FUNCTORIAL PHILOSOPHY FOR FORMAL PHENOMENA
PRELIMINARY DRAFT
Neil P. Strickland
2
Contents
1. Introduction *
* 5
2. Notation *
* 5
3. Morava K-Theory and E-Theory *
* 5
4. Schemes *
* 7
5. Formal Rings and Modules *
* 10
6. Formal Schemes *
*11
7. Schemes in Algebraic Topology *
* 13
8. Points and Sections *
* 14
9. Zariski Spectra and Geometric Points *
* 17
10. Sheaves, Modules and Vector Bundles *
* 18
11. Polarised Schemes *
* 20
12. Faithful Flatness and Descent *
* 20
13. Constant and Etale Schemes *
* 22
14. Formal Groups *
*24
15. More about Morava E-Theory *
*27
16. Differential Geometry of Formal Groups *
* 28
17. Divisors and Bundles *
* 29
18. Classification of Divisors *
* 31
19. Cohomology of Thom Spectra *
*34
20. Norms and Full Sets of Points *
* 34
21. Subgroup Divisors *
* 38
22. Cohomology of Abelian Groups *
* 39
23. Deformations of Formal Groups *
* 40
24. The Action of the Morava Stabiliser *
* 41
25. Quotient Groups as Deformations *
* 43
3
4 CONTENTS
26. Level Structures *
* 44
27. Category Schemes *
*47
28. Cohomology Operations *
*48
29. The Ando Orientation *
* 51
30. Cartier Duality *
* 52
31. Barsotti-Tate Groups *
* 53
32. Nilpotents, Idempotents and Connectivity *
* 53
33. The Weierstrass Preparation Theorem *
* 54
34. Dictionary *
* 56
Bibliography *
*57
1. INTRODUCTION *
* 5
1.Introduction
The purpose of this paper is to introduce the "schematic viewpoint" in algebr*
*aic topology. This seems to
be the most natural framework in which to discuss the algebraic structures whic*
*h arise from complex-oriented
cohomology theories. Many of the parts which are original are joint work with M*
*ike Hopkins and Matthew
Ando.
We give a definition of (formal) schemes which is well adapted to the particu*
*lar technicalities which arise
in the study of Morava K-theory and completed E(n)-theory. We show how to inte*
*rpret the generalised
(co)homology of CP1 , Z x BU, Bpm, projective bundles and Thom spaces of comple*
*x vector bundles, and
various other spaces, using the language of formal group theory.
While we use many ideas from algebraic geometry, our examples are rather diff*
*erent from those usually
considered by the algebraic geometer on the Clapham omnibus. It is thus diffic*
*ult to extract from the
literature an adequate set of foundations for our work, which cover the situati*
*ons which we need to cover
without prolonged discussion of phenomena which we will never encounter. This p*
*aper makes some attempt
to remedy this. The reader should be warned, however, that very few of our defi*
*nitions will be precisely
equivalent to those used in algebraic geometry.
The sections are mostly arranged in a pedagogical order, with some more techn*
*ical pieces of algebra placed
at the end.
2. Notation
Given a category C, we usually write C(X; Y ) for the set of C-morphisms from*
* X to Y .
Given a spectrum E we write E* for E*S0. If E* has period 2 and is concentrat*
*ed in even degrees we
write E for E0 and E(X) for E0(X). Often, E and K will be Morava E-theory and K*
*-theory, as explained
in section 3. The prime p and the height n will be omitted from the notation. I*
*n this context, we write
E_Z = ss0LK (E ^ Z+) (where Z is a space).
All vector bundles over spaces will be complex. (We will also consider vector*
* bundles over schemes, see
section 10
If R is a ring then R{ai| i 2 I} means the free R-module on the indicated ele*
*ments. If the aiare already
elements of an R-module M then use of this notation implicitly claims that they*
* generate a free submodule.
3. Morava K-Theory and E-Theory
In this section, we define the cohomology theories which will provide the cen*
*tral examples for the rest of
this paper.
Let p be a prime, and n > 0 an integer (called the height). We shall say tha*
*t we are working at the
chromatic prime pn, and omit p and n from the notation almost everywhere. In pa*
*rticular, K and E will
refer to spectra closely related to those usually called K(n) and [E(n)_ detail*
*s are_given below.
We will write = Fpnfor_the finite field of order pn. This has the form Fp[__*
*!]=h(__!), for a suitable monic
irreducible factor h of the cyclotomic polynomial pn-1(x) in Fp[x]. This can be*
* lifted uniquely to give a
monic irreducible factor h of pn-1in Zp[x], and we can define the Witt ring W =*
* WFpnas Zp[!]=h(!).
The ring W is a free module of rank n over Zpand is a complete discrete valua*
*tion ring. Any element
x 2 W can be written uniquely as pvy with v 0 and y 2 Wx . In particular, W is*
* local with maximal ideal
(p) and residue field W=p = .
There is a unique map o : -! W (the Teichm"uller map) satisfying o(ab) = o(a)*
*o(b) and o(a) = a
(mod p). Indeed, if "a2 W is any lift of a, then the sequence "apnkconverges p*
*-adically to o(a) as m -!1.
We also write ^afor o(a). P
P Any element a 2 W can be writtenPuniquely as a = k 0o(ak)pk, for suitable a*
*k 2 . If also b =
k 0o(bk)pk and c = a + b = k0o(ck)pk then the ck are essentially given in t*
*erms of the ak and bk by
the Witt addition formula. However, this fact is rarely useful in the present c*
*ontext.
6 CONTENTS
There is a unique automorphism OE of W (the Frobenius automorphism) satisfyin*
*g OE(^a) = ^apfor all a 2 .
This also has OEn = 1. The fixed ring of Cn = acting on W is jus*
*t Zp. In fact, W ' Zp[Cn] as
Zp[Cn]-modules (but not as rings).
We define an ungraded ring
E = E0 = W[[u1; : :;:un-1]]
and a graded ring
E* = E[u; u-1] |u| = -2
We also take u0= p and un = 1 and uk = 0 for k > n.
The coefficient ring of the Brown-Peterson spectrum is
BP* = Z(p)[vk | k > 0] |vk| = -2(pk- 1)
We define a map BP* -! E* sending vk to upk-1uk. Using this, we define a funct*
*or from spectra to
E*-modules by
E*(X) = E*BP*BP*(X)
The BP*-module E* is Landweber exact, so this functor is a homology theory, rep*
*resented by a spectrum
which we shall also call E. We shall refer to this as Morava E-theory. The ring*
* E has a pleasant interpretation
in terms of deformations of formal groups, which will be discussed later.
__
Remark 3.1.Given two spectra E and E0and a natural isomorphism f:E*(-) -!E0*(*
*-), there is an
isomorphism f :E -!E0of representing spectra, unique up to addition of a phanto*
*m map. In the present
case, we shall see shortly that E can be written as the homotopy inverse limit *
*of a tower of spectra with finite
homotopy groups. It follows that there are no phantom maps to E, and thus that *
*E is unique up to unique
isomorphism.
This spectrum has been constructed by pure homotopy theory, so only homotopic*
*al methods (such as
obstruction theory) are available to analyse it. Nonetheless, Hopkins and Mille*
*r have shown that can be made
canonically into an E1 ring spectrum.
The ring E is a complete, regular local ring, with maximal ideal mE = (p = u0*
*; u1; : :;:un-1). By iterated
cofibrations, preferably carried out in the derived category of E1 E-modules, o*
*ne can construct an algebra-
spectrum K over E with K* = E*=mE = [u1 ]. We shall refer to this as Morava K-t*
*heory. It is a finite
wedge of suspensions of the spectrum usually called K(n).
More generally, given ff = (ff0; : :;:ffn-1) we define Iff= (uff00; : :u:ffn-*
*1n-1) C E0. We can construct E-
algebra spectra Eff= E=Ial with E0ff= E0=Iff. For a cofinal family of ideals If*
*fthere is a finite ring spectrum
Mffsuch that Eff= E ^ Mff(see [3]). The original spectrum E can be recovered as*
* E = holimE-ff.
*
* ff
If n = 1 then E = KUp, the p-adic completion of complex K-theory. Moreover, K*
* = KU=p.
We make E into a topological ring by declaring {mk | k 0} to be a base of ne*
*ighbourhoods of zero. This
actually makes E into a formal ring (see section 6). We also consider K as a fo*
*rmal ring with the discrete
topology.
Rather than think about E and related rings directly, we shall consider the r*
*epresented functors spf(E).
This is the functor from a suitable class of topological rings to sets defined *
*by
spf(E)(R) = Homcts(E; R)
Details are in section 6.
4. SCHEMES *
* 7
We shall use the following notation:
X = spf(E)
G = spf(E(CP1 ))
X1 = spf(W) = V (u1; : :u:n-1) < X
G1 = G xX X1
X0 = spf() = V (mE) < X1
G0 = spf(K(CP1 )) = G xX X0
4. Schemes
Let Ringsbe the category of rings. Given a ring R we consider the functor
spec(R):Rings -!Sets
spec(R)(S) = Rings(R; S)
An affine scheme is a functor X :Rings-! Setssuch that X ' spec(R) for some R. *
*We shall often say
"scheme" instead of "affine scheme". Non-affine schemes do occur in topology (f*
*or example, in the theory
of the period mapping or of elliptic spectra), but we will have quite enough to*
* do without considering them
here.
Example 4.1.
Gm (S) = Sx = the group of unitsSof
Gm (S) ' Rings(Z[x; x-1]; S) so Gm ' spec(Z[x1 ])
One might say that the scheme Gm is a "more natural" object than the representi*
*ng ring Z[x1 ]. This is true
to a much greater extent of many of the rings which arise in topology.
The group Gm (S) = Sx is usually called the multiplicative group of S, so we *
*simply refer to Gm as "the
multiplicative group". It arises, incidentally, in equivariant topology: Gm = s*
*pec(K0S1).
Example 4.2.A formal group law over a ring S is a formal power series
X
F(x; y) = aklxkyl
k;l0
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(S) = { formal group lawsSover}
To see thatPFGL is a scheme, we consider the ring L0 = Z[akl| k; l 0] and the *
*formal power series
F(x; y) = aklxkyl2 L0[[x;.y]]We then let I be the smallest ideal of L0 such t*
*hat the formal group law
conditions for F are satisfied modulo I. For example, the first condition says *
*that a00- 1 2 I and ak02 I for
k > 0, and the second says that akl- alk2 I. Finally, set L = L0=I. It is easy *
*to see that FGL = spec(L).
The ring L is called the Lazard ring. It is usual in algebraic topology to iden*
*tify L with MU*. We shall take
a slightly different point of view (explained at the end of section 14) which t*
*akes the grading into account.
8 CONTENTS
We also define a pre-scheme to be an arbitary functor X :Rings-!Sets. We shal*
*l usually only use this
language when we intend to prove later that X is a scheme.
Note that specis a functor Ringsop-!Schemes. In fact, by Yoneda's lemma, it i*
*s an equivalence of
categories:
Schemes(spec(S); spec(R)) ' spec(R)(S) ' Rings(R; S)
We write A1 for the affine line:
A1(S) = S A1= spec(Z[x])
We write OX = Schemes(X; A1) so (by Yoneda again) we have OX = R iff X = spec(R*
*). We refer to OX
as the ring of functions on X. We shall often write X for OX and HomX for HomOX*
* .
Example 4.3.There is a map ff:Gm x FGL -!FGL defined by
ff(u; F) = Fu Fu(x; y) = uF(x=u; y=u)
It is best to define this map as above, and work as far as possible with the de*
*scription given, rather than
trying to work out the representing map ff*:OFGL -!OGm OFGL. Sometimes one cann*
*ot avoid calculating
the representing map, so we shall do this case as an example. We think of aijas*
* a natural map FGL(R) -!R,
defined implicitly by X
F(x; y) = aij(F)xiyj
ij
Thus X X
Fu(x; y) = u aij(F)(x=u)i(y=u)j= u1-i-jaij(F)xiyj
This shows that aij(Fu) = u1-i-jaij(F), in other words ff*(aij) = u1-i-jaij.
Example 4.4.A strictly invertible power series over a ring S is a formal powe*
*r series f 2 S[[x]]such that
f(x) = x + O(x2). This implies, of course, that f has a composition-inverse g =*
* f-1, so that f(g(x)) = x =
g(f(x)). We write IPS(S) for the set of such f, which is easily seen to be a sc*
*heme. It is actually a group
scheme, in that IPS(S) is a group (under composition), functorially in S.
The group IPSacts on FGL by
(f; F) 7! Ff Ff(x; y) = f(F(f-1x; f-1y))
A strict isomorphism between formal group laws F and G is a strictly invertib*
*le series f such that
f(F(a; b)) = G(f(a); f(b)). Let SIbe the following scheme:
SI(S) = {(F; f; G) | F; G 2 FGL(S) andf :F -!G is a strict}iso
There is an evident composition map
SIxFGLSI-! SI ((F; f; G); (G; g; H)) 7! (F; gf; H)
Moreover, there is an isomorphism
IPSx FGL-! SI (F; f) 7! (F; f; Ff)
Again, one can write implicit formulae in the representing rings, but this shou*
*ld be avoided where possible.
The category of schemes is quite "geometric". It has an initial object ; = s*
*pec(0) and a final object
1 = spec(Z). It has coproducts and pullbacks:
X t Y = spec(OX x OY)
X xZ Y = spec(OX Z OY)
As functors, we have
(X xZ Y )(R) = X(R) xZ(R)Y (R)
4. SCHEMES *
* 9
but
(X t 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 (X t Y )(R) is (by Yoneda) a map spec(*
*R) -!X t Y . This will be
given by a decomposition spec(R) = spec(S) t spec(T) and maps spec(S) -!X and s*
*pec(T) -!Y .
More general colimits do exist in the category of schemes, but the geometric *
*interpretation is typically
bad. Part of the problem is that we consider only affine schemes. Algebraic geo*
*meters do have an extensive
theory of non-affine schemes, of course, but they seem not to be very relevant *
*in topology. Even if we allowed
non-affine schemes, many problems with colimits would remain. Some of these can*
* be resolved using the
ideas of faithfully flat descent and stack theory, which we will discuss later.
An important class of delicate colimit problems which we will have to conside*
*r involves taking the quotient
of a scheme X by the action of a finite group G. The functor S 7! X(S)=G is unl*
*ikely to be a scheme. The
obvious candidate for X=G is spec(OGX). This gives a map X(S)=G -!(X=G)(S), whi*
*ch is iso when S is an
algebraically closed field, but not in general.
We can also do a number of things with subschemes. A closed subscheme of X is*
* a scheme of the form
V (I) = spec(OX =I) for an ideal I OX . An open subscheme is one of the form D*
*(a) = spec(OX [a-1]) for
some a 2 OX , and a locally closed subscheme has the form D(a) \ V (I) = spec(O*
*X [a-1]=I).
Example 4.5.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 subscheme V ((x - )(x - )) corresponds to the pair of poin*
*ts {; }. If = , this is
to be thought of as the point with multiplicity two, or as an infinitesimal th*
*ickening 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 open subschemes and still have an affine sc*
*heme. Again, it would
be easy enough to consider non-affine schemes, but it rarely seems to be necess*
*ary. Moreover, a closed
subscheme V (a) determines the complementary open subscheme D(a) but not conver*
*sely; D(a) = D(a2) but
V (a) 6= V (a2) in general. *
* p _
We say that a scheme X is reduced iff OX has no nonzero nilpotents, and write*
* Xred= spec(OX = 0),
which is the largest reduced closed subscheme of X. Moreover, if Y X is closed*
* then Yred= Xrediff
X(k) = Y (k) for every field k.
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 "union with multiplicity" _ in par*
*ticular, V (I) + V (I) 6= V (I)
in general. In the previous example, we have
{} [ {} = V ((x - )2)
which is a thickening of {}. Note that V (IJ)red= V (I \ J)red.
We shall say that X is connected iff it cannot be split nontrivially as Y t Z*
*, iff there are no idempotents
in OX other than 0 and 1. There is more information about this sort of question*
* in section 32.
Example 4.6.Let E be Morava E-theory, and G a finite group. Write X = spec(p-*
*1(E0BG)). Warning:
this is not the same as (p-1E)0BG ' (p-1E)0(point). Work of Hopkins, Kuhn and R*
*avenel [8] implies that
X has one component for each conjugacy class of Abelian p-subgroups of G. On th*
*e other hand, spec(K0BG)
is connected (where K is Morava K-theory).
We shall say that a scheme X is integral iff OX is an integral domain, and th*
*at X is irreducible iff Xred
is integral. We also say that X isSNoetherian iff the ring OX is Noetherian. If*
* so, then Xredcan be written
in a unique way as a finite union iYiwith Yian integral closed subscheme. The *
*schemes Yiare called the
irreducible componentsSof Xred; they are precisely the schemes V (pi) for pia m*
*inimal prime ideal of OX . We
can also write X = iXiwith (Xi)red= Yi, but this decomposition is not quite un*
*ique. See [11, section 6]
for this material.
10 CONTENTS
S
Suppose that X is NoetherianSand reduced, say XT= i2 SYias above for some fi*
*nite set S. Suppose
S = S0t 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. This construction occurs in the*
* Greenlees-May theory of
local cohomology and Tate spectra [4, 5].
Example 4.7.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 x-axis, Y is the y-axis and X0is an infinitesimal thickening of X*
*. The schemes X and Y are
integral, and X0is irreducible because X0red= X. The scheme Z is reducible, and*
* its irreducible components
are X and Y .
__
Example 4.8.Let G be a finite group, and X = spec(H 2*(BG; Fp)). Then work of*
* Quillen shows that X
has one irreducible component for each maximal conjugacy class of elementary Ab*
*elian p-subgroups.
Example 4.9.Let E be Morava E-theory (with height n), and suppose that A is a*
* finite Abelian p-
group. Write A* = Hom(A; S1) and EA = E(BA*). We shall show in section 26 that *
*Y = spec(EA) has
one irreducible component YB for each quotient A=B of rank at most n, and deriv*
*e many properties of the
schemes YB.
5. Formal Rings and Modules
Let R be a ring, and M an R-module. A linear topology on M is a topology such*
* that the collection of
open submodules forms a base of neighbourhoods of zero. We shall write N O M to*
* indicate that N is an
open submodule. Note that if N is open and N L M then L is a union of transla*
*tes of N and thus
also open. Similarly, M \ N is open, so N is closed. Note also that the ring op*
*erations are automatically
continuous for a linear topology on R. Any directed family of submodules gives *
*rise to a linear topology in
an obvious way. We say that M is complete with respect to a given linear topolo*
*gy iff
M = lim -M=N
NOM
Suppose that M is a complete linearly-topologised A-module, and that N is a s*
*ubmodule. Then N inherits
a linear topology in an obvious way. The closure of N is given by
__ "
N = N + L
LOM
Moreover, N is closed iff it is complete. If so, the quotient M=N also inherits*
* a complete linear topology.
