MULTICURVES AND EQUIVARIANT COBORDISM
N. P. STRICKLAND
Abstract.Let A be a finite abelian group. We set up an algebraic framewor*
*k for studying
Aequivariant complexorientable cohomology theories in terms of a suitab*
*le kind of equivariant
formal groups. We compute the equivariant cohomology of many spaces in th*
*ese terms, including
projective bundles (and associated Gysin maps), Thom spaces, and infinite*
* Grassmannians.
1.Introduction
Let A be a finite abelian group. In this paper, we set up an algebraic framew*
*ork for studying
Aequivariant complexorientable cohomology theories in terms of a suitable kin*
*d of formal groups.
In part, this is a geometric reformulation of earlier work of Cole, Greenlees, *
*Kriz and others on
equivariant formal group laws [3, 4, 5, 10]. However, the theory of divisors, r*
*esidues and duality
for multicurves is new, and forms a substantial part of the present paper. Alth*
*ough we focus on
the finite case, many results can be generalised to compact abelian Lie groups.*
* On the other hand,
we have evidence that nonabelian groups will need a completely different theory.
We now briefly recall the nonequivariant theory, using the language of formal*
* schemes de
veloped in [15]. Let E be an even periodic cohomology theory, and put S = spec*
*(E0) and
C = spf(E0CP+1) = lim!spec(E0CP+n). The basic facts are
n
(a)C is a formal group scheme over S.
(b) If we forget the group structure, then C is isomorphic to the formal affi*
*ne line bA1Sas a
formal scheme over S; in other words, C is a formal curve over S.
(c)For many interesting spaces X, the formal scheme spf(E0X) has a natural d*
*escription
as a functor of C; for example, we have spf(E0BU(d)) = Cd= d = Div+d(C), *
*the formal
scheme of effective divisors of degree d on C.
Now let U = UA be a complete Auniverse, and let SA be the category of Aspec*
*tra indexed on
U (as in [11]). Consider an Aequivariant commutative ring spectrum E 2 SA that*
* is periodic and
orientable in a sense to be made precise later. In this context, the right anal*
*ogue of CP 1 is the
projective space P U. This has an evident Aaction. We put S = spec(E0) and C =*
* spf(E0P U).
This is again a formal group scheme over S, but it is no longer a formal curve.*
* This appears to
create difficulties with (c) above, because we no longer have a good hold on Cd*
*= d or a good
theory of divisors on C.
Our first task is to define the notion of a formal multicurve over S, and to *
*show that C is an
example of this notion. Later we will develop an extensive theory of formal mul*
*ticurves and their
divisors, and show that many statements about generalized cohomology can be mad*
*e equivariant
by replacing curves with multicurves.
2.Multicurves
Definition 2.1. Let X = spf(R) be a formal scheme, and let Y be a subscheme of *
*X. We say
that Y is a regular hypersurface if Y = spf(R=J) for some ideal J = IY R that*
* is a free module
of rank one over R. Equivalently, there should be a regular element f 2 R such *
*that the vanishing
locus V (f) = spf(R=f) is precisely Y .
Let S = spec(k) be an affine scheme.
Definition 2.2. A formal multicurve over S is a formal scheme C over S such that
(a)C = spf(R) for some formal ring R
___________
Date: November 4, 2002.
1991 Mathematics Subject Classification. 55N20,55N22,55N91,14L05.
Key words and phrases. formal group, equivariant cohomology .
Partially supported by the Centre de Recerca Matem`atica and the Isaac Newton*
* Institute.
1
2 N. P. STRICKLAND
(b) There exists a regular element y 2 R such that for all k 0, the ideal R*
*yk is open in R,
and R=yk is a finitely generated free module over OS, and R = limR=yk.
k
(c)The diagonal subscheme C xS C is a regular hypersurface.
A generator d for the ideal I will be called a difference function for C (beca*
*use d(a, b) = 0 iff
(a, b) 2 iff a = b). We will choose a difference function d, but as far as po*
*ssible we will express
our results in a form independent of this choice. An element y as in (b) will *
*be called a good
parameter on C.
Remark 2.3. If S is a formal scheme, then we can write S = lim!Sfffor some fil*
*tered system of
ff
affine schemes, and formal schemes over S are the same as compatible systems of*
* formal schemes
over the Sffby [15, Proposition 4.27]. In the rest of this paper, we will gener*
*ally work over an
affine base but will silently use this result to transfer definitions and theor*
*ems to the case of a
formal base where necessary.
The formal affine line bA1S= spf(k[[x]]) is a formal multicurve, and the cate*
*gory of formal
multicurves is closed under disjoint union. Conversely, condition (c) implies *
*that the module
1C=S= I =I2 is free of rank one over R = OC, so formal multicurves may be tho*
*ught of as being
smooth and onedimensional. Similarly, if y is a good parameter then R is a fin*
*itely generated
projective module over k[[y]], which means that C admits a finite flat map to b*
*A1S, again indicating
a onedimensional situation. If k is an algebraically closed field, we shall se*
*e later that every small
formal multicurve over S is a finite disjoint union of copies of bA1S.
Remark 2.4. Note that I is the kernel of the multiplication map ~: Rb R !R, w*
*hich is split
by the map a 7! a 1. It follows that I is topologically generated by elemen*
*ts of the form
a b  ab 1. We also see by similar arguments that for any ideal J R, the*
* kernel of the
multiplication map (R=J) (R=J) !R=J is just the image of I and thus is gene*
*rated by d.
Definition 2.5. A formal multicurve group over S is a formal multicurve over S *
*with a commu
tative group structure.
In the presence of a group structure, axiom (c) can be modified.
Definition 2.6. Let C be a commutative formal group scheme over S. A coordinate*
* on C is a
regular element x 2 OC whose vanishing locus is the zerosection. Clearly, such*
* an x exists iff the
zerosection is a regular hypersurface.
Remark 2.7. If x is a coordinate, then so is the function __xdefined by __x(a) *
*= x(a).
Proposition 2.8. Let C be a formal group scheme over S satisfying axioms (a) an*
*d (b) in
Definition 2.2. Then C is a formal multicurve iff the zerosection S !C is a r*
*egular hypersurface.
More precisely, if x is a coordinate on C, then the function d(a, b) = x(b  a)*
* defines a difference
function, and if d is a difference function, then the function x(b) = d(0, b) i*
*s a coordinate.
The proof relies on the following basic lemma.
Lemma 2.9. Let C be a formal multicurve, and let f :X !C be any map of schemes*
*. Then the
function d0(x, b) = d(f(x), b) on X xS C is regular in OXxSC .
Proof.We have a short exact sequence as follows:
Rb R xd!Rb R ~!R.
We regard Rb R as a module over R via the map t 7! t 1. The map ~ is then Rl*
*inearly split
by the map t 7! t 1, so the sequence remains exact after applying the functor*
* OX bR(). The
resulting sequence is just 0
OXxSC xd!OXxSC ! OX ,
which proves the lemma.
Corollary 2.10. Let C q!S be a formal multicurve, and let S u!C be a section.*
* Then the
subscheme uS C is a regular hypersurface, and the ideal IuS is generated by t*
*he function
d0(c) = d(u(q(c)), c), or equivalently
d0= (C = S xS C ux1!C xS C d!A1).
Proof.Take X = S in the lemma.
MULTICURVES AND EQUIVARIANT COBORDISM 3
Proof of PropositionF2.8.irst suppose that the zero section is a regularQhypers*
*urface, so we can
choose a coordinateQx. It follows easily from axiom (b) that R ' 1k=0OS as to*
*pological OS
modules, so Rb R = 1k=0R as Rmodules, so 1 x is a regular element in R R. If *
*we regard x as
a function on C, this says that the function x1:(a, b) 7! x(b) is a regular ele*
*ment of OCxSC , whose
vanishing locus is precisely the closed subscheme where b = 0. The map s: (a, b*
*) 7! (a, b  a) is an
automorphism of C xS C, and s*x1 is the function d(a, b) = x(b  a). As s is an*
* automorphism,
we see that d is regular and its vanishing locus is the subscheme where a = b, *
*or in other words
the diagonal.
The converse is the case u = 0 of Corollary 2.10.
To formulate the definition of an equivariant formal group, we need some basi*
*c notions about
divisors.
Definition 2.11. A divisor on C is a scheme of the form D = spec(OC=J), where J*
* is an open
ideal generated by a single regular element, and OC=J is a finitely generated p*
*rojective module
over OS. Thus D is a regular hypersurface in C and is finite and very flat ove*
*r S. Strictly
speaking, we should refer to such subschemes as effective divisors, but we will*
* have little need for
more general divisors in this paper.
If Di = spf(R=Ji) is a divisor for i = 0, 1 then we put D0 + D1 := spf(R=(J0J*
*1)), which is
easily seen to be another divisor.
The degree of D is the rank of OD over k. Note that this need not be constant*
*, but that S can
be split as a finite disjoint union of pieces over which D has constant degree.
If T is a scheme over S, then a divisor on C over T means a divisor on the fo*
*rmal multicurve
T xS C over T .
Note that if D is an effective divisor of degree one, then the projection D ß*
*!S is an isomorphism,
1
so the map S ß! D C is a section of C. Conversely, if u: S ! C is a secti*
*on, then (by
Corollary 2.10) the image uS is a divisor of degree one, which is conventionall*
*y denoted by [u].
In the case of ordinary formal curves, it is wellknown that there is a modul*
*i scheme Div+d(C)
for effective divisors of degree d on C, and that it can be identified with the*
* symmetric power
Cd= d. Analogous facts are true for multicurves, but much more difficult to pro*
*ve. We will return
to this in Section 14.
Let A be a finite abelian group (with the group operation written additively)*
*. We write A* for
the dual group Hom (A, Q=Z).
Definition 2.12. Let X = spf(R) be a formal scheme, and let Y = spf(R=J) be a c*
*losed formal
subscheme. We say that X is a formal neighbourhood of Y if R is isomorphic to l*
*im R=Jm as a
*
* m
topological ring, or equivalently X = lim!spf(R=Jm ), which essentially means *
*that every point
m
in X is infinitesimally close to Y .
Definition 2.13. An Aequivariant formal group or Aefg over a scheme S is a fo*
*rmal multi
curve group C over S, together with a homomorphism OE: A* !C, such that C is t*
*he formal
neighbourhood of the divisor X
[OE(A*)] := [OE(ff)] C.
ff2A*
Remark 2.14. The notation OE: A* !C really means that OE is a homomorphism fro*
*m A* to the
group of sections of the projection C !S. Equivalently, we have a group scheme
a Y
A*x S = S = spec( OS)
ff2A* ff2A*
over S, and OE gives a homomorphism A*x S !C of group schemes over S.
Now choose a coordinate x on C, and put d(a, b) = x(ba). For any ff 2 A* we *
*have a function
xffon C defined by xff(a) = x(a  OE(ff)) = d(OE(ff), a). More precisely, xffis*
* the composite
C = S xS C OE(ff)xS1!C xS C subtract!C x!A1.
*
* Q
The vanishing locus of xffis the divisor [OE(ff)], so the vanishing locus of th*
*e product y := ffxff
is the divisor [OE(A*)]. We see using Corollary 2.10 that y is a regular elemen*
*t in OC. The final
condition in Definition 2.13 says that y is topologically nilpotent. It is not *
*hard to deduce that y
is a good parameter on C.
4 N. P. STRICKLAND
Proposition 2.15. Let f be a monic polynomial of degree d > 0 over OS, and let *
*R be the
completion of OS[x] at f. Then the scheme C = spf(R) = lim!V (fk) A1Sis a fo*
*rmal multicurve.
k
Proof.Condition (b) is clear, because {xi i < dj} is a basis for R=(fj) over O*
*S. Next, observe
that
R ' OS[[y]][x]=(f(x)  y) = OS[[y]]{xi i < d},
so
Rb R ' OS[[y0, y1]][x0, x1]=(f(x0)  y0, f(x1)  y1) = OS[[y0, y1]]{xi0xj*
*1 i, j < d}.
It is clear that y1 y0 is not a zerodivisor in this ring, and x1 x0 divides *
*y1 y0 so it is also
not a zerodivisor. It is not hard to check that the multiplication map Rb R *
*! R induces an
isomorphism (Rb R)=(x1  x0) ' R, and it follows that x1  x0 generates I , an*
*d thus that (c)
holds.
Definition 2.16. We say that a formal multicurve C over S is embeddable if it h*
*as the form
lim!V (fk) as above for some monic polynomial f.
k
Lemma 2.17. Suppose that k = OS is an algebraically closed field, and that C is*
* a formal
multicurve over S. Then C is a finite disjoint union of copies of bA1S, and is *
*embeddable.
__
Proof.Let y 2 R be a good parameter. Then the ring R := R=y is a finitedimensi*
*onal algebra
over the field k, so it splits as a finite product of local algebras. As R is c*
*omplete at (y) we can lift
this splitting to R, which splits C as a disjoint union, say C = C1q . .q.Cr. I*
*t is easy to see that
each Ci is a formal_multicurve._Put_Ri=_OCi, so R_= R1 x . .x.Rr._Let yi be the*
* component
of y in Ri and put Ri = Ri=yi, so R = R1 x . .x.Rr. Moreover,_Ri is local, wit*
*h maximal
ideal mi say. As k is algebraically closed we see that Ri=mi = k. This gives an*
* augmentation
u*i:Ri! k, or equivalently a section ui:S !Ci. It follows from Corollary 2.10*
* that the kernel_
of u*iis generated_by a single regular element, say xi. This means that the im*
*age of xi in Ri
generates mi, so Ri' k[xi]=xmifor some m, and thus yidivides xmi. On the other *
*hand, we clearly
have u*(yi) = 0 so xidivides yi. It is now easy to check that Ri= k[[xi]], so C*
*i' bA1S.
Finally, as k is algebraically closed,Qit is certainly infinite, so we can ch*
*oose distinct elements
~1, . .,.~r 2 kQsay. If we put f(x) = i(x  ~i) we find that the completion *
*of k[x] at (f) is
isomorphic to ri=1k[[x]] and thus to OC. This proves that C is embeddable.
3.Differential forms
We next recall some basic ideas about differential forms, and record some for*
*mulae that will be
useful later in our study of residues.
Given a formal multicurve C over S, we put
= 1C=S= I =I2 ,
and call this the module of differential forms on C.
We also put 2 = spf(OCxSC =I2 ), and regard this as the secondorder infinit*
*esimal neigh
bourhood of in C xS C. In these terms, is the module of functions on 2 tha*
*t vanish on
.
Given a difference function d 2 I , we let ff be the image of d in ; this g*
*enerates freely as
a module over OC, so we can regard as a trivialisable line bundle on C.
For any function f 2 OC, we write df for the image of 1 f  f 1 in , or *
*equivalently the
function (a, b) 7! f(b)  f(a) on 2. As usual, we have the Leibniz rule
d(fg) = fd(g) + gd(f).
Now suppose that C has a commutative group structure. In particular, this giv*
*es a zerosection
Z C, and we write Z2 = spec(OC=I2Z) and
! = IZ=I2Z= { functions on Z2 that vanish}on.Z
The map b 7! (0, b) gives an inclusion Z2 ! 2 and thus a map !!, which in t*
*urn gives an
isomorphism Z = ! of line bundles on S. The image of df under this map is the*
* element d0f
corresponding to the function b 7! f(b)f(0) on Z2. If x is a coordinate on C, *
*then d0x generates
! freely as a module over OS.
Next, for any function f 2 OC we define a function Df on 2 by
(Df)(a, b) = f(b  a)  f(0).
MULTICURVES AND EQUIVARIANT COBORDISM 5
This construction gives a map D: OC ! . If x is a coordinate then Dx is the re*
*striction of the
usual difference function d(a, b) = x(b  a) to 2, so it is a generator of .
It is easy to see that Df depends only on d0f, and thus that D induces an OS*
*linear inclusion
! ! , right inverse to the restriction map ! Z = !. A differential form*
* is said to be
invariant if it lies in the image of this map.
By extension of scalars, we obtain an OClinear map OC OS ! ! , sending f *
* d0g to fDg.
In particular, it sends f d0x to fDx, and so is an isomorphism.
4.Equivariant projective spaces
We now start to build a connection between multicurves and Aequivariant topo*
*logy (where A
is a finite abelian group). Naturally, this involves the generalised cohomology*
* of the projective
spaces of representations of A. In this section, we assemble some facts about t*
*he homotopy theory
of such projective spaces.
For ff 2 A* = Hom (A, Q=Z) we write Lfffor C with A acting by a.z = e2ßiff(a)*
*z. In particular,
L0 has trivial action, and Lff Lfi= Lff+fi. For any finitedimensional represe*
*ntation V , we put
V [ff] = {v 2 V  av = e2ßiff(a)v for alla 2 A}.
L L
It is wellknown that V = ffV [ff] and Hom C[A](V, W ) = ffHomC(V [ff], W [*
*ff]). It follows that
if there exists an equivariant linear embedding V ! W , then the space of such*
* embeddings is
connected, giving a canonical homotopy classLof maps P V !P W of projective sp*
*aces.
We write U[ff] = Lff C1 , and U = UA = ffU[ff], so U is a complete Aunive*
*rse. We
write P U for the projective space associated to U, which has a natural Aactio*
*n. By the previous
paragraph, for any finitedimensional representation V , there is a canonical m*
*ap P V !P U up to
homotopy. Similarly, the space of equivariant linear isometries U U !U is co*
*ntractible, which
gives a canonical homotopy class of maps P U x P U !P U, making P U an abelian*
* group up to
equivariant homotopy. We can choose a conjugatelinear equivariant automorphism*
* Ø: U ! U,
and the resulting map P U !P U is the negation map for our group structure.
It is wellknown that P U is the classifying space for equivariant complex li*
*ne bundles. More
precisely, for any Aspace X, we write PicA(X) for the group of isomorphism cla*
*sses of equivariant
complex line bundles over X. Let T denote the tautological line bundle over P U*
*, so T 2 Pic(P U).
Then for any Aspace X, the construction [f] 7! [f*T ] gives a group isomorphis*
*m [X, P U]A '
PicA(X). Note that we regard T as the universal example; some other treatments *
*in the literature
use the dual bundle T *= O(1) instead.
Note that A acts by scalars on U[ff], and thus acts as the identity on P U[ff*
*] P U. Moreover,
the map L 7! Lff L gives a homeomorphism CP 1 = P (C1 ) ! P U[ff]. Using this*
*, we have
a homeomorphism (P U)A = A* x CP 1, and thus a bijection ß0((P U)A) = A*, which*
* is easily
seen to respect the natural group structures. Thus, the group structure on P U *
*gives a translation
action (up to homotopy) of A* on P U. We write øff:P U ! P U for translation b*
*y an element
ff 2 A*.
For various purposes we will need to use an Afixed basepoint in P U. We have*
* embeddings
Lff!U[ff] !U, and P Lffis an Afixed point. Any other fixed point lies in the*
* same component
of (P U)A as P Lfffor some ff, so it can be replaced by P Lfffor most purposes.*
* Moreover, the
map øffgives a homotopy equivalence of pairs (P U, P Lfi) !(P U, P Lff+fi). Wh*
*ere not otherwise
stated, we use P L0 as the basepoint.
Proposition 4.1. Let V , W and X be unitary representations of A, where V and W*
* have finite
dimension and X is a colimit of finitedimensional subrepresentations. Put U =*
* V W X.
Then there is a homotopycommutative diagram as follows, in which the maps mark*
*ed q are the
obvious collapses, the maps marked j are the obvious inclusions, and ffi is the*
* diagonal map.
P U ________________________________________________wPfUfxiP U
 
