torsor
over Map (G; Gm ). One can check that this structure arises from our geometric
coaction of Z x BU on MP .
8.7. Homology operations. Let G be an ordinary formal group over S, and let
H be an ordinary formal group over T . Let ssS and ssT be the projections from
S x T to S and T . We write Hom (G; H) for Hom SxT (ss*SG; ss*TH), which is a
scheme over S x T by Proposition 6.15. Recall that Hom (G; H)(R) is the set of
triples (s; t; u) where s 2 S(R) and t 2 T (R) and u: Gs !Ht is a map of formal
groups over spec(R). We write Iso(G; H)(R) for the subset of triples for which *
*u is
an isomorphism. If we choose coordinates x and y on G and H, then for any u we
have y(u(g)) = OE(x(g)) for some power series OE 2 R[[t]]with OE(0) = 0, and u *
*is an
isomorphism if and only if OE0(0) is invertible. It follows that Iso(G; H) is a*
*n open
subscheme of Hom (G; H).
Proposition 8.39.Let E and E0 be even periodic ring spectra. Then there is a
natural map SE^E0 ! Iso(GE ; GE0) of schemes over SE x SE0. This is an isomor
phism if E or E0 is Landweber exact over MP .
Proof.We write S0= SE0 and0G0= GE0. The evident ring maps E ! E ^E0 E0
give maps S q SE^E0 q!S0, and pullback squares
G u_____GE^E0______wG0
  
  
  
  
u u u
S u______SE^E0q______S0wq0
This gives an isomorphism v :q*G ! (q0)*G0. Using this, we easily construct t*
*he
required map.
Now consider the case E0 = MP , so that S0 = FGL . Then Iso(G; G0)(R) is
the set of triples (s; F; x), where s 2 S(R) and F is a formal group law over R
and x: Gs !spec(R) x bA1is an isomorphism over spec(R) such that x(g + h) =
F (x(g); x(h)). In other words, x is a coordinate on Gs and F is the unique for*
*mal
group law such that x(g + h) = F (x(g); x(h)). Thus, we find that Iso(G; G0) =
Coord(G) = MP E= spec(ss0MP ) (see Section 8.6). It follows after a comparison
of definitions that our map SE^E0 ! Iso(G; G0) is an isomorphism.
Now suppose that E00is Landweber exact over E0, in the sense that there is
a ring map E0 !E00which induces an isomorphism E000E00E00Z = E000Z for all
spectra Z. We then find that G00= G0xS0S00and that
SE^E00= SE^E0 xS0S00= Iso(G; G0) xS0S00= Iso(G; G00);
as required. ___
88 NEIL P. STRICKLAND
Remark 8.40. If there are enough K"unneth isomorphisms, then E01 E0will be a
Hopf ring over E0 and thus the *indecomposables Ind(E01 E0) will be an algebra
over E0 using the circle product. The procedure described in [15] will then giv*
*e a
map spec(Ind(E01 E0)) !Hom (G; G0), which is an isomorphism in good cases.
Definition 8.41.Let G and G0 be formal groups over S and S0, respectively. A
fibrewise isomorphism from G to G0is a square of the form
f
G _____wG0
 
 
 
u u
S ______S0wg
such that the induced map G !f*G0is an isomorphism of formal groups over S.
Definition 8.42.We write OFG for the category of ordinary formal groups over
affine schemes and fibrewise isomorphisms, and EPR for the category of even pe
riodic ring spectra. We thus have a functor EPR op ! OFG sending E to GE .
We write LOFG for the subcategory of OFG consisting of Landweber exact formal
groups, and LEPR for the category of those E for which GE is Landweber exact.
Proposition 8.43.If E 2 EPR and E02 LEPR then the natural map
EPR (E0; E) !OFG (GE ; GE0)
is an isomorphism. Moreover, the functor LEPR op !LOFG is an equivalence of
categories.
Proof.Using [13, Proposition 2.12 and Corollary 2.14], we see that there is a c*
*ofi
bration P ! Q ! E0 !P , in which P and Q are retracts of wedges of finite
spectra with only even cells, and the connecting map E0! P is phantom. If W is
an even finite spectrum then we see from the AtiyahHirzebruch spectral sequence
that E1W = 0 and E0W is projective over E0 and [W; E] = Hom (E0W; E0) and
[W; E] = 0. It follows that all these things hold with W replaced by P or Q.
Using the cofibration we see that E1E0= 0, and there is a short exact sequence
E0P ae E0Q i E0E0:
Now consider the diagram
0 _______________[E0;wE]_____________[Q;wE]_____________[P;wE]
  