Definition 5.1.A formal ring is a ring with a given linear topology, with res*
*pect to which it is complete.
Definition 5.2.An ideal of definition for a formal ring R is an open ideal I *
*such that {Ik | k 0} is a
base of neighbourhoods of zero. Note that such a thing may or may not exist, an*
*d that in general a power of
an open ideal need not be open.
Example 5.1.Consider K(BU) = K0BU = K[[ck | k >,0]]where K is Morava K-theory*
*. We give this a
topology by declaring that ker(K(BU) -!K(Z)) be open for any finite complex Z a*
*nd any map Z -!BU. In
particular, the augmentation ideal J is open. Note that every open ideal contai*
*ns ck for k 0. In particular,
J2 is not open. This shows that there is no ideal of definition.
6. FORMAL SCHEMES *
* 11
Definition 5.3.A formal module over a formal ring R is a complete linearly to*
*pologised R-module M
such that the action map R x M -!M is continuous. (Note that this has nothing t*
*o do with the Lubin-Tate
theory of formal A-modules, which are formal groups G with a map A -!End(G)).
If M and N are formal modules, the symbol HomR(M; N) will always refer to con*
*tinuous homomorphisms.
Note that
Hom R(M; N) = liml-im-!HomR(M=Mff; N=Nfi)
fi ff
where Mffand Nfirun over open submodules. We give this module the topology of u*
*niform convergence, which
is defined by the family of submodules {Hom (M; Nfi)}; this makes it into a for*
*mal module. In particular, we
define the topological dual M_ = Hom(M; R).
If S is a set, we write F(S; M) for the formal R-module of functions a:S -!R,*
* topologised as a product of
copies of M. We say that a 2 F(S; M) is nullconvergent iff for all M0O M we hav*
*e as2 M0for almost all
s 2 S. We write F0(S; M) for the set of nullconvergent functions. We give F0(S;*
* M) the topology defined by
the submodules F(S; M0) \ F0(S; M), where M0O M. One can then check that F(S; M*
*)_ is topologically
isomorphic to F0(S; M_). If M is finitely generated then F0(S; M)_ is algebraic*
*ally isomorphic to F(S; M_)
but has a different topology. Moreover, for all M and N there are topological i*
*somorphisms
F(S; M)b RF(T; N) = F(S x T; M bRN)
F0(S; M)b RF0(T; N) = F(S x T; M bRN)
We say that M is pro-free iff it is topologically isomorphic to F(S; R) for s*
*ome S. If so, the functor
N 7! M bRN ' F(S; N)
is exact and preserves infinite products.
Example 5.2.Suppose that Z is a spectrum and E0Z is free over E0 on generator*
*s indexed by a set S.
Then E(Z) ' F(S; E) and E_ Z = ss0LK (E ^ Z) ' F0(S; E).
Remark 5.1.Let R be a Noetherian formal ring. Suppose that there is an ideal*
* of definition I, and
that I m for every maximal ideal m C R. Then every finitely generated module i*
*s complete under the
topology defined by the submodules IkM. Let f :M -!N be a map of finitely gener*
*ated modules. Then f
is continuous, if it is injective it is a closed embedding, and if it is surjec*
*tive it is open (and thus a quotient
map). The key point here is the Artin-Rees lemma, see [11, section 8].
6. Formal Schemes
The definitions in this section are not the usual ones in algebraic geometry,*
* but they appear to be what is
required for our applications.
Definition 6.1.If R is a formal ring, we define a functor
spf(R):Rings -!Sets spf(R)(S) = lim-!Rings(R=I; S)
ICOR
A formal scheme is a functor of the form spf(R) for some formal ring R. We writ*
*e Formalfor the category
of formal schemes.
Note that any ordinary scheme is a formal scheme. We shall say that a formal *
*scheme is discrete iff it is
actually an ordinary scheme. Any formal scheme is a directed colimit of closed *
*inclusions of discrete schemes.
Considered as a functor Rings-!Sets, it preserves finite limits.
Example 6.1.Suppose X is a scheme and Y = V (I) is a a closed subscheme. Let *
*ObXbe the completion
of OX at I, and give it the linear topology defined by the powers of I. Then bX*
*= spfObXis a formal scheme,
called the formal completion of X along Y . It is the colimit of the schemes V *
*(Ik) in the category of formal
schemes.
12 CONTENTS
Example 6.2.Consider the ring R = Z[[x]]with the formal topology defined by p*
*owers of x. Then
spf(R)(S) = bA1(S) = Nil(S) = { nilpotent elementsSof}
Example 6.3.If Z is a reasonable space, with finite subspectra Zff, then E(Z)*
* will be the same as
limE-(Zff), and this will be a formal ring. More details are given in section 7.
Example 6.4.spfZpis a formal scheme.
Example 6.5.Write K for p-adic complex K-theory, and consider
K(CP1 ) = limK-(CPk) = limK-(BZ=pl) = Zp[[x]]
k l
This ring has three different formal topologies, defined by ideals
Jk;l= ([pk](x); pl)
k
Ik = ker(K(CP1 ) -!K(BZ=pk)) = ((1 + x)p - 1)
Km = ker(K(CP1 ) -!K(CPm-1 )) = (xm )
The first of these seems most useful. It is also defined by the ideals (xk; pl).
Given a formal scheme X, we can form the ring
OX = Formal(X; A1)
A point x 2 X(A) gives a map ^x:OX -!A; we write Jx for the kernel. These ideal*
*s form a directed family,
and thus give rise to a linear topology on OX . Moreover, OX is complete with t*
*his topology, so OX is a
formal ring. One can show that
Formal(X; Y ) = FormalRings(OY; OX )
Ospf(R)= R
so that the category of formal schemes is dual to that of formal rings.
Definition 6.2.Let R be a formal ring. An element x 2 R is topologically nil*
*potent iff xn -! 0 as
n -!1, iff it is nilpotent in R=I for every I CO R. We write Nil(R) for the ide*
*al of topologically nilpotent
elements. An ideal J C R is topologically nilpotent iff J Nil(R), and strongly*
* topologically nilpotent iff for
all I CO R we have JN I for N 0. Note that a finitely generated ideal which i*
*s topologically nilpotent
is strongly so.
Note that
Formal(X; bA1) = FormalRings(Z[[x]]; OX ) = Nil(OX )
The following is proved later as proposition 32.1.
Proposition 6.1.Nil(R) is the intersection of all open prime ideals, and is t*
*hus closed.
We can define products and coproducts of formal schemes, with
OXtY = OX x OY
OXxY = OX bOY = limO-Xff OXfi
ff;fi
Let X be a formal scheme. If J C OX is a closed ideal then OX =J is a formal*
* ring and we write
V (J) = spf(OX =J). A formal scheme of this kind will be called a closed formal*
* subscheme of X.
Most constructions with schemes can be carried over to formal schemes, by req*
*uiring rings and modules
to be topologised and maps to be continuous.
7. SCHEMES IN ALGEBRAIC TOPOLOGY *
* 13
We will also consider formal schemes as representable functors on the categor*
*y of formally-topologised
rings:
X(R) = FormalRings(OX ; R) = Formal(spfR; X)
7.Schemes in Algebraic Topology
Let E be a spectrum like Morava E-theory. The precise list of properties we r*
*equire is as follows. E must
be a complex-orientable two-periodic ring spectrum with Eodd= 0. We must be giv*
*en a directed system of
ideals IffC E0 and algebra-spectra Effsuch that E*ff= E*=Iffand E = holimE-ff. *
*It follows that the Iff
ff
define a formal topology on E0. We also require that E0 be Noetherian, and that*
* there be a subsequence of
the Iffwhich is cofinal. This will allow us to apply the usual theory of limand*
* lim1for towers. Much of the
theory would go through with less restrictive assumptions, of course.
Example 7.1.Anything obtained from Morava E-theory by killing some generators*
* qualifies. In particu-
lar, Morava K-theory is an example, as are p-adic or mod p complex K-theory. An*
*y two-periodic even ring
spectrum E with the discrete topology also qualifies. If we take Morava E-theor*
*y at height n, invert um
(with m < n) and complete at (u0; : :u:m-1), we get another example.
We write E(Z) for E0(Z) and E for E0.
Definition 7.1.Let Z be a CW-spectrum, and let Zfirun over the finite subspec*
*tra. Write Jfffi=
ker(E(Z) -!Eff(Zfi)). We say that Z is tolerable iff E(Z) = lim -E(Z)=Jfffiand *
*E1(Z) = 0. If so, we give
ff;fi
E(Z) the formal topology defined by the submodules Jfffi. We say that Z is dece*
*nt if E0Z is a free E-module
and E1Z = 0 _ this implies that Z is tolerable.
A finite spectrum Z is tolerable iff E1Z = 0. Indeed, [kZ; E=Iff] is finite, *
*by induction on the number of
cells. Thus the tower is Mittag-Leffler and [kZ; E] = lim[-kZ; E=Iff] by the Mi*
*lnor exact sequence.
ff
As remarked earlier, if Z is decent then E(Z) is pro-free. It follows that E(*
*Z)b EE(W) is a cohomology
theory of W, and thus that E(Z ^ W) = E(Z)b EE(W).
Definition 7.2.Let Z be a tolerable CW complex. We then write
ZE = spfEZ = spf(E(Z))
This is a formal scheme, covariantly functorial in Z. It is also naturally pola*
*rised (see section 11).
For any finite complex and many infinite complexes, the ring E(Z) will be Noe*
*therian. The Artin-Rees
lemma will then apply, and many questions about formal topologies will simplify*
* greatly (c.f. remark 5.1).
There are various obvious possible modifications. For example, we can think a*
*bout H*Z = H*(Z; F2) as an
ungraded commutative ring, and define ZH to be its spectrum. We can define an a*
*ction of Gm = spec(F2[u1 ])
on ZH by the map
ff*:H *Z -!OGm H*Z
X X
ff*( ak) = akuk (ak 2 HkZ)
k k
This action will then keep track of the grading.
In what follows, we will give (some) details of the two-periodic case, but fe*
*el free to present examples from
analogous cases.
Write X = pointE= spf(E0), so there is a canonical map ZE -! X. We now have a*
* covariant functor
from the homotopy category of (some) spaces to the (geometric) category of pola*
*rised formal schemes over
X. This preserves quite a lot of structure:
(Y t Z)E = YE t ZE
(Y x Z)E = YE xX ZE if Z is decent
14 CONTENTS
Example 7.2.Let K denote p-adic complex K-theory, and let A be a finite Abeli*
*an p-group, with classi-
fying space BA. Then we have
(BA)K = Hom(A*; Gm )
In other words, for any formal Zp-algebra R we have
Hom(K(BA); R) = Hom(A*; Rx)
This is just a paraphrase of the well-known fact that
K(BA) = Zp[A*]
W
Example 7.3.If E = MU[u1 ] = k2Z2kMU then MU* ' E0 (by a 7! u-|a|=2a). Thu*
*s, (by a
fundamental result of Quillen) we have X ' spec(MU*) ' FGL.
Many more examples will be given when we have a little more language with whi*
*ch to talk about them.
Note also that there are a number of rings (and therefore schemes) of topolog*
*ical origin which do not quite
fit into the framework discussed above. For example, if Z is an H-space then we*
* can consider spec(E0Z). It
turns out to be more useful to modify this slightly, and consider
E_Z = ss0holimE-ff^ Z+
ff
In the case of Morava E-theory, this is the same as ss0LK (E ^ Z) (see [9]). *
*Note that as usual we are
suppressing the height n from the notation, so LK = LK(n).
If Z is decent then E_ Z = E0Z^ = E(Z)_. On the other hand, if G is a finite *
*group then the typical
situation seems to be as follows. E0BG is a free module of finite rank over E0 *
*and EoddBG = 0. However,
E0BG is I-torsion and has an odd-dimensional part. On the other hand, ss*LK (E*
* ^ BG+) is just the
continuous dual of E*BG. I am not sure what is the strongest theorem one can pr*
*ove along these lines.
More generally, given a K-local E-algebra spectrum F we can consider spf(F0) *
*as a scheme over X =
spf(E0). By taking F to be the function spectrum F(Z+; E) or the localised smas*
*h product LK (E ^ Z+) we
recover the previous examples.
8.Points and Sections
Let X be a scheme. An R-valued point of X is an element a 2 X(R). Such a thin*
*g corresponds naturally
to a map a0:Y = spec(R) -!X. Explicitly, a0is the natural transformation
Y (S) = Rings(R; S) -!X(S)
defined as follows. If Y (S) 3 b: R -! S then a0(b) = X(b)(a), the image of a *
*2 X(R) under the map
X(b): X(R) -!X(S). From now on we shall not distinguish notationally between a *
*and a0. We shall also
refer to R-valued points as Y -based points, or points defined over Y (where Y *
*= spec(R) still), and write
a 2R X or a 2Y X. Such a thing should be thought of as a family of points of X *
*indexed by Y .
We shall often talk about "a point a 2 X" without specifying R. In that conte*
*xt, the word "scalar" means
an element of the unspecified ring R, in other words a point of A1 over R. Give*
*n a point a of X and an
element f 2 OX we write f(a) for the image of a under the map f :X(R) -!A1(R) =*
* R. If X = spec(S)
then we can identify OX with S and so consider f as an element of S, and a as a*
*n element of Rings(S; R).
With these identifications, we have f(a) = a(f).
In these terms, we have
D(f)(R) = {a 2R X | f(a) is invertible}
V (I)(R) = {a 2R X | f(a) = 0 for allf 2 I}
8. POINTS AND SECTIONS *
* 15
Example 8.1.Let F be a point of FGL, in other words a formal group law over s*
*ome 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
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 for any F over any ring R that v(F) is the image of a211+ 8a12under *
*the map L -!R classifying
F.
Example 8.2.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 A1(R) -!FGL (R), in oth*
*er words a map of schemes
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 single formal group law over Z[a] = OA1. This map A1-! FGL is a*
*ctually a closed embedding,
in other words an isomorphism of A1 with the closed subscheme V (J) of FGL, whe*
*re J = (akl| k + l > 1).
Example 8.3.The point of view described above allows for some slightly schizo*
*phrenic constructions,
such as regarding the two projections ss0; ss1:X x X -!X as two points of X ove*
*r 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 analo*
*gous to thinking of the
identity map of K(Z; n) as a cohomology class u 2 HnK(Z; n); this is of course *
*the universal example of a
space with a given n-dimensional cohomology class.
Often, we will have a given base scheme X and consider various schemes Y = sp*
*ec(R) with given maps
Y -!X. We refer to such a scheme Y as a scheme over X, or just an X-scheme, and*
* write SchemesXfor the
category of X-schemes. Given another X-scheme Z, a section of Z over Y is a map*
* a:Y -!Z such that the
composite Y -a!Z -!X is the given map Y -! X. Such sections biject with OX -alg*
*ebra maps OZ -!OY.
We write a 2Y=X Z to indicate that a is a section, and let (Y; Z) or (R; Z) den*
*ote the set of sections.
Often, we will describe a scheme Y over X by describing (R; Y ) as a functor *
*of OX -algebras R. More
precisely, we have equivalences between the following categories:
(1)Schemes Y over X.
(2)Representable functors Y 0:OX -Alg-!Sets
(3)Representable functors Y 00:SchemesopX-!Sets
(4)OX -algebras R
The equivalences are given by
R = OY
Y 0(S) = OX -Alg(R; S) = (S; Y )
Y 00(Z) = SchemesX(Z; Y ) = (Z; Y )
Y (S) = Rings(R; S) = {(x; y) | x 2 X(S); y 2 Y 0(S)}
In the last equation, S is regarded as an OX -algebra via x*:OX -!S.
16 CONTENTS
Example 8.4.Suppose that Y and Z are schemes over X, and that OY is a finitel*
*y generated free module
over OX . We can define a functor MapX(Y; Z) from schemes over X to sets by
(W; MapX(Y; Z)) = SchemesW(W xX Y; W xX Z)
This is in fact representable. To see this, observe that an element of (W; MapX*
*(Y; Z)) is just an OW -algebra
map
OW X OZ -!OW X OY
or equivalently, just an OX -algebra map
OZ -!OW X OY
Write O_Y= HomX (OY; OX ) and A = SymX[O_YX OZ]. Then
OX -Alg(A; OW ) = OX -Mod(O_YX OZ; OW ) = OX -Mod(OZ; OW X OY)
A suitable quotientPB of A will pickPout the algebra maps. To be more explicit,*
* let {ei} be a basis for OY
over OX , with 1 = ibieiand eiej= kcijkek. Let {ffli} be the dual basis for*
* O_Y. Then B is A mod the
relations X
fflk ab = (ffli a)(fflj b)
i;j
ffli 1 = di
More abstractly, B is the largest quotient of A such that the following diagram*
*s commute:
Z1
OZ OZ O_Y _____OZw|O_Y
_ | ||
1Y | | _
|u | O_ ______wOXj
OZ OZ O_Y O_Y || Y |
| | |
twi|st | 1j | |j
| | | |
|u | ||u |
OZ O_Y OZ O_Y || _ _____ |u
| OY OZ wB
| |
| |
|u |u
B B ___________wBB
We conclude that MapX(Y; Z) = spec(B). We also write Schemes_X(Y; Z) for this s*
*cheme.
Example 8.5.Now suppose that G and H are group schemes over X, and that OG is*
* a finitely generated
free module over OX . We can then define a closed subscheme Groups_X(G; H) Sch*
*emes_X(G; H) such that
(W; Groups_X(G; H)) = GroupsW(W xX G; W xX H)
In particular, take X = spec(Z), and
G = p = specZ[x]=(xp- 1) H(R) = {r 2 Rx | rp = 1}
H = Z=p_= spec(F(Z=p; Z))
This is the constant group scheme corresponding to Z=p. The representing ring *
*is the ring of functions
Z=p -!Z under pointwise multiplication, made into a Hopf algebra in the usual w*
*ay. We write ek for the
function having value 1 at k 2 Z=p and 0 elsewhere, so the ek form a basis for *
*F(Z=p; Z).
K = Groups_X(G; H) = Groups_Z(p; Z=p_)
9. ZARISKI SPECTRA AND GEOMETRIC POINTS *
* 17
A point of K over a ring A is just a map of Hopf algebras
F(Z=p; A) -!A[x]=(xp- 1)
There is a trivial point of K over Z, defined by ek 7! ffik;0. Moreover, if ! i*
*s a primitive p'th root of unity
then there is a point of K over the ring B = Z[1=p; !] defined by
p-1X
ek 7! ( !klxl)=p
l=0
Together, these points give a map
spec(Z x B) -!K
This can be shown to be an isomorphism.
We conclude this section with some remarks about open mappings. We have to ma*
*ke 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 F X is closed. We can then define G = {y 2 Y | f-*
*1y F} = f(Fc)c.