qV W qV^qW
 
u u
P U=P (V W )______wP_(V X)=P V ^ P (W X)=P_W___wP U=P V ^ P U=P W
ffi j^j
_
Moreover, if dim(X) = 1 then ffiis just the standard homeomorphism
SHom(X,V W)= SHom(X,V )^ SHom(X,W).
6 N. P. STRICKLAND
All maps and homotopies are natural for isometric embeddings of V , W and X.
Remark 4.2. The above diagram gives a map
_* * * *
ffi:E (P (V X), P V ) E (P (W X), P W ) !E (P U, P (V W )).
_*
In his unpublished thesis [3], Cole writes a * b for ffi(a b). The idea of us*
*ing this construction
seems to be original to that thesis; our approach differs only in being somewha*
*t more geometric.
Proof.Assume for the moment that X is finitedimensional. We start by defining *
*a map
__fl:P U=P (V W ) !P U=P V ^ P U=P W,
_
which will be homotopic to (j ^ j) O ffi. For u = (v, w, x) 2 Ux := U \ {0} we *
*put
s = s(u) = (kwk  kvk)=(kvk + kwk + kxk).
Note that s(u) 2 [1, 1], and s(~u) = s(u) for all ~ 2 Cx, and s(u) 0 iff kwk*
* kvk. We next
define ff, fi :Ux !U by
(
ff(v, w,=x)((1  s)v, sw, x)ifs 0
(v, 0, x) ifs 0
(
fi(v, w,=x)(0, w, x) ifs 0
(sv, (1 + s)w, x)ifs 0.
Note that ff(~u) = ~ff(u) and similarly for fi.
We claim that when u 6= 0, the line joining u to ff(u) never passes through 0*
* (so in particular
ff(u) 6= 0). Indeed, if s 0, then the points on the line have the form (v, tw*
*, x) for 0 t 1. Thus,
the line can only pass through zero if v = x = 0. The relation s 0 means that*
* kwk kvk = 0,
so w = 0 as well, contradicting the assumption that u 6= 0. In the case s > 0, *
*the points on the
line have the form ((1  ts)v, (1  t + ts)w, x). As s > 0 and 0 t 1 we hav*
*e 1  t + ts > 0.
For the line to pass through zero we must thus have x = w = 0, and the relation*
* s 0 means
that kvk kwk = 0, again giving a contradiction. Similarly, the line from u to*
* fi(u) never passes
through 0.
It follows that ff and fi induce selfmaps of P U that are homotopic to the i*
*dentity, so the map
fl = (ff, fi): P U !P U x P U is homotopic to the diagonal map ffi.
Next, note that if u 2 V W , then for s 0 we have fl(u) 2 U x W , and for*
* s 0 we have
fl(u) 2 V x U. It follows that the induced map on projective spaces has
fl(P (V W )) (P U x P W ) [ (P V x P U),
so there is an induced map
__fl:P U=P (V W ) !P U=P V ^ P U=P W.
As fl is homotopic to ffi,_we see that __flO qV W ' (qV ^ qW ) O ffi.
To construct the map ffi, we need a slightly different model. Clearly
P U \ P (V W ) = (V x W x Xx )=Cx = (V x W x S(X))=S1,
and P U=P (V W ) is the onepoint compactification of this. Similarly, P (V *
* X)=P V ^ P (W
X)=P W is the_onepoint compactification of the space (V x S(X))=S1x (W x S(X))*
*=S1. We can
thus define ffiby giving a proper map
V x W x S(X) !V x S(X) x W x S(X)
with appropriate equivariance. The map in question just sends (v, w, x) to (v, *
*x, w, x).
If X is onedimensional and (v, x) 2 V x S(X) then we have a linear map ff: X*
* ! V given
by ff(x) = v, which does not change if we multiply (v, x) by an element of S1. *
* This gives a
homeomorphism (V x S(X))=S1 = Hom (X, V_), and thus P (V X)=P V = SHom(X,V.)I*
*t is easy
to see that with this identification, ffiis just the standard homeomorphism
SHom(X,V W)= SHom(X,V )^ SHom(X,W).
_ __
We now show that (j ^ j) O ffi' fl. Put
T = {((v0, w0, x0), (v1, w1, x1)) 2 U2  k(w0, x0)k = k(v1, x1)k = *
*1},
so that P U=P V ^ P U=P W is the onepoint compactification of T=(S1 x S1). Def*
*ine maps
`t:V x W x S(X) !T
MULTICURVES AND EQUIVARIANT COBORDISM 7
for 0 t 1 by 8i j
< ((1st)v,stw,x)_, (0,iw,fx)s 0
`t(v, w, x) = :i k(stw,x)k(stv,(1+st)w,x)j
(v, 0, x), ____________k(stv,x)kifs 0,
where s = (kwk  kvk)=(kvk + kwk + kxk) as before. We claim that these maps are*
* proper. To
see this, put
((v0, w0, x0), (v1, w1, x1)) = max(kv0k, kw1k),
and Tk = {t 2 T  (t) k}. It is easy to see that every compact subset of T i*
*s contained in some
Tk, so it will be enough to show that `1tTk is compact. In the case s 0 we h*
*ave 0 1  st 1
and k(stw, x)k kxk = 1 so k((1  st)v=k(stw, x)k)k kvk kwk, so (`t(v, w,*
* x)) = kwk.
Similarly, when s 0 we have (`t(v, w, x)) = kvk, so in general (`t(v, w, x)*
*) = max(kvk, kwk).
It follows immediately that `tis proper, and we get an induced family of maps
`t:P U=P (V W ) !P U=P V ^ P U=P W.
_ __
It is easy to see that `0 = (j ^ j) O ffiand `1 = fl. The proposition follows e*
*asily (for the case where
X has finite dimension).
If X has infinite dimension, we apply the above to all finite dimensional sub*
*representations of
X. We see by inspection that all constructions pass to the colimit, so the conc*
*lusion is valid for
X itself.
By an evident inductive extension, we obtain the following:
CorollaryL4.3. Let L1, . .,.Ld be onedimensional representations of A, and let*
* X be as above.
Put Y = iLiand U = Y X. Then there is a homotopycommutative diagram as fol*
*lows:
P U ________________________________wPfUrfi
 q
q 
u ^ ^ u
P U=P Y______w_ P (Li X)=P Li_____w P U=P Li
ffi i j i
_
Moreover, if dim(X) = 1 then ffiis just the standard homeomorphism
^
SHom(X,Y )= SHom(X,Li).
i
We conclude with some further miscellaneous observations about the space P U.
Proposition 4.4. The space P U is equivariantly equivalent to F (EA+, CP 1) (wh*
*ere CP 1 is the
usual space with trivial Aaction). Equivalently, P U is the second space in th*
*e Borel cohomology
spectrum F (EA+, H), so [X, P U]A = H2(XhA) for any Aspace X. Moreover, the sp*
*ace P U is
equivariantly equivalent to S1 with the trivial action.
Proof.There is an evident inclusion CP 1 = P (UA ) !(P U)A !P U. This is a no*
*nequivariant
equivalence, and so gives an equivariant equivalence F (EA+, CP 1) ! F (EA+, P*
* U). On the
other hand, the collapse map EA+ !S0 gives a map j :P U !F (EA+, P U) ' F (EA*
*+, CP 1).
We claim that this is an equivalence. Indeed, if we take fixed points for a sub*
*group A0 A we
get a map A*0x CP 1 !F ((BA0)+, CP 1) of commutative Hspaces. It is clear that
8
>Z ifk = 2
:0 otherwise.
On the other hand, we have
ßkF ((BA0)+, CP 1) = [ k(BA0)+, K(Z, 2)] = H2kBA0.
This clearly vanishes for k > 2 and gives Z for k = 2. Standard arguments with *
*the coefficient
sequence Z !Q !Q=Z give H1BA0 = 0 and H2BA0 = A*0, showing that ß*F ((BA0)+, *
*CP 1) is
abstractly isomorphic to ß*(A*0x CP 1). With a little more work one sees that t*
*he isomorphism
is induced by j, and the first part of the proposition follows.
We now see that
P U ' F (EA+, CP 1) = F (EA+, CP 1) = F (EA+, S1).
8 N. P. STRICKLAND
As above we find that (
ßk(F (EA+, S1)A0) = H1kBA0 = Z ifk = 1
0 otherwise.
It follows that the obvious map S1 !F (EA+, S1) is an equivariant equivalence.
Proposition 4.5. Let T be the tautological line bundle over P U, and let S(T n)*
* be the unit circle
bundle in the n'th tensor power of T . Then S(T n) is equivariantly equivalent *
*to F (EA+, B(Z=n)).
Proof.It is wellknown that in the case A = 0 we have S(T n) = B(Z=n) = K(Z=n, *
*1). In the
general case, note that S(T n) consists of pairs (L, u) where L 2 P U and u 2 L*
*n and kuk = 1.
Suppose that (L, u) is fixed by a subgroup A0 A. We see that A0 acts on L by *
*some character
ff 2 A*0, so A0 acts on u by nff, but u is fixed so nff = 0. Given that nff = 0*
*, we see that every
point in Ln is fixed by A0. Using this, we see that S(T n)A0 = A*0[n]xB(Z=n), w*
*here A*0[n] denotes
the subgroup of points of order n in A*0. Using this, we find that ß*S(T n)A0 =*
* H1*(BA0; Z=n),
and the claim follows by the same method as in the previous proposition.
Proposition 4.6. Put F = {B S1 x A  B \ S1 = {1}}, which is a family of subg*
*roups of
S1 x A. Then the unit sphere S(U) is a model for EF, and so P U = (EF)=S1.
Proof.First, we let S1 Cx act on S(U) by multiplication, and let A act in the*
* usual way.
These actions commute and so give an action of S1 x A. We need only check that *
*S(U) has the
characterizing property of EF, or in other words that S(U)B is contractible for*
* B 2 F and empty
for B 62 F. If B 2 F then B \ S1 is trivial so B is the graph of a homomorphism*
* OE: A0 !S1 for
some subgroup A0 A. Put
V = {v 2 U  a.v = OE(a)1v for alla 2 A0},
so S(U)B = S(V). As U is a complete A0universe, we see that V is infinite dime*
*nsional, and so
S(V) is contractible as required. On the other hand, as S1 acts freely on S(U),*
* it is clear that
S(U)B = ; whenever B 62 F.
5.Equivariant orientability
Now let E be a commutative Aequivariant ring spectrum. We next need to formu*
*late suitable
notions of orientability and periodicity for E, and deduce consequences for the*
* rings E*P V . Our
results differ from those of [3] only in minor points of technical detail. We s*
*tart by introducing
some auxiliary ideas.
Definition 5.1. Let R be an Ealgebra spectrum, and M a module spectrum over R.*
* We
say that M is a free Rmodule if it is equivalent as an Rmodule to a wedge of *
*(unsuspended)
copiesLof R, or equivalently, there is a family of elements ei2 ß0M such that t*
*he resulting maps
i[ nA=B+, R] ![ nA=B+, M] are isomorphisms for all n 2 Z and all B A. We s*
*ay that such
elements eiare universal generators for for ß0M over ß0R. We will often leave t*
*he identification of
R and M implicit. For example, if we say that an element e is a universal gener*
*ator for E0(X, Y )
over E0X, we are referring to the case R = F (X+, E) and M = F (X=Y, E).
Definition 5.2. Let E an Aequivariant ring spectrum, and consider a class x 2 *
*E0(P U, P L0).
For any ff 2 A* we can embed Lff L0 in U, and thus restrict x to get a class u*
*Lff2 E0(P (Lff
L0), P L0) = eE0SLff. This in turn gives an Emodule map mff: LffE !E.
We say that x is a complex coordinate for E if for all ff the map mffis an eq*
*uivalence, or
equivalently uLffgenerates LffE as an Emodule. We say that E is periodically*
* orientable if it
admits such a coordinate. We say that E is evenly orientable if in addition, we*
* have ß1E = 0.
From now on, we assume that E is periodically orientable. We choose a complex*
* coordinate x,
but as far as possible we state our results in a form independent of this choic*
*e. We write __x= Ø*x,
where Ø: P U ! P U is the negation map for the group structure. It is easy to *
*see that this is
again a coordinate.
Recall that for any line bundle L over X, there is an essentially unique map *
*fL :X !P U with
f*T ' L (where T is the tautological bundle over P U). We define the Euler clas*
*s of L by
e(L) = f*L*(x) = f*L(__x).
Thus, the element x 2 E0P U is the Euler class of T *, and __xis the Euler clas*
*s of T .
MULTICURVES AND EQUIVARIANT COBORDISM 9
Remark 5.3. There is some inconsistency in the literature about whether e(L) sh*
*ould be f*L(x)
or f*L(__x). The convention adopted here is the opposite of that used in [15], *
*but I believe that it is
more common in other work and has some technical advantages. The conventions us*
*ed elsewhere
in this paper are fixed by the following requirements.
(a)We have e(V W ) = e(V )e(W ).
(b) The Euler class of V is the restriction of the Thom class in eE0XV to the*
* zero section
X XV .
Our substitute for the nonequivariant theory of Chern classes will be more abst*
*ract,Pso we will
not need sign conventions. The r^ole normally played by theLChern polynomialQ *
*i+j=dim(V )cixj
will be played by a certain element fV ; if A = 0 and V = iLithen fV = i(x +*
*F e(Li)).
Next note that we can define
xff:= ø*ffx 2 E0(P U, P Lff).
Because
(ø*ffT )* = (Lff T )* = Lff T *= Hom (T, Lff),
we have xff= e(Hom (T, Lff)). If L is a onedimensional representation isomorph*
*ic to Lff, we also
use the notation xL for xff. We can identify E0(P (Lfi Lff), P Lff) with eE0SL*
*fiff, and we find
that xffrestricts to ufiff, which is a universal generator.
Now consider a finitedimensional representation V of A. We have a canonical *
*homotopyLclass of
embeddings P V !P U, and thus a welldefined group E0(P U, P V ). We can write*
* V as di=1Li,
and Corollary 4.3 gives a map ^
P U=P V ! P U=P Li
i
compatible with the diagonal.QUsing this, we can pull back xL1 ^ . .^.xLd to ge*
*t a class xV 2
E0(P U, P V ) that maps to ixffiin E0P U. Note that for any representation W *
*containing V we
can choose an embedding W !U and pull back xV along the resulting map P W !P *
*U to get a
class in E0(P W, P V ), which we again denote by xV .
Lemma 5.4. Let V W be complex representations of A, with dim(W=V ) = 1. Then *
*xV is a
universal generator for E0(P W, P V ).
Proof.Write VV= L1 . . .Ld as before, and X = W V , so W = V X and P W=P *
*V =
SHom(X,V )= iSHom(X,Li). Because x is a complex coordinate, we know that xLi2 *
*E0(P W, P Li)Q
restricts to a universal generator viof SHom(X,Li). It follows from Corollary 4*
*.3 that xV = ivi2
eE0SHom(X,V )= E0(P W, P V ), and this is easily seen to be a universal generat*
*or.
Corollary 5.5. Let 0 = U0 < U1 < . .<.Ud = U be representations of A with dim(U*
*i) = i. Then
{xUi i < d} is a universal basis for E0P U over E0.
Proof.This follows by an evident induction from the lemma.
Remark 5.6. As __xis another coordinate, it gives rise to another universal bas*
*is {__xUi i < d} for
E0P U, which is sometimes more convenient.
We record separately some easy consequences that are independent of the choic*
*e of flag {Ui}:
Proposition 5.7. Let U be a ddimensional representation of A. Then
(a)F (P U+, E) is a free module of rank d over E.
(b) If U = V W then the restriction map F (P U+, E) !F (P V+, E) is split *
*surjective. The
kernel is a free module of rank one over F (P W+, E), generated by xV .
We now put S = spec(E0) and R = E0P U and C = spf(R). We must show that C is*
* an
equivariant formal group over S.
We first exhibit a topological basis for R. We can list the elements of A* as
A* = {ff0 = 0, ff1, . .,.ffn1}
(where n = A), and then define ffk for all k 0 by ffni+j= ffj. We then hav*
*e an evident
filtration
0 = V0 < V1 < V2 < . .<.U = lim!Vk
k
10 N. P. STRICKLAND
L
where Vk = j 0.*
* In particular it
divides y, which is a regular element in R, so x is also a regular element. It *
*is also now easy to
see x generates the ideal E0(P U, P L0), which is just the augmentation ideal i*
*n the Hopf algebra
R, so the vanishing locus of x is the zerosection in C. Thus x is a coordinate*
* on C, showing (via
Proposition 2.8) that C is in fact a formal multicurve group.
Next, recall that ß0((P U)A) = A*, which gives a map A* !P U of groups up to*
* homotopy, and
thus a map OE: A* !C of formal group schemes. By working through the definitio*
*ns, we see that
the imagePof the section OE(ff) is the closed subscheme spec(E0P Lff) = spec(R=*
*xff), so the divisor
D := ff[OE(ff)] is Y
spec(R= xff) = spec(R=y) = E0P C[A].
ff
As y is topologically nilpotent, we see that any function on C that vanishes on*
* D is topologically
nilpotent, so C is a formal neighbourhood of D. We have thus proved the followi*
*ng result:
Theorem 5.8. Let E be a periodically orientable Aequivariant ring spectrum. Th*
*en the scheme
C := spf(E0P U) is an Aequivariant formal group over S := spec(E0).
Remark 5.9. We have I0 = {f 2 OC  f(0) = 0} = E0(P U, P C), and thus I20= E0(P*
* U, P (C
C)), and thus
! = I0=I20= E0(P (C C), P C) = eE0S2 = ß2E.
6. Simple examples
Let bCbe a nonequivariant formal group over a scheme S, so bCis the formal ne*
*ighbourhood of
its zero section. For any finite abelian group A, we can of course let OE: A* *
*!bCbe the zero map,
and this gives us an Aequivariant formal group. More generally, any homomorphi*
*sm A* !Cb
will give an Aefg, although often there will not be any homomorphisms other th*
*an zero.
Now suppose that bCis the formal group associated to a nonequivariant even pe*
*riodic ring
spectrum bE. We then have an Aequivariant ring spectrum E = F (A+, bE) (which *
*the Wirthmüller
isomorphism also identifies with A+ ^ bE). This satisfies E*X = bE*res(X), wher*
*e res:SA !S0 is
the restriction functor. It follows easily that E is periodically orientable, a*
*nd that the associated
equivariant formal group is just bC, equipped with the zero map OE: A* !bCas a*
*bove.
For a slightly more subtle construction, suppose we allow S to be a formal sc*
*heme, and assume
that some prime p is topologically nilpotent in OS. Suppose also that the forma*
*l group bChas
finite height n. Put S0= Hom (A*, bC); it is wellknown that OS0is a free modul*
*e of rank An over
OS, so S0is finite and flat over S. By definition, S0is the universal example o*
*f a formal scheme T
over S equipped with a homomorphism from A* to the group of maps T !Cbof forma*
*l schemes
over S, or equivalently the group of sections of T xS C over T . If we put C0= *
*S0xS bC, there is
thus a tautological map OE: A* !C0. Here C0is an ordinary formal group over S0*
*and thus is the
formal neighbourhood of its zero section. It follows that (C0, OE) is automatic*
*ally an Aequivariant
formal group over S0.
Now suppose we have a K(n)local even periodic ring spectrum bE. We give the*
* ring ß0bE
the natural topology as in [8, Section 11] _ in most cases of interest, this is*
* the same as the
Inadic topology. We then put S = spf(ß0bE) and bC= spf(Eb0CP 1), which gives *
*an ordinary
formal group of height n over S. Let EA denote a contractible space with free A*
*action, and put
E = F (EA+, bE). This is a commutative Aequivariant ring spectrum, with E*X = *
*bE*XhA, where
MULTICURVES AND EQUIVARIANT COBORDISM 11
XhA denotes the homotopy orbit space or Borel construction. In particular, we h*
*ave E0(point) =
bE0BA, and it is wellknown that this is canonically isomorphic to OHom(A*,Cb).*
* Next, observe
that we have an Aequivariant inclusion P U[0] ! P U, which is nonequivariantl*
*y a homotopy
equivalence, so the map EAxP U[0] !EAxP U is an equivariant homotopy equivalen*
*ce. It follows
that E*P U = E*P U[0] = bE*(BA x CP 1) = bE*BA Eb*bE*CP 1, and thus that spf(E*
*0P U) =
Hom(A*, bC)xCb. This shows that the equivariant formal group associated to E is*
* just the pullback
C0= S0xS bCas discussed above.
7.Formal groups from algebraic groups
We now show how to pass from algebraic groups (in particular, elliptic curves*
* or the multiplica
tive group) to equivariant formal groups.
7.1. The multiplicative group. Let S = spec(k) be a scheme, and consider the gr*
*oup scheme
Gm xS = spec(k[u, u1]) over S. Suppose we are given a homomorphism OE from A*x*
*S to Gm xS
of group schemes over S, or equivalently a homomorphism OE: A* !kx of abstract*
* groups. We
can then form the divisor X
D = [OE(ff)] = spec(k[u 1]=y),
Q ff
where y = ff(1  u=OE(ff)). It is convenient to observe that u is invertible *
*in k[u]=y and thus in
k[u]=ym for all m, so D can also be described as spec(k[u]=y). We then define C*
* to be the formal
neighbourhood of D in Gm x S, so
C = lim!spec(k[u]=ym ) = spf(k[u]^y),
m
which is an embeddable formal multicurve. It is easy to see that this is a subg*
*roup of Gm x S and
is an equivariant formal group, with coordinate x = 1  u.
The universal example of a ring with a map A* !kx is k = Z[A*], which can be*
* identified with
the representation ring R(A). Thus, the universal example of a scheme S with a *
*map A* x S !
Gm xS as above is S = Hom (A*, Gm ) = spec(R(A)). We can apply the above constr*
*uction in this
tautological case to get an equivariant formal group C over Hom (A*, GmQ). Expl*
*icitly, if we let
vff2 Z[A*] be the basis element corresponding to ff 2 A* and put y = ff(1  uv*
*ff) 2 Z[A*][u],
then C = spf(Z[A*][u]^y).
Theorem 7.1 (ColeGreenleesKriz). The Aefg associated to the equivariant comp*
*lex Ktheory
spectrum KA is isomorphic to the Aefg C over Hom (A*, Gm ) constructed above.
Proof.This is just a geometric restatement of [4, Section 6]. It is proved by i*
*dentifying K*AP U
with K*AxS1EF+ (where F = {B A x S1  B \ S1 = {1}} as in Proposition 4.6) an*
*d applying
a suitable completion theorem.
7.2. Elliptic curves. We now carry out the same program with the multiplicative*
* group replaced
by an elliptic curve (with some technical conditions assumed for simplicity). S*
*uppose that we are
given a ring k and an element ~ 2 k, and that 2, ~ and 1  ~ are invertible in *
*k. Let eCbe
the elliptic curve given by the homogeneous cubic y3 = x(x  z)(x  ~z), so the*
* zero element is
O = [0 : 1 : 0], and the points P := [0 : 0 : 1], Q := [1 : 0 : 1] and R := [~ *
*: 0 : 1] are the three
points of exact order two in eC. Define rational functions t and r on eCby t([x*
* : y : z]) = x=y and
r([x : y : z]) = z=y. One checks that the subscheme U = eC\ {P, Q, R} is the af*
*fine curve with
equation r = t(t  r)(t  ~r), and that on U, the function t has a simple zero *
*at O and no other
poles or zeros.
NowTlet A be an abelian group of odd order n, and let OE: A* !eCbe a homomor*
*phism. Define
V = ff(U + OE(ff)), which is an affine open subscheme of U.
Lemma 7.2. For each fi 2 A*, the section OE(fi): S !Ceactually lands in V .
Proof.We first show that for all fl 2 A*, the section OE(fl) lands in U. Put D *
*= [P ] + [Q] + [R],
so U = eC\ D. Let T be the closed subscheme of points s 2 S where OE(fl)(s) 2 D*
*; we must show
that T = ;. As n is odd and D is the divisor of points of exact order 2, we see*
* that multiplication
by n is the identity on D, but of course n.OE(ff) = O. We conclude that over T *
*we have O 2 D.
As 2 is invertible in k we know that O and D are disjoint, so T = ; as required.
We now apply this to fl = fi  ff to deduce that OE(fi) 2 U + OE(ff). This ho*
*lds for all ff, so
OE(fi) 2 V as claimed.
12 N. P. STRICKLAND
P
We nowQdefine C to be the formal neighbourhood of the divisor D = ff[OE(ff)*
*] in V . If we put
s(a) = fft(aOE(ff)) then s 2 OV and the vanishing locus of s is just D, so we*
* have OC = (OV )^s.
Using this, we see that C is an equivariant formal group, with coordinate t and*
* good parameter s.
Now suppose instead that we are given a curve eCover S as above, but not the *
*map OE: A* !eC.
We can then consider the scheme S1 = Hom (A*, eC), which is easily seen to be a*
* closed subscheme
of Map (A*, U) and thus affine. We can thus pull back eCto get a curve eC1over*
* S1 equipped
with a tautological map OE: A* !Ce1, and we can carry out the previous constru*
*ction to get an
equivariant formal group C1 over S1. This should be associated to some kind of *
*Aequivariant
elliptic cohomology theory. It is not hard to construct a suitable theory if O*
*S is a Qalgebra;
see [18], for example. For more general base schemes, little is known.
8.Equivariant formal groups of product type
A simple class of Aefg's can be constructed as follows. Let bCbe an ordinary*
*, nonequivariant
formal group, and let B be a subgroup of A. We then have a formal multicurve C *
*:= B* x bCand
a homomorphism
OE := (A* res!B* inc!B* x bC= C),
giving an Aefg. Equivariant formal groups of this kind are said to be of produ*
*ct type.
Proposition 8.1. An Aefg (C, OE) is of product type iff for every character ff*
* 2 A* with OE(ff) 6= 0
in C (or equivalently, x(OE(ff)) 6= 0 in OS), the element x(OE(ff)) is invertib*
*le in OS. (This is easily
seen to be independent of the choice of coordinate.)
Proof.First suppose that for all ff with OE(ff) 6= 0, the element x(OE(ff)) is *
*invertible. The kernel
of OE is a subgroup of A*, so it necessarily has the form ann(B) for some B A*
*, so OE factors as
A* res!B* _!C for some _. By assumption, x(_(fi)) is invertible for all fi 2*
* B* \ {0}.
Let bC= {a 2 C  x(a) is nilpotent} be the formal neighbourhood of 0 in C, an*
*d define
oe :B* x bC!C by oe(fi, a) = _(fi)Q+ a. We need to show that oe is an isomorph*
*ism. For this, we
define xfi(a) = x(a_(fi)) and y = fi2B*xfiand R = OC. From the definition of *
*an equivariant
formal group, we know that R = R^y, and it is clear that
Y
OB*xCb= R^xfi.
fi
It will thus suffice to show that the natural map
Y
R^y! R^xfi
fi
is an isomorphism. This will follow from the Chinese Remainder Theorem if we ca*
*n check that
the ideal (xfi(a), xfl(a)) contains 1 whenever fi 6= fl. This is clear because *
*modulo that ideal, we
have _(fi) = a = _(fl), so _(fi  fl) = 0, so x(_(fi  fl)) = 0, but x(_(fi  f*
*l)) is invertible by
assumption. Thus, C is of product type, as claimed.
Conversely, suppose that C is of product type. The vanishing locus of x is co*
*ntained in {0}xCb,
so x must be invertible on (B* \ {0}) x bC. It follows immediately that when OE*
*(ff) 6= 0 we have
OE(ff) 2 (B* \ {0}) x bCand so x(OE(ff)) is invertible, as required.
Corollary 8.2. Every Aequivariant formal group over a field is of product type.
Proof.This is immediate from the proposition.
We next show how groups of product_type occur in topology. For this we need *
*to use the
geometric fixed point functors OEB:SA ! S0 for B A. The definition and prope*
*rties of these
functors will be recalled in Section 10.
Theorem 8.3. Let bKbe a nonequivariant even periodic cohomology theory, with as*
*sociated formal
group bCover S, and let B be a subgroup of A. Define a cohomology theory K* on *
*SA by K*X =
bK*_OEBX. Then K is evenly periodic, and the associated equivariant formal grou*
*p is just B* x bC
over S.
MULTICURVES AND EQUIVARIANT COBORDISM 13
__B B *
*__B
Proof.Note that OESV = SV for any virtual complex representation V , and that *
*OE 1 X =
1 XB for any based Aspace X. It follows that ß1K = ß1bK = 0 and that the per*
*iodicity
isomorphism F (S2n, bK) = bKgives an isomorphism
B *B *
K*(X+ ^ SV ) = bK*(XB+^ SV ) = bKX+ = K X+
of modules over K*X+. This implies that K is evenly periodic, with K0(point) = *
*bK0(point) and
thus spec(K0(point)) is the base scheme S for bC. We also have
K0P U = bK0(P U)B = bK*(B* x CP 1) = OB*xCb,
so the equivariant formal group associated to K is just B* x bCas claimed.
Example 8.4. Let bK= bK(p, n) be the twoperiodic version of Morava Ktheory at*
* a prime p,
with height n. We define an equivariant theory K = K(p, n, B) as above; this is*
* called equivariant
Morava Ktheory. In [17] we present evidence that these theories deserve this *
*name, because
they play the expected r^ole in equivariant analogues of the HopkinsDevinatzS*
*mith nilpotence
theorems, among other things. The same paper also explains the representing obj*
*ect for the theory
K, and shows that we have natural isomorphisms as follows:
K*(X ^ Y )= K*(X) K* K*(Y )
K*X = Hom K*(K*X, K*).
9.Equivariant formal groups over rational rings
We next prove the equivariant analogue of the wellknown fact that all formal*
* groups over a
Qalgebra are additive. We write bGafor the ordinary additive formal group over*
* S. If we consider
formal schemes over S as functors in the usual way, this sends an OSalgebra R *
*to the set Nil(R)
of nilpotents in R. Given a free module L of rank one over OS (or equivalently,*
* a trivialisable
line bundle over S), we can instead consider the functor R 7! L OS Nil(R), whi*
*ch we denote by
L bGa. This gives a formal group over S, noncanonically isomorphic to bGa. If*
* C is a formal
multicurve group over S, then the cotangent spaces to the fibres give a trivial*
*isable line bundle
!C on S. This is easily seen to be the same as !Cb, where bCis the formal neigh*
*bourhood of zero,
as usual. From now on we just write ! for this module. If S lies over spec(Q) t*
*hen the theory of
logarithms for ordinary formal groups gives a canonical isomorphism bC!!1 b*
*Ga.
Theorem 9.1. Let (C, OE) be an Aequivariant formal group over a scheme S, such*
* that`the
integer n = A is invertible in OS. Then there is a canonical decomposition S *
*= B ASB, and
a corresponding decomposition a
C ' SB xS bCx B*,
where bCis the formal neigbourhood of 0 in C. Moreover, if OS is a Qalgebra th*
*an bC' !1C bGa
and so a
C ' SB xS (!1C bGa) x B*.
Proof.Put n = A, and choose a coordinate x on C. For formal reasons we have *
*x(a + b) =
x(a) + x(b) (mod x(a)x(b)) as functions on C2, and it follows that x(na) = f(a)*
*x(a) for some
function f on C with f(0) = n. Let C[n] denote the closed subscheme of points o*
*f order n in
C, so C[n] = {a 2 C  f(a)x(a) = 0} = spf(OC=(f.x)). Note that S is embedded as*
* the zero
section in C with OS = OC=x, so in OS we have f = f(0) = n 2 OS Q, so f is in*
*vertible mod
x, so 1 2 (f) + (x). By the Chinese remainder theorem, the scheme C[n] splits a*
*s S q T , where
T = spf(OC=f). Note that x is zero on S and invertible on T .
Now consider the map OE from A* to the group of sections of C[n] over S. Supp*
*ose that for
each ff 2 A* we either have OE(ff)(S) T (and so x(OE(ff)) is invertible) or O*
*E(ff)(S) S (so
x(OE(ff)) = 0)); it then follows immediately from Proposition 8.1 that C is of *
*product type. In
general, however, it is not true that OE(ff)(S) T or OE(ff)(S) S; instead, *
*we can just pull back
the splitting C[n] = S q T along the map OE(ff): S ! C to get a splitting S = *
*Sffq Tffwith
OE(ff)(Sff) S and OE(ff)(Tff) T . Next, for U A* we put
" "
MU = Sff\ Tff.
ff2U ff62U
14 N. P. STRICKLAND
`
It is clear that S = UMU, and that MU`= ; unless U is a subgroup of A*. Thus,*
*`if we put
SB = Mann(B), we have a splitting S = B SB, and a corresponding splitting C = *
* B CB, where
CB is an Aefg over SB. It is now easy to see that CB = SB xS bCx B* as require*
*d. The rational
statement now follows from the nonequivariant theory.
Remark 9.2. Nonequivariantly, one knows that rational spectra are determined by*
* their homo
topy groups. This gives a classification of rational even periodic cohomology t*
*heories, as follows.
Let E denote the category of pairs (S, L), where S is an affine scheme over Q a*
*nd L is a trivialisable
line bundle over S. The morphisms from (S0, L0) to (S1, L1) are pairs (f, g) wh*
*ere f :S0 !S1
and g is an isomorphism L0 !f*L1 of line bundles over S0. Let E0be the categor*
*y of pairs (S, bC),
where S is as before and bCis a (nonequivariant) formal group over S, with morp*
*hisms defined
in the analogous way. Let E00be the category of even periodic rational ring spe*
*ctra. Then there
is a contravariant equivalence E00! E0 sending E to (spec(E0), spf(E0CP 1)), a*
*nd a covariant
equivalence E0! E sending (S,QbC) to (S, !Cb), so the composite sends E to (sp*
*ec(E0), E2). If E
maps to (S, L) then Em X = n Hm+2n(X; Ln).
*
* __B
Now let QSA denote the categoryQof rational Aspectra. One knows that the fun*
*ctors OE:SA !
S0 induce an equivalence QSA ! B A QS0. (Note here that because A is abelian,*
* there are no
nontrivial Weyl groups or conjugacies between subgroups; we have used this to s*
*implify the usual
statement.) In particular, any evenly periodic rational equivariant cohomology *
*theory E* has the
form Y __ Y __
Em X = EmBOEBX = Hm+2n(OEBX; !nB)
B B,n
for some family {E*B}B A of nonequivariant even periodic rational theories, wit*
*h associated formal
groups bCBand line bundles`!B over SB = spec(E0B). By taking X = 1 and then X =*
* P U we find
that S := spec(E0) = B SB and
a a
C := spf(E0P U) = B* x bCB= SB x (!1B bGa) x B*.
B
In other words, the topological picture is perfectly parallel to the algebraic *
*one.
The following slight extension can easily be proved in the same way.
Corollary 9.3. Let (C, OE) be an Aequivariant formal group over a scheme S, su*
*ch that OS is an
algebra over Z(p). There is of course a unique splitting A = A0x A1, where A0 i*
*s a pgroup and p
does not divide A1. Let C0 C be the formal neighbourhood of [OE(A*0)],`and *
*let OE0:A*0!C0 be
the restriction of OE. Then there is a canonical decomposition S = B A1 SB, an*
*d a corresponding
decomposition a
C ' SB xS C0x B*,
such that over SB, the map OE is the product of OE0 and the restriction map A*1*
*!B*.
10.Equivariant formal groups of pushout type
We next consider a slightly different generalization of the notion of a group*
* of product type.
Definition 10.1. Suppose we have a subgroup B A and a formal multicurve group*
* C0, with a
map OE0:(A=B)* !C0making it an A=Bequivariant formal group. There is an evide*
*nt embedding
(A=B)* !A*, which we can use to form a pushout
OE0
(A=B)* _____C0w
v 
 