ffE0 ffQ ffP
  
u u u
0 ___________Homw(E0E0; E0)_____wHom (E0Q; E0) _____Homw(E0P; E0):
The short exact sequence above implies that the bottom row is exact. The top
row is exact because of our cofibration and the fact that [P; E] = 0. We have
seen that ffP and ffQ are isomorphisms, and it follows that ffE0 is an isomorph*
*ism.
Thus, [E0; E] is the set of maps of E0modules from E0E0 to E0. One can check
that the ring maps E0 ! E biject with the maps of E0algebras from E0E0 to
E0 (using [13, Proposition 2.19]). We see from Proposition 8.39 that these maps
biject with sections of SE^E0 = Iso(GE ; GE0) over SE , and these are easily se*
*en
to be the same as fibrewise isomorphisms from GE to GE0. Thus EPR (E0; E) =
FORMAL SCHEMES AND FORMAL GROUPS 89
OFG (GE ; GE0), as claimed. This implies that the functor LEPR op! LOFG is fu*
*ll
and faithful, so we need only check that it is essentially surjective. Suppose *
*that
G is a Landweber exact ordinary formal group over an affine scheme S. Define a
graded ring E* by putting E2k+1= 0 and E2k = !kG=Sfor all k 2 Z, so in particul*
*ar
E0 = OS. A choice of coordinate on G gives a formal group law F over OS = E0
and thus a map S ! FGL or equivalently a map u: MP0 = OFGL ! E0. If G0 =
GMP is the evident formal group over FGL then one sees from the construction
that S xFGL G0 = G. Given this, we see that our map u extends to give a map
MP* !E*. We define a functor from spectra to graded Abelian groups by
E*Z = E* MP* MP*Z = E* MU* MU*Z;
where we have used the map MU ! MP of ring spectra to regard E* as a mod
ule over MU*. One can also check that E0Z = E0 MP0 MP0Z. The classical
Landweber exact functor theorem implies that this is a homology theory, repre
sented by a spectrum E. The refinements given in [13, Section 2.1] show that E
is unique up to canonical isomorphism, and that it admits a canonical commuta
tive ring structure, making it an even periodic ring spectrum. It is easy to ch*
*eck
that E0CP 1 = E0b MP0MP 0CP 1 and thus that GE = S xFGL G0 = G, as __
required. __
References
[1]J. F. Adams. A variant of E. H. Brown's representability theorem. Topology,*
* 10:185198,
1971.
[2]J. F. Adams. Stable Homotopy and Generalised Homology. University of Chicag*
*o Press,
Chicago, 1974.
[3]M. Demazure. Lectures on pdivisible groups. SpringerVerlag, Berlin, 1972.*
* Lecture Notes in
Mathematics, Vol. 302.
[4]M. Demazure and P. Gabriel. Groupes algebriques. Tome I: Geometrie algebriq*
*ue,
generalites, groupes commutatifs. Masson & Cie, Editeur, Paris, 1970. Avec u*
*n appendice
Corps de classes local par Michiel Hazewinkel.
[5]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules and *
*Algebras in
Stable Homotopy Theory, volume 47 of Amer. Math. Soc. Surveys and Monographs*
*. American
Mathematical Society, 1996.
[6]A. Fr"olich. Formal Groups, volume 74 of Lecture Notes in Mathematics. Spri*
*ngerVerlag,
1968.
[7]J. P. C. Greenlees and N. P. Strickland. Varieties and local cohomology for*
* chromatic group
cohomology rings. 40 pp., To appear in Topology, 1996.
[8]A. Grothendieck and J. L. Verdier. Expose I in Theorie des topos et cohomol*