Clearly f is open iff (F closed implies G closed). Now suppose that f :X -! Y *
*is a map of schemes,
and that F X is a closed subscheme. For any point y :T -! Y of Y , so we have*
* a closed inclusion
Fy = T xY F -!Xy = T xY X. We can thus define a sub-pre-scheme G of Y by
G(T) = {y 2 Y (T) | Fy = Xy}
Definition 8.1.A map f :X -!Y is open iff for every closed subscheme F X the*
* prescheme G defined
above is a closed subscheme of Y .
Proposition 8.1.Suppose that f makes OX into a free module over OY. Then f is*
* open.
Proof. Write A = OX and B = OY, and choosePa basis A = B{eff}. Suppose that F*
* = V (I) is a closed
subscheme of X with I = (gfi) and gff= figfiffeff. Consider a C-valued point *
*y*:B -!C of Y . This will
lie in G(C) iff C B AP= C B (A=I), iff the image of I in C B A = C{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
G = V (J), where J = (gfiff). __|_|
9.Zariski Spectra and Geometric Points
If A is a ring, we define the associated Zariski space to be
zar(A) = { prime idealsp < A}
If X is a scheme, we write Xzar= zar(OX ). Note that
V (I)zar= zar(OX =I) = {p 2 Xzar| I p}
D(a)zar= zar(OX [a-1]) = {p 2 Xzar| a 62 p}
(X t Y )zar= Xzart 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 th*
*en induces a continuous
map Xzar-!Yzar.
18 CONTENTS
Suppose that R is an integral domain, and that x 2R X. 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 arbitary
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 Q-algebra, or that some prime p is nilpotent in OX . Let k be an algebr*
*aically closed field of the
appropriate characteristic, with transcendence degree at least the cardinality *
*of R. Then it is easy to see that
X(k) -!Xzaris epi.
A useful feature of the Zariski space is that it behaves quite well under col*
*imits. The following proposition
is an example of this.
Proposition 9.1.Suppose that a finite group G acts on a scheme X. Then (X=G)z*
*ar= Xzar=G, and
(X=G)(k) = X(k)=G when k is an algebraically closed field.
A number of interesting things can be detected by looking at Zariski spaces. *
*For example, Xzarsplits as
a disjoint union iff X does _ see corollary 32.5.
Example 9.1.In this example, all rings are Q-algebras. Let X(R) be the set o*
*f n x n matrices M
over R with M2 = M; this is a closed subscheme of An2Q. Let X(m) denote the clo*
*sed subscheme where
trace(M) = m. For any field K Q, elementary linear algebra gives
an
X(K) = X(m)(K)
m=0
It follows by corollary 32.5 that
an
X = "X(m) X"(m)zar= X(m)zar
m=0
We also use the space Xzarto define the Krull dimension of X.
Definition 9.1.If there is a chain p0< : :<:pn in Xzar, but no longer chain, *
*then we say that dim(X) =
n. If there are arbitarily long chains then dim(X) = 1.
Example 9.2.dim(Zp) = 1 _ the unique maximal chain is (0) < (p).
Example 9.3.If E = W[[u1; : :;:un-1]](as in Morava E-theory) and X = spf(E) t*
*hen dim(X) = n.
Example 9.4.dim(FGL ) = 1.
The appropriate generalisation to formal rings is not entirely clear. The pro*
*ofs of the results in section 26
use Zariski spaces, but only for formal schemes with OY a complete Noetherian l*
*ocal ring. In this context
all prime ideals are closed, and only the maximal ideal is open. As yet I know *
*no applications for a more
general theory.
We ought really to say something here about rigid analytic spaces (as used in*
* [6]), but I'm not sure what.
10. Sheaves, Modules and Vector Bundles
A sheaf over a formal scheme X will simply mean a formal module over OX , in *
*other words a complete,
linearly topologised R-module such that the action R x M -!M is continuous.
A vector bundle or locally free sheaf will mean a finitely generated projecti*
*ve OX -module, with the obvious
topology. If M and N are vector bundles then linear maps M -! N are automatica*
*lly continuous and
M bRN = M R N. The dual module M_ is also a vector bundle and M__ = M. If the e*
*valuation map
M X M_ -!OX is iso, we say that M is a line bundle or invertible sheaf. If L an*
*d M are line bundles, we
often write LM for L X M and L-1 for L_.
The most common situation coming from algebraic topology is that vector bundl*
*es are actually free modules
and line bundles are free of rank one.
10. SHEAVES, MODULES AND VECTOR BUNDLES *
* 19
Example 10.1.Let Z be a space, and V a vector bundle over Z with Thom space Z*
*V . Then L(V ) =
E"(ZV ) is a line bundle over ZE.
Given a vector bundle M over X, we define formal schemes A(M) and bA(M) over *
*X by
(R; A(M)) = M bXR
(R; bA(M)) = M bXNil(R)
To see that these are indeed representable, write
M
SymX[M_] = SymkOX[M_]
k
Sym0X[M_] = lim -SymOX=I[M_]=I
ICOOX
Y
Sym00X[M_] = SymkOX[M_]
k
Note that Sym00is the completion of Sym or Sym0at J. Suppose that a_2 Sym00. Th*
*en a_2 Sym iff ak = 0
for k 0. Moreover, a_2 Sym0iff ak -!0 as k -! 1. More precisely, given I CO OX*
* we require that
al2 I SymlX[M_] for l 0. In particular, if OX is discrete then Sym0= Sym.
It is now not hard to check that
A(M) = spf(Sym 0X[M_])
bA(M) = spf(Sym 00X[M_])
More generally, if M is pro-free then the functor
(R; A(M)) = M bXR
is again a scheme, represented by Sym0X[M_]. Here we have to build the symmetr*
*ic algebra using the
completed tensor product, of course.
Example 10.2.spf(E_BU) = A(E"(CP1 )).
If M is a sheaf over a scheme X and x 2 X(R) is an R-valued point of X then w*
*e write Mx = M OX R.
Here R is considered as an OX -module via the map x:OX -! R. Thus Mx is an R-mo*
*dule, which should
be thought of as the fibre of M at x.
Suppose that P is a vector bundle finitely generated projective module over A*
* = OX . We shall say that
P has constant rank m if Px ' km for any field-valued point x:A -!k. The follow*
*ing proposition is partial
justification for the name "vector bundle" (see also example 12.8) .
`n
Proposition 10.1.We can canonically write X = m=0 Xm such that P has constan*
*t rank m on Xm ,
and Xm is an open and closed subscheme.
Proof. We have P = E:An for some matrix E 2 Mn(A) with E2 = E. For sets S; T *
* {1; : :n:} with
|S| = |T| = m we write EST for the m x m minor of E indexed by S x T and E0STfo*
*r the complementary
(n - m) x (n - m) minor of 1 - E. We also write aST = det(EST) det(E0ST). We le*
*t Im denote the ideal
generated by the elements aST for which |S| = |T| 6= m, and write Xm = V (Im ).*
* Suppose that f :A -!k
where k is a field. Elementary linear algebra applied to the`matrix f(E) assure*
*s us that f(aST) 6= 0 for some
S; T with |S| = |T| = rank(f(E)). It follows that X(k) = m Xm (k). We deduce u*
*sing corollary 32.5 that
there are ideals I0mwith the same radical as Im , such that the corresponding s*
*chemes X0mpartition X. It is
easy to see that P has constant rank m on X0m. (see also example 9.1. __|_|
20 CONTENTS
11.Polarised Schemes
Definition 11.1.A polarised scheme is a scheme X equipped with a free line bu*
*ndle L, in other words
a free module L of rank one over OX . A morphism (X; L) -!(Y; M) is a map f :X *
*-!Y together with an
isomorphism L ' f*M. Equivalently, a morphism is a pullback square
"f
L _____Mw
| |
| |
| |
|u |u
X _____Ywf_w
If X is a polarised scheme, then there is a noncanonical isomorphism u:L -!OX*
* (so u 2 L-1).
The category of polarised schemes is equivalent to that of graded rings R* su*
*ch that Rodd= 0 and there
exists a unit u 2 R-2. The equivalence is just R2k= Lk , where the tensor produ*
*ct is taken over R0 = OX
and L(-k) = HomR0(Lk ; R0).
In particular, if E is a two-periodic ring spectrum then X = spf(ss0E) has a *
*natural polarisation. If E
is p-adic K-theory then L-1 = Zpu, and the action of the Adams operations is k*
*u = k-1u (for k 2 Zpx).
This shows that although L is a trivial line bundle, it is not equivariantly tr*
*ivial for an important group of
automorphisms.
12.Faithful Flatness and Descent
Let f :A -!B be a map of rings. We say as usual that f is flat iff B is flat *
*as an A-module via f. We
say that f (or B) is faithfully flat iff it is flat and also satisfies the foll*
*owing equivalent conditions:
(1)B A M = 0 implies M = 0.
(2)If C* is a complex of A-modules and B A C* is acyclic then C* is acyclic.
(3)The map of Zariski spaces zar(B) -!zar(A) is surjective.
Let f :X -!Y be a map of affine schemes. We say that f is (faithfully) flat i*
*ff f*:OY -!OX is.
Example 12.1.An open`inclusion D(a) -!X (where a 2 OX ) is always flat. If a1*
*; : :;:am 2 OX generate
the unit ideal then k D(ak) -!X is faithfully flat.
Example 12.2.If D is a divisor on G over Y (see section 17) then D -!Y is fai*
*thfully flat.
Example 12.3.Looking forward to section 14, suppose that G and H are formal g*
*roups over X. Suppose
that OX is a complete local ring, and write X0 = spf(OX =m). Suppose that q :G *
*-!H is not zero on X0.
We shall see later that q is faithfully flat.
Example 12.4.Let G be a finite group, X a G-space and V a complex vector bund*
*le over X. Suppose
that Flag(V )G -!XG is surjective. Then (Flag(V )G)E -!(XG )E is faithfully fla*
*t [8].
Proposition 12.1.The composite of two faithfully flat maps is faithfully flat*
*. If X -!Y is faithfully flat
and Z -!Y is arbitary then X xY Z -!Z is faithfully flat. __|_|
Proposition 12.2.If X -!Y is faithfully flat, then the diagram
X xY X -!-!X -!Y
is a coequaliser in the category of affine schemes, and this remains true after*
* pulling back along an arbitary
map Z -!Y .
Proof. Write A = OY and B = OX , so we need to show that the following diagra*
*m is an equaliser:
A -!B -!-!B A B
ae
a -!f(a) b -! b1 1 b
12. FAITHFUL FLATNESS AND DESCENT *
* 21
As B is faithfully flat over A, it is enough to check that the diagram is an eq*
*ualiser after tensoring by B over
A. Indeed, this makes it a split equaliser:
B = B A A - B A B - B A B A B
bb0- b b0 bb0 b00- b b0 b00
For the last part, we need only recall that Z xY X -!Z is again faithfully flat*
*. __|_|
Now suppose that f :X -!Y is faithfully flat, and that M is a sheaf over X. W*
*e will need to know when
M descends to a sheaf over Y , in other words when there is a sheaf N on Y such*
* that M ' f*N. There is an
entirely parallel theory for schemes U over X; one asks whether they have the f*
*orm V xY X for some scheme
V over Y .
Definition 12.1.Descent data for a sheaf M over X consists of a collection of*
* maps a;b:Ma -!Mbfor
any pair of points a; b of X with f(a) = f(b). These maps are required to be na*
*tural in (a; b), and to satisfy
the cocycle conditions a;a= 1 and a;c= b;cO a;b.
Remark 12.1.The universal example of such a pair a; b consists of the two pro*
*jections ss0; ss1:X xY X -!
X. It is thus enough to specify a map : ss*0M -! ss*1M. Similarly, the cocycle *
*conditions can be checked
over X and X xY X xY X respectively.
Remark 12.2.Note also that the cocycle conditions imply a;bO b;a= 1, so a;bis*
* iso.
See section 27 for another interpretation of this.
Definition 12.2.Descent data as above are effective iff there is a sheaf N ov*
*er Y and an isomorphism
OE:M ' f*N such that
-1
a;b= (Ma OE-!Nf(a)= Nf(b)OE--!Mb)
Proposition 12.3 (Faithfully Flat Descent).If f is faithfully flat, then desc*
*ent data are always
effective. Moreover, the functor f* gives an equivalence between sheaves over Y*
* and sheaves over X with
given descent data. The inverse sends M to the equaliser N of the following dia*
*gram:
N -!M -!-!OY X M
ae *
m -! sss1ms*
0m
In other words, the sections of N biject with sections s of M such that s(b) = *
*a;bs(a) whenever f(a) = f(b).
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 fidelement plat et quasi-compact;*
* the compactness condition is
automatic for affine schemes). Suppose that a certain statement S is true whene*
*ver it holds fpqc-locally. We
then say that S is an fpqc-local statement.
Remark 12.3.Let X beSa topological space. We say that a statement`S holds loc*
*ally on X iff there is
an open covering X = iUisuch 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 coveri*
*ng map". We would then say
that S holds locally iff it holds after pulling back along a disjoint covering *
*map. One can get many analogous
concepts varying the class of maps in question. For example, we could use cover*
*ing 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.
22 CONTENTS
Example 12.5.Suppose that N is a sheaf on Y which vanishes fpqc-locally. This*
* means that there is a
faithfully flat map f :X -!Y such that f*N = OX Y N = 0. By the very definition*
* of faithful flatness,
this implies that N = 0. In other words, the vanishing of N is an fpqc-local co*
*ndition. Similarly, suppose
that n 2 N vanishes fpqc-locally, so that f*n = 0 2 M = f*N. By proposition 12.*
*3, we know that there is
an equaliser diagram
N -!M -!-!ss*1M
In particular, f*:N -!M is mono and thus n = 0. Thus, the vanishing of n is als*
*o an fpqc-local condition.
Example 12.6.Flatness is itself an fpqc-local property. Indeed, suppose that*
* X -! Y is such that
X xY Z -!Z is flat for some faithfully flat map Z -!Y . One can then show easil*
*y from the definitions that
X -!Y is flat.
*
* `
Example 12.7.Any open inclusion D(a) -!X is clearly flat, as is any finite co*
*product i2ID(ai) -!X
of such. Using the third criterion forSfaithful flatness, we see that this map *
*is faithfully flat iff no maximal
ideal contains all the ai, iff X(k) = iD(ai)(k) whenever k is a field, iff (ai*
*| i 2 I) = A.
Example 12.8.Suppose that P is a vector bundle over X = spec(A). We keep the *
*notation of proposi-
tion 10.1 (and its proof). Suppose`that P has constant rank m. The claim is tha*
*t P is fpqc-locally free of
rank m. To see this, write Y = |S|=|T|=mD(aST). This is clearly flat over X, a*
*nd P is free over D(aST)
(becauseSAm -jS!An E-!P is iso) and hence over Y . Moreover, the map Y -! X is *
*faithfully flat because
X(k) = S;TD(aST)(k) when k is a field.
Example 12.9.Any monic polynomialPf 2 A[x] can be factored as a product of li*
*near terms, locally in the
flat topology. Indeed, suppose f = m0(-1)m-kam-kxk with a0= 1. It is well kno*
*wn that B = Z[x1;Q: :x:m]
is free of rank m! over C = Bm = Z[oe1; : :o:em ]. A basis is given by the mon*
*omials xff= xffkkfor which
ffk < k. We can map C to A by sending oekQto ak, and then observe that D = B C *
*A is free and thus
faithfully flat over A. Clearly f(x) = k(x - xk) in D[x], as required.
Example 12.10.Let f :G -!H be a homomorphism of group schemes over X. Suppose*
* that the kernel
K is faithfully flat over X, and that f is fpqc-locally surjective. This means *
*that the identity point 1 2X X is
in the image of f after pulling back along a faithfully flat map Y -!X. In othe*
*r words, the homomorphism
G xX Y -! H xX Y admits a non-additive section, so that G xX Y ' H xX K xX Y as*
* schemes over
Y . It follows that G xX Y -! H xX Y is faithfully flat, and finally that f its*
*elf is faithfully flat. This has
applications to various maps of the connective covers BU.
13. Constant and Etale Schemes
Let X be a formal scheme and S a set. We define F(S; OX ) to be the ring of f*
*unctions S -!OX , with the
product topology. We write
S_X= spf(F(S; OX ))
Such a scheme is called a constant scheme over X. It is the S-fold coproduct of*
* copies of X, in the sense that
Schemes(S_X; Y ) = Map(S; Schemes(X; Y ))
SchemesX(S_X; Y ) = Map(S; (X; Y ))
Moreover, we have Y xX S_X= S_Y. In particular, S_X= S_x X, where S_= spf(F(S; *
*Z)).
We shall say that a map f :X -!Y is anetale covering iff it becomes constant *
*after pulling back along a
faithfully flat map Z -!Y .
13. CONSTANT AND ETALE SCHEMES *
* 23
Example 13.1.Let K -! L be a finite separable field extension; then spec(L) -*
*! spec(K) is anetale
covering. Indeed, by the theorem of the primitive element, we can write L = K[x*
*]=f(x) for some irreducible,
separable polynomialQf(x). Let K0be the splitting field of f over K, so that K0*
*is clearly faithfully flat over
K. Moreover, f(x) = i2I(x - ai) 2 K0[x] with ai6= aj for i 6= j. It follows b*
*y the Chinese remainder
theorem that
K0K L = K0[x]=f(x) ' F(I; K0) g(x) 7! (i 7! g(ai))
We can also interpret I as K-Alg(L; K0); in this picture, the map
K0K L -!F(I; K0)
sends a b to (oe 7! aoe(b)).
Example 13.2.More generally, let A be an arbitary ring and f(x) a monic polyn*
*omial of degree m over
A. We would likeQto know when A[x]=f(x) isetale over A. Let B be a faithfullyQf*
*lat extension of A in which
f factors as m-1k=0(x - bk). We define the discriminant of f as = k6=l(bk *
*- bl). This is a symmetric
polynomial in the roots bk, and thus lies in A and is independent of the choice*
* of B and the factorisation.
Suppose that is invertible. As in the previous example, we have a map B[x]=f(x*
*) -!Bm . If we use the
basis {xk | 0 k < m} for B[x]=f(x) and the obvious basis for Bm then the matri*
*x of this map is the
Vandermonde matrix M = (alk)0k;l=(Sn - p; Sa - aOES)
(with S acting as F). Any element a of Dn can be written uniquely as
1X n
a = akSk ak 2 W apk= ak
k=0
or
n-1X
a = blSl bk 2 W
l=0
This is invertible iff a06= 0 iff b02 Wx . The corresponding automorphism aG of*
* G0 is given by
X k X l
aGfl0(t) = fl0(aktp ) = blfl0(tp )
k l
28 CONTENTS
The group = Dx ' Aut(G0) is the (non-strict) Morava stabiliser group.
16. Differential Geometry of Formal Groups
A formal group should be thought of as the analogue of a Lie group in the cat*
*egory of schemes, so we
need to understand some differential geometry. The cotangent bundle is just the*
* module G=X of K"ahler
differentials. To define this, we let I be the kernel of the multiplication map
OG X OG -!OG
We regard this as an OG-module using the left factor. We set
G=X = I=I2
If f 2 OG we define
df = f 1 - 1 f + I2 2 G=X
and check easily that
d(fg) = d(f)g + fd(g)
It follows that if x is a coordinate then G=X is a free module over OG on one g*
*enerator.
This construction can be interpreted as follows. The closed subscheme V (I) *
*GxX G is just the diagonal
G=X, in other words the set of pairs of points (a; b) of G in the same fibre su*
*ch that a = b. The slightly
thicker subscheme V (I2) is the set of pairs (a; b) where a and b are "infinite*
*simally close to first order". A
form ! 2 G=X is just a function on V (I2) which vanishes on . The form df corre*
*sponds to the function
(a; b) 7! f(a) - f(b).
We write TG for the dual bundle _G=X. The Lie algebra of G is the pullback of*
* TG along the zero-section:
LG = 0*TG. One checks that LG_ is just J=J2, where J = {f 2 OG | f(0) = 0} is t*
*he augmentation ideal.
We next need to understand how to define invariant differentials on G, and sh*
*ow that the space !G of such
is isomorphic to LG_ (as in the case of Lie groups).
If we are prepared to work "synthetically" with infinitesimal neighbourhoods,*
* this is easy. A form ! on G
is a function on V (I2) such that !(a; a) = 0. We say that such a form is invar*
*iant iff !(a + c; b + c) = !(a; b)
for any a; b; c where a and b are close to first order. It is clear that such a*
*n ! is freely and uniquely determined
by the function !0 on V (J2) sending c to !(0; c). This is just the image of ! *
*in J=J2 = LG_, or equivalently
the value of ! at the zero section.
To make this more concrete, choose a coordinate x on G and write F for the re*
*sulting formal group law. We
need to find the invariant form ! on G with !0= d0x 2 LG_. As !G = OG{dx}, we m*
*ust have ! = g(x)dx
for some g 2 OX [[x]]with g(0) = 1. As a function on V (I2) we have
!(a; b) = g(x(a))(x(a) - x(b))
Write D2F for the partial derivative of F with respect to the second variable. *
*If c 2 V (J2) we have x(c)2= 0
so
!(a; a + c) = g(x(a)) [x(a) - F(x(a); x(c))]= -g(x(a)) D2F(x(a); 0*
*) x(c)
On the other hand, we are supposed to have
!(a; a + c) = !(0; c) = -x(c)
It follows that
g(x) = D2F(x; 0)-1 ! = dx=D2F(x; 0)
Proposition 16.1.Suppose that OX is torsion-free, and that G and H are formal*
* groups over X. Then
L: Hom(H; G) -!Hom X(LH; LG) is injective.
17. DIVISORS AND BUNDLES *
* 29
Proof. Suppose that f :G -! H has Lf = 0. Let x be a coordinate on G, and ! *
*= g(x)dx the
corresponding invariant differential. Let y be a coordinate on H, so f*x = u(y)*
* for some u 2 OX [[y]]with
u(0) = 0. Then f*! is an invariant differential on H which vanishes (because Lf*
* = 0) at zero, so it vanishes
everywhere. Thus
0 = f*(g(x)dx) = g(u(y))u0(y)dy
Because g(x) is invertible, this means that u0= 0. Because OX is torsion-free, *
*this means that u = 0 and
thus f = 0 __|_|
17. Divisors and Bundles
Let G=X be a formal group. An effective divisor of degree m on G over an X-sc*
*heme Y is a closed formal
subscheme D < G xX Y such that OD is a free module of rank m over OY.
Let x be a coordinate on G, and let D be such a divisor. Note that OD is a qu*
*otient of OGX OY = OY[[x]].
Write m
X
f(t) = fD(t) = cm-ktk c0= 1
k=0
for the (monic) characteristic polynomial of the OY-linear endomorphism of OD g*
*iven by multiplication by
x. By Cayley-Hamilton, we see that f(x) = 0 in OD. It follows that
mX
OD = OY[[x]]=f(x) = OY[[x]]= cm-kxk
k=0
for uniquely determined elements ck 2 OY. We refer to f(x) as the equation of D.
Lemma 17.1.ck is topologically nilpotent for k > 0.
Proof. Let p be an open prime ideal in OX . By lemma 32.1, we need only chec*
*k that ck 2 p, or
equivalently that f(t) becomes tm over OX =p. As x 2 Nil(OG) and OG -! OD is c*
*ontinuous, we know
that multiplication by x is a nilpotent endomorphism of OX =p X OD ' (OX =p)m .*
* Some standard linear
algebra over the field of fractions of OX =p assures us that the characteristic*
* polynomial can only be tm , as
required. __|_|
Example 17.1.If E is complex oriented then CPmE= spf(OG=xm+1) is a divisor on*
* G over X.
Example 17.2.Suppose that E is Morava K-theory or E-theory of height n at a p*
*rime p. We have a
long fibration sequence
Z=pm -!S1 f-!S1 -!BZ=pm -!CP1 -Bf-!CP1
where f(z) = zpm. We pick out from this the circle bundle
S1 -!BZ=pm -!CP1
The Euler class is just [pm ](x) = (pmG)*x. This can be written (by the Weierst*
*rass preparation theorem) as
g(x)u(x)nwheremu(x) is an invertible power series and g(x) is a monic polynomia*
*l of degree pnm, congruent
to xp modulo the maximal ideal of E. In particular, [pm ](x) is not a zero-di*
*visor, so the Gysin sequence
for our circle bundle is just a short exact sequence
m](x)
E(BZ=pm ) - E(CP1 ) x[p------E(CP1 )
It follows that
E*BZ=pm = E*[[x]]=[pm ](x) = E*[[x]]=g(x)
This is the cokernel in the category of Hopf algebras of the map (pmG)*:E(CP1 )*
* -!E(CP1 ). We conclude
that G(m) = (BZ=pm )E is a divisor of degree pnm on G over X, and is also the k*
*ernel of pmG:G -!G. This
map is in fact an isogeny. This means that it is about as surjective as a map o*
*f schemes can be without being
30 CONTENTS
split, so G should be thought of as a divisible sort of group. In fact, (Qp=Zp)*
*n is a good model to have in
mind.
Example 17.3.Let Z be a space, and V a complex vector bundle of dimension m o*
*ver Z. We can define
the associated projective bundle
P(V ) = {(z; L) | z 2 Z ; L Vz a line}
(where "line" means "one-dimensional subspace"). There is a tautological line *
*bundle L(V ) over P(V ),
defined by
L(V )(z;L)= L
This is classified by a map P(V ) -!CP1 . We thus obtain a map P(V ) -!CP1 x Z *
*and hence P(V )E -!
G xX ZE. It is a standard fact that
X
E*P(V ) = E*Z[[x]]= cm-kxk
where the coefficients cm-k are the Chern classes of V (and x is identified wit*
*h the Euler class of L(V )).
This shows that D(V ) = P(V )E is a divisor of degree m on G over ZE.
Example 17.4.Consider the diagonal subscheme < G xX G. This can be considere*
*d as a divisor on
G defined over G, or as the family of one-point divisors [a] parametrised by a *
*2 G. If we write OGxXG =
OX [[x;iy]]n the obvious way, then J = (x - y). Let L -!CP1 be the universal b*
*undle. Then D(L) = .
Example 17.5.Let a1; : :;:am be sections of G over an X-scheme Y , so that x(*
*ak) 2 OY. We can define
ideals
Jk < OGxXY Jk = {f | f(ak) = 0} = (x - x(ak))
and thus a divisor Y Y
D = OGxXY = Jk = OGxXY = (x - x(ak))
k k *
* P
Note that this is actually independent of the choice of coordinate x. We write *
*[a1; : :a:m] or k[ak] for this
divisor. Note that {ak} is not really a single point, but should be thought of *
*as
{ak} = {(ak(y); y) | y 2 Y } G xX Y
Given a divisor D, a list of sections giving rise to it as above is called a fu*
*ll set of points for D. If i is a
vector bundle of dimension m over Z then a splitting of i as a sum of line bund*
*les gives rise to a full set of
sections of the associated divisor. See section 20 for some related but more ge*
*neral definitions.
We would like to be able to define virtual divisors, to be compared with virt*
*ual bundles. We can do this
as follows. Let Y be a formal scheme, and H a formal group over Y . If x and y *
*are two coordinates on H
then x = uy for a unit u 2 OxH. Thus, the ring
MH = lim -x-1(OY=I bYOH )
ICOOY
P
is invariantly defined. Any element f 2 MH has the form k2Zakxk with ak 2 OY *
*and ak -!0 as k -!-1.
Definition 17.1.A Cartier divisor on H over Y is an element of MxH=OxH.
If we need to distinguish between Cartier divisors and divisors as defined pr*
*eviously, we shall refer to the
latter as effective Weil divisors. The effective Weil divisors form an Abelian *
*semigroup with cancellation.
This can be embedded in a group in the usual way. We refer to elements of this *
*group as Weil divisors.
If D is an effective Weil divisor of degree m, then JD = ker(OH -! OD) is a C*
*artier divisor. This
construction gives a homomorphism from the group of all Weil divisors to that o*
*f Cartier divisors. This is
iso if Y is a connected informal scheme, but not in general.
18. CLASSIFICATION OF DIVISORS *
* 31
Suppose that Y is a connected formal scheme, so that OY has no nontrivialPide*
*mpotents. We say that
g(x) 2 MH is holomorphic at infinity if it can be written as g(x) = k0akxk, a*
*nd then we write g(1) for
a0.
Using corollary 33.3, we see that any Cartier divisor has a unique representa*
*tive of the form xng(x), where
n 2 Z, g(x) is holomorphic at infinity and g(1) = 1. We refer to n as the degre*
*e of the divisor.
18.Classification of Divisors
N m
Let G be a formal group over X. Consider the m'th tensor power R = c OXOG and*
* the symmetric subring
S = Rm . If x is a coordinate on G then
R = OX [[x1; : :;:xm ]]
S = OX [[c1; : :;:cm ]]
where the ck are (up to sign) the elementary symmetric functions of the xk (and*
* we take c0= 1).
It is clear that spf(R) is the m-fold product GmX= G xX : :G:. The k'th proj*
*ection ak:GmX-! G is
a section of G overPGmX. As in example 17.5, these sections give a divisor D0 *
*on G overPGmX. In fact,
D0 = spfR[[x]]= cm-kxk, so D0 is obtained by pulling back a divisor D = spfS[*
*[x]]= cm-kxk over
Y = spf(S).
Now let D0be a divisor on G over an arbitary formal X-scheme Y 0. Then
X
OD0= OY 0[[x]]= c0m-kxk
for uniquely determined topologically nilpotent coefficients c0l2 Nil(OY 0). T*
*here is a map OY -! OY 0
sending clto c0l, and thus a corresponding map Y 0-!Y . This is clearly the uni*
*que map Y 0-!Y for which
the pullback of D is D0. This construction gives a natural bijection
{divisors of degree m on G over}Y=0FormalX(Y; Y 0)
It follows that the functor
Div+m(R) = {(f; D) | f :spf(R) -!X ; D a divisors of degree m on G}o*
*ver R
is actually a formal scheme; in fact Div+m= Y .
Topologically, of course, we have
Div+m= BU(m)E
Moreover, BU(m) classifies bundles, Div+mclassifies divisors, we know how to co*
*nstruct divisors from bundles,
and everything is compatible in the evident sense.
It is tempting to interpret the above construction as saying that Div+m= GmX=*
*m . This is true in the sense
that Div+mis the categorical quotient of GmXby the action of m in the category *
*of formal schemes, and this
is useful for constructing maps out of Div+m(e.g. proposition 18.1, or the defi*
*nition of convolution below).
However, the functor Div+mis rather poorly related to the functor R 7! G(R)mX(R*
*)=m .
We can make {Div+m}m0 into a graded semiring in the category of schemes, as *
*follows. Given a divisor
D we write JD for the ideal such that D = spf(OGxXY =JD). Given a coordinate x,*
* we know that JD = (f)
for some monic polynomial f, so JD is actually a free module of rank one over O*
*GxXY . We define addition
of divisors by JD+D0 = JDJD0. This defines a map
Div+mxX Div+n-!Div+m+n
Suppose now that we have full sets of sections {a1; : :a:m} and {b1; : :b:n} *
*for D and D0. It is easy to see
that D + D0= {a1; : :a:m; b1; : :b:n}. In this context, we can also define the *
*convolution
D * D0= {ak+ bl| 1 k m; 1 l n}
32 CONTENTS
Here ak + blrefers to the addition in the group G, of course. Using the descrip*
*tion Div+m= GmX=m and
working with the universal case, one can construct a unique map Div+mxX Div+n-!*
*Div+mngiving rise to this
convolution.
This semiring structure also arises from the maps BU(m)xBU(n) -!BU(m+n) and B*
*U(m)xBU(n) -!
BU(mn) classifying direct sum and tensor product of vector bundles.
Arguments similar to the above show that
BUE = Div0= { Cartier divisors of degree}0
(Z x BU)E = Div= { Cartier divisors}
Note that the natural inclusion is
BU(n) -!{n} x BU -!Z x BU
Another interpretation of BUE is the following. Further explanation of the st*
*atement is given in the proof.
Proposition 18.1.Div0= BUE the free group scheme over X on the underlying poi*
*nted X-scheme of
G = CP1E. Similarly, Divis the free ring scheme over X generated by the group s*
*cheme G. Moreover, the
above remains true after pullback along an arbitary map Y -!X.
Proof. First, let us make the claim more concrete. There is a map j :G -!Div0*
*sending a point a of G to
the divisor [a] - [0]. This corresponds to the usual map CP1 -! {0} x BU, class*
*ifying the reduced canonical
bundle L - 1. Let H be a commutative group in the category of formal schemes ov*
*er X (not necessarily of
dimension one), and let f :G -!H be a map of formal schemes with f(0) = 0. The *
*claim is that there is a
unique factorisation f = g O j with g :Div0-!H a group map. To prove this, obse*
*rve that the map
m sum
Gm -f-!Hm ---! H
factors through a map
gm :Gm =m = Div+m-!H :
There are maps
jm :Div+m-!Div0 D 7! D - m[0]
corresponding to the usual maps BU(m) -!BU. It is easy to see that E(BU) = limE*
*-(BU(m)) as formal
rings, so Div0= lim-!Div+mas formal schemes. The maps gm thus fit together to g*
*ive a map g :Div0-!H as
required. By merely changing the notation, we see that the same holds after bas*
*e change to Y . For the second
claim, we note that Divis a ring scheme over X under addition and convolution o*
*f divisors, corresponding to
direct sum and tensor product of bundles. The unit element is the one-point div*
*isor [0]. The map j0:G -!Div
sending a to [a] is a homomorphism from G to Div(considered as a semigroup unde*
*r convolution). Suppose
that R is another ring scheme over X, and that f0:G -! R is a homomorphism in t*
*he same sense. The
claim is that f0= g0O j0for a unique map g0:Div -!R of ring schemes. To see thi*
*s, define f :G -!R by
f(a) = f0(a) - 1 and use the first part to get a map g :Div0-!R of (additive) g*
*roup schemes. The map
Div0-! Divmsending D to D + m[0] is iso, so we can define g0:Div -!R by g0(D + *
*m[0]) = g(D) + m.
One can check that this works. __|_|
Remark 18.1.There is a certain confusion among the people about the above res*
*ult. It is widely held
that "E*BU is the Witt Hopf algebra" and that "the Witt Hopf algebra classifies*
* curves". Both of these
things are true up to unnatural isomorphism. If we choose a basic curve fl :bA1*
*' G, then the above gives an
isomorphism
GroupsX(BUE; H) = Map0X(G; H) ' Map0X(bA1; X) = Curves(H)
However, because of the arbitary choice of fl we cannot expect this identificat*
*ion to commute with most
interesting operations. Moreover, it is telling us about curves on any group H,*
* and not especially about
curves on G.
18. CLASSIFICATION OF DIVISORS *
* 33
Example 18.1.We could, nonetheless, take H = G, and ask for a map of groups D*
*iv0-!G extending
the inclusionPj0 : a 7! [a]P- [0]. ThePresulting map is just (B det)E :BUE -! B*
*S1E. It sends a degree zero
divisor ini[ai] (with ini= 0) to iniai.
There is an evident partial order on divisors, defined by D0 D iff D = D0+ D0*
*0for some (necessarily
unique) D00, iff fD0 divides fD. This order is particularly useful in the light*
* of the following proposition.
Proposition 18.2.Suppose that D and D0are divisors on G defined over an X-sch*
*eme Y . There is then
a closed subscheme Z Y which is universal for the condition D0 D. In other wor*
*ds, a map w: W -!X
has w*D0 w*D iff w factors through Z.
Proof. Suppose D has degree m. We can then write
m-1X
fD0(x) = akxk (mod fD)
k=0
for uniquely determined coefficients ak. It follows that
m-1X
fw*D0(x) = w*akxk (mod fw*D)
k=0
Thus, we can take I = (ak | 0 k < m) and Z = V (I) = spf(OY=I). __|_|
See section 26 for an interesting application.
The scheme of divisors is functorial in two different ways. First, suppose th*
*at q :G -!H is an map of
formal groups over X. Using the description Div+m(G) = GmX=m , we see immediate*
*ly that q induces a map
q*: Div+m(G) -!Div+m(H). Moreover, this gives a map of ring schemes.
Now suppose that q is an isogeny of degree d, and that D is a divisor on H of*
* degree m. We can then
form the pullback
q*D _____Dww
y| y|
| |
| |
|u |u
G ______Hwwq
We see that q*D -!D is free of degree d, and thus that q*D is a divisor of deg*
*ree md on H. This gives a
map q*: Div(H) -!Div(G), which is q*-linear in the sense that
q*(D * q*D0) = (q*D) * D0
Given a divisor D over Y , the ideal JD is free of rank one over OGxXY . It c*
*an thus be thought of as a
line bundle over G xX Y ; in this guise we shall refer to it as O(D).
Remark 18.2.There are apparently two possible meanings for q*O(D). On the one*
* hand, O(D) is a line
bundle over H, so we can form the pullback L = OG H O(D). On the other hand, q**
* is a map OH -!OG
and O(D) is an ideal in OH so we can form the image M = q*(O(D)), which is an O*
*H -submodule of OG.
Applying flatness of q to the inclusion O(D) -!OH , one can see that L ' OGM. N*
*ow let f be a generator
of O(D). Thinking of f as a map H -!Ab1, we see that D = f*[0]. It follows that*
* q*D = (f O q) * [0] and
thus that f O q = q*f 2 M generates O(q*D). In conclusion,
*
q*O(D) = OG: image(O(D) q-!O(G)) = O(q*D)
34 CONTENTS
19.Cohomology of Thom Spectra
We also have an algebraic analogue of the Thom space of a vector bundle. Let *
*D be a divisor on G over
Y . Pulling back the line bundle O(D) on G xX Y by the zero section 0: Y -! G x*
*X Y , we obtain a line
bundle over Y :
L(D) = 0*O(D) = JD=J{0}+D
It is not hard to show that L(D + D0) = L(D) L(D0). If Y = ZE and D = D(V ) *
*= P(V )E then
L(D) = "E*ZV , the reduced cohomology of the Thom space of V . This follows fro*
*m the cofibration
P(V ) -!P(V C) -!ZV
We also write L(V ) for L(D(V )).
We write for the diagonal divisor on G xX G (see example 17.4). In terms of*
* topology, we have
= D(L), where L is the universal line bundle over CP1 . *
* __
We also write * for convolution of divisors (so that D(V W) = D(V ) * D(W)) *
*and D for the image of
D under the negation map (-1)G :G -!G.
__
Proposition 19.1.L(D * ) = O(D )
Proof. First note that if f(x) 2 OGxXY generates JD then f(x -F y) 2 OGxXY xX*
*G generates JD* .
This implies that
L(D * ) = OY[[x;_y]]f(x_-F_y)O
xf(x -F y) Y[[x; y]]
Next, consider the embedding Y xX G -! G xX Y xX G by (y; b) 7! (0; y; b). Thi*
*s induces a map
OGxXY xXG-! OY xXG, sending x to 0. It is easy to see that this induces L(D * )*
* ' (f([-1]F(y))) = J__D
as required. __|_|
There is a small point of compatibility to be dealt with here._Pulling back t*
*he above statement along
the zero map Y -! Y xX G, we obtain an isomorphism L(D) ' 0*O(D ). On the othe*
*r hand, we_have
L(D) = 0*O(D) by definition. One can check that the resulting isomorphism 0*O(D*
*) ' 0*O(D ) is just
induced by the negation map (-1)G.
The Thom module for the trivial line bundle over a point is just J0=J20= !G. *
*This is of course compatible
the isomorphism "E(1C) = "E(CP1) = E-2 = L-1 = !G.
20.Norms and Full Sets of Points
Suppose that f :X -! Y is a finite free map, in other words it makes OX into *
*a finitely generated free
module (of rank m, say) over OY. We can then define a (nonadditive) norm map Nf*
* = NX=Y: OX -!OY,
by letting Nf(u) be the determinant of multiplication by u, considered as an OY*
*-linear endomorphism of OX .
Proposition 20.1.
(1)NX=Y(uv) = NX=Y(u)NX=Y(v)
(2)NY=ZO NX=Y = NX=Z
(3)NX=Yf*(w) = wn
Moreover suppose we have a pullback diagram
V _____Xwa
| |
g | |f
| |
|uu |uu
W _____Ywb
If f is a finite free map then so is g, and NgO a* = b*O Nf. __|_|
Example 20.1.Suppose C = B[t]=g(t), for some monic polynomial g of degree m. *
*By using the basis
{1; t; : :t:m-1} for C over B, we find that NC=B(t) = (-1)m g(0).
20. NORMS AND FULL SETS OF POINTS *
* 35
More generally, given Z -!Y we define
NZX=Y= NXxYZ=Z:OX Y OZ -!OZ
In particular, we can consider Z = A1Y= specOY[t], take u 2 OY and consider the*
* characteristic polynomial
PX=Y(u; t) = NZX=Y(t - u) 2 OX [t]
Similarly, we define PZX=Y(u; t) for arbitary Z -!Y and u 2 OX Y OZ.
Example 20.2.Let B and C be as in example 20.1. I claim that NB[s]C=B(t - s) *
*= g(s). To see this, take
u = t - s, so that
B[s] B C = C[s] = B[s; t]=g(t) = B[s][u]=g(s + u)
Note that h(u) = g(s + u) is a monic polynomial of degree m in u over B[s], so *
*example 20.1 gives
NB[s]C=B(t - s) = NC[s]=B[s](u) = h(0) = g(s)
Definition 20.1.Suppose Z is a scheme over Y . We say that a list (a1; : :a:n*
*) 2 (Z; X)n is a full set
of points of X (over Z) iff for all g :W -!Z and u 2 OWxYX we have
Y
NWX=Y(u) = u(ai)
i
It is equivalent to require that for all W and u we have
Y
PWX=Y(u; t) = (t - u(ai))
i
In the last two equations, aireally means g*ai2 (W; X).
Lemma 20.2.Suppose C = B[t]=g(t), for some monic polynomial g of degreeQm, an*
*d write X = spec(C)
and Y = spec(B). Then {b1; : :b:m} is a full set of sections iff g(s) = i(s - *
*b*it).
Proof. NecessityQfollows immediately from exampleP20.2. For sufficiency, writ*
*e bifor b*it and suppose
that g(s) = i(t - bi). Consider anQelement f(t) = m-1i=0aiti2 A B C, for som*
*e B-algebra A. We are
required to show that NAC=B(f) = if(bi). By an evident naturality argument, we*
* may assume that
A = B = Z[a0; : :;:am-1; b1; : :;:bm ]
We only need to verify that certain polynomial expressions in the a's and b's a*
*re equal, so we are free replace
B by a larger ring. In particular, we can invert the discriminant of g (c.f. ex*
*ample 13.2). In this context, the
map
B[t]=g(t) -!Bm t 7! (b1; : :b:m)
is iso, and the claim is trivial. __|_|
Example 20.3.Suppose that D is an effective divisor of degree m on a formal g*
*roup G=X, and that
a1; : :;:am are sections of G. By choosing a coordinatePon G and appealing to *
*lemma 20.2, we see that
the aiform a full set of sections for D iff D = i[ai]. This shows that defini*
*tion 20.1 generalises that in
example 17.5.
Next, consider the X-prescheme defined by
(T; FSP(D=X)) = { full sets of points for}D over T
This is actually a scheme, in fact a closed subscheme of the m-fold fibre produ*
*ct DmX. Put xk = x(ak) and
let
dk 2 ODmX= OX [x1; : :;:xm ]=(f(x1); : :f:(xm ))
Q
be the coefficient of tm-k in i(t - x(ai)). If we write J = (d1- c1; : :d:m- c*
*m ), then FSP(D=X) = V (J).
36 CONTENTS
Example 20.4.Let Z be a space and V a vector bundle over Z. Then
FSP(D(V )=ZE) = Flag(V )E
This follows easily from the universal case, in which Z = BU(m) and V is the ca*
*nonical bundle. We take
EU(m) to be the space of m-frames in C1 , so that BU(m) = EU(m)=U(m) is the Gra*
*ssmannian of m-
dimensional subspaces. Let T(m) = (S1)m U(m) be the maximal torus. With this m*
*odel it is clear that
Flag(V ) = EU(m)=T(m) = BT(m) ' (CP1 )m . Thus ZE = Div+m= GmX=m , D(V ) is the*
* universal divisor
over Div+m, and Flag(V )E = GmX= FSP(D(V )).
Example 20.5.Let q :G -! H be an isogeny of formal groups over X, with kernel*
* K. Suppose that
{a1; : :;:an} is a full set ofQsections for K (they need not form a group). Wri*
*te oifor the translation map
b 7! b + aiof G. Then q*Nqf = io*if.
Proof. Let oe :K xX G -!G be the addition map, so we have a pullback square
K xX G _____Gwoe
| |
| |
ss| |q
| |
| |
|uu |uu
G _______wHq
Write "ai= aixX 1G :G -!K xX G. These maps form a full set of sections for ss *
*:K xX G -!G, and
oe O "ai= oi. Thus
Y Y
q*Nqf = Nssoe*f = "ai*oe*f = o*if
i i
__|_|
Example 20.6.This example uses concepts defined in section 26 below. Consider*
* the finite free map
Level(A; G) -! X, where rank(A) n . This has no sections. Suppose that pm A *
*= 0, so there exist
monomorphisms ff:A -!m . Consider the pulled-back map
Level(A; G) xX Level(m ; G) -!Level(m ; G)
For each monomorphism ff as above, we get a section
(OE O ff; OE) - OE
These sections form a full set, although the corresponding map
a
Level(m ; G) -!Level(A; G) xX Level(m ; G)
ff
is only an isomorphism rationally.
Example 20.7.We can generalise the part of example 20.3 as follows. Let Y -! *
*X be an arbitary finite
free map, and define the X-prescheme FSP(Y=X) by
(T; FSP(Y=X)) = { full sets of points for}Y over T
This is again a closed subscheme of YXm.
20. NORMS AND FULL SETS OF POINTS *
* 37
Proof. The condition for an m-tuple a_of sections to be a full set involves a*
*n element f of OY 0xXY. The
universal example of a scheme Y 0over X with m sections and a function f is "Y=*
* YXmxX A(OY). In other
words, for any point x 2 X
Y"x= {(a_; f) | a_2 Yxm; f :Yx -!A1}
Note that O"Yis free over OYXm, so that the projection "Y-!YXmis an open map by*
* proposition 8.1. We have
Q
a function g = NY"Y=X(f) - f(ai) 2 O"Y. The fibre of V (g) over a point a_2 Y*
*xmis
Y
V (g)a_= {f :Yx -!A1| N(f) = f(ai)}
i
Thus
(T; FSP(Y=X)) = {a_2 (T; Yxm) | V (g)a_= "Ya_}
It follows (from the definition 8.1 of an open map) that this is a closed subsc*
*heme as claimed. To make this
more explicit, write
A = OX B = OY
C = BAm = OYXm
D = C A SymA[B_] = O"Y
The inclusion B_ -!SymA [B_] gives by adjunction an element of BSym A[B_] and t*
*hus (under the obvious
inclusion) an element f 2 B A D. Taking the determinant of this as D-linear end*
*omorphism of B A D
gives an element Nf 2 D. On the other hand, we have m inclusions ak:B -!C, each*
* giving rise to a map
B A D = B A C A SymA[B_] -!C A SymA[B_]
Q
The image of f is naturally denoted f(ak). We can now form g = Nf - f(ak) 2 D*
*. Expanding this in
terms of a basis for D over C, we get various coefficients gff2 C. Finally, FSP*
*(Y=X) = spec(C=(gff)). __|_|
Our next task is to extend the norm construction to line bundles and their se*
*ctions. First, we describe the
motivating example. Let X be a scheme, S a finite set, and Y = S xNX. A line bu*
*ndle L over Y just consists
of a family {Ls}s2Sof line bundles over X. We can define NY=XL = sLs, conside*
*red as a bundle over X.
Given a section a = (as)s2Sof L, we define a section NY=Xa of NY=XL in the evid*
*ent way.
Now let f :Y -! X be an arbitary finite free map, of degree m say. Given a li*
*ne bundle L over Y , we
define V V
NfL = NY=XL = HomX ( mXOY; mXL)
Let l be a section of L. By regarding it as an OX -linear map l: OY -!L, we get*
* an section Nf(l) of NfL.
Here are some properties of the functor on line bundles; analogous things are t*
*rue for the map on sections.
(1)NfL is a line bundle over X.
(2)NfOY = OX and Nf(L Y M) = Nf(L) X Nf(M).
(3)Nf(f*N) = Nm . N
(4)Suppose that {a1; : :;:am } is a full set of sections. Then NfL ' ka*kL.
Proposition 20.3.Let q :G -! H be an isogeny of formal groups over X, with ke*
*rnel K 2 Div(G).
Suppose D 2 Div(G). Then NqO(D) = O(q*D), and q*NqO(D) = q*O(q*D) = K * D.
Proof. This is an equality between two ideals in OH . As q is faithfully flat*
*, it suffices to show that
q*NqOD = q*O(q*D). Write K for the kernel of q. By making a faithfully flat bas*
*e change to give K a full
set of sections, and using a pullback diagram as in example 20.5, we see that
q*NqO(D) = O(K * D)
Now suppose D = [a]. Note that K = q*[0], so
q*[q(a)] = q*o*-q(a)[0] = o*-aq*[0] = [a] * K
38 CONTENTS
After giving D a full set of points and extending this in the evident way, we c*
*onclude that q*O(q*D) =
O(K * D) also. __|_|
21.Subgroup Divisors
A subgroup divisor on G over Y is a divisor H on G over Y which is a subgroup*
*-scheme of G xX Y over
Y . It is equivalent to require that OH be a quotient Hopf algebra of OG X OY o*
*ver OY, which is free of
finite rank over OY.
Example 21.1.Let G0 be the formal group arising from Morava K-theory, so OG0 *
*= [[x]]and OG20=
[[x; y]]. This is a Hopf algebra with (x) = F(x; y) = x + y (mod xy). Note eve*
*ry element of OG0 has the
form xm u for a unit u 2 OxG0, so that every divisor defined over has the form*
* spf([[x]]=xm ) for some m.
For this to be a subgroup divisor, we must have
(xm ) = F(x; y)m = 0 (mod xm ; ym )
If we work mod (x; y)m+1 then F(x; y)m = (x + y)m and this will only vanish mod*
* (xm ; ym ) if m is a power
of p. On the other hand, as the coefficients of F lie in Fp , we have
k pk pk pk pk
F(x; y)p = F(x ; y ) = 0 (mod x ; y )
It follows that the subgroup divisors defined over are precisely the divisors *
*spf([[x]]=xpk).
Example 21.2.Let Ga be the (informal) additive group over Fp, so that
Ga = spec(Fp[x]) G2a= spec(Fp[x; y]) (x) = x + y
A degree m divisor defined over an Fp-algebra R has the form spec(R[x]=f(x)) wh*
*ere f(x) is monic of
degree m and f(x + y) = 0 (mod f(x); f(y)), say f(x + y) = g(x; y)f(x) + h(x; y*
*)f(y). We can make this
representation unique by requiring that g have degree less than m in y. Conside*
*ring the coefficient of xkyl
with l m we conclude that h is constant. A similar argument shows that g is co*
*nstant. By working mod
x or y we find that g = h = 1, andPthus f(x + y) = f(x) + f(y). From this we fi*
*nd that it is neccessary and
sufficient for f to have the form rk=0akxpkwith m = pr and ar= 1.
Example 21.3.Let Gm be the (informal) multiplicative group, so that
Gm = spec(Z[u1 ]) G2m= spec(Z[u1 ; v1 ]) (u) = uv
Let H be a closed subgroup of Gm over R such that OH is free of rank r over R. *
*Let f be the characteristic
polynomial of u on OH , so f is a monic polynomial of degree r and f(u) = 0 in *
*OH . Moreover, f(0) is the
determinant of u on OH and thus a unit. It follows that
OH = R[u1 ]=f(u) = R{1; u; : :u:r-1}
The condition for H to be a subgroup scheme is that f(uv) = 0 (mod f(u); f(v)).*
* Using the obvious bases,
it is not hard to conclude that f(u) = ur- 1. In particular, H is already defin*
*ed over Z rather than R.
Example 21.4.If E is Morava E-theory then the divisor G(r) = (BZ=pr)E = spfOG*
*=[pr](x) = ker(prG)
is actually a subgroup divisor _ see example 17.2.
Example 21.5.Consider the Morava K theory formal group G0=X0. It is a finite *
*(but arduous) com-
putation to classify subgroup divisors on G0 over X0-schemes. For example, if n*
* = p = 2, then a subgroup
divisor of degree 4 over a -algebra R has equation
x4+ b5x3+ bx2+ (b3+ b6)x
with b7 = 0. I have Mathematica programs to assist with such calculations, whi*
*ch I would be happy to
distribute or discuss.
22. COHOMOLOGY OF ABELIAN GROUPS *
* 39
Suppose that H is a subgroup divisor on G of degree r; let us say for simplic*
*ity that it is defined over X.
We can define a quotient group G=H as follows. There are two maps
ff; fi :OG -!OGxXH
(fff)(g; h) = f(g + h) (fif)(g; h) = f(g)
We define OG=H to be the equaliser, i.e. OG=H = {f 2 OG | fff = fif}. Finally, *
*we set G=H = spf(OG=H),
and write q :G -!G=H for the projection.
Suppose that x is a coordinate on G. We can define a function y on G=H as fol*
*lows. Observe that fi makes
OGxXH into a free module of rank r over OG. Multiplication by ff(x) is an OG-en*
*domorphism of OGxXH
with this module structure. Let "y2 OG be the determinant of this endomorphism.*
* It can be shown that
ff"y= fi"y, so "ycomes from an element y 2 OG=H. Suitably interpreted (see sect*
*ion 20), one can say that
Y
y(qa) = x(a + b)
b2H
Y
y(c) = x(c)
qa=c
It can also be shown that OG=H = OX [[y]], and that this is a sub-Hopf algebr*
*a of OG. In other words, G=H
is a quotient formal group of G. Moreover, OG is free of rank r over OG=H, so t*
*hat the map G -!G=H is
faithfully flat.
Another useful fact is that the "multiplication by r" map H -r!H is always ju*
*st the zero map. This is
reasonable, because r should be thought of as the order of the finite group H. *
*Moreover, if the base ring OX
is p-local then r is a power of p.
Recall (see section 18) that the divisors of degree m themselves form a schem*
*e Div+m= spfOX [[c1; : :;:cm.]]
It is not hard to see that there are certain power series fi(c_) in the variabl*
*es ck such that a divisor with
parameters ck is a subgroup divisor iff fi(c_) = 0. It follows the the functor *
*on OX -algebras
(Subm; R) = { subgroup schemes of degree m on G}over R
is a scheme, represented by the OX -algebra OX [[c1; : :;:cm=]](fi). Of course*
*, it is the empty scheme unless
m = pk for some k, by one of the remarks above.
We have a good understanding of this scheme in the case of Morava E-theory. T*
*o explain this, consider
the ring Rk = E(Bpk). A partition subgroup means a subgroup of the form ix j w*
*ith i + j = pm
and i; j > 0. We write I for the ideal in Rk generated by transfers from parti*
*tion subgroups. We also
let V denote the standard permutation representation of pk; note that this has *
*a trivial summand after
restriction to a partition subgroup. The Euler class e(V - 1) thus restricts to*
* zero on any such subgroup,
implying that I J = ann(e(V - 1)). A construction involving the E1 structure*
* on E gives a map
Y = spf(EBpm=I) -!Subpk. We shall publish elsewhere a proof of the following re*
*sult (the algebraic input
is provided by [12]).
Theorem 21.1.With notation as above, we have I = J and Y = Subpk. Moreover, *
*OY = R=I is a
finitely generated free module over OX .
22. Cohomology of Abelian Groups
In this section, we suppose we keep the notations of section 3.
Let A be a finite Abelian p-group. We write I = Qp=Zp < S1 and
A* = Hom(A; I) = Hom(A; S1)
Note that A = A**. We define a formal pre-scheme Hom(A; G) over X by
(Y; Hom(A; G)) = Hom(A; (Y; G))
40 CONTENTS
Proposition 22.1.The pre-scheme Hom(A; G) is a scheme. It is represented by t*
*he ring E(BA*), which
is a free module over E of rank |A|n.
Proof. Suppose a 2 A, and let f :E(BA*) -!R be a map of E-algebras. The eleme*
*nt a gives a map
A* -!S1 and*thus a map Ba:BA* -!CP1 . Let OE(a) 2 (R; G) = E-Alg(E(CP1 ); R) be*
* the composite
E(CP1 ) Ba--!E(BA*) f-!R. This construction gives a map
OE:A -!(R; G)
or in other words a section of Hom(A; G) over R. This gives a map of pre-schemes
ffA :spf(E(BA*)) -!Hom (A; G)
It is easy to see directly that
Hom(A A0; G) = Hom(A; G) xX Hom(A0; G)
Recall from example 17.2 that E(BZ=pm ) = E[[x]]=[pm ](x) is free of finite ran*
*k over E. This implies (by a
K"unneth argument) that
spfE(B(A A0)*) = spfE(BA*) xX spfE(BA0*)
It also implies that
(Y; (spfEBZ=pm )) = {a 2 (Y; G) | pa = 0} = (Y; Hom(Z=pm ; G))
It follows that ffA is iso for all A. __|_|
23.Deformations of Formal Groups
Consider the Morava formal group G0=X0. The scheme X0 is about as small as a *
*scheme can get, so we
can think about the problem of extending G0 over a larger base Y . More general*
*ly, we can pull back G0
along a map Y0 -!X0 and then try to extend it to a "thicker" scheme Y Y0. A de*
*formation of G0=X0 is
such an extension; we shall be more precise in a moment.
We shall initially define deformations only over a restricted class of base s*
*chemes Y . We shall then show
that there is a universal example in this restricted setting. We can if we wish*
* extend the definition by requiring
that this example remain universal (as one does in extending K-theory to infint*
*e complexes).
We shall say that a formal scheme Y is local iff OY is a complete local ring,*
* topologised by powers of the
maximal ideal. If so, we write Y0 = spec(OY=m), and call this the special fibre*
* of Y . We shall say that Y is
semilocal iff it is a finite disjoint union of local formal schemes. If so, we *
*define Y0 in the obvious way. If H
is a formal group over a semilocal formal scheme, we define H0= H xY Y0.
Definition 23.1.A deformation of G0=X0consists of a semilocal formal scheme Y*
* together with a formal
group H=Y and a fibrewise isomorphism as follows.
"f0
H0 _____G0w
| |
| |
| |
|u |u
Y0 _____X0wf0
A morphism of deformations is a fibrewise isomorphism of formal groups making *
*the obvious diagram
commute.
Note that the Morava E-theory formal group G=X is by construction a deformati*
*on of G0=X0.
Proposition 23.1.An endomorphism of a deformation H=Y which is the identity o*
*n Y is the identity.
24. THE ACTION OF THE MORAVA STABILISER *
* 41
Proof. We may assume that Y is local (rather than just semilocal). The claim *
*is true (by the definition
of a map of deformations) over Y0 = spf(OY=m). Choose a coordinate y on H compa*
*tible in the evident
sense with the usual coordinate x0 on G0. Thus y(pa) = y(a)q+ v(y(a)) where q =*
* pn and v 2 m[[y]]. Let f
be the endomorphism, so y(f(a)) = y(a) + u(y(a)) for some u 2 m[[y]]. Suppose u*
* 2 mk[[y]]. We find that
y(pf(a))= (y(a) + u(y(a))q+ h(y(a) + u(y(a)))
= y(a) + h(y(a))
= y(pa) (mod mk+1)
On the other hand, y(pf(a)) = y(f(pa)) because f is a homomorphism. Thus y(f(pa*
*)) = y(pa) over V (mk+1),
but a 7! pa is faithfully flat (see example 12.3) so y(f(b)) = y(b) over V (mk+*
*1). __|_|
The above is a prerequisite for a strong classification theory for deformatio*
*ns, and it at least makes plausible
the following theorem.
Theorem 23.2.G=X is terminal in the category of deformations of G0=X0. In ot*
*her words, given a
deformation H=Y , there is a map g :Y -! X and an isomorphism H ' g*G compatibl*
*e with the given
isomorphism (H0; Y0) ' (G0; X0).
Because of this, we refer to G as the universal deformation of G0, and to X a*
*s the (Lubin-Tate) universal
deformation space.
We can reinterpret the above slightly as follows. Let Y be a semilocal formal*
* scheme, and consider the
category of deformations of G0=X0with base Y . We can write a typical object as*
* (H; f0; "f0). The morphisms
are required to be the identity on Y . Write Def(Y ) for the set of isomorphism*
* classes. There is then a natural
isomorphism
Formal(Y; X) ' Def(Y )
24. The Action of the Morava Stabiliser
In this section (as in the last) we use the notation of the "Morava situation*
*". We shall show that there is a
unique action of the Morava stabiliser group ' Aut(G0) on the universal deform*
*ation G=X extending the
action on G0=X0. The action is (highly) nontrivial on X, so this does not contr*
*adict our earlier claim that
End(G) = Zp.
The topological significance is as follows. As mentioned in section 3, Hopkin*
*s and Miller have shown that
E can be made into an E1 ring spectrum in such a way that acts by E1 ring maps*
* on the nose. This
gives an action of on G=X, extending the action on G0=X0. The algebraic argume*
*nt given below shows
that this characterises it uniquely, so that calculation of the action becomes *
*a problem in pure algebra.
Suppose a 2 , and write aG0 for the corresponding automorphism of G0. Then a *
*acts on the functor
Y 7! Def(Y ) by
(H; f0; "f0) 7! (H; f0; "f0O a-1G0)
It therefore acts also on the representing object X.
We can be slightly more explicit and also more precise. We can present G=X as*
* a deformation of G0=X0
in a twisted way, using the embedding
a-1G0
G0 _____G0w_____wG
| | |
| | |
| | |
|u |u |u
X0 _____X0w1____wX
42 CONTENTS
Because G=X (with the usual deformation structure) is terminal, the above fits*
* into an expanded diagram
as follows.
aG
G|___________________________Goe|||
|@I |
|@ |
| @ |
| @ a-1G0 |
|| G0|___________G0_-| ||
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| |? 1 |? |
| X0 ___________X0_- |
| |
| @ |
| @ |
| @ |
|? @R |?
X ___________________________XoeaX (*
*1)
or equivalently
aG
G|___________________________G-|||
|@I |
|@ |
| @ |
| @ a |
|| G0|___________G0_-|G0 ||
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| |? 1 |? |
| X0 ___________X0_- |
| |
| @ |
| @ |
| @ |
|? @R |?
X ___________________________X-aX (*
*2)
It follows that acts on G=X as claimed.
We can give more explicit descriptions of this action in a variety of cases.
(1)The subgroup Zpx (which is actually the centre) acts trivially on X. The*
* action of Zpx on G
is just the obvious one discussed in section 15.
(2)Using the universal property of G=X, one can show that the action of Wx *
* extends the action
on G1-! X1 discussed in section 15. In fact, Wx is the largest subgroup o*
*f which preserves X1,
and it acts as thePidentity on X1.
(3)For general a = kakSk we have
X pm
uk = a-10 al a*X(um ) (mod m2)
k=l+m
25. QUOTIENT GROUPS AS DEFORMATIONS *
* 43
so
0 1 0 1-10 1
u0 a0 0p 0 0 u0
B u1 C BB a1 a0 0 0 CC B u1C
a*XB@u C = a0B p p2 C B C (mod m2)
2 A @ a2 a1 a0 0 A @ u2A
u3 a3 ap2ap21ap30 u3
(4)There are uniquely determined functions tk: -!OX such that
a*XFX k
a*G(x) = tk(a)xp
k
These functions satisfy
a*GFX k 0a*XFX l1
tk(a)([p]F(x))p = [p]a*XF@ tl(a)xp A
k l
In principle one can determine a*X(uk) and tk(a) by expanding this out an*
*d comparing coefficients
of x.
Further information can be obtained from the Dieudonne module of G (see [6]).
Because acts on G=X, it also acts on the line bundle !G of invariant differe*
*ntials. On the other hand,
because acts on the spectrum E, it acts on E-2 = "E0CP1 = !G. It can be shown *
*that these two actions
are the same.
25. Quotient Groups as Deformations
Let Y be a semilocal formal scheme. Suppose we have a point a 2 Def(Y ). This*
* means that we have an
isomorphism class of deformations H=Y of G0=X0. By the classification theory, t*
*his corresponds to a map
a: Y -! X; each representative H is uniquely isomorphic to Ga = a*G. We shall m*
*ostly use the notation
Ga. The picture is that G is a bundle of groups over the scheme X = Def, that a*
* is a point of X, and that
Ga is the fibre of G over the point a.
By assumption, Y is a semilocal formal scheme, but we shall argue as though i*
*t were local as this will
involve no loss of generality. This means that OY0is field; we are given a map *
*Y0 -!X0, so it must have
characteristic p. There is thus a Frobenius map FY0:Y0 -!Y0, given by F*Y0u = u*
*p for u 2 OY0. Similarly,
there is a map FH0 of formal groups covering FY0.
Now suppose we also have a subgroup divisor K < H, of degree pm say, so we ca*
*n form the quotient group
H=K. The claim is that this group can alsombe considered as a deformation of G0*
*. Indeed, example 21.1
tells us that K0 = Y0xK is just spfOH0=(yp ). This implies that OH=K0 = OY0[[z*
*]], where z can be
identified with ypm under the embedding OH=K0 -! OH0. This in turn means that *
*FmH0induces a map
(H=K)0= H0=K0-! H0 covering FmY0. This gives the required fibrewise isomorphism:
m f"0
(H=K)0= H0=K0 _____H0wF_________wG0
| | |
| | |
| | |
| | |
|u |u |u
Y0 ___________Y0wFm_________X0wf0
We need to check that the left hand square is indeed a pullback,mwhich is ess*
*entially clear once the notation
is straight. Write L = OY0, and write :L -!L for the map a 7! ap induced by *
*FmY0. The claim is now
that the map m m
L L; L[[y]]-!L[[yp ]] a u 7! aup
44 CONTENTS
is iso. This is a special case of the fact that an embedding L -!M together wit*
*h the map y -!ypm induces
an isomorphism
m
M L L[[y]]' M[[yp ]]
The fact that L = L just confuses the issue.
Anyway, the above diagram exhibits H=K as a deformation of G0. We therefore g*
*et a new map b:Y -!X,
and a uniquely determined isomorphism H=K = Ga=K -!Gb. In other words, we get a*
*n exact sequence of
group schemes over Y :
K -!a*G q-!b*G
On the special fibre, we have b0= Fm O a0 and q0= a*0(Fm ).
Conversely, suppose we have a semilocal scheme Y and two points a; b of X def*
*ined over Y . We say
that a map q :a*G -!b*G is an Fm -isogeny if it behaves as above on the special*
* fibre. Using Weierstrass
preparation, we see that q is indeed an isogeny. Moreover, there is a subgroup *
*divisor K < a*G such that
q factors as a*G -!a*G=K -~!b*G. Using the classification theorem for deformati*
*ons, we see that (a; K)
determines (a; q; b) and vice-versa. We conclude that the following functor is *
*isomorphic to Subpm, and hence
is a formal scheme:
IsogFm(Y ) = {(a; q; b) | a; b 2 Def(Y ); q :Ga -!Gb an Fm -isog*
*eny}
Remark 25.1.Note that FnX0= 1 and pG0 = Fn :G0 -!G0 (because the formal group*
* law for K has
[p](x) = xpn). This implies that pG :G -!G is an Fn-isogeny. This fact is somew*
*hat accidental, however.
All formal groups of height n over an algebraically closed field, but the choic*
*e of a representative formal
group over Fp or Fpn is arbitary. Some choices satisfy pG0 = Fn, but most do no*
*t. In particular, if n = 2
and G0 is the formal group associated to a supersingular elliptic curve over Fp*
*2then this need not hold. It
does hold if n = 1 and G0 is the formal multiplicative group, however.
We also define a
IsogF*= IsogFm
m0
This gives a category scheme, i.e. a functor from schemes to categories. The o*
*bject-scheme is X = Defand
the morphism-scheme is IsogF*. We will develop a little general theory of such *
*categories in section 27, and
apply these ideas to cohomology operations in section 28.
26. Level Structures
Let A be a finite Abelian p-group, and OE a point of Hom(A; G) defined over a*
*n X-scheme Y , so OE:A -!
(Y; G). We need a good notion of what it means for OE to be injective. The naiv*
*e definition has bad technical
properties; in particular, it does not give a subscheme of Hom(A; G).
To formulate a better notion, we need some definitions. First, we set A1 = {a*
* 2PA | pa = 0}. For each
a 2 A we have a section OE(a) of G and hence a divisorP[OE(a)]. We write [OE] =*
* a2A[OE(a)], which is a divisor
of degree |A| on G over Y . Similarly, we set [OE]1 = a2A1[OE(a)]. It turns o*
*ut that the following conditions
are equivalent:
(1)[OE] is a subgroup divisor.
(2)[OE]1 G(1) = ker(G p-!G).Q
(3)[p](x) is divisible by a2A1(x - x(OE(a))).
We say that OE is a level structure or level-A structure iff these conditions a*
*re satisfied. More generally, we
say that OE is a pk-fold level structure iff the following equivalent condition*
*s hold:
(1)pk[OE]1 G(1) Q
(2)[p](x) is divisible by a2A1(x - x(OE(a)))pk.
26. LEVEL STRUCTURES *
* 45
We write
Levelk(A; G) Level0(A; G) = Level(A; G) Hom(A; G)
for the corresponding subfunctors. Using the second criterion for a level stru*
*cture and proposition 18.2,
we see that they are closed subschemes: there are quotient rings DADkAof E(BA*)*
* = OHom(A;G)such that
Levelk(A; G) = spf(DkA). These rings will be described in more detail later.
First, however, we make some remarks which will relate to power operation in *
*Morava E-theory. Write a
for the map Level(A; G) -!X. There is thus a level structure OE (the universal *
*example of such a thing) on
a*G. This gives rise to a subgroup divisor K = [OE]. As in section 25, we have *
*a map b: Level(A; G) -!X
classifying G=K, and thus an isogeny q :a*G -!b*G with kernel K.
To return to the construction of DA, recall that [p](x) can be factored as g(*
*x)u(x), where g(x) is a monic
polynomial of degree pn and u(x) is invertible. Using this, we see that Levelk(*
*A; G) = ; if rank(A) > n - k
(where rank(A) is the number of cyclic factors).
Example 26.1.Suppose Y = spf(R), where R is an integral domain in which p 6= *
*0. It turns out that a
map OE:A -!(Y; G) is a level structure iff it is injective.
Example 26.2.Suppose Y = X0n= spf(). Then (Y; G) = 0, so the only possible OE*
* is the zero map.
Moreover, we have [p](x) = xp in [[x]]. This implies that OE is a pk-fold level*
* structure iff rank(A) n - k.
Exampler26.3.Suppose n = 1, so that G = G^m is the formal multiplicative grou*
*p, and [pr](x) =
(1 + x)p - 1. We know that Level(A; G) = ; if rank(A) > 1, which leaves only th*
*e case A = Z=pr. In this
case, a map OE: A -!(Y; G) sends the generator to a root ff of [pr](x). One can*
* check that this is a level
structure iff ff is a root of [pr](x)=[pr-1](x) = f(x). Thus Level(A; G) = spf(*
*Zp[x]=f(x)).
Example 26.4.Suppose n = p = 2 and A = {0; a; b; c = a + b} ' (Z=2)2. Suppose*
* that p = u1 = 0 in
the E-algebra OY (so that Y lies entirely over the special fibre X0 X). Suppos*
*e OE:A -!(Y; G). Write
ff = x(OE(a)) and so on. One can show that [2](x) = x4 and x +F y = x + y + x2y*
*2 (mod x4; y4) over OY.
It follows that ff4 = [2](ff) = x(OE(2a)) = 0 and similarly fi4 = fl4 = 0. More*
*over, c = a + b implies that
fl = ff +F fi = ff + fi + ff2fi2. For OE to be a level structure, we require th*
*at x4 be divisible by
x(x - ff)(x - fi)(x - fl)
It follows that this product must actually be equal to x4, and hence that (x - *
*ff) divides x3, and hence that
ff3= 0. Similarly, fi3 = 0 and so the product can be expanded as
fffi(ff + fi)x + (ff2+ fffi + fi2)x2+ ff2fi2x3+ x4= x4
This says that fffi(ff + fi) = (ff2 + fffi + fi2) = ff2fi2 = 0. In fact, it is*
* equivalent to require only that
ff2+ fffi + fi2 = 0, as the other relations follow easily from this. We conclud*
*e that OE is a level structure iff
ff3= fi3 = ff2+ fffi + fi2 = 0
This means that
Level(A; G) xX X0= spf([ff; fi]=(ff3; fi3; ff2+ fffi + fi2))
We next turn to the definition and properties of the rings DA and DkA. We fir*
*st give a simple definition
and then a more complicated one. Unfortunately it seems necessary to use the mo*
*re complicated version to
establish the properties of DA and hence justify the simple version.
Because spf(EA) = Hom (A; G) we have a universalPmap OE: A -! (EA; G). We ca*
*n thus think of A
as a subset of EA via a 7! x(OE(a)). Write J = a6=0ann(a) C EA; then DA = EA*
*=J. Unfortunately,
this definition is hard to work with, because annihilators are not stable under*
* change of base; if a 2 R and
f :R -!S then ann(f(a)) may be much larger than f(ann(a)).
For the more complicated version, we choose generators {e0; : :e:r-1} for A, *
*so that A =
with m0 m1: : :mr-1. We may assume wlog that r n, as otherwise DA = 0. We w*
*rite A(k) =
and define DA(k)recursively.QSet A(k)0= {a 2 A(k) | pmka = 0}.*
* Over DA(k)it works out
that [pmk](x) is divisible by a2A(k)(x - x(OE(a))). Moreover, the quotient ha*
*s a Weierstrass factorisation
46 CONTENTS
as g(x)u(x), where g(x) is a monic polynomial of degree pnmk- |A(k)0|. We take *
*DA(k+1)= DA(k)[ff]=g(ff),
where ff = x(OE(ek)).
The ring DA has many good properties. It is a complete regular local ring of *
*Krull dimension n, and thus
a unique factorisation domain, and thus is integrally closed in its field of fr*
*actions. It is a quotient of
W[[e0; : :e:r-1; ur; : :u:n-1]]
by a principal prime ideal. The prime p lies in the square of the maximal ideal*
* of DA (provided A 6= 0), and
the role of the single relation is to write p in terms of the generators eiand *
*uj. Moreover, DA is a finitely
generated free module over E. S
As an importantSspecial case, take m = (Z=pm)n and = (Qp=Zp)n = m m . We wr*
*ite Dm = Dm
and D1 = m Dm . Note that Dm is obtained from E by adjoining a full set m of r*
*oots for [pm ](x) (or
equivalently, for the associated Weierstrass polynomial).
We can do a certain amount of Galois theory with these rings. Write K and KA *
*for the fields of fractions
of E and DA. Write also Mon(A; B) for the set of monomorphisms from A to B.
(1)Hom E(DA; DB) = HomK (KA; KB) = Mon(A; B).
(2)Km is Galois over K with group Aut(m ).
(3)The rank of DA over E is | Mon(A; )|.
Similar constructions produce the rings DkA. Suppose that OE: A -!(Y; G) is a*
* pk-fold level structure.
It is immediate from the definition that xpkdivides [p](x) = expF(px) +F u1xp+F*
* : :u:n-1xpn-1+F xpnin
OY[[x]], and thus that p = u0= u1: :=:uk-1= 0 in OY. Thus, to construct DkA, we*
* start with E0= E=Ik =
[[uk; : :u:n-1]], where Ik = (u0; : :u:k-1). We then adjoin pk'th roots of the *
*remaining generators to get a
new ring E00(thus making a totally inseparable extension). Over E00we have [pm *
*](x) = f(x)pkfor some
power series f, and we can adjoin roots of f to get DkAjust as we adjoined root*
*s of [p](x) to get DA. The
resulting ring DkAis again a complete regular local ring.
For each subgroup B A we have closed subschemes
Level(A=B; G) Hom(A=B; G) Hom(A; G)
These are the irreducible components of Hom(A; G), and
[
Hom (A; G) = Hom(A; G)red= Level(A=B; G)
B
If we look only at what happens above the open subscheme D(p) = spec(p-1E) < X,*
* then this union is
disjoint:
a
Hom (A; G) xX D(p) = Level(A=B; G) xX D(p)
B
In terms of rings, we have a monomorphism
Y
EA -! DA=B
B
which is an isomorphism after inverting p.
More generally, consider the subschemes X(k) = spf(E=Ik) X and X(k)0= X(k)\D*
*(uk). We find that
[
(Hom (A; G) xX X(k))red= Levelk(A=B; G)
B
Moreover, the union is disjoint over X(k)0. The schemes Levelk(A=B; G) for vary*
*ing k and B are precisely
the irreducible closed subschemes of Hom (A; G) invariant under the action of t*
*he Morava stabiliser group
= Aut(G0).
27. CATEGORY SCHEMES *
* 47
27.Category Schemes
A category scheme is a category object in the category of schemes, or equival*
*ently, a functor C from rings
to categories, such that obj(C) and mor(C) are schemes. We shall also refer to *
*such a thing as an internal
category in the category of schemes.
Example 27.1.The category of formal group laws and strict isomorphisms (more *
*precisely, the functor
R 7! (FGL (R); SI(R)) is an internal category.
Example 27.2.Let q :X -!Y be a map of schemes. We can define an internal cate*
*gory C = C(q) by
obj(C) = X and mor(C) = X xY X. Thus, given points a; b 2 X there is a unique m*
*ap a -!b if qa = qb,
and no maps otherwise.
Example 27.3.We can define four different category schemes with object scheme*
* the Lubin-Tate defor-
mation space X = Def:
(1)mor(DefIso)(a; b) = { isomorphismsGa -!Gb}
(2)mor(DefHom)(a; b) = { homomorphismsGa -!Gb}
(3)mor(IsogF*)(a; b) = {F*-isogeniesq :Ga -!Gb}
(4)mor(Isog)(a; b) = { all isogeniesGa -!Gb}
We need to see that these really are representable. Firstly, let denote the Mo*
*rava stabiliser group. It can
be shown that
DefIso= __x Def= spf(C(; E)) = spf(E_E)
Next, DefHomcan be constructed as a closed subscheme of MapX(G; G) by technique*
*s which should by now
be familiar. We conjecture that it arises in topology as follows. Let E_0be the*
* 0'th space in the -spectrum
for E. Then A = ss0LK (E ^ E_0) is supposed to be a Hopf ring, so that the *-in*
*decomposables Ind(A) form
a ring under the circle product. We are supposed to get
DefHom= spf(Ind(A))
This can`be proved modulo technicalities about formal topologies. Next, we kno*
*w from section 25 that
IsogF*' kSubpkis a scheme, and it was explained in theorem 21.1 how it arises*
* in topology. We can
rephrase this a little to make it look more like the case of DefHom. Consider E*
*(DS0), where
_ a
DS0 = DkS0 = Bk
k k
This can also be made into a Hopf ring, as explained in section 28 below. We ha*
*ve
IsogF*= spf(IndE(DS0))
Finally, let f :Ga -!Gb be an isogeny of deformations, with kernel K say. Argui*
*ng much as in section 25,
we get a unique Fm -isogeny Ga -!Gc with kernel K and an isomorphism Gc-! Gb, w*
*hose composite is f.
This shows that Isog= IsogF*xX DefIso, which is a scheme as required.
Definition 27.1.An internal functor on an internal category C is a scheme F o*
*ver obj(C) with the
following extra structure. For any points a; b 2Y obj(C) and any C-morphism u:a*
* -!b we are given a map
Fu:Fa -!Fb. These maps satisfy F1= 1 and Fuv= FuFv. Moreover, if we pull a,b an*
*d u back along a map
f :Y 0-!Y (so that Ff*a= u*Fa automatically) then Ff*u= f*Fu. By analysing the *
*universal example, it
is equivalent to require a map F xobj(C)mor(C) -!F satisfying some not-quite-so*
*-obvious conditions.
Example 27.4.In a suitable technical setting, spec(MU*Z) becomes an internal *
*functor on FGL.
Example 27.5.An internal functor on C(q) is just a map Z -! X with descent da*
*ta. Thus, if q is
faithfully flat, then internal functors on C(q) are equivalent to schemes over *
*Y .
48 CONTENTS
28. Cohomology Operations
Let E and K be Morava E-theory and K-theory of height n, say. Let Z be a tole*
*rable space, so that ZE
is a formal scheme over X = Def. We would like to understand the extra structur*
*e which this has because of
the action of various kinds of operations on E(Z). More generally let F be a K-*
*local E-algebra spectrum.
For example, we could have F = F(Y+; E) where Y is a tolerable space, or F = LK*
* (E ^ MU). Write
Z = spf(F0), which is a formal scheme over X.
Conjecture 28.1.Assuming various things about flatness, projectivity, and com*
*pleteness, we have ac-
tions as follows.
(1)Z is an internal functor for DefIso.
(2)If F = F(Y+; E) then Z is an internal functor for DefHom.
(3)If F is an E1 E-algebra then Z is an internal functor for Isog.
(4)If F = LK (E ^ Y+) and Y is a decent infinite loop space then Z has a cov*
*ariant action of Isogand
a contravariant action of DefHom.
All of these things are almost certainly true, with a suitable choice of tech*
*nical details about formal
topologies. However, I have not yet pinned these details down. The last stateme*
*nt follows from the others.
There should be some sort of compatibility statement (which would be the analog*
*ue of the Nishida relations)
but I'm not sure what it should say. Note that none of the above accounts for n*
*onadditive unstable operations,
although we shall say something about nonadditive operations when we outline th*
*e construction the actions
mentioned above.
Example 28.1.The scheme G itself is an internal functor for DefHom, as are th*
*ings constructed from it
like Div+m= BU(m)E and Hom(A; G) = (BA*)E. The scheme Orient(G) = spf(E_MU) has*
* (as predicted)
no contravariant action of DefHom, because of the requirement that an orientati*
*on be an isomorphism G -!
bA(L). It does have a covariant action of Isog, however, given by the norm cons*
*truction discussed in section 29
below. Divisors can be pushed forwards by an arbitary map, or pulled back by an*
* isogeny; this gives two
actions on Div= (Z x BU)E, as predicted.
We next indicate the construction of the action of Isog. Because DefIsoalrea*
*dy acts, it is enough to
construct an action of IsogF*. We shall ignore some technicalities about topolo*
*gies.
It is traditional to construct power operations one symmetric group at a time*
*, giving a (nonadditive) map
Pk:F0 -!F(Bk). We will instead outline an approach which uses all symmetric gro*
*ups simultaneously,
and which compares very nicely with the Boardman-Johnson-Wilson [1] approach to*
* unstable operations.
Unfortunately, this approach does introduce extra technicalities about formal t*
*opologies, which we have not
yet got around to resolving. W
There is a total extended power functor DX = k0 DkX from spectra to spectra,*
* given on spaces by
DkX = Ek^k X^k. In particular, DkS0 = (Bk)+. One also knows that D(X _ Y ) ' DX*
* ^ DY . The
obvious maps S0 -!S0_ S0 -!S0 thus give maps
DS0 OE-!DS0^ DS0 -!DS0
The components can be described as follows.
OEk;l:Bkx Bl-! Bk+l
is the usual map, and
k;l:Bk+l-!Bkx Bl
is the transfer (this is not obvious, but it is stated in [2] and proved in [10*
*]). We also consider the diagonal
map
k:Bk -!Bkx Bk
It turns out that E(DS0) becomes a Hopf ring with star and circle products give*
*n by OE and , and coproduct
induced by . Moreover, the E1 structure of E gives a map : DE -!E with various *
*good properties. To
28. COHOMOLOGY OPERATIONS *
* 49
be more precise, D is a monad on spectra and makes E into a D-algebra. Given u*
* 2 E (i.e. u:S0 -!E)
we can define [u] = O Du 2 E(DS0). It turns out that [u] * [v] = [u + v] and [*
*u] O [v] = [uv], so that E(DS0)
is a Hopf ring over the ring-ring E[E].
We now dualise. It can be shown that E(DS0) is pro-free and E_ DS0 = E(DS0)_.*
* We thus discover
that T = spf(E_ DS0) is an E-algebra object in the category of schemes over X =*
* spf(E), in other words
a functor from formal E-algebras to E-algebras. It should be possible to introd*
*uce a natural topology on
T(R), in a way which will make T into a comonad.
Now let F be a K-local, E1 E-algebra spectrum. Using the fact that F is K-loc*
*al, one can show that
F0(DS0) = E(DS0)b EF0. We propose to define a coaction of our comonad on F0, in*
* other words a map
of E0-algebras
F0 -!E-Alg(E_DS0; F0)
The right hand side is contained in
E-Mod(E_DS0; F0) = E(DS0)b EF0 ' F(DS0)
In this guise, the map F0 = F(S0) -!F(DS0) just sends u:S0 -!F to O D(u).
We can now "linearise" this construction. For any ungraded Hopf ring A over E*
*, we define
GL(A) = GLE(A) = {group-like elements} = {a 2 A | ffl(a) = 1 and (a) *
*= a a}
This is a ring, with addition given by the star product and multiplication by t*
*he circle product. Moreover,
spf(A_) is a ring scheme, so E-Alg(A_; R) is a ring; this is just the same as G*
*LR (Rb EA). Also, the
star-indecomposables in A form a ring under ordinary addition and circle produc*
*t, and Ind(Rb E A) =
Rb EInd(A). There is a ring homomorphism
GL (A) -!Ind(A) a 7! a - [0] = a - 1
Combining this with our power operation, we get a map
F0 -!T(F0) = GL(F0b EE(DS0)) -!F0b EInd(E(DS0))
This turns out to be a ring map. If we recall that the star product in E(DS0) i*
*s given by transfers, and use
theorem 21.1, we conclude that spfInd(E(DS0)) = IsogF*. We have thus constructe*
*d a map
spf(F0) xX IsogF*-!spf(F0)
After chasing many more diagrams, we conclude that this makes spf(F0) into an i*
*nternal functor as claimed.
The Boardman-Johnson-Wilson theory, suitably adapted, should make the functor
T0:R 7! E-Alg(E(E_0); R)
into a comonad in the category of formal E-algebras. Recall that G is a functor*
* from formal E-algebras to
groups. The results of [7], suitably adapted, should show that T0 is the initia*
*l example of a representable
functor from formal E-algebras to formal E-algebras equipped with a map G -!GOT*
*0. One can show purely
algebraically that such an example exists, and is a comonad. If Z is a decent s*
*pace then E(Z) is T0-coalgebra.
Linearising this coaction makes ZE into an internal functor for DefHom. This pa*
*rt of the theory is joint work
with Paul Turner.
We now present a different version of power operations which is in some ways *
*less satisfying, but which
does have the advantage that more details are in place.
Let A be a finite Abelian p-group, with dual A*. We write DA*Z for the A*-ext*
*ended power of a spectrum
Z. We also write DA = OLevel(A;G)as in section 26, and apologise profusely for *
*any confusion caused by
this. If Z is a space then *
DA*Z = EA*xA*Z^(A )
The usual construction gives a map
F0 = [S0; F] -![DA*S0; DF] -![DA*S0; F] = F(BA*)
50 CONTENTS
This is additive mod transfers from proper subgroups of A*. Using the fact that*
* F is K-local, one can show
that the right hand side is just E(BA*) E F0. Combining this with the map E(BA**
*) -!DA (which kills
the transfers) we get a ring map
F0 -!DA E F0
or equivalently
spf(F0) - Level(A; G) xX spf(F0)
Here, the fibre product is formed using the obvious map Level(A; G) -!X, which *
*we shall call a. Performing
the above construction with F = E and with F = F(CP1+; E), we get a new map b: *
*Level(A; G) -!X and
a map of schemes q :a*G -!b*G. One can check that this is an isogeny of formal *
*groups, that the kernel
is the subgroup divisor K defined by the level structure, and that the map is a*
* power of Frobenius mod the
maximal ideal of DA. This is enough to prove that q and b are the same as the m*
*aps constructed algebraically
in section 26.
More generally, suppose we have a short exact sequence A -! B -! C. This giv*
*es a dual sequence
C* -!B* -!A* and hence a map B* -!A*o C*. Now, given an element u 2 F(BC*) = [D*
*C*S0; F] we can
form the composite
B(B*)+ -!B(A*o C*)+ = DA*DC*S0 DA*u----!DA*F -!F
This gives rise to a ring map DC E F -!DB E F.
To interpret this, define a new category LevelDef(Y ) as follows. The objects*
* are triples (a; A; OE), where
a 2 Def(Y ) and OE:A -!Ga is a level structure. The morphisms are diagrams
f
A ______Bww
v| v|
O|E |
| |
|u |u
Ga _____Gbwwq
in which q is an F*-isogeny, and ker(q) = [OE(ker(f))] as divisors. This defin*
*es a category scheme, provided
that we restrict to a small skeleton of the category of finite Abelian p-groups*
* (or avoid the company of
obsessive set-theorists).
Write Z = spf(F0) as before, and "Z= LevelDefxDefZ. The extended power constr*
*uction defined above
makes "Zinto an internal functor on LevelDef. In other words, given a morphism *
*(f; q) in LevelDefas above,
we get a map
Zf;q:Za -!Zb
satisfying the obvious conditions.
Let Z be an internal functor for IsogF*and D a divisor of degree m on G over *
*Z. Thus D < Z xX G is
a subscheme of an internal functor, so it makes sense to require it to be a sub*
*functor. We shall say that D
is an equivariant divisor if this is so. Let us make this condition more explic*
*it. Write ss for the map Z -!X
and Ga for Gssa. Suppose that a; b 2 Z, that q :Ga -!Gb is an F*-isogeny, and t*
*hat Zq(a) = b. In future,
we shall simply write q :a 7! b for this. We then have two degree-m divisors Da*
* < Ga and Db < Gb, so it
makes sense to ask that q*Da = Db. This holds for all q :a 7! b iff D is an equ*
*ivariant divisor. If so, the
map q*:OGb-! OGa restricts to give a map (c.f. proposition 20.3)
O(Db) -!q*O(Db) = q*O(q*Da) = O(K * Da)
This kind of structure arises in topology as follows. Let V be the standard p*
*ermutation representation of
m , which we can regard as a vector bundle over Bm . Let W be another vector bu*
*ndle over a tolerable
space Z. A small generalisation of our previous construction gives a (nonadditi*
*ve) map
"E(ZW ) -!"E((Bm x Z)V W )
29. THE ANDO ORIENTATION *
* 51
After passing to the quotient by proper transfers, we hope to find that D(V ) =*
* K is the universal subgroup
defined over Subm(G). Assuming this, we find that D(V W) = K *D(W) and that D(W*
*) is an equivariant
divisor on ZE. The power operation on
"E((Z x CP1 )WL ) = L(D(W) * ) = O(_____D(W))
__ __
can be identified with the map q*:O(D b) -!O(K * Da). The operation on "E(ZW ) *
*itself can be obtained by
pulling back along the zero section.
29. The Ando Orientation
Now suppose that y is an orientation of the universal deformation G=X. Then y*
* corresponds to a ring
map MU -!E. One would like to know when this is a map of H1 ring spectra.
Let us first reinterpret the notion of an orientation. Let G ss-!X be a polar*
*ised formal group.
Definition 29.1.A k-rigid line bundle on G is a line bundle M together with a*
*n isomorphism :Lk '
0*M. A rigid section of M is an isomorphism : ss*Lk ' M extending . We write ri*
*g(M) for the set
of rigid sections. Let N be a line bundle over X and m: ss*N -! M an isomorphi*
*sm. We then write
(m) = m(0)-1m:ss*Lk -!M. This is clearly a rigid section.
Now let J = O([0]) be the augmentation ideal in OG. The polarisation makes t*
*his into a 1-rigid line
bundle, and rig(J) = Orient(G).
Consider again an orientation y of the universal deformation. Let q :Ga -!Gbm*
*be an Fm -isogeny. We
thus have orientations ya and yb. We can think of Nqya as a map from ss*Lp to
NqJa = NqOGa[0] = OGb(q*[0]) = Jb
It thus makes sense to consider (Nqya) 2 Orient(Gb).
Definition 29.2.y is an Ando orientation iff (Nqya) = yb for all a; b and q a*
*s above.
Theorem 29.1.y is an Ando orientation iff the map y :MU -!E is H1 .
There is a simpler and closely related notion for coordinates.
Definition 29.3.A coordinate x on the universal deformation is an Ando coordi*
*nate iff Nqxa = xb for
all a; b and q as above.
Theorem 29.2.It is equivalent to require that NpGx = x. Any coordinate x0 on *
*G0 extends uniquely to
an Ando coordinate x on G. In fact, if x is any coordinate on G then NkpGx conv*
*erges to the Ando coordinate
y which agrees with x on G0.
Remark 29.1.We have used here the fact that pG0 = Fn and thus pG is an Fn-iso*
*geny (c.f. remark 25.1).
The theorem becomes a little more complex without this.
If x is a coordinate on G then (x) is an orientation. If x is an Ando coordin*
*ate then (x) is an Ando
orientation. Conversely, any Ando orientation is (x) for some Ando coordinate x.
Example 29.1.Let G be the formal multiplicative group (defined over Zp), and *
*x the usual coordinate
x(u) = u - 1. Let i be a primitive p'th root of 1, and write A = Zp[i], which i*
*s faithfully flat over Zp. After
pulling back to A, we have a full set of points {1; i; : :i:p-1} for G(1) = ker*
*(pG). This gives
p-1Y
(NpG)f(up) = f(uik)
k=0
In particular (if p > 2), Y
(NpG)x(up) = (uik - 1) = up- 1 = x(up)
Thus NpGx = x.
52 CONTENTS
Example 29.2.Let G0 be the Morava K-theory formal group over X0n= spf(), wher*
*e = Fpn. Let
x = x0 be thenusual p-typical coordinate,Pso that x(pa) = x(a)p by definition.*
*nIt is then easy to see that
f(pa) = f(a)p for any function f = kffkxk 2 OG. Moreover, G0(1) = spf([x]=xpn*
*) has {0; : :0:} (with
pn entries) as a full set of points. It follows that (NpGf)(pa) = f(a)p = f(pa)*
* for any f 2 OG.
Example 29.3.More generally, suppose that G is a formal group over a complete*
* local ring A, of residue
characteristic p > 2. Suppose that a coordinate x has the property that x(pa) =*
* f(x(a)) for some (monic)
Weierstrass polynomial f, of degree pn say. Then NpG(x) = x. To see this, write*
*ny = f(x) and B = A[[y]]
and g(t) = f(t) - y 2 B[t]. Then C = A[[x]]= B[x]=g(x), so NC=B(x) = (-1)p g(0)*
*. Because f(0) = 0 and
p is odd, we get NC=B(x) = y. To compute NpG(x), we have to identify C with B v*
*ia x 7! x O pG = y, so
NpG(x) = x as claimed.
30.Cartier Duality
Let H be a commutative formal group scheme over a formal scheme X. We do not *
*assume that H is
a one-dimensional formal group. Suppose that OH is a pro-free OX -module. The d*
*uality then works well
enough to make O_Hinto a topological Hopf algebra (in the obvious sense involvi*
*ng the completed tensor
product). Thus DH = spf(O_H) is again a formal group scheme over X.
Let bGmbe the formal multiplicative group:
Gbm(R) = 1 + Nil(R) Rx
i j
bGm= spfZ[u1 ]^(u-1)= spf(Z[[x]]) x = u - 1
Proposition 30.1.DH = Groups_(H; bGm). In other words, for any formal X-schem*
*e Y we have
(Y; DH) = GroupsY(H xX Y; bGmx Y )
Proof. A map of schemes over Y f :H xX Y -!GbmxY is just the same as a map of*
* schemes H xX Y -!
Gbm A1, which is just the same as an element f02 OH bXOY such that f0- 1 is top*
*ologically nilpotent.
Note that OH bXOY is a Hopf algebra over OY. It is easy to see that f is a map *
*of group schemes iff ffl(f0) = 1
and (f0) = f0bf0. Because OH is pro-free, we have
OH bXOY ' OX -Mod(O_H; OY) = OX -Mod(ODH ; OY)
Write f00for the map ODH -! OY corresponding to f0. Again, one sees that this i*
*s a ring map iff ffl(f0) = 1
and (f0) = f0bf0. This gives the required bijection. __|_|
We shall often just write DH = Hom(H; bGm).
Example 30.1.Take X = spf(Zp) and H = bGmx X, so that DH = End(H). There is a*
* continuous
function O:Zp -!OH = Zp[[x]]defined by O(n) = (1+x)n. Using C(Zp; OX ) = C(Zp; *
*Zp)b ZpOX , we obtain
an adjoint map :O_H= ODH -! C(Zp; Zp). This can be shown to be a topological i*
*somorphism.
Example 30.2.Let H be the constant group A_= spf(F(A; Z)), where A is a finit*
*e Abelian group. Then
DH = spf(Z[A]).
Example 30.3.Let X be the Lubin-Tate deformation space, and H = Div0= BUE. Th*
*en
D(Div0) = spf(E_BU) = Hom(Div0; bGm) = Map0(G; bGm)
Here Map0(G; bGm) means the scheme of maps preserving the zero section. The las*
*t equality comes from
proposition 18.1.
31. BARSOTTI-TATE GROUPS *
* 53
31. Barsotti-Tate Groups
Let G be a formal group over a formal scheme X. Suppose that pG :G -! G is a*
*n isogeny of degree
pn > 1. It follows that p is topologically nilpotent in OX . We write G(m) = ke*
*r(pmG); this is a finite free
group scheme over X of degree pnm. These groups fit together into an inductive *
*system of closed inclusions
G(*) = (G(1) -!G(2) -!G(3) -!: :):
This is called the Barsotti-Tate group (or BT-group) associated to G.
More generally, one can define a Barsotti-Tate group over X to be a system H(*
**) = (H(1) -!H(2) -!: :):
of closed inclusions of finite group schemes, such that there are short exact s*
*equences
k
H(k) -!H(k + l) p-!H(l)
The point is that G(*) behaves differently from G under change of base. We s*
*tarted with a formal
scheme X, with ring of functions OX . We can forget the topology on OX and cons*
*ider the informal scheme
X" = spec(p-1OX ). This is crude, of course. There is no sensible linear topo*
*logy on p-1OX , but there
is a sensible topology of a more general kind (consider the case OX = Zp). Howe*
*ver, we have not as yet
developed the relevant theory. We can pull back G(*) to get a BT-group "G(*) ov*
*er "X(provided we allow
ourselves a little latitude with the algebraic geometers' definition of a BT-gr*
*oup). It turns out that "G(m) is
anetale torsion group. If G=X is the Lubin-Tate universal deformation, then "G(*
*m) becomes isomorphic to
the constant group m = (Z=pm )n after a faithfully flat base change.
Suppose, on the other hand, that we want to pull back G to get a formal group*
* over "X. We cannot just
take p-1OG, because this is not a formal power series ring over p-1OX = OX". In*
*stead, we have to complete
again at (x) after rationalising. This gives a formal group "G. This is now a f*
*ormal group over a rational ring,
so it has a logarithm and is isomorphic to the additive group. It therefore has*
* no torsion subgroups, and the
associated BT-group is zero.
Consider again the universal deformation G=X, of height n. Write Im = (u0; :*
* :u:m-1) C OX and
R = (u-1mR)^Im. We make this a formal ring, with ideal of definition Im , and w*
*rite Y = spf(R). We can pull
back G(*) to get a BT-group H(*) over Y . It turns out that this fits into an e*
*xtension
H(*)inf-!H(*) -!H(*)et
Here H(l)infis local (or "infinitesimal") and H(l)etisetale. (The extension is *
*analogous to the sequence
G1 -!G -!ss0G where G is a Lie group.) The infinitesimal part has degree plm. T*
*heetale part has degree
pl(n-m), and becomes isomorphic to (Z=pn-m)lafter faithfully flat extension.
It seems that the BT-group is the natural home of "chromatic fringe phenomena*
*" such as the Greenlees-May
generalised Tate cohomology and the root invariant.
32.Nilpotents, Idempotents and Connectivity
Recall that for a formal ring R, Nil(R) is the set of topologically nilpotent*
* elements.
Proposition 32.1.Nil(R) is the intersection of the open prime ideals of R, an*
*d thus is closed.
Proof. It is easy to reduce the first statement to the discrete case, which i*
*s theorem 1.2 of [11]. The
second statement follows as open ideals are closed. __|_|
Proposition 32.2 (Idempotent Lifting).Suppose that e 2 R= Nil(R) is idempoten*
*t. Then there is a
unique idempotent "e2 R lifting e.
Proof. It is enough to prove this mod each open ideal I, so we may assume tha*
*t R is discrete. 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
54 CONTENTS
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
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. __|_|
Proposition 32.3.Suppose that e 2 R is idempotent, and f = 1 - e. Then
eR = R=f = R[e-1] = {a 2 R | fa = 0}
This ring has the same topology as a subspace or as a quotient of R and R is ho*
*meomorphic to eR x fR.
Proof. This is all fairly trivial. __|_|
Theorem 32.4 (Chinese Remainder Theorem).Suppose that {Iff} is a finite famil*
*y 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. [11, Theorems 1.3,1.4] __|_|
`
Corollaryp32.5.Suppose_thatQzar(R) = ffzar(R=Iff) (a finite coproduct). Then*
* there are unique ideals
Jff Iff Jffsuch that R ' ffR=Jff.
T
Proof. Proposition 32.1 implies that ffIffis nilpotent. If ff 6= fi then no *
*prime ideal contains Iff+ Ifi,
so Iff+ Ifi= R. Now use the chinese remainder theorem, followed by idempotent l*
*ifting. __|_|
33. The Weierstrass Preparation Theorem
In this section, we assemble some results about the structure of rings of for*
*mal power series.
P
Theorem 33.1 (Weierstrass Preparation).Let R be a formal ring, and f(x) = k*
*ckxk 2 R[[x]]a
formal power series. Suppose that ck is topologically nilpotent for k < n, and *
*that cn is a unit. Regard R[[x]]
as an algebra over R[[y]]via y 7! f(x). Then the map
n-1X
ff:R[[y]]n-!R[[x]] a_7! akxk
k=0
is a homeomorphism.
Proof. Write A = R[[y]]and B = R[[x]]. Set I = (ak | k < n). This is a topolo*
*gically nilpotent ideal,
so that for any J CO R we have IN J for N 0. It is also finitely generated,*
* which implies that
Jm A = Jm [[x]](and similarly for B). Without loss of generality, we may assume*
* that cn = 1. For 0 k < n
and l 0 write zkl= xkf(x)l, so that zkl= xk+nl(mod J; xk+nl+1). Given an R-mod*
*ule M consider the
map Y X
M -!M[[x]] a_7! aklzkl
k;l k;l
It is clear that this is iso if JM = 0, given the form of zklmod J. It then fol*
*lows inductively for M = R=Jk.
As J is topologically nilpotent and R is complete, we have R = limR-=Jk, so our*
* map is iso for M = R.
k
This implies that our original map ff is also iso. We still need to prove that *
*it is a homeomorphism. Suppose
I CO R and m 0, so that U = (xm ; I)B is a basic open neighbourhood of zero in*
* B. If Jl I then
ff((ylm; I)An) U, so ff is continuous. Next, observe that xn is a unit multipl*
*e of f(x) in B=JB. It follows
that x is nilpotent in B=(I; ym ) for any I CO R and m 0; say xl2 (I; ym ). Mo*
*reover, taking M = I above,
we have ff(IAn) = IB. It follows that ff((I; ym )An) (I; xl)B, and thus ff is *
*an open map. __|_|
33. THE WEIERSTRASS PREPARATION THEOREM *
* 55
Corollary 33.2.If f is as above then there is a unique way to write f(x) = g(*
*x)u(x) where g(x) is a
monic polynomial of degree n and u(x) is invertible. Moreover, g(x) = xn (mod J*
*) and neither f nor g is a
zero-divisor in R[[x]].
Proof. It is immediate from the theorem that R[[x]]=f(x) = R{xk | 0 k < n}. *
* Thus, there is a
unique way to write xn mod f(x) as an R-linear combination of xk for k < n, in *
*other words there is a
unique monic polynomial g(x) of degree n dividing f(x). Next, observe that xn d*
*ivides f(x) mod J; by
an obvious uniqueness argument, g(x) = xn (mod J). Thus, g(x) is invertible in *
*x-1R=J[[x]], and thus also
in x-1R=Jm [[x]]. It follows easily that g(x) is not a zero-divisor in R[[x]],*
* so there is a unique u(x) such
that f(x) = g(x)u(x). Moreover, u(x) is visibly invertible mod the topologicall*
*y nilpotent ideal J, so it is
invertible. __|_|
Next, we consider the ring
C = (x-1R[[x]])^ = limx--1R=I[[x]]
I
P
This is the ring of series f(x) = k2Zakxk such that ak -!0 as k -! 1. We sha*
*ll say that f 2 C is
holomorphic at zero (resp. infinity) if ak = 0 for k < 0 (resp. k > 0).
P
Corollary 33.3.Suppose that f(x) = akxk 2 C has an 2 Rx, and ak 2 Nil(R) fo*
*r k < n. Then
f 2 Cx . Moreover, f(x) can be written uniquely as xng(x)u(x) with
(1)g(x) holomorphic at infinity
(2)g(1) = 1
(3)u(x) holomorphic at zero
(4)u(0) 2 Rx.
Proof. Without loss of generality, n = 0. Suppose I C R is open, and write fI*
*(x) for the image of f
in x-1R=I[[x]]. Then theorem 33.1 applies to xm fI(x) for some m 0. This gives*
* a unique factorisation
fI = gIuI as described above (with gI equal to x-m times the polynomial provide*
*d by the theorem). These
pass to the limit as required. __|_|
Proposition 33.4.Suppose that R is connected, in other words it has no idempo*
*tents other than 0 and
1. Suppose that f 2 Cx . Then f is as in corollary 33.3
P P
Proof. Suppose f = kakxk and f-1 = g = lblxl. Write Jn = (ak | k < n) + (*
*bl| l < -n). Suppose
that I CO R. There exists K > 0 such that ak; bk 2 I for k < -K. Suppose that p*
* C R is prime, and that
p I. For some n we have ak 2 p for k < n but an 62 p. Similarly bl2 p for l < *
*m but bm 62 p. This
means that mod p we have fg = anbm xn+m plus higher terms, with anbm 6= 0. This*
* means that m = -n
and |n| K, and that Jn p; but Jl6 p for l 6= n. It follows that the ideals Jn*
* are pairwise coprime mod I.
Their intersection is contained in all prime ideals p I, hence is nilpotent mo*
*d I. The Chinese remainder
theorem together with idempotent lifting gives a canonical splitting
Y
R=I = R(I; n)
n
such that anb-n is invertible in R(I; n) and nilpotent in all the other factors*
*. This splitting is actually finite,
because R(I; n) = 0 when |n| > K. In the limit, we get
Y
R = limR(I; n)
n I-
As R is connected, we have R = limR-(I; n) for some n. It follows that anb-n is*
* invertible in R, hence in
I *
* __
R=I, so that R=I = R(I; n) for all I. This implies that ak is topologically nil*
*potent for k < n. |_|
Of course, if R is not connected, we may get different n's on different compone*
*nts.
56 CONTENTS
34. Dictionary
____________________________________________________________
| spf(E) T|he universal deformation space Def= X |
| | |
| spf(K) T|he special fibre X0< X |
| | |
| CP1 |The universal deformation G over X |
| E | |
| (BZ=pm )E G|(m) = ker(pm :G -!G) |
| V| G |
| K(Z=pm ; l)E l|G(m) |
| | |
| (BA*)E |Hom(A; G) |
| P | |
| spf(E(BA*)= ann(x(a)))L|evel(A; G) |
| a6=0 | |
| BU(m)E |Div+(G) |
| | m |
| BUE |Div0(G) |
| | |
| (Z x BU)E D|iv(G) |
| | |
| spf(E_CP1 ) H|om(G; bGm) |
| | |
| spf(E_(Z x BU)) Ma|p(G; bGm) |
| | |
| spf(E_BU) |Map(G; bGm) |
| | 0 |
| spf(E_MU) |Orient(G) |
| | |
| E_E |The pro-constant stabiliser group __= DefIso|
|________________________|__________________________________ |
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*re of unstable BP-algebras, preprint,
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*spectra and their applications, Lecture
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[3]Ethan S. Devinatz, Small ring spectra, Journal of Pure and Applied Algebra 8*
*1 (1992), 11-16.
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*, Memoirs Of The American Mathematical
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57
Index
D(a), 9 Brown-Peterson spectrum, 6
D * D0, 31
DkA, 45 Cartier divisor, 30, 32
DA, 45 category scheme, 47
E1 , 41 Chinese remainder theorem, 54
F(S;M), 11 closed formal subscheme, 12
Fm -isogeny, 44 closed ideal, 12
F0(S;M), 11 closed subscheme, 9
M_, 11 complete, 10
Mx, 19 complex-orientable, 26
Nf, 34 components, 9
NX=Y, 34 connected, 9
PX=Y(u;t), 35 constant group scheme, 16
V (I), 9 constant rank, 19
Xzar, 17 constant scheme, 22
DefHom, 47 convolution, 31
DefIso, 47 coordinate, 24
Div, 32 cotangent bundle, 28
Div+m, 31 cotangent space, 25
FGL, 7 curve, 25
Hom(A;G), 39
HomR(M;N), 11 decent, 13
IPS, 8 deformation, 40
IsogF*, 44, 47 degree, 31
L(D), L(V ), 34 descent data, 21
Level(A;G), 45 descent data, effective, 21
LevelDef, 50 discrete, 11
Levelk(A;G), 45
Nil(R), 12
OFG, 26 effective divisor, 29
O(D), 33 equation, 29
OX, 8, 12 equivariant divisor, 50
G=X, 28 etale covering, 22
SI, 8
, 28 faithfully flat, 20
Subm, 39 fibrewise isomorphism, 25
A(M), 19 flat, 20
dim(X), 18 formal completion, 11
Formal, 11 formal group, 24
bA(M), 19 formal group law, 7, 25
!G,_25 formal module, 11
D, 34 formal ring, 10
Schemes, 7 formal scheme, 11
spf, 11 fpqc locally, 21
Groups_X(G;H), 16 Frobenius automorphism, 6
Schemes_X(Y;Z), 16 Frobenius map, 43
zar(A), 17 full set of points, 30, 35
k-rigid line bundle, 51
geometric point, 18
affine line, 8 graded rings, 20
affine scheme, 7 group scheme, 24
Ando coordinate, 51
Ando orientation, 51 height, 5
Artin-Rees lemma, 11, 13 holomorphic at infinity, 31
base change, 24 ideal of definition, 10
basic curve, 25 idempotent lifting, 53
58
INDEX *
*59
integral, 9 topological dual, 11
internal category, 47 topologically nilpotent, 12
internal functor, 47 transfer, 39
invariant differential, 28, 43
invertible sheaf, 18 uniform convergence, 11
irreducible, 9
irreducible components, 9 vector bundle, 18, 19, 22
isogeny, 25 virtual divisors, 30
Krull dimension, 18 Weierstrass preparation theorem, 54
Lazard ring, 7, 15 Witt ring, 5
level structure, 44
line bundle, 18 Zariski space, 17
linear topology, 10
local, 40
locally closed subscheme, 9
locally free sheaf, 18
locally in the flat topology, 21
logarithm, 27
minimal prime, 9
Morava stabiliser group, 28, 41
multiplicative group, 7
Noetherian, 9
norm map, 34
open mapping, 17
open subscheme, 9
orientation, 26
partition subgroup, 39
point, 14
polarised formal group, 26
polarised scheme, 20
pre-scheme, 8
pro-free, 11
projective bundle, 30
proper action, 24
quotient scheme, 9
reduced, 9
rigid section, 51
ring of functions, 8
schematic union, 9
section, 15
semilocal, 40
sheaf, 18
spec, 7
special fibre, 40
strict isomorphism, 8, 26
strongly topologically nilpotent, 12
subgroup divisor, 38
Teichm"uller map, 5
Thom space, 34
tolerable, 13