 
 
u u
A* _______C.wOE
*
* `
If we choose a transversal T to (A=B)* in A*, then the underlying scheme of C i*
*s just ff2TC.
This implies that the formation of the pushout is compatible with base change, *
*and that C is an
Aequivariant formal group. Formal groups constructed in this way are said to b*
*e of pushout type.
(The case where OE0= 0 evidently gives groups of product type.)
MULTICURVES AND EQUIVARIANT COBORDISM 15
We next examine how formal groups of this kind can arise in equivariant topol*
*ogy. For this, we
need to recall the various different change of group functors and fixedpoint f*
*unctors for Aspectra.
Given a homomorphism i :B !A, there is a pullback functor i*: SA !SB, which*
* preserves
smash products and function spectra. (Note that if i is not injective, then i*U*
*A is not a complete
Buniverse, so the definition of i* contains an implicit change of universe.) I*
*f i is the inclusion of
a subgroup then i* is called restriction and written resAB. This functor has a *
*left adjoint written
X 7! A ^B X, and a right adjoint written X 7! FB(A+, X). These two adjoints ar*
*e actually
isomorphic, by the generalized Wirthmüller isomorphism [11, Theorem II.6.2].
If i is the projection A ! A=B then i* is called inflation. This has a right*
* adjoint functor
~B :SA ! SA=B, which we call the LewisMay fixed point functor. The adjunction*
* is discussed
in [11, Section II.7]; there ~BX is written XB . One can check that the followi*
*ng square commutes
up to natural isomorphism:
C
SA _____wSA=C~
 
A  A=C
resB resB=C
 
u u
SB _____wSB=C.~C
It will be convenient to write
__B A=B B B A
~ = res0 ~ = ~ resB:SA !S0.
__B
The usual_equivariant homotopy groups of X are defined by ßB*X = ß*~ X. The fun*
*ctors ~B
and ~B do not preserve smash products, and there is no sense in which ~B acts a*
*s the identity on
Bfixed objects.
Lewis and May also introduce another functor OEB :SA !SA=B, called the geome*
*tric fixed point
functor. To explain the definition, let V be a complex representation of A. We *
*write ØV for the
usual inclusion S0 !SV , which can be regarded as an element of the R(A)grade*
*d homotopy
ring ß*S0 in dimension V . It is easily seen to be zero if V A6= 0, but it tur*
*ns out to be nonzero
otherwise. It is also clear that ØV W = ØV ØW .
By dualizing the standard cofibration S(V )+ !S0 ØV!SV , we see that D(S(V*
* )+) deserves to
be called S0=ØV . On the other hand, we have
S0[Ø1V] = lim!(S0 ØV!SV ØV!S2V !. .).= S1V .
It follows that for any X 2 SA, the spectrum X[Ø1V] = X ^ S1V is a Bousfield *
*localization of X,
or more specifically, the finite localization away from the thick ideal generat*
*ed by S0=ØV . There
is another0characterization as follows. Let F be the family of those subgroups *
*A0 A such that
V A 6= 0, and let C be the thick ideal generated by {A=A0+ A02 F}. It is not h*
*ard to see that
(S1V )A0 is contractible for A02 F, and equivalent to S0 for A062 F. It is well*
*known that up to
homotopy there is a unique space with these properties, denoted by eEF, and tha*
*t X ^ eEF is the
finite localization of X away from C. It follows that C is the same as the thic*
*k ideal generated by
S0=ØV .
Now fix a subgroup B A, and take
M
V = VA,B := C[A] (C[A]B) = Lff.
ff2A*\ann(B)
In this context, we write ØA,B for ØV , and we also write ØA for ØA,A. We also *
*put F[B] = {C
A  B 6 C}, and note that eEF[B] = S1V . The geometric fixedpoint functor OEB*
* :SA !SA=B is
defined by
OEBX = ~B(X[Ø1A,B]) = ~B(X ^ eEF[B]).
(In [11] the functor OEB is actually defined in a different way, but the above *
*description is proved
as Theorem II.9.8). Let ß :A !A=B be the projection. One can check that OEB pr*
*eserves smash
products [11, Proposition 9.12], the composite
* OEB
SA=B ß!SA ! SA=B
16 N. P. STRICKLAND
is the identity [11, Proposition 9.10], and the following diagram commutes:
OEC
SA _____wSA=C
 
A  A=C
resB resB=C
 
u u
SB _____wSB=C.OEC
Moreover, for any Aspace X we have OEB 1 X = 1 XB [11, Corollary 9.9]. It wil*
*l be convenient
to write __B
OE = resA=B0OEB = OEB resAB:SA !S0.
This_again preserves smash products, and it is known that a spectrum X 2 SA sat*
*isfies X = 0 iff
OEBX = 0 in S0 for all B A. We will also need the following property:
Lemma 10.2. Suppose that B A, and write Ø = ØA,B. Then for X, Y 2 SA there ar*
*e natural
equivalences
~BF (X, Y [Ø1]) = ~BF (X[Ø1], Y [Ø1]) = F (OEBX, OEBY ).
Proof.First note that the map W ! W [Ø1] is an equivalence iff W is concentra*
*ted over B as
defined in [11, page 109]. Let C be the category of such W , so we have functor*
*s OEB = ~B :C !SA=B
and _ :SA=B !C given by _(Z) = (ß*Z)[Ø1]. We see from [11, Corollary II.9.6] *
*that OEB and _
are mutually inverse equivalences, and it follows that
[X, Y [Ø1]]B*= [X[Ø1], Y [Ø1]]B*= [OEBX, OEBY ]A=B*.
Now consider W 2 SA=B and replace X by (ß*W ) ^ X in the above. We deduce that
[W, ~BF (X, Y [Ø1])]A=B*= [W, ~BF (X[Ø1], Y [Ø1])]A=B*= [W, F (OEBX, OE*
*BY )]A=B*.
The claim now follows by the Yoneda lemma.
Theorem 10.3. Let E0 be an A=Bequivariant periodically orientable ring spectru*
*m, with as
0
sociated equivariant formal group (A=B)* OE!C0. Let ß :A ! A=B be the project*
*ion, and put
E = (ß*E0)[Ø1A]. Then E is an Aequivariant periodically orientable ring spect*
*rum, and for all
X 2 SA we have
E*X = E0*OEBX
E*X = (E0)*OEBX.
Moreover, the formal group associated to E is the pushout of C0along the inclus*
*ion (A=B)* !A*.
Proof.Because ß* preserves smash products, it is clear that ß*E0is a commutativ*
*e Aequivariant
ring spectrum, and so the same is true of E. We saw earlier that OEBß* = 1, so *
*OEBE = E0. Also,
we have E ^ X = E ^ X[Ø1A], so
~B(E ^ X) = OEB(E ^ X) = OEB(E) ^ OEB(X) = E0^ OEB(X).
We can apply ~A=B to this to see that ~A(E ^ X) = ~A=B(E0^ OEB(X)), and by appl*
*ying ß* we
deduce that E*X = E0*OEBX.
For the corresponding statement in cohomology, we see using Lemma 10.2 that ~*
*BF (X, E) =
F (OEBX, OEBE) = F (OEBX, E0). We again apply the functor ß*~A=B() to see tha*
*t E*X =
E0*OEBX, as claimed.
In particular, if X is an Aspace we have OEB 1 X = 1 XB and so E*X = E0*XB *
*. Thus,
if we put S = spec(E00(point)), then S is also the same as spec(E0(point)). We *
*next consider
the space P V , whereLV is a representation of A. We can split V into isotypic*
*al parts for the
action of B, say`V = fiV [fi], where V [fi]Qis a sum of representations Lffwi*
*th ffB = fi. We then
have (P V )B = fiP V [fi], and so E*P V = fiE0*P V [fi]. Using this,`it is *
*easy to see that E
is periodically orientable. Next, consider the space P UA, so (P UA)B = fiP (*
*UA[fi]). The space
P (UA[0]) is canonically identified with P UA=B, so spf(E00P UA[0]) = C0. For f*
*i 6= 0, we can choose
~fi2 A* extending fi, and then tensoring with L~figives an equivalence ` :P UA*
*[fi] ' P UA=B. If we
change ~fiby an element fl 2 (A=B)*, then ` changes by the automorphism`øflof *
*P UA=B. Using
this, it is not hard to identify the curve C = spf(E0P UA) = spf(E00P UA[fi])*
* with the pushout
of C0along the map (A=B)* !A*.
MULTICURVES AND EQUIVARIANT COBORDISM 17
11.Equivariant Morava Etheory
Let bC0be the standard ptypical formal group of height n over S0 = spec(Fp).*
* We write bKfor
the twoperiodic Morava Ktheory spectrum whose associated formal group is bC0,*
* so bK*= Fp[u 1]
with u = 2. This formal group has a universal deformation bC1over S1 := spf(Z*
*p[[u1, . .,.un1]]).
We write bEfor the corresponding Landweberexact cohomology theory, and refer t*
*o it as Morava
Etheory. Now suppose we have a finite abelian group A and a subgroup B. We d*
*efine C0 =
B*xCb0, which_is an Aefg of product type over S0, associated to the equivarian*
*t Morava Ktheory
K*X := bK*OEBX. We can also define an A=Bequivariant cohomology theory by X 7!*
* bE*Xh(A=B),
as in Section 6. The associated equivariant formal group is bC2= bC1xS1S over S*
*, where S =
Hom((A=B)*, bC1). We then perform the construction in Section 10. This gives an*
* Aequivariant
theory E = E(p, n, B), defined by
E*X = bE*((OEBX)h(A=B)),
whose associated equivariant formal group is the pushout of bC2along the inclus*
*ion (A=B)* !A*.
We write C for this pushout, and we refer to E as equivariant Morava Etheory. *
* In [17] we
give some evidence that this name is reasonable, related to the theory of Bousf*
*ield classes and
nilpotence. Here we give a further piece of evidence, based on formal group the*
*ory.
We first note that S0 is a closed subscheme of S1, which is in turn a closed *
*subscheme of
S = Hom ((A=B)*, bC1) (corresponding to the zero homomorphism). The restriction*
* of C to S1 is
just B* x bC1, and the restriction of this to S0 is just C0. The inclusion C0 *
*!C corresponds to
a ring map OC !OC0, or equivalently E0P U !K0P U. It can be shown that this c*
*omes from
a natural map E*X !K*X of cohomology theories. Indeed, there is certainly a no*
*nequivariant
map q :bE!bK. Moreover, up to homotopy there is a unique map A=B !E(A=B) of A*
*=Bspaces,
which gives a natural map
res(Y ) = (A=B+ ^ Y )=(A=B) !(E(A=B)+ ^ Y )=(A=B) = Yh(A=B)
__B
for A=Bspectra Y . If Y = OEBX then res(Y ) = OEX and so we get a map
__B q* * __B *
E*X = bE*(OEBX)h(A=B)!bE*OE! Kb OEX = K X,
as required.
Definition 11.1. A deformation of the Aefg C0 over S0 consists of an Aefg C0 *
*over a base S0
together with a commutative square ~
f
C0 _____C0w
 
 
 
u u
S0 _____S0wf
such that
(a)f is a closed inclusion, and S0is a formal neighbourhood of f(S0)
(b) ~finduces an isomorphism C0 !f*C of Aefg's over S0.
If C0and C00are deformations, a morphism between them means a commutative square
~g
C0_____wC00
 
 
 
u u
S0_____wS00g
such that ~ginduces an isomorphism C0 !g*C00of Aefg's over S0. A universal d*
*eformation
means a terminal object in the category of deformations.
As mentioned previously, the formal group bC1associated to bEis the universal*
* deformation of
the formal group bC0associated to bK. Equivariantly, we have the following anal*
*ogue.
Theorem 11.2. The Aequivariant formal group C (associated to equivariant Morav*
*a Etheory)
is the universal deformation of C0 (associated to equivariant Morava Ktheory).
18 N. P. STRICKLAND
Proof.Suppose we have an Aefg (C0, OE0) over S0equipped with maps (f, ~f) maki*
*ng it a deforma
tion of C0. We will identify S0 with f(S0) and thus regard it as a closed subsc*
*heme of S0. Similarly,
we regard C0 as the closed subscheme C0S0 of C0. Note that S0is a formal neigh*
*bourhood of S0,
and it follows that C0 is a formal neighbourhood of C0. We choose a coordinate *
*x0on C0, and
note that it restricts to give a coordinate on C0.
Now let bC0denote the formal neighbourhood of the zero section in C0. We have*
* (Cb0)S0= bC0,
so we can regard bC0as a deformation of the ordinary formal group bC0. As bC1is*
* the universal
deformation of bC0, this gives us a pullback square
bC0____wbC1~g
 
 
 
u u
S0_____wS1.g
Next, suppose we have ff 2 (A=B)* A*, giving a section OE0(ff) of C0and an el*
*ement x0(OE0(ff)) 2
OS0. As C0S0= C0 = B* x bC0and ffB = 0 we have OE0(ff)S0= 0, so x0(OE0(ff)) *
*maps to 0 in OS0.
As S0is a formal neighbourhood of S0, it follows that x0(OE0(ff)) is topologica*
*lly nilpotent in OS0,
and thus that OE0(ff) is actually a section of bC0. Thus, ~gO OE0gives a map (A*
*=B)* !bC1, which is
classified by a map h: S0! Hom((A=B)*, bC1) = S. The maps ~gand h combine to g*
*ive a map
~h:bC0!h*bC= h*(Cb1xS1S) = g*Cb1.
This can be regarded as an isomorphism of`A=Bequivariant formal groups.
Next,Pthe decomposition C0 = B* x bC0= fi2B*bC0gives orthogonal idempotents*
* efi2 OC0
with fiefi= 1. As C0 is a formal neighbourhood of C0, these can be lifted to*
* orthogonal
`
idempotents in OC0, giving a decomposition C0= fiC0fisay. One can check that C*
*0fi= OE0(ff)+Cb0
for any ff 2 A*with ffB = fi, and it follows that C0is just the pushout of the*
* map OE0:(A=B)* !bC0
and the inclusion (A=B)* ! A*. It follows in turn that ~hextends to give an i*
*somorphism
C0 !g*C, and thus a morphism C0 !C of deformations. All steps in this constr*
*uction are
forced, so one can check that this morphism is unique. This means that C is th*
*e universal
deformation of C0, as claimed.
12.A completion theorem
Suppose we have an Aequivariant formal group (C, OE), and a subgroup B A, *
*giving a
subgroup (A=B)* A*. Let S0 be the closed subscheme of S where OE((A=B)*) = 0.*
* Equivalently,
if we put eff= x(OE(ff)) and J = (eff ff 2 (A=B)*), then S0 = V (J) = spec(OS*
*=J). If we put
C0 = S0 xS C then OE induces a map _ :B* x S0 !C0 making C0 into a Bequivaria*
*nt formal
group over S0. Next, we put S1 = lim!spec(OS=Jm ) = spf((OS)^J), the formal ne*
*ighbourhood of
m
S1 in S, and C1 = S1xS C. This is an Aequivariant formal group over S1 for whi*
*ch OE((A=B)*)
is infinitesimally close to 0.
Now suppose that C comes from an Aequivariant periodically orientable theory*
* E. We would
like to interpret C0 and C1 topologically.
Proposition 12.1. Let E0 be the Bspectrum resAB(E), representing the theory E**
*(AxB Y ) for B
spaces Y . Let C00=S00be the associated Bequivariant formal group. Then there *
*is a map S00!S0
(which may or may not be an isomorphism) and an isomorphism C00= C0xS0S00.
Proof.We have S00= spec(ß0E0) = spec(E0A=B), so there is a natural map S00!S. *
*Moreover,
we have P UB ' resABP UA, which gives an isomorphism A xB P UB ' A=B x P UA and*
* thus
E00P UB ' E0(A=B x P UA) = E0(A=B) E0 E0P UA.
This shows that the formal group for E0 is just C00:= C xS S00. All that is lef*
*t is to check that the
map S00!S factors through S0, so C00can also be described as C0xS0S00. To see *
*this, note that
OE comes from the inclusion j :A* = ßA0P U ! P U, so the corresponding map OE0*
*0over S00comes
from the map 1 x j :(A=B) x A* !(A=B) x P U. Using the isomorphism
[(A=B) x A*, (A=B) x P U]A= [A*, (A=B) x P U]B
= Map(A*, ßB0((A=B) x P U))
= Map(A*, (A=B) x B*)
MULTICURVES AND EQUIVARIANT COBORDISM 19
we see that the restriction of (1 x j) to (A=B) x (A=B)* is null, so that OE00(*
*(A=B)*) = 0 as
claimed.
If E is the complex Ktheory spectrum KUA, then we saw earlier that S = Hom (*
*A*, Gm ) and
so
S0 = {OE 2 Hom (A*, Gm )  OE((A=B)*) = 0} = Hom (B*, Gm ).
On the other hand, it is wellknown that KU*A(A xB Y ) = KU*BY so E0 = KUB so *
*S00=
Hom(B*, Gm ) = S0. A similar argument works for theories of the form E*X = bE*X*
*hA where bE
is K(n)local as in Section 6, in which case we have S = Hom (A*, bC) and S0 = *
*S00= Hom (B*, bC).
At the other extreme, for theories of the form E*X = bE*(resA0(X)), we have S0 *
*= S and S00=
(A=B) x S.
We next consider C1. Recall that there is an Aspace E[ B] characterised by *
*the property
that E[ B]C is contractible for C B and empty for C 6 B. The first approxim*
*ation would be
to consider the ring spectrum F (E[ B]+, E). However, as S1 is a formal scheme*
* rather than an
affine scheme, we need a prospectrum rather than a spectrum. The solution is t*
*o define Fo(X+, E)
to be the prosystem of ring spectra F (Xff+, E), where Xffruns over finite sub*
*complexes of X,
and to put E1 = Fo(E[ B]+, E). The desired description of E*1P U is a kind of*
* completion
theorem in the style of AtiyahSegal, so we expect to need finiteness hypothese*
*s. However, with
these hypotheses, we have an exact result rather than an approximate one as in *
*the previous
proposition.
Theorem 12.2. Suppose that E*(point) is a Noetherian ring, and that E*(A=C) is *
*finitely gen
erated over it for all C A. Then the Aequivariant formal group associated to*
* E1 is C1.
Proof.This is essentially taken from [7]. Choose generators ff1, . .,.ffr for (*
*A=B)*, let Libe the
onedimensional representation corresponding to ffi and let Øi denote the inclu*
*sion S0 ! SLi.
There is a canonical Thom class uiinQE0SLi, and Ø*i(ui) is the Euler class ei= *
*x(OE(ffi)). One
checksQeasily that the space P := iS(1Li) is a model for E[ B], and the spac*
*es T (m) :=
iS(mLi) form a cofinal system of finite subcomplexes, so E1 is equivalent to t*
*he tower of ring
spectra F (T (m)+, E) = D(Tm(m)+) ^ E. Next, by taking the SpanierWhitehead d*
*ual of the
cofibration S(mLi)+ !S0 Øi!SmLi, we see that D(S(mLi)+) deserves to be calle*
*d S=Ømi, and
so D(T (m)+) deserves to be called S=(Øm1, . .,.Ømr). This suggests that ß*(E ^*
*D(T (m)+)) should
be E*=Jm , where Jm = (um1, . .,.umr) E*. Unfortunately, there are correctio*
*n terms. More
precisely, the cofibration displayed above gives a twostage filtration of D(S(*
*mLi)+) for each i,
and by smashing these together we get a (r + 1)stage filtration of D(T (m)+), *
*and thus a spectral
sequence converging to ß*(E ^ D(T (m)+)). The first page is easily seen to be t*
*he Koszul complex
for the sequence em1, . .,.emr, so the bottom line of the second page is E*=Jm *
*, and the remaining
lines are higher Koszul homology groups. The filtrations are compatible as m va*
*ries, so we get a
spectral sequence in the abelian category of progroups converging to ß*E1. In *
*the second page,
the bottom line is the tower {E*=Jm }m 0, and the remaining lines are protrivi*
*al by [7, Lemma
3.7]. It follows that ß*E1 ' {E*=Jm } as progroups, and so the formal scheme c*
*orresponding to
ß0E1 is lim!spec(E0=Jm ) = lim spec(E0=Jm ) = S1. We now replace E by F (P (n.*
*C[A])+, E)
m !m
and then take the limit as n tends to infinity to conclude that spf(E01P U) = C*
* xS S1 = C1 as
claimed.
Remark 12.3. Using the same circle of ideas one proves that the kernel of the m*
*ap E0=J !
E0(A=B) is nilpotent, so the map S00!S0 is dominant; compare [7, Theorem 1.4].
13.A counterexample
Here we exhibit a Z=2equivariant formal group C with a number of unusual pro*
*perties, which
are only possible because the base scheme S is not Noetherian. The phenomena de*
*scribed here
are the main obstruction to our understanding of the equivariant Lazard ring.
For any Aequivariant formal group (C, OE), there is a natural map _ :A* x bC*
*!C given by
_(ff, a) = OE(ff) + a. As C is a formal neighbourhood of [OE(A*)],Qit is natura*
*l to expect that _
should be an epimorphism, or equivalently that the map _*:OC ! ffOCbshould be*
* injective.
The key feature of the example to be constructed here is that _* is not in fact*
* injective.
Start with k0 = F2[e], let M be the module F2[e 1]=Fp[e], and let k be the sq*
*uarezero extension
k0 M. More explicitly, k is generated over k0 by elements u1, u2, . .s.ubject *
*to eui+1= ui(with
u0 interpreted as 0) and uiuj = 0. Put S = spec(k).
20 N. P. STRICKLAND
Next, let R be the completion of k[x] at the element y = x2+ ex, so R = k[[y]*
*]{1, x}, and put
C = spf(R) = {x 2 A1S x2+ ex is nilpotent}.
This is a subgroup of A1Sunder addition. In the corresponding Hopf algebra stru*
*cture on R, the
elements x and y are both primitive. There is a homomorphism OE: Z=2 !C sendin*
*g 0 to 0 and
1 to e. The corresponding divisor is just R=y, and as R is complete at y, we de*
*duce that (C, OE) is
an equivariant formal group.
Next, we can define maps ~0, ~a:R !k[[t]] by
~0(x)= t
~a(x)= t + e
~0(y)= ~a(y) = t2+ te.
Q
The map _*:OC ! ffOCbis just the map (~0, ~a): R !k[[t]] x k[[t]]. Now consi*
*der the element
X k
f = u12k+1y2 2 R.
k 0
We then have X
k
~0(f)= u12k+1(t2+ et)2
k 0
X k+1 X k k
= u12k+1t2 + u12k+1e2 t2
k 0 k 0
X k+1 X k
= u12k+1t2 + u12kt2
k 0 k 0
0
= u120t2 = u0t = 0.
We also have ~a(f) = 0 by the same argument, so _*(f) = 0.
14. Divisors
We now return to the purely algebraic theory of formal multicurves and their *
*divisors.
Recall that a divisor on C is a regular hypersurface D C such that OD is a *
*finitely generated
projective module over OS, which is discrete in the quotient topology. We also *
*make the following
temporary definition; one of our main tasks in this section is to show (in Prop*
*osition 14.15) that it
is equivalent to the preceeding one. (For divisors of degree one, this follows *
*from Corollary 2.10.)
Definition 14.1. A weak divisor on C is a closed subscheme D C that is finite*
* and very flat
over S (so OD is a discrete finitely generated projective module over OS). Thus*
*, a weak divisor
D = spf(R=J) is a divisor iff the ideal J is open and generated by a regular el*
*ement. If y is a
good parameter on C, we note that J is open iff yN 2 J for N 0.
If D0 = spf(R=J0) is a divisor and D1 = spf(R=J1) is a weak divisor then one *
*checks that the
scheme D0+ D1 := spf(R=(J0J1)) is again a weak divisor.
Definition 14.2. Now suppose we have a map q :T ! S of schemes which is finite*
* and very
flat, so that OT is a discrete finitely generated projective module over OS. I*
*f g 2 OT then
multiplication by g gives an OSlinear endomorphism ~g of OT, whose determinant*
* we denote by
Nq(g) or NT=S(g).
Definition 14.3. Fix a difference function d on C. For any weak divisor D on C *
*over, we can
regard d by restriction as a function on D xS C. We also have a projection q :D*
* xS C !C, and
we put
fD = Nq(d) = N(DxSC)=C(d) 2 OC.
We will eventually show that D = spf(OC=fD ).
Remark 14.4. Consider the case where C is an ordinary formal group, with coordi*
*nate x and
associated formal group law FP. We then have OC = OS[[x]] and OCxC = OS[[x0, x1*
*]], and d =
x1 F x0. If D has the form i[ui] forQsome family of sections ui, then we hav*
*e elements ai=
x(ui) 2 OS and weQwill see that fD = i(x F ai). This is a unit multiple of*
* the Chern
polynomial gD = i(x  ai), and it is familiar that D = spf(OC=gD ), so D = sp*
*f(OC=fD ) also.
In the multicurve case, one can still define gD (as the norm of the function (a*
*, b) 7! x(b)  x(a))
and we find that it is divisible by fD , but gD =fD need not be invertible so O*
*C=gD 6= OD .
MULTICURVES AND EQUIVARIANT COBORDISM 21
Lemma 14.5. Let R be a ring, P a finitely generated projective Rmodule, and ff*
* an automorphism
of P . Then ff is injective iff det(ff) is a regular element.
Proof.After localising we may assume that P = Rd for some d, and ff is represen*
*ted by a d x d
matrix A. If det(A) is regular, the equation adj(A)A = det(A)Id implies immedia*
*tely that ff is
injective. Conversely, suppose that ff is injective. As P is flat, it follows t*
*hat ff d: P d!P dis
also injective. It is easy to check with bases that ~dP is naturally isomorphic*
* to the image of the
antisymmetrisation map P d!P d. In particular, it embeds naturally in P d, *
*and it therefore
follows that ~dff is injective. On the other hand, ~dP is an invertible Rmodul*
*e, so End(~dP ) = R,
and ~d(ff) = det(ff) under this isomorphism. It follows that det(ff) is regular*
* as claimed.
Corollary 14.6. For any weak divisor D on C, the element fD 2 OC is regular.
Proof.Take R = OC and P = ODxSC and ff = ~d. We know from Lemma 2.9 that ff is *
*injective,
and the claim follows.
Lemma 14.7. Let q :T ! S be finite and very flat, and let g be a function on T*
* . If there is a
section u: S !T such that g O u = 0 then Nq(g) = 0.
Proof.Put J*= ker(u*:OT ! OS), so g 2 J. We have a short exact sequence of OS *
*modules
J ! OT u! OS, which is split by the map q*: OS ! OT. The sequence is preser*
*ved by ~f,
and ~f(OT) = OT.f J so the induced map on the cokernel is zero. Zariskilocal*
*ly on S we
can choose bases adapted to the short exact sequence and it follows easily that*
* det(~f) = 0 as
claimed.
Corollary 14.8. The function fD 2 OC vanishes on D.
Proof.We have fD D = Nq0(d), where q0:D xS D !D is the projection on the seco*
*nd factor.
The diagonal map ffi :D !D xS D is a section of q0with d O ffi = 0, so Nq0(d) *
*= 0.
Lemma 14.9. If D = D0+ D1 (where D0, D1 are divisors) and g 2 OD then
ND=S(g) = ND0=S(g)ND1=S(g).
Proof.Put R = OC, and let the ideals corresponding to Di be Ji = (fi) for i = 0*
*, 1. We then
have a short exact sequence of OD modules as follows:
OD0 = R=f0 xf1!OD = R=(f0f1) !OD1 = R=f1.
This is splittable, because OD1 is projective over OS. The map ~g preserves the*
* sequence, and it
follows easily that det(~g) = det(~gOD0) det(~gOD1), as required.
Corollary 14.10. If D = D0+ D1 as above then fD = fD0fD1.
Proof.Just change base to C and take g = d.
Lemma 14.11. Suppose that D is a weak divisor of degree r, that D0 is a divisor*
* of degree r0,
and that D0 D. Then D = D0+ D00for some weak divisor D00of degree r  r0.
Proof.Put J = ID and J0= ID0. As D0is a genuine divisor, we have J0= Rf0 for so*
*me regular
element f0 2 R. As D0 D, we have J J0. Put J00= {g 2 R  f0g 2 J} J. We th*
*en have a
short exact sequence
0
R=J00xf!R=J !R=J0.
As R=J and R=J0 are projective modules of ranks r and r0 over k, it follows tha*
*t R=J00is a
projective module of rank r  r0. Thus, the scheme D00:= spf(R=J00) is a weak d*
*ivisor. From the
definition of J00we have J0J00 J. Conversely, if h 2 J then certainly h 2 J0 =*
* Rf0 so h = gf0
for some g 2 R. From the definitions we have g 2 J00, so h 2 J00J0. This shows *
*that J = J0J00
and so D = D0+ D00.
Definition 14.12. Let D be a weak divisor of constantPdegree r. A full set of p*
*oints for D is a
list u1, . .,.ur of sections of S such that D = i[ui]. If there exists a full*
* set of points, it is clear
that D is actually a genuine divisor. (This concept is due to Drinfeld, and is *
*explained and used
extensively in [9].)
*
* Q
Proposition 14.13. If u1, . .,.ur is a full setQof points for D, then ND=S(g) =*
* ig(ui) for any
function g on D. Moreover, we have fD (a) = id(a, ui), and so OD = OC=fD .
22 N. P. STRICKLAND
Proof.As the projection [ui] ! S is an isomorphism, we see that N[ui]=S(g) = g*
*(ui). The
first claim follows easily usingQfrom Lemma 14.9 by induction on r. It follows*
* similarly from
Corollary 14.10 that fD (a) = id(a, ui). As d is a difference function we have*
* O[ui]= OC=d(a, ui)
and so OD = OC=fD as claimed.
Lemma 14.14. Let D be a weak divisor of constant degree r. Then there is a fini*
*te, very flat
scheme T over S such that the weak divisor T xS D on T xS C has a full set of p*
*oints (and so is
genuine).
Proof.By an evident induction, it suffices to show that after very flat base ch*
*ange we can split D
as [u]+D00for some section u and some weak divisor D00. It is enough to find a *
*section u: S !D,
for then [u] D and we can apply the previous lemma. For this we can simply pu*
*ll back along the
projection map D !S (which is very flat by assumption) and then the diagonal m*
*ap D !DxSD
gives the required ät utological" section.
Proposition 14.15. Every weak divisor is a genuine divisor.
Proof.Let D = spf(R=J) be a weak divisor. We may assume without loss that it ha*
*s constant
degree r. We know from Corollary 14.6 and Corollary 14.8 that fD is regular in *
*R and lies in J;
we need only show that it generates J. It is enough to do this after faithfully*
* flat base change, so
by Lemma 14.14 we may assume that we have a full set of points. Proposition 14.*
*13 completes
the proof.
15.Embeddings
Let C be a nonempty formal multicurve over a scheme S. In this section we stu*
*dy embeddings
of S in the affine line A1S= A1 x S. If q is the given map C ! S, then any map*
* C ! A1Sof
schemes over S has the form (x, q) for some x: C !A1, or equivalently x 2 OC.
Now choose a difference function d on C. Given x 2 OC, we can define x0:C xS *
*C !A1 by
x0(a, b) = x(b)  x(a). Equivalently, x0 is the element 1 x  x 1 in OCxSC*
* = OC bOSOC.
It is clear that x0vanishes on the diagonal, and thus is divisible by d, say x0*
*= `(x)d for some
`(x) 2 OCxSC . This element `(x) is unique, because d is not a zerodivisor.
Proposition 15.1. Let C q!S be a nonempty formal multicurve. A map (x, q): C *
*! A1S
is injective if and only if `(x) is invertible. If so, then (x, q) induces an *
*isomorphism C !
lim!V (fk) A1Sfor some monic polynomial f 2 OS[t], showing that C is embedda*
*ble.
k
Proof.Put X = {(a, b) 2 C xS C  x(a) = x(b)} = V (x0) = V (`(x)d). We see that*
* x is injective if
and only if V (x0) = = V (d), if and only if d = ux0= u`(x)d for some u 2 OCx*
*SC . As d is not
a zero divisor, this holds if and only if `(x) is invertible.
If so, we may assume without loss that d = x0. Choose a good parameter y, so*
* OC=y has
constant rank r over OS for some r. Put D = spec(R=y), let p: C xS D !C be the*
* projection,
and put z = Np(x0). The proof of Proposition 14.15 shows that z is a unit multi*
*ple of y.
We next claim that {1, x, . .,.xr1} is a basis for R=y = R=z over k, and tha*
*t z = f(x) for a
unique monic polynomial f of degree r. It is enough to check this after faithfu*
*lly flat base change,
so we may assumeQthat D = [u0]+. .+.[ur1]Qfor some list of sections uiof D. If*
* we put ai= x(ui)
we see that z = i(x  ai). If we put ei= j 0 with Js A.JG .
Now apply this with J = Im ; we see that A.(Im )G contains Ims for some s, an*
*d thus is open.
This shows that AG is neat in A.
Now suppose that A is faithfully flat over AG. We claim that (IG )m is open i*
*n AG. Indeed,
the above shows that for large j we have Ij A.IG . It follows that Ijm (IG *
*A)m = A.(IG )m .
It is also clear that A.(Ijm)G Ijm, so A.(Ijm)G A.(IG )m . By faithful fla*
*tness, for any
ideals J, J0 AG we have A.J A.J0 iff J J0. We deduce that (Ijm)G (IG )m*
* . The ideal
(Ijm)G = Ijm \ AG is open in the subspace topology, so the same is true of (IG *
*)m . We also have
(IG )m (Im )G and the ideals (Im )G form a basis of neighbourhoods of 0; it f*
*ollows that the same
is true of the ideals (IG )m .
Corollary 16.9. Let A, I and G be as in the lemma, and let H be a subgroup of G*
*. Suppose that
the inclusion AH !A is faithfully flat, and that IH is finitely generated. The*
*n AG is neat in AH .
Proof.The lemma (with G replaced by H) tells us that {(IH )m  m 0} is a basi*
*s of open ideals
in AH . As IH is finitelyQgenerated, the same is true of (IH )m , say (IH )m = *
*(b1, . .,.bn). Consider
the polynomial OEbi(t) = g(t  g.bi) as in the proof of the lemma. As bi2 (IH*
* )m Im , we see
that OEbi(t) 2 tr+ (Im )G[t]. Using the relation OEbi(bi) = 0 we see that bri2 *
*(Im )GAH , so
(IH )m(n(r1)+1) (br1, . .,.brn) (Im )GAH ,
so (Im )GAH is open in AH . As the ideals (Im )G are a basis of open ideals in *
*AG, we deduce that
AG is neat as claimed.
Lemma 16.10. Suppose that A = k[y1, . .,.yr], with the evident action of G = r*
*, and with
topology determined by the powers of the ideal I = (y1, . .,.yr). Let H be a su*
*bgroup of G of the
form r1x . .x. rk, with r = r1+ . .+.rk. Then
(a)A is topologically free of rank G = r!Qover AG
(b) A is topologically free of rank H = iri! over AH
(c)AH is topologically free of rank G=H over AG
(d) The topology on AH (resp. AG) is determined by powers of the ideal IH (re*
*sp. IG ), which
is finitely generated.
Proof.It is wellknown that AG = k[u1, . .,.ur], where uiis the i'th elementary*
* symmetric function
in the variables yi. Similarly, we have AH = k[v1, . .,.vr], where v1, . .,.vr1*
*are the elementary
symmetric functions of y1, . .,.yr1, and vr1+1, . .,.vr1+r2are the elementary s*
*ymmetric functions
of yr1+1, . .,.yr1+r2and so on. By considering the maps
AG !AH !A !A=I = k,
we see that IG = (u1, . .,.ur) and IH = (v1, . .,.vr), so in particular these *
*ideals are finitely
generated.
We next claim that A is algebraically free of rank H over AH . Everything i*
*s compatible with
base change, so it will be enough to prove this when k = Z. In this case, all t*
*he rings involved
are Noetherian domains with unique factorisation and the claim is a standard pi*
*ece of invariant
theory. Similarly, we see that A and AH are algebraically free of the indicated*
* ranks over AG, and
so the inclusions AG !AH !A are faithfully flat.
Using Lemma 16.8 and Corollary 16.9 we deduce that the inclusions AG !AH !A*
* are neat.
A neat extension that is an algebraically free module is always topologically f*
*ree, which proves (a),
(b) and (c). We have seen that IG and IH are finitely generated, and the rest o*
*f (d) follows from
Lemma 16.8.
Proof of Proposition 16.6.Claim (a) is clear. Claim (b) follows from part (a) o*
*f Lemma 16.10 by
passing to completions, and part (c) follows immediately from (a) and (b).
For claim (d), put
A0 = {ff 2 A  ffi< n for alli}
B0 = A0= r = {fi 2 B  fij = 0 for allj n}.
26 N. P. STRICKLAND
Q *
* __
For fi 2 B0we put Hfi= i fii r, so ø1{fi} ' r=Hfi. As A0is a basis for Rr *
*over Rr, we
L __Hfi __
deduce that Sr = Rrr is isomorphic to fiRr as a module over Sr. It will thus *
*suffice to show
that k[[y1, . .,.yr]]Hfiis topologically free of rank  r=Hfi over k[[y1, . .,*
*.yr]] r, and this follows
from part (c) of Lemma 16.10 by passing to completions. __
For part (e), note that Rr is finitely generated over Srand thus is certainly*
* finitely generated
over the larger ring Sr. Neatness follows from Lemma 16.8.
Finally, we must show that for each_of our rings there is a finitely generate*
*d_ideal J whose
powers determine the topology._For Rr, we can obviously_take_J to be the ideal *
*Ir:= (y1, . .,.yr).
Lemma 16.8 tells us that_for_Sr we can use the ideal Jr:= Irr= (u1,_. .,.ur). F*
*or Sr (which is
topologically free_over Sr) we can therefore use the ideal Jr := JrSr. Similarl*
*y, for Rr we can use
the ideal Ir = IrRr.
Lemma 16.11. If the curve C is embeddable, then Rr is topologically free of ran*
*k r! over Sr.
Proof.We may assume that
C = spf(k[x]^f(x)) = lim!spec(k[x]=f(x)m )
m
for some monic polynomial f(x). Put A = k[x1, . .,.xr], and give this the topol*
*ogy determined by
the powers of the ideal I = (f(x1), . .,.f(xr)), so Cr = spf(A^I). The evident *
*action of G := r on
A is continuous, and A is free of rank r! over AG. We see from Lemma 16.8 that *
*AG is neat in A,
so A is topologically free over AG of rank r!, and the claim follows by passing*
* to completions.
Lemma 16.12. Let A be a ring, M a finitely generated Amodule, and B a faithful*
*ly flat A
algebra. Suppose that B A M is a free Bmodule of rank s. Then M is a projecti*
*ve Amodule of
the same rank.
Proof.First, we claim that if m is a maximal ideal in A with residue field K = *
*A=m, then
dimK(K A M) = s. Indeed, by faithful flatness there exists a prime ideal n B*
* with n \ A = m.
Using Zorn's lemma we can find a maximal element of the set of all such ideals *
*n, and this is easily
seen to be a maximal ideal in B. It follows that the residue field L = B=n is a*
* field extension of
K, so
dimK(K A M) = dimL(L K K A M) = dimL(L B (B A M)),
which is evidently equal to s.
We now choose a finite generating set {m1,P. .,.mt} for M. For each subset S *
* {1, . .,.t} with
S = s, we let fS: AS !M be the map a_7! sasms, and we let PS and QS be the*
* kernel and
cokernel of fS.
Next, we put IS = ann(QS) A. If m is maximal as before, we claim that there*
* exists S such
that IS 6 m. Indeed, as dimK(K M) = s, we can certainly choose S such that {m*
*i i 2 S} gives
a basis for K A M. It follows that K A QS = 0, or equivalently that mQS = QS.*
* The module
QS is generatedPby the elements mj for j 62 S, so we can find elements ujk2 m f*
*or each j, k 62 S
such that mj = kujkmk. Let U be the square matrix with entries ujkand put u =*
* det(I  U).
As in [13], we see that u = 1 (modPm) and u 2 IS, so IS 6 m as claimed. *
* P
It follows from this claim thatP SIS is not contained in any maximal ideal, *
*so SIS = A.
We can thus choose aS 2 IS with SaS = 1. It follows that spec(A) is the union*
* of the basic
open subschemes D(aS) = spec(A[a1S]).
We have aSQS = 0 and so QS[a1S] = 0, so the map fS becomes surjective after *
*inverting
aS. It follows that the resulting map 1 fS: B[a1S]S !B[a1S] A M is also s*
*urjective. Here
both source and target are free modules of the same finite rank over B[a1S], s*
*o our map must
in fact be an isomorphism. As B[a1S] is faithfully flat over A[a1S], we deduc*
*e that fS actually
gives an isomorphism A[a1S]s !M[a1S]. This shows that M is locally free of r*
*ank s, and thus is
projective.
Corollary 16.13. Let k be a ring, and let A be a formal kalgebra whose topolog*
*y is defined by
the powers of a single open ideal J (so A = lim A=Jm ). Let M be a finitely ge*
*nerated Amodule
m
such that M = lim M=Jm M. Let k0 be a faithfully flat kalgebra, and put A0 =*
* k0bkA and
m
M0 = k0bkM = A0bAM. Suppose that M0 is a free module of rank s over A0; then M*
* is a
projective module of rank s over A.
MULTICURVES AND EQUIVARIANT COBORDISM 27
Proof.First, note that the map A=Jm ! A0=Jm A0= k0 k A=Jm is a faithfully fla*
*t extension
of discrete rings. We can thus apply the lemma and deduce that M=Jm M is a fini*
*tely generated
projective module of rank s over A=Jm .
Next, as M is finitely generated, we can choose an epimorphism f :At !M for *
*some t. Let
Xm be the set of Amodule maps g :M=Jm M !(A=Jm )tsuch that the induced map
M=Jm M g!(A=Jm )tf!M=Jm M
is the identity. As M=Jm M is projective over A=Jm , we see that Xm is nonempty*
*. There is an
evident projection ßm :Xm !Xm1, which we claim is surjective. Indeed, given g*
* 2 Xm1 we can
use the projectivity of M=Jm M again to see that there exists a map h: M=Jm M *
*!(A=Jm )tlifting
g. Let ffi be the determinant of the resulting map fh: M=Jm M !M=Jm M, so ffi *
*2 A=Jm . Because
g 2 Xm1, we see that ffi maps to 1 in A=Jm1 . As the kernel of the projection*
* A=Jm1 !A=Jm
is nilpotent, it follows that ffi is a unit, so fh is an isomorphism. After rep*
*lacing h by h(fh)1 we
may assume that fh = 1, so h 2 Xn and ß(h) = g. It follows that lim Xm 6= ;, a*
*nd this gives a
m
map g :M !Atwith fg = 1. Thus, M is a retract of a free module, and hence is p*
*rojective.
Corollary 16.14. Rr is a projective module of rank r! over Sr, so the projectio*
*n Cr !Cr= r is
a finite, faithfully flat map of degree r!.
Proof.In Corollary 16.13, we take A = Sr and M = Rr. We know from Proposition 1*
*6.6 that
the topology on Sr is determined by powers of the ideal Jr = (u1, . .,.ur), and*
* that Sr is neat
in Rr. This means that the given topology on Rr is determined by the ideals Jmr*
*Rr. As Rr is
complete, we deduce that Rr = lim Rr=JmrRr. We next take k0 = OS0 to be a fai*
*thfully flat
r
extension of k such that the curve C0= S0xS C is embeddable; this is possible b*
*y Corollary 15.3.
Using Lemma 16.11, we see that M0 is topologically free of rank r! over A0, so *
*we can apply
Corollary 16.13 and deduce that Rr is projective over Sr.
17. Classification of divisors
Our main task in this section is to prove the following result.
Theorem 17.1. Let C be a formal multicurve over a scheme S. Then for formal sch*
*emes S0over
S, there is a natural bijection between divisors of degree r on S0xS C and maps*
* S0! Cr= r over
S.
Construction 17.2. We must first construct a universal example. We start by put*
*ting i =
{(a1, . .,.ar, b) 2 Cr+1  b = ai}, which is a divisor ofQdegree one on C over *
*Cr.P If we define
di(a_, b) = d(ai, b) then O i = Rr+1=di. Now put ffir = idi 2 Rr+1 and Der= *
* i i =
spf(Rr+1=ffir), which is a divisor of degree r on C over Cr+1. On the other han*
*d, we note that
ffir 2 Rrr+1= SrbkR, so we can define Dr = spf((SrbR)=ffir), which is a closed *
*formal subscheme
of Cr= r xS C. It is clear that Rr SrODr = ODer, which is free of rank r over *
*Rr. We know
from Corollary 16.14 that Rr is faithfully flat over Sr, and it follows from Le*
*mma 16.12 that ODr
is a projective module of rank r over Sr. Moreover, the relevant ideal is gener*
*ated by the regular
element ffir, so Dr is a divisor on C over Cr= r.
Now put Qr = Cr= r for brevity. As in Section 16, we choose a topological ba*
*sis {ei} for
OC, and use it to construct a topologicalPbasis {e0fi fi 2 B} for OQr. We the*
*n put M =
spec(Z[t0, t1, . .].), and put g = itiei, regarded as a function on M x Qrx C*
*. We then put
Y
h = NMxDr=MxQr (g) 2 OMxQr = OS[ti i 0]e0fi.
fi
P
We claim that h is actually equal to fitfie0fi. Indeed, although this is an e*
*quation in OMxQr , it will
suffice to prove it in the larger ring OMxCr . In that context, we can describe*
* h asQNMxDer=MxCr(g).
NowQletPßj:Cr !C be the j'th projection. Using Proposition 14.13 we see that h*
* = jß*jg =
j itiß*jei. Expanding this out gives
0 1
X X X X
h = tø(ff)eff= @ tfi effA= tfie0fi
ff2A fi2B ø(ff)=fi fi2B
as claimed.
28 N. P. STRICKLAND
Now suppose we have a map c: S0! Qr over S, and D = c*Dr over S0. We deduce *
*easily that
X
NMxD=MxS0 (g) = tfic*(e0fi).
fi
This shows that c*(e0fi) depends only on D, and {e0fi fi 2 B} is a topological*
* basis for Sr, so the
ring map c*:Sr !OS0depends only on D, so the map c: S0! Cr= r depends only on*
* c. We
record this formally as follows:
Proposition 17.3. Suppose we have two maps c0, c1:S0! Cr= r over S, and that c*
**0Dr = c*1Dr
as divisors over S0. Then c0 = c1.
Proof of Theorem 17.1.Let S0be a scheme over S, and let A be the set of maps S0*
*! Cr= r over
S, and let B be the set of divisors of degree r on C over S0. The construction *
*c 7! c*Dr gives
a map OE: A !B, which is injective by Proposition 17.3. To show that OE is sur*
*jective, suppose
we have a divisor D 2 B. We can choose a faithfully flat map q :T ! S0 such th*
*at q*D has a
full set of points, say u_= (u1, . .,.ur). We deduce that q*D is the pullback o*
*f eDralong the map
T u_!Cr, and thus is the pullback of Dr along the composite c = (T u_!Cr !Cr*
*= r). Now let
q0, q1:T xS0T !T be the two projections, so qq0 = qq1. Note that
(cq0)*Dr = q*0c*Dr = q*0q*D = (qq0)*D,
and similarly
(cq1)*Dr = q*1c*Dr = q*1q*D = (qq1)*D.
As qq0 = qq1 we see that (cq0)*D = (cq1)*D, and so (by Proposition 17.3) we hav*
*e cq0 = cq1.
By faithfully flat descent, we have c = _cq for a unique map _c:S0 !Cr= r. We*
* then have
q*_c*Dr = c*Dr = q*D, and using the faithful flatness of q, we deduce that D = *
*_c*Dr = OE(_c).
This shows that OE is also surjective, and thus a natural bijection.
Definition 17.4. In the light of Theorem 17.1, it makes sense to write Div+r(C)*
* for Cr= r. The
evident projection
Cr= rxS Cs= s = Cr+s=( rx s) !Cr+s= r+s
gives a map
oer,s:Div+r(C) xS Div+s(C) !Div+r+s(C).
It is easy to check that this classifies addition of divisors, in the following*
* sense: if we have divisors
D = u*Dr and D0= v*Ds on C over S0, then D + D0= w*Dr+s, where
w = (S0(u,v)!Div+r(C) xS Div+s(C) oe!Div+r+s(C)).
` +
We put Div+(C) = rDivr(C). As one would expect, this is the free abelian mono*
*id scheme
generated by C; see [15, Section 6.2] for technical details.
Definition 17.5. Now suppose that C has an abelian group structure, written add*
*itively. We
can then define ~~:Cr xS Cs !Crsby
~~(a0, . .,.ar1; b0, . .,.bs1)i+rj= ai+ bj
(for 0 i < r and 0 j < s). The composite
Cr xS Cs ~~!Crsq!Crs= rs
is invariant under rx s, so we get an induced map
~r,s:Divr(C) xS Divs(C) !Divrs(C).
If we have divisors D = u*Dr and D0= v*Ds on C over S0, then we define D *D0to *
*be the divisor
w*Drs, where
w = (S0(u,v)!Div+r(C) xS Div+s(C) ~!Div+rs(C)).
We call this the convolution of D andPD0. This operationPmakes Div+(C) into aPs*
*emiring. If we
have full sets of points, say D = i[ai] and D0= j[bj] then D * D0is just *
*i,j[ai+ bj].
Proposition 17.6. Let D and D0be divisors on C over S. Then there exists a clos*
*ed subscheme
T S such that for any scheme S0 over S, we have S0xS D S0xS D0 iff the map *
*S0 !S
factors through T .
MULTICURVES AND EQUIVARIANT COBORDISM 29
Proof.As OD is finitely generated and projective over OS, we can choose an embe*
*dding i: OD !
ONSof OSmodules, and a retraction r :ONS !OD . We then have i(fD0) = (a1, . *
*.,.aN ) for
some elements aj 2 OS, and we put J = (a1, . .,.aN ) and T = spec(OS=J). We fi*
*nd that a
map S0 !S factors through T iff J maps to 0 in OS0, iff fD0 maps to 0 in OS0 O*
*S OD , iff
S0xS D S0xS D0.
Proposition 17.7. Let D be a divisor on C over S, and suppose that r 0. Then *
*there is a
scheme Subr(D) over S such that maps S0! Subr(D) over S biject with divisors D*
*0 S0xS D
of degree r.
Proof.Over the formal scheme Div+r(C) we have both the originally given divisor*
* D and the
universal divisor Dr. We let Subr(D) denote the largest closed subscheme of Div*
*+r(C) where Dr
is contained in D (which makes sense by Proposition 17.6). It is formal to chec*
*k that this has the
required property.
Proposition 17.8. Let D be a divisor on C over S, and suppose that r 0. Then *
*there is a
scheme Pr(D) over S suchPthat maps S0! Pr(D) over S biject with lists (u1, . .*
*,.ur) of sections
of C over S0 such that i[ui] S0xS D.
Proof.Over the formal scheme Cr we have both the originally given divisor D and*
* the divisor
eDr. We let Pr(D) denote the largest closed subscheme of Cr where eDris contain*
*ed in D (which
makes sense by Proposition 17.6). It is formal to check that this has the requi*
*red property.
*
* P
Remark 17.9. Suppose that D has degree r. Then Pr(D) classifiesPrtuples for wh*
*ich i[ui] D,
but by comparing degrees we see that this means that i[ui] = D. Thus, Pr(D) c*
*lassifies full sets
of points for D.
Lemma 17.10. Suppose we have ring maps A !B !C, and C is a projective module *
*of degree
m > 0 over B, and also a projective module of degree nm > 0 over A. Then B is a*
* projective
module of degree n over A.
Proof.We can use the second copy of B to make HomA (B, B) into a Bmodule. For *
*any Bmodule
N there is an evident map Hom A(B, B) B N !Hom A(B, N). This is evidently an *
*isomorphism
if N is a free module of finite rank, and thus (by taking retracts) also when N*
* is projective of finite
rank over B. In particular, we have Hom A(B, B) B C = Hom A(B, C). As C is als*
*o projective
over A, the same kind of argument shows that
Hom A(B, C) = Hom A(B, A) A C = (Hom A(B, A) A B) B C.
It follows that (Hom A(B, A) A B) B C = Hom A(B, B) B C. More precisely, the*
*re is a natural
map
ff: Hom A(B, A) A B !Hom A(B, B),
given by ff(OE b)(b0) = OE(b0)b. By working through the above argument more *
*carefully, we
see that ff B 1C is an isomorphism. However, C is faithfully flat over B soPff*
* itself must be an
isomorphism. In particular, we see that 1B lies in the image of ff, so 1B = PN*
*i=1ff(OEi bi) for some
maps OEi:B !A and elements bi2 B. This means that for all b 2 B we have b = *
*iOEi(b)bi. We
can use the elements OEito give a map OE: B !AN , and the elements bito give a*
* map fi :AN !B.
We find that fiOE = 1, which proves that B is projective. It is now clear that *
*the rank must be
n.
Proposition 17.11. Let D be a divisor of degree s on C over S, and suppose that*
* 0 r s. Then
there are natural maps Pr(D) p!Subr(D) q!S which are finite and very flat, wi*
*th deg(p) = r!
and deg(q) = s!=(r!(s  r)!) (so deg(qp) = s!=(s  r)!).
*
* P
Proof.Over Pr(D) we have tautological sections u1, . .,.ur of C giving a diviso*
*r D0r:= i[ui]
on C. This is contained in (qp)*D, so we can form the divisor D00r:= (qp)*D  D*
*0r, which has
degree s  r over Pr(D). It is easy to identify Pr+1(D) with D00r, so deg(Pr+1*
*(D) ! S) =
(s  r) deg(Pr(D) !S). By an evident induction, we see that the map pq is fini*
*te and very flat,
with degree s!=(s_ r)!, as claimed.
Next, let D be_the tautological divisor of degree r on C over_Subr(D). We ca*
*n then form
the_scheme_Pr(D ), which classifies full sets of points on D . As above, we_se*
*e that the map
Pr(D ) !Subr(D)_is finite and very flat, with degree r!. We claim that Pr(D ) *
*= Pr(D). Indeed,_
a map S0! Pr(D ) over S corresponds to a map S0! Subr(D), together with a lif*
*ting to Pr(D ).
Equivalently, it corresponds to a divisor of degree r contained in S0xS D, toge*
*ther with sections
30 N. P. STRICKLAND
u1, . .,.ur:S0 !C giving a full set of points for that divisor.PThe full set o*
*f points determines
the divisor, so it is equivalent to just give sections ui with i[ui] S0xS D*
*, or equivalently, a
map S0! PrD over S. The claim follows by Yoneda's lemma. It follows that the m*
*ap p is finite
and very flat, with degree r!. We can now apply Lemma 17.10 to see that q is fi*
*nite and very flat,
with degree s!=(r!(s  r)!).
Proposition 17.12. For the universal divisor Ds over Div+s(C) we have
Subr(Ds)= Div+r(C) xS Div+sr(C)
Pr(Ds)= Cr xS Div+sr(C).
Proof.Let S0 be a scheme over S. Then a map S0 !Subr(Ds) over S corresponds t*
*o a map
S0! Div+r(C), together with a lifting to Subr(Ds). Equivalently, it correspond*
*s to a divisor D of
degree s on C over S0, together with a subdivisor D0 D of degree r. Given such*
* a pair (D, D0),
we have another divisor D00= D  D0, which has degree s  r. There is evidentl*
*y a bijection
between pairs (D, D0) as above, and pairs (D0, D00) where D0 and D00are arbitra*
*ry divisors of
degrees r and s  r. These pairs correspond in turn to maps S0! Div+r(C) xS Di*
*v+sr(C) over
S. The first claim follows by Yoneda's lemma, and the second claim can be prove*
*d in the same
way.
Corollary 17.13. We have Ds = Div+s1(C) xS C = Cs= s1.
Proof.Take r = 1, and observe that P1(Ds) = Sub1(Ds) = Ds and Div+1(C) = C.
18. Local structure of Div+d(C)
Let C be a formal multicurve over a base S. In the nonequivariant case, we k*
*now that
Div+n(C) ' spf(OS[[c1, . .,.cn]]) = bAnS, so Div+n(C) is a formal affine space *
*of dimension n over
S. Equivariantly, this is not even true when n = 1. However, we will show in th*
*is section that
Div+n(C) is still a öf rmal manifold", in the sense that the formal neighbourho*
*od of any point is
isomorphic to bAnS, at least up to a slight twisting. Later we will apply this *
*to calculate E0BU(V ),
where BU(V ) is the simplicial classifying space of the unitary group of a repr*
*esentation V of A.
We state the result more formally as follows.
Theorem 18.1. Let C = spf(R) be a formal multicurve over S = spec(k), with a di*
*fference
function d. Let s: S ! Div+n(C) be a section, classifying a divisor D = spf(R=*
*J) C. Then
the formal neighbourhood of sS in Div+n(C) is isomorphic to the formal neighbou*
*rhood of zero in
MapS(D, A1S) (by an isomorphism that depends on the choice of d).
The rest of this section constitutes a more detailed explanation and a proof.
We first examine the two formal schemes that are claimed to be isomorphic. W*
*e put A0 =
(Rbn) n and X0 = spf(A0) = Div+n(C). The section s corresponds to a kalgebra m*
*ap A0 !k,
with kernel K say. We put A = (A0)^Kand X = spf(A). This is the formal neighbou*
*rhood of sS
in Div+n(C).
Now consider the scheme Y0 = Map S(D, A1S). For any scheme T over S, the maps*
* T ! Y0
over S are (essentially by definition) the maps D xS T ! A1 of schemes, or equ*
*ivalently the
elements in the ring OD k OT. These biject with the maps O_D= Hom k(OD , k) !*
* OT of k
modules, or with the maps B0 = Symk[O_D] !OT of kalgebras. Thus, we have Y0 =*
* spec(B0).
We let B be the completion of B0 at the augmentation ideal, and put Y = spf(B),*
* which is the
formal neighbourhood of the zero section in Y0. Of course B0 is just the direc*
*t sum of all the
symmetric tensor powers of O_D, and B is the direct product of the same terms. *
*If OD is free over
k (rather than just projective) then B is isomorphic to k[[c1, . .,.cn]]; in th*
*e general case, it should
be regarded as a slight twist of this. Note that maps T ! Y over S biject with*
* klinear maps
O_D!Nil(OT), or equivalently elements of OD kNil(OT). Note also that a choice*
* of generators
x1, . .,.xr for O_Dgives a split surjection k[[x1, . .,.xr]] !B.
There is an evident map
ff: Div+n(C) = X0 !Y0 = MapS(D, A1S),
sending the section s0classifying a divisor D0to the function (fD0)D :D !A1. *
*This clearly sends
s itself to zero, so it sends the formal neighbourhood of s to the formal neigh*
*bourhood of zero, so
it gives a map ff: X !Y . We shall show that this is an isomorphism.
MULTICURVES AND EQUIVARIANT COBORDISM 31
Note that because D and D0have the same degree, we have s0= s if and only if *
*fD0 is divisible
by fD , if and only if ff(s0) = 0. This shows that the kernel K of the map s*:A*
*0 !k is generated
by the image under ff* of the augmentation ideal in B0. In particular, we see t*
*hat K is finitely
generated.
Because OD = R=J is projective over k, we can choose a ksubmodule P R such*
* that
R = P J. It follows that the map P !R !OD is an isomorphism, with inverse ,*
* say.
Lemma 18.2. Let I k be a finitely generated ideal with Im = 0, and let g 2 R *
*be such that
g = fD (mod IR). Then g is a regular element, the ideal Rg is open, and we have*
* R = Rg P .
Q
Proof.A standard topological basis for R gives an isomorphismQR = ik, and usin*
*g the fact that
I is finitelyQgenerated we see that IjR = (IR)j = iIj. We thus have a finite f*
*iltration of R with
quotients iIj=Ij+1.
Now consider the klinear selfmap of R given by ~(qfD + r) = qg + r for q 2 *
*R and r 2 P .
This is easily seen to induce the identity map on the quotients of the above fi*
*ltration, so it is an
isomorphism. It follows easily that g is regular and R = Rg P .
As D is a divisor, we know that RfD is open. Thus, for any good parameter y w*
*e have yl2 RfD
for large l, say yl= ufD . We also know that fD = g + h for some h 2 IR, so yl=*
* uh (mod g).
As Im = 0 we have ylm = um hm = 0 (mod g), so Rg is also open.
We now define a map fi from sections of Y to sections of X. A section of Y i*
*s an element
r 2 Nil(k)OD . As OD is finitely generated we have r 2 IOD for some finitely ge*
*nerated ideal
I Nil(k), and by finite generation this satisfies Im = 0 for some m. We can*
* thus apply the
lemma to the function g = fD + ,(r) 2 R and conclude that the subscheme D0= spe*
*c(R=g) is
a divisor of degree n, classified by a section s0 of X0 say. Over the subschem*
*e spec(k=I) S
it clearly coincides with s, so (s0)*(K) I, so (s0)*(Km ) = 0. This shows tha*
*t s0is actually a
section of X, as required. We can thus define fi(r) = s0.
In order to define a map fi :Y !X of formal schemes over S, we need to defin*
*e maps fiT from
sections of Y over T to sections of X over T , naturally for all schemes T over*
* S. For this we just
replace C by T xS C, P by OT k P and follow the same procedure.
We now define another map ff0:X ! Y . It will again be sufficient to do thi*
*s for sections
defined over S. Let s0be a section of X, classifying a divisor D0. Put I = (s0)*
**K k; this is
finitely generated because K is, and nilpotent because s0 lands in X. Over spe*
*c(k=I) we have
D0= D, so fD0 = fD (mod IR). The lemma tells us that R = RfD0 P , so there ar*
*e unique
elements h 2 R and p 2 P such that fD = hfD0  p. By reducing modulo I we see t*
*hat h = 1
(mod IR) and p 2 IP . We let r be the image of p in R=fD = OD , so r 2 IOD and *
*,(r) = p. The
map ff0:X ! Y is defined by ff0(s0) = r. Note that h is invertible so fD0 is a*
* unit multiple of
fD + p = fD + ,(r), so D0= spec(R=(fD + ,(r))) = fi(r). This shows that fiff0= *
*1.
In the other direction, suppose we start with r 2 IOD and put D0 = spec(R=(fD*
* + ,(r)))
(corresponding to fi(r)). There is then a unique element p 2 P congruent to fD*
* modulo fD0,
and ff0fi(r) is by definition the image of p in OD . It is clear that fD is co*
*ngruent to ,(r) modulo
fD + ,(r), which is a unit multiple of fD0, so p = ,(r) and ff0fi(r) = r. This *
*shows that ff0fi = 1,
so ff0and fi are isomorphisms.
We actually started by claiming that the (slightly more canonical) map ff is *
*an isomorphism.
As fi is an isomorphism, it suffices to check that the map fffi :Y !Y is an is*
*omorphism, or that
(fffi)* is an automorphism of OY = B. As B is the completed symmetric algebra o*
*f a finitely
generated projective module, it will suffice to show that (fffi)* is the identi*
*ty modulo the square of
the augmentation ideal. By basechange to the universal case, it will suffice t*
*o show that fffi(r) = r
whenever r 2 IOD with I2 = 0. Given such an r, we form the divisor D0= spec(R=(*
*fD + ,(r)))
corresponding to fi(r), and observe that fD0 = u(fD + ,(r)) for some u 2 Rx. A*
*s fD0 = fD
(mod IR) we must have u = 1 (mod IR). As ,(r) 2 IR and I2 = 0 we have u,(r) = ,*
*(r) and so
fD0 = ,(r) (mod fD ), so fffi(r) = r as claimed.
19.Generalised homology of Grassmannians
Consider a periodically orientable theory E with associated equivariant forma*
*l group C =
spf(E0P`U) over S = spec(E0). Let GrU be the space of rdimensional subspaces o*
*f U, and put
GU = 1r=0GrU. Here we reprove the following result from [5].
32 N. P. STRICKLAND
Theorem 19.1 (Cole, Greenlees, Kriz). There are natural isomorphisms
E*GrU = (E*P U) rr
E*GrU = ((E*P U)br) r
spf(E0GrU)= Cr= r.
We first introduce some additional structure. For any complex inner product s*
*pace V , we put
`
R0(V ) = V GV+ = V GrV+.
r
Using the evident maps GrU x GsV !Gr+s(U V ) we get maps ~U,V:R0(U) ^ R0(V *
*) !
R0(U V ). We also have inclusions jU :SU = UG0U+ ! R0(U). These maps make R*
*0 into
a commutative and associative ring in the category of orthogonal prespectra [12*
*]. All this works
equivariantly in an obvious way. The weak homotopy type of R0 is
_ !
a
R0 ' lim! U 1 R0(U) = lim! 1 GU+ = 1 GU+ = 1 GrU .
U U U U r +
W
We write QrR0 for the subfunctor V 7! V GrV+, so that R0 = rQrR0 and QrR0 ' G*
*rU+. In
particular, we have Q0R0 = S0 and Q1R0 = P U+. This gives a map E*P U !E*R0 an*
*d thus a
ring map SymE* E*P U !E*R0. The theorem says that this is an isomorphism. For *
*the proof,
we need some intermediate spectra. For any representation W , we put
QrRW (V )= V GrV Hom(T,W)
`
RW (V )= QrRW (V ) = V GV Hom(T,W).
r
This again gives a commutative orthogonal ring spectrum, with weak homotopy typ*
*e RW '
GUHom(T,W). In the case W = 0 we recover R0 as before. An inclusion W !W 0give*
*s a ring map
i: RW ! RW0. In particular, we have a ring map R0 !RW , whose fibre we denote*
* by JW . This
is weakly equivalent to the stable fibre of the zero section GU+ !GUHom(T,W)+,*
* and thus is the
sphere bundle of the bundle Hom (T, W ) over GU.
Next, recall that there is an isometric embedding U W ! U, and that the sp*
*ace of such
embeddings is connected. We have
Q1RW = P UHom(T,W)= P (U W )=P W ' P U=P W.
Using this, we have a diagram as follows, in which the rows are cofibrations:
P W+ ______PwU+ _____PwU=P W
  
  
u u u
JW ________R0w________RWw.
The map P U=P W !RW gives a ring map
`W : SymE*E*(P U, P W ) !E*RW .
Theorem 19.2. The above maps `W are isomorphisms.
The proof will be given after some preparatory results.
First, suppose we have representations W and L with dim(L) = 1. We put W 0= W*
* L and
investigate the fibre of the map RW ! RW0. We may assume that W 0 U, and then*
* we have a
map
SHom(L,W)= P LHom(T,W) P UHom(T,W)!RW ,
which we denote by bW,L. Multiplication by bW,L gives a map Hom(L,W)RW !RW ,*
* which we
again denote by bW,L. (Note that this sends Hom(L,W)QrRW into Qr+1RW , or in *
*other words,
it increases internal degrees by one.)
Proposition 19.3. The sequence
Hom(L,W)RW bW,L!RW ! RW L = RW0
is a cofibration.
MULTICURVES AND EQUIVARIANT COBORDISM 33
Proof.This is a special case of the following fact. Suppose we have a space X w*
*ith vector bundles
U and V . Let S(U) for the unit sphere bundle in U, and D(U) for the unit disc *
*bundle, so XU is
homeomorphic to D(U)=S(U).*We can pull back V along the projection q :S(U) !X *
*and thus
form the Thom space S(U)q V. It is not hard to see that there is a cofibration
*V V U V
S(U)q ! X !X .
We will apply this with X = GU and V = Hom (T, W ) and U = Hom (T, L),*so that *
*XV = RW
and XU V = RW0. To prove the proposition, we need to identify S(U)q V with Hom*
*(L,W)RW .
To do this, observe that S(U) is the space of pairs (M, ff) where M is a fini*
*tedimensional
subspace of U and ff: M !L is a linear map of norm one. As L has dimension one*
*, we find that
ff can be written as the composite of the orthogonal projection M !M ker(ff) *
*with an isometric
isomorphism M ker(ff) !L. Using this, we identify S(U) with the space of pai*
*rs (N, fi), where
N is a finitedimensional subspace of U and fi :L ! U N is an isometric embe*
*dding; the
correspondence is that M = N fi(L) and
1
ff = (N fi(L) proj!fi(L) fi!L).
We can thus define a map k :S(U) !GU by k(N, fi) = N (or equivalently, k(M, ff*
*) = ker(ff)).
This makes S(U) into an equivariant fibre bundle over GU. The fibre over a poin*
*t N 2 GU is the
space L(L, U N), which is wellknown to be contractible, and the contraction c*
*an be chosen to be
equivariant with respect to the stabilizer of N in A. It follows that k is an e*
*quivariant equivalence.
To understand the inverse of k, recall that L W 0 U, so we can put
Y = {N 2 GU  N is orthogonalLto} = G(U L).
Define j :Y !X by j(N) = N L, and then define ~_:Y !S(U) by
~_(N) = (N L, proj:N L !L),
so q~_= j. Clearly k~_:Y = G(U L) ! GU = X is just the map induced by the i*
*nclusion
U L !U. As the space of linear isometries between any two complete Auniverses*
* is equivariantly
contractible, we see that this inclusion is an equivariant equivalence. As the *
*same is true of k,
we deduce that ~_is also an equivariant equivalence. We can thus identify S(U)*
* with Y and
q :S(U) !X with j :Y !X. It follows that we can identify q*V with j*V , but t*
*he fibre of j*V
over a point N 2 Y is Hom (j(N), W ) = Hom (L, W ) Hom(N, W ), so
*V Hom(L,W) Hom(T,W) Hom(L,W)
Y j = Y ' RW .
This gives a cofibration Hom(L,W)RW !RW !RW0, and one can check from the d*
*efinitions
that the first map is just multiplication by bW,L.
Now choose a complete flag
0 = W0 < W1 < . .<.U,
where dimCWi= i and U = lim!Wi. Put R(i) = RWi, so we have maps
i
R0 = R(0) !R(1) !R(2) !. ...
Put Li= Wi+1 Wiand Ui= Hom (Li, Wi) and bi= bWi,Li, so we have a cofibration
UiR(i) bi!R(i) !R(i + 1).
L
Lemma 19.4. Suppose that B A, and split W as fi2B*W [fi] in the usual way. *
*Then
__B ^
OE RW = RW[fi],
fi2B*
where as before
W [fi] = {w 2 W  bw = e2ßifi(b)w for allb 2 B},
__B
and so the connectivity of (OE RW )=S0 is at least minfi(2 dimC(W [fi])  1).
Proof.We have __
OEBGU = (GU)B = { BinvariantLsubspaces of}U.
Any complex representation U of B splits as fiU[fi], so a subspace U U is i*
*nvariant iff it is
the direct sum of its intersections with the subspaces U[fi]. It follows that
Y Y
(GU)B = GU[fi] ' GC1 .
fi fi
34 N. P. STRICKLAND
We have a tautological bundle T [fi] over GU[fi], and the bundle Hom CB(T, W ) *
*over (GU)B is the
external direct sum of the bundles Hom C(T [fi], W [fi]). The Thom complex GU[f*
*i]Hom(T[fi],W[fi])is
just RW[fi], and it follows that (GU)Hom(T,W)B is just the smash product of th*
*ese factors, as
claimed.
For the last statement, note that if X is a space and U is a vector bundle of*
* real dimension d
over X, then XU is always (d  1)connected. Now let V be a complex universe, a*
*nd V a complex
vector space of finite dimension d. The bundle Hom (T, V ) over GrV has real di*
*mension 2rd, so
conn(QrRV ) 2rd  1, and
`
conn(RV =S0) = conn( QrRV ) 2d  1.
r>0
The claim follows easily.
Corollary 19.5. lim!R(i) = S0.
i
Proof.The unit map S0 ! Q0R(i) is an isomorphism for all i, so lim!Q0R(i) = S*
*0. It will
i _*
*_B
thus suffice to show that lim!R(i)=S0 = 0, or equivalently that the spectrum O*
*E(lim R(i)=S0) =
__B i *
* ! i
lim!((OE R(i))=S0) is nonequivariantly contractible for all B. As U is a compl*
*ete universe, we have
i __B
dimWi[fi] !1 as i !1 for all fi, so conn(OE R(i)=S0) !1, and the claim follo*
*ws.
We now let E be a periodically oriented theory, with orientation x say. This *
*gives a universal
generator uifor eE0SUi, and a basis {ci i 0} for eE0P U. Put
ER(i) = SymE*E*(P U, P Wi) = E*[cj  j i] = ER(0)=(ck  k < i),
and let QrER(i) be the submodule generated by monomials of weight r (where each*
* generator cj
is considered to have weight one). We then have maps
`i= `Wi: ER(i) !E*R(i),
which restrict to give maps
`ir:QrER(i) !E*QrR(i).
The elements uiand ciare related as follows: the inclusion P Li! P U gives an *
*inclusion SUi !
P UHom(T,Wi)' P U=P Wi, and the image of uiunder this map is the same as the im*
*age of ciunder
the evident quotient map P U ! P U=P Wi. It follows that the cofibration UiR(*
*i) bi!R(i) !
R(i + 1) gives rise to a cofibration
E ^ R(i) ci!E ^ R(i) !E ^ R(i + 1),
which restricts to give a cofibration
E ^ Qr1R(i) ci!E ^ QrR(i) !E ^ QrR(i + 1).
Proposition 19.6. The maps `irare isomorphisms for all i and r.
Proof.The maps `j0and `j1are visibly isomorphisms, so we may assume inductively*
* that `j,r1
is an isomorphism for all j. The cofibration displayed above gives a diagram D(*
*i) as follows:
Qr1ER(i) v______QrER(i)wci____QrER(iw+w1)
  
  
`i,r1' `i,r `i+1,r
  
u u u
E*Qr1R(i) ______E*QrR(i)wci__wE*QrR(iq+i1)r
We first prove that `iris surjective for all i. Let (i) be the image of `ir, s*
*o the claim is that
(i) = E*QrR(i). For j i we write K(j) for the kernel of the map E*QrR(i) !E*
**QrR(j).
Clearly K(i) = 0 (i), and by chasing the diagram D(j) we see thatSif K(j) *
* (i) then
K(j + 1) (i) also. Corollary 19.5 now tells us that E*QrR(i) = jK(j) (i,*
* r) as required.
We now see that in D(j), the vertical maps are surjective, so qjris surjectiv*
*e. As the bottom
row is part of a long exact sequence and the right hand map is surjective, we c*
*onclude that the
bottom row is actually a short exact sequence. Using the snake lemma, we concl*
*ude that the
induced map ker(`jr) !ker(`j+1,r) is an isomorphism. It follows that for any m*
* > j, the map
MULTICURVES AND EQUIVARIANT COBORDISM 35
ker(`jr) ! ker(`mr) is an isomorphism. However, we have ker(`jr) QrER(j), a*
*nd it is also
clear that when r > 0, any element of QrER(j) maps to zero in QrER(m) for m 0*
*. It follows
that ker(`jr) must be zero, so `jris an isomorphism as claimed.
Proof of Theorem 19.2.Given any subrepresentation W < U, we can choose our flag*
* {Wi} such
that W = Wifor some i. The theorem then follows from Proposition 19.6.
20.Thom isomorphisms and the projective bundle theorem
Let E be a periodically orientable Aequivariant cohomology theory, with asso*
*ciated equivariant
formal group (C, OE) over S. For any Aspace X, we will write XE = spf(E0X).
Now let V be an equivariant complex vector bundle over X. We write P V for th*
*e associated
bundle of projective spaces, and XV for the Thom space (so XV = P (V C)=P V ).*
* In this section,
we will give a Thom isomorphism and a projective bundle theorem to calculate eE*
**XV and E*P V .
First, it is wellknown that equivariant bundles of dimension r over X are cl*
*assified by homotopy
classes of Amaps X ! GrU. We saw above that E0GrU = Sr, and moreover the sta*
*ndard
topological basis {e0fi} for Sr is dual to a universal basis for E0GrU. It foll*
*ows that (GrU)E =
Cr= r = Div+r(C).
Now let T denote the tautological bundle over GrU. It is not hard to identify*
* the projective
bundle P T !GrU with the addition map
Gr1U x P U = Gr1U x G1U !GrU,
and thus to identify E0P T with Sr1bR = OCr= r1, so P TE = Cr= r1. On the ot*
*her hand,
we can use Corollary 17.13 to identify Cr= r1 with the universal divisor Dr ov*
*er Cr= r.
Now suppose we have a vector bundle V over X, classified by a map c: X !GrU,*
* so c*T ' V .
The map c is then covered by a map ~c:P V !P T , which gives a map ~c*:ODr = E*
*0P T !E0P V .
We can combine this with the evident map E*X !E*P V to get a map
`X,V:ODr SrE*X !E*P V.
Theorem 20.1. For any X and V as above, the map `X,V is an isomorphism (and so *
*E*P V is
a projective module of rank r over E*X).
Proof.We first examine the simplifications that occur when V admits a splitting*
* V = L1 . . .Lr,
where each Liis a line bundle. In this case, the classifying map X !GrU factor*
*s through P Ur, so
the map Sr !E0X factors through Rr. As ODr SrRr = ODer, we see that `X,V is th*
*e composite
of an isomorphism with a map
`0X,V:ODer Rr E*X !E*P V.
Next, choose a coordinate x on C and define a difference function d(a, b) = x(b*
*  a) as usual.
Define a function dion Cr+1 by
di(a1, . .,.ar, b) = d(ai, b) = x(b  ai),
Q
as in Construction 17.2. We then put ci= j 0, whereas*
* for j = 0 we
get a0, which is 1 because the polynomial f(x) = i+j=rajxj is monic. On the o*
*ther hand, we
also have ij(1) = ffi0jby definition, so res((1 ij)(e)dx=f(x)) = ij(1) as req*
*uired.
Given this, it would be reasonable to define residues on multicurves using th*
*e maps ffl. To make
this work properly, we need to check that these maps are compatible for differe*
*nt divisors.
Proposition 21.30. Suppose we have divisors D0 D1, corresponding to ideals K1*
* K0 R.
Let j :K10=R !K11=R be the evident inclusion, and let q :A1 = R=K1 !R=K0 = *
*A0 be the
projection. Define ffii:A_i!k by ffii(OE) = OE(1). Then the following diagram *
*commutes:
ffi0 Ø0`0
ku______A_0_____K10=Rw'
v v
 
q_ j
u u
k u_________A_1ffi1_K11=Rw'Ø0`0
Proof.As q(1) = 1, it is clear that the left hand square commutes. For the righ*
*t hand square,
choose generators fifor Kiand put ei= ,(fi), so that
Ø0`0(OE) = (1 OE)(ei)=fi ff
for OE 2 A_i.
As D0 D1, we have f1 = gf0 for some g, and so ,(f1) = (g 1),(f0) + (1 f*
*0),(g), or in
other words e1 = (g 1)e0+ (1 f0),(g).
Now suppose we have OE 2 A_0, so (q*OE)(f0) = 0, so (1 q*OE)((1 f0),(g)) *
*= 0. We then have
Ø0`0q*(OE)= ((1 q*OE)(e1)=f1) ff
= (g(1 OE)(e0))=(gf0) ff
= (1 OE)(e0)=f0 ff
= jØ0`0(OE)
as claimed.
Definition 21.31. Define ffi :Hom OS(OD , OS) !OS by ffi(OE) = OE(1). We let r*
*es:MC OC !
OS be the unique map whose restriction to I1D OC is the composite
I1D OC !(I1D=OC) OC ' Hom OS(OD , OS) ffi!OS.
(This is welldefined, by Proposition 21.30, and compatible with the classical *
*definition, by Propo
sition 21.29.)
Proposition 21.32. For any g, f 2 OC, if f is divisorial than
res((g=f)df) = trace(OC=f)=OS(g).
MULTICURVES AND EQUIVARIANT COBORDISM 47
In particular, we have res((1=f)df) = dimOS(OC=f). Moreover, we also have
` '
res(d(g=f)) = res fd(g)__gd(f)_f2= 0.
Proof.Both facts are wellknown for residues in the classical sense, so they ho*
*ld whenever C is
embeddable. Using Corollary 15.3, we deduce that they hold for a general multic*
*urve C. We will
give a more direct and illuminating proof for the first fact; we have not been *
*able to find one for
the second fact.
We use abbreviated notation as before, with K = Rf so that A = R=f. The multi*
*plication
map ~: A2 !A restricts to give an Alinear map ~: J !A, or in other words an *
*element of J*.
The trace map ø :A !k can be regarded as an element of A_. We claim that the e*
*lements ø, ~
and (df)=f correspond to each other under our standard isomorphisms
A_ ' J* ' (K1=R) .
To see this, note that (1 ø)(u) = traceA2=A(u) for all u 2 A2. Using the spli*
*ttable short exact
sequence
I !A2 ~!A,
we see that
(1 ø)(u) = trace(I xu!I) + ~(u).
If u 2 J then multiplication by u is zero on I and we deduce that (1 ø)(u) = *
*~(u). This shows
that `0(ø) = ~ as claimed.
Next, let e = ,(f) be the standard generator of J, and let j be the dual gene*
*rator of J*, so
j(e) = 1. Using Proposition 21.25, we see that ~ corresponds to the element (~*
*(e)=f) ff =
(1=f) (~(e)ff) in (K1=R) . Now, the module = eI=eI2is originally a modu*
*le over R2 that
happens to be annihilated by ker(~) = eI, and so is regarded as a module over R*
* via ~. Thus,
~(e)ff is just the same as eff. Moreover, ff is just the image of d in , so e*
*ff is the image of
ed = ,(f)d = 1 f  f 1, and this image is by definition just df. Thus, ~ 2 *
*J* corresponds to
(1=f) df as claimed.
As the isomorphism A* !(K1=R) is Alinear, we see that gø maps to (g=f)*
*df, so
res((g=f)df) = (gø)(1) = ø(g),
as claimed.
Remark 21.33. It is useful and interesting to reconcile this result with [16, P*
*roposition 9.2].
There we have a pdivisible formal group bC= spf(R) of height n over a formal s*
*cheme S = spf(k),
where k is a complete local Noetherian ring of residue characteristic p, and we*
* will assume for
simplicity that k is torsionfree. Fix m 1 and let _ :bC!bCbe pm times the i*
*dentity map. In
this context the subgroup scheme D := bC[pm ] = ker(_) is a divisor of degree p*
*nm , so the ring OD
is selfdual (with a twist) as before. Given any coordinate x, we note that OD *
*= R=_*x, so the
meromorphic form ff = D(x)=(_*x) is a generator of the twisting module (K1=R) *
* . We claim
that ff is actually independent of x. Indeed, any other coordinate x0has the fo*
*rm x0= (x + x2q)u
for some u 2 kx and q 2 R. It follows that d0(x0) = ud0(x), so that D(x0) = uD*
*(x). We
also have _*(x0) = u_*(x) (mod _*(x)2), and it follows that _*(x0)1 = u1_*(x)*
*1 (mod R),
so D(x0)=_*(x0) = D(x)=_*(x) mod holomorphic forms, as claimed. Thus, we have a*
* canonical
generator for (K1=R) and thus a canonical generator for A_, giving a Frobe*
*nius form on
OD . The cited proposition says that this is the same as the Frobenius form com*
*ing from a transfer
construction. As discussed in the preamble to that proposition, pm times the tr*
*ansfer form is the
same as the trace form. In view of Proposition 21.32, this means that pm ff = d*
*(_*x)=(_*x). In
fact, this is easy to see directly. We know that D(x) generates and agrees wi*
*th d(x) at zero, so
d(x) = (1 + xr)D(x) for some function r 2 R. It follows that
d(_*x) = _*(d(x)) = (1 + _*(x)_*(r))_*(D(x)).
As D(x) is invariant we have _*D(x) = pm D(x). It follows that d(_*x)=(_*x) = p*
*m D(x)=(_*x) =
pm ff in (K1=R) , as claimed.
Remark 21.34. It should be possible to connect our treatment of residues with t*
*hat of Tate [19].
However, Tate assumes that the ground ring k is a field, and it seems technical*
*ly awkward to
remove this hypothesis.
48 N. P. STRICKLAND
21.3. Topological duality. Consider a periodically orientable theory E, an Asp*
*ace X, and an
equivariant complex bundle V over X. To avoid some minor technicalities, we wil*
*l assume that
X is a finite ACW complex; everything can be generalised to the infinite case *
*by passage to
(co)limits. Let C be the multicurve spf(E0(P U x X)) over S := spec(E0X). We *
*then have a
divisor D = D(V ) on C, to which we can apply all the machinery in the previous*
* section. In
particular, this gives us a residue map
res:(I1D=OC) OC !OS.
On the other hand, if we let ø denote the tangent bundle along the fibres of P *
*V , then there is a
stable PontrjaginThom collapse map X+ !P V ø, giving a Gysin map
p!:eE0P V ø!E0X = OS.
Theorem 21.35. There is a natural isomorphism eE0P V ø= (I1D=OC) OC , which*
* identifies
the Gysin map with the residue map.
This is an equivariant generalisation of a result stated by Quillen in [14]. *
*Even in the nonequiv
ariant case, we believe that there is no published proof. The rest of this sect*
*ion constitutes the
proof of our generalisation. (The case of nonequivariant ordinary cohomology is*
* easy, and is a
special case of the result proved in [6].)
We retain our previous notation for rings, and write P 2V = P V xX P V , so
k= OS = E0X
R= OC = E0(P U x X)
A= OD = E0(P V )
R2= Rb R = E0(P U x P U x X)
A2= A A = E0(P 2V ).
Next, observe that P2V is a subspace of P 2V , and Proposition 20.10 tells us t*
*hat E0P2V =
OP2D = A2=J, so J = E0(P 2V, P2V ). On the other hand, there is another natural*
* description
of E0(P 2V, P2V ), which we now discuss. Let T be the tautological bundle on P *
*V , consider the
vector bundles T ?= V T and U = Hom (T, T ?), and let BOU denote the open uni*
*t ball bundle
in U. A point in BOU is a triple (x, L, ff) where x 2 X and L 2 P Vx and ff: L*
* ! Vx L,
such that kff(u)k < kuk for all u 2 L \ {0}. We can thus consider graph(ff) and*
* graph(ff) as
onedimensional subspaces of L x (Vx L) = Vx, or in other words points of P V*
*x, so we have a
map ffi0:BOU !P 2V given by
ffi0(x, L, ff) = (graph(ff), graph(ff)).
Proposition 21.36. The map ffi0is a diffeomorphism BOU !P 2V \ P2V .
Proof.First, we must show that ffi0(x, L, ff) 62 P2V , or in other words that g*
*raph(ff) is not perpen
dicular to graph(ff). For this, we choose a nonzero element u 2 L, so v0 = u +*
* ff(u) 2 graph(ff)
and v1 = u  ff(u) 2 graph(ff). It follows that = kuk2  kff(u)k2; th*
*is is strictly pos
itive because kffk < 1, so the lines are not orthogonal, as required. We there*
*fore have a map
ffi0:BOU !P 2V \ P2V .
Any element of P 2V \ P2V has the form (x, M0, M1) where x 2 X and M0, M1 2 P*
* Vx and M0
is not orthogonal to M1. This means that we can choose elements ui 2 Mi with ku*
*ik = 1 and
such that t := is a positive real number. One checks that the pair (u0*
*, u1) is unique up to
the diagonal action of S1. Put v = u0+ u1 and w = u0 u1. By CauchySchwartz we*
* have t 1,
and by direct expansion we have
= 0
= 2(1 + t) > 0
= 2(1  t) < .
We can thus put L = Cv 2 P Vx and define ff: L ! L? by ff(zv) = zw; these are *
*clearly
independent of the choice of pair (u0, u1). As kwk < kvk we have kffk < 1. As v*
* + ff(v) = 2u0
we have graph(ff) = M0, and as v  ff(v) = 2u1 we have graph(ff) = M1. It foll*
*ows that the
construction (x, M0, M1) 7! (x, L, ff) gives a welldefined map i :P 2V \P2V !*
*BOU, with ffi0i = 1.
One can check directly that iffi0is also the identity, so ffi0is a diffeomorphi*
*sm as claimed.
MULTICURVES AND EQUIVARIANT COBORDISM 49
Corollary 21.37. The bundle U is the normal bundle to the diagonal embedding ff*
*i :P V !P 2V ,
there is a homeomorphism P 2V=P2V = P V U, and the quotient map P 2V ! P 2V=P2*
*V can be
thought of as the PontrjaginThom collapse associated to ffi.
Remark 21.38. There are easier proofs of this corollary if one is willing to be*
* less symmetrical.
Now, the above corollary together with Proposition 21.22 and Section 3 gives
eE0P V U= E0(P 2V, P2V ) = J = K=K2 A __*= K=K2 k !_.
On the other hand, we have
U = Hom (T, T ?) = Hom (T, p*V ) C,
so (using the case W = V of Proposition 20.13)
*V )2
P V U= 2P V Hom(T,p = P (V V )=P V.
Remark 5.9 tells us that eE0(S2) = !_, and it is clear that E0(P (V V ), P V*
* ) = K=K2. We
thus obtain
Ee0P V U= K=K2 k !_
again. These two arguments apparently give two different isomorphisms eE0P V U*
*! K=K2 k!_,
but one can show (using Remark 20.16) that they are actually the same.
We next recall some ideas about Gysin maps. We discuss the situation for man*
*ifolds, and
leave it to the reader to check that everything works fibrewise for bundles of *
*manifolds, at least
in sufficient generality for the arguments below. Let f :M ! N be an analytic*
* map between
compact complex manifolds. (It is possible to work with much less rigid data, b*
*ut we shall not
need to do so.) Let øM and øN be the tangent bundles of M and N, and let f be*
* the virtual
bundle f*øN  øM over M. Then for any virtual bundle U over N, a wellknown*va*
*riant of the
PontrjaginThom construction*gives a stable map T (f, U): NU !Mf U+ f, and th*
*us a map
f! = T (f, U)*:Ee0Mf U+ f !Ee0NU . Using the ring map f*: E0N ! E0M we regar*
*d the
source and target of T (f, U)* as E0Nmodules, and we find that T (f, U)* is E0*
*Nlinear. We also
find that T (f, U)* can be obtained from T (f, 0)* by tensoring over E0N with e*
*E0NU . Finally, we
have a composition formula: given maps M f!N g!P , we have gf= f + f* g and
*V +
T (f, g) O T (g, 0) = T (gf, 0): P V! M(gf) .gf
Now consider the maps M ffi!M2 1xß!M, where ß is the constant map from M *
*to a point.
We have ffi= øM and ß = øM , so the transitivity formula says that the compo*
*site
M+ 1^T(ß,0)!M+ ^ Mø T(ffi,1xß)!M+
is the identity. Assuming a Künneth isomorphism, we get maps
E0M ffi!!E0M eE0Mø 1ß!!E0M,
whose composite is again the identity.
Now specialise to the case M = P V . As before we put A = E0P V and identify *
*eE0Mø with J,
and the map ffi!= T (ffi, 0)*:Ee0Mø !E0(M2) with the inclusion J !A A. We k*
*now that the
map
ffi!= T (ffi, 1xß)*:A = E0M !E0M eE0Mø = A J*
is obtained from T (ffi, 0)* by tensoring over A A with A J*. It follows e*
*asily that ffi!(1) =
u 2 A J*, where u is as in Construction 21.8. The equation (1 ß!)ffi!= 1 no*
*w tells us that
(1 ß!)(u) = 1. Proposition 21.9 now tells us that ß!= ffl: J* !k. This prove*
*s Theorem 21.35.
22.Further theory of infinite Grassmannians
Recall from Section 19 that GU denotes the space of finitedimensional`subspa*
*ces of U, which
is the natural equivariant generalization of the space GC1 = d 0BU(d). We kno*
*w from The
orem 19.1 that E0GU is the symmetric algebra over E0 generated by E0P U = O_C. *
* It follows
that
spec(E0GU)= Map(C, A1)
spf(E0GU)= Div+d(C).
In this section, we obtain similar results for spaces analogous to Z x BU, BU a*
*nd BSU.
50 N. P. STRICKLAND
Definition 22.1. For any finitedimensional Auniverse U, we put 2U = U U. We*
* write U+
for U 0 and U for 0 U so 2U = U+ + U. We put
2dim(U)a
Ge(U) = G(2U) = Gd(2U);
d=0
a point X 2 eG(U) should be thought of as a representative of the virtual vecto*
*r space X  U.
We embed G(U) in eG(U) by X 7! X U = X+ + U. We define gdim:eG(U) !Z by gdi*
*m(X) =
dim(X)  dim(U), and eGd(U) = {X  gdim(X) = d}. Given an isometric embedding j*
* :U ! V ,
we define j*:Ge(U) ! eG(V ) by j*(X) = (j j)(X) + W, where W = V j(U). T*
*here
is an evident map oe :eG(U) x eG(V ) ! eG(U V ) sending (X, Y ) to X Y ; o*
*ne checks that
gdim(X Y ) = gdim(X) + gdim(Y ) and that the map oe is compatible in an obvio*
*us sense with the
maps j*.
If U is an infinite Auniverse, we define 2U = U U as before, and put eG(U)*
* = lim!eG(U),
*
* U
where U runs over finitedimensional subspaces. Equivalently, eG(U) is the spac*
*e of subuniverses
V < 2U such that the space V \ U has finite codimension in V and also has fini*
*te codimension in
U. This is a natural analogue of the space Z x BU.
Proposition 22.2. For any B A we have
Y a
(GU)B = G(U[fi]) = Map(B*, BU(d))
fi2B* d
Y
(GeU)B= eG(U[fi]) = Map(B*, Z x BU)
fi2B*
where
U[fi] = {u  b.u = exp(2ßifi(b))u for allb 2 B}
is the fiisotypical part of U. In each case, the first equivalence is A=Bequi*
*variant, but the second
is not.
L
Proof.For the first isomorphism, just note that U splits Aequivariantly as f*
*iU[fi], and a sub
space V < U is Binvariant iff it is the direct sumQof its intersections with t*
*he subspaces U[fi].
This gives an A=Bequivariant isomorphism (GU)B = fiG(U[fi]), and it is clear *
*that G(U[fi]) is
` `
nonequivariantly a copy of dBU(d) so (GU)B = Map(B*, dBU(d)). The argment fo*
*r (GeU)B
is essentially the same.
We next write R+A = N[A*] = ßA0(GU) for the additive semigroup of honest repr*
*esentations
of A, and RA = Z[A*] = ßA0(GeU) for the additive group of virtual representatio*
*ns. It is clear
that the semigroup ring E0[R+A] is a polynomial algebra over E0 with one genera*
*tor ufffor each
character ff, and the group ring E0[RA] is the Laurent series ring with the sam*
*e generators. In
other words, we have
E0[R+A] = E0[uff ff 2 A*]
E0[RA]= E0[uff, u1ff ff 2 A*] = E0[R+A][v1]
Q
where v = ffuff. Note that
spec(E0[R+A])= Map(A*, A1)
spec(E0[RA])= Map(A*, Gm ),
and the isomorphisms R+A = ßA0GU and RA = ßA0eGU give maps E0[R+A] ! E0GU and
E0[RA] !E0eGU.
Now let OE be the obvious isomorphism
C[A] U = C[A] C[A]1 ! C[A]1 = U,
and define OE0:GdU !Gd+AU by OE0(X) = OE(C[A] X).
Proposition 22.3. The space eGU is the telescope of the selfmap OE0of GU. We t*
*hus have
E0eGU = v1E0GU = E0[RA] E0[R+A]E0GU,
and so spec(E0eGU) = Map(C, Gm ).
MULTICURVES AND EQUIVARIANT COBORDISM 51
Proof.Put U0 = C[A][z, z1], and identify this with 2U by sending (ek, 0) to zk*
* and (0, ek) to
zk1. The standard embedding GU ! eGU now sends X to X U. It is easy to ch*
*eck that
eGU = lim zkGU on the nose, and that the inclusion zkGU !zk1GU is isomorph*
*ic to the
!k
map OE0. The first claim follows, and the second claim is just the obvious cons*
*equence in homology.
The tensor product description of E0eGU gives us a pullback square
spec(E0eGU)______________spec(E0GU)w= Map(C, A1)
 
 
 
 
u u
Map (A*, Gm ) = spec(E0[RA])__wspec(E0[R+A]) = Map(A*, A1).
As C is a formal neighbourhood of the image of OE, we see that a map C f!A1 la*
*nds in Gm if
and only if the composite A* x S OE!C f!A1 lands in Gm . Given this, we see t*
*hat the pullback
is just Map(C, Gm ) as required.
We next introduce the analogue of BU.
Proposition 22.4. There is a natural splitting eGU = Z x eG0U, and we have spec*
*(E0eG0U) =
Map0(C, Gm ) (the scheme of maps f :C !Gm with f(0) = 1).
Proof.We have already described an equivariant map gdim:eGU !Z, and defined eG*
*0U = ker(gdim).
We also have (GeU)A = Map (A*, Z x BU) so ßA0(GeU) = Map (A*, Z) = RA, which gi*
*ves an
equivariant map i: RA !eGU (where RA has trivial action). The composite gdimO *
*i: RA !Z is
just the usual augmentation map ffl sending a virtual representation to its dim*
*ension. Thus, if we
let j :Z !RA be the unit map, then iOj is a section of gdim. As eGU is a commu*
*tative equivariant
Hspace, we can define a map ffi :eGU ! eG0U by x 7! (i(j(gdim(x)))  x), and *
*we find that the
resulting map (gdim, ffi): eGU !Z x eG0U is an equivalence. This is easily see*
*n to be parallel to the
splitting Map(C, Gm ) = Gm x Map0(C, Gm ) given by f 7! (f(0), f(0)=f), which g*
*ives the claimed
description of spec(E0GU).
Remark 22.5. There are two other possible analogues of BU. Firstly, one could t*
*ake the colimit
of the spaces GdU using the maps V 7! V C; the scheme associated to the corre*
*sponding space
is then Map 0(C, A1), which classifies maps f :C ! A1 with f(0) = 1. Alternati*
*vely, one could
take the colimit of the spacesQGdAU using the maps V 7! V C[A]. This gives *
*the scheme of
maps f :C !Gm for which ff2A*f(OE(ff)) = 1. However, the space eG0U describe*
*d above is the
one that occurs in Greenlees's definition of the spectrum kUA, and is also the *
*one whose Thom
spectrum is MUA.
We next introduce the analogue of BSU. For this, we need an analogue of the m*
*ap B det:BU !
CP 1.
Definition 22.6. Given a universe U of finite dimension d, we put eFU = Hom (~d*
*U, ~d(2U)).
We make this a functor as follows. Given an isometric embedding j :U !V , we p*
*ut W = V jU
and e = dim(W ). As j :U ! jU is an isomorphism and ~eW is onedimensional, we*
* have an
evident isomorphism
eFU = Hom (~djU ~eW, ~d(2jU) ~eW).
The isomorphism V = jU W gives an isomorphism ~djU ~eW = ~d+eV and an embe*
*dding
~d(2jU) ~eW !~d+e(2V ). Putting this together gives the required map j*:FeU*
* !FeV .
There are also obvious maps eF(U) Fe(V ) !eF(U V ), compatible with the abo*
*ve functorality.
This gives maps P eF(U) x P eF(V ) !P eF(U V ) of the associated projective *
*spaces.
Next, recall that a point of eG0U is a ddimensional subspace X 2U. We defi*
*ne
fdet(X) = Hom (~dU, ~dX) 2 P eFU.
One can check that this gives a natural map fdet:eG0!P eF, with fdet(X Y ) =*
* fdet(X) fdet(Y ).
Finally, for our complete universe U we put eFU = lim!eFU, where U runs over*
* the finite
U
dimensional subuniverses. It is easy to check that this is again a complete Au*
*niverse, and thus
52 N. P. STRICKLAND
is unnaturally isomorphic to U. The maps fdetpass to the colimit to give an Hm*
*ap fdet:eG0U !
P eFU ' P U. We write SGe0U for the pullback of the projection S(FeU) !P eFU a*
*long the map
fdet, or equivalently the space of pairs (V, u) where V 2 eG0U and u is a unit *
*vector in the one
dimensional space fdet(V). As S(FeU) is equivariantly contractible, this is jus*
*t the homotopy fibre
of fdet.
Proposition 22.7. There is a natural splitting eG0U = SGe0U x P U (which does n*
*ot respect the
Hspace structure).
Proof.It is enough to give a section of the Hmap