*ogieetale des
schemas, volume 269 of Lecture Notes in Mathematics. SpringerVerlag, 1972. *
*(SGA 4, second
edition).
[9]R. H. Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathem*
*atics.
SpringerVerlag, 1977.
[10]P. Henrici. Applied and computational complex analysis. Vol. 1. Wiley Class*
*ics Library. John
Wiley & Sons Inc., New York, 1988. Power series_integration_conformal mappin*
*g_location
of zeros, Reprint of the 1974 original, A WileyInterscience Publication.
[11]M. J. Hopkins, M. Ando, and N. P. Strickland. Elliptic spectra, the Witten *
*genus, and the
theorem of the cube. 65 pp., Submitted to Inventiones, 1997.
[12]M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel. Generalised group characters *
*and complex
oriented cohomology theories. Preprint (various editions, since the late 198*
*0's).
[13]M. Hovey and N. P. Strickland. Morava Ktheories and localisation. Mem. Ame*
*r. Math. Soc.,
139(666):104, 1999.
[14]P. T. Johnstone. Stone spaces, volume 3 of Cambridge Studies in Advanced Ma*
*thematics.
Cambridge University Press, 1982.
90 NEIL P. STRICKLAND
[15]T. Kashiwabara, N. Strickland, and P. Turner. The Morava Ktheory Hopf ring*
* for BP. In
Algebraic topology: new trends in localization and periodicity (Sant Feliu d*
*e Guixols, 1994),
Progr. Math., 136, pages 209222. Birkh"auser, Basel, 1996.
[16]N. M. Katz and B. Mazur. Arithmetic Moduli of Elliptic Curves, volume 108 o*
*f Annals of
Mathematics Studies. Princeton University Press, 1985.
[17]M. Lazard. Commutative formal groups. SpringerVerlag, Berlin, 1975. Lectur*
*e Notes in
Mathematics, Vol. 443.
[18]H. Matsumura. Commutative Ring Theory, volume 8 of Cambridge Studies in Adv*
*anced
Mathematics. Cambridge University Press, 1986.
[19]D. G. Quillen. On the formal group laws of unoriented and complex cobordism*
*. Bulletin of
the American Mathematical Society, 75:12931298, 1969.
[20]D. G. Quillen. Elementary proofs of some results of cobordism theory using *
*Steenrod opera
tions. Advances in Mathematics, 7:2956, 1971.
[21]D. G. Quillen. The spectrum of an equivariant cohomology ring, I and II. An*
*nals of Mathe
matics, 94:549602, 1971.
[22]D. C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres. Aca*
*demic Press,
1986.
[23]D. C. Ravenel. Nilpotence and Periodicity in Stable Homotopy Theory, volume*
* 128 of Annals
of Mathematics Studies. Princeton University Press, 1992.
[24]D. C. Ravenel, W. S. Wilson, and N. Yagita. BrownPeterson cohomology from *
*Morava K
theory. KTheory, 15(2):147199, 1998.
[25]N. P. Strickland. Functorial philosophy for formal phenomena. 59 pp., Prepr*
*int, 1994.
[26]N. P. Strickland. Finite subgroups of formal groups. Journal of Pure and Ap*
*plied Algebra,
121:161208, 1997.
[27]N. P. Strickland. Morava Etheory of symmetric groups. Topology, 37(4):757*
*779, 1998.
[28]N. P. Strickland. Products on MUmodules. Trans. Amer. Math. Soc., posted o*
*n March 1,
1999, PII:S 00029947(99)024368 (31 pp., to appear in print), 1999.
[29]N. Yagita. The exact functor theorem for BP*=Intheory. Proceedings of the *
*Japan Academy,
52:13, 1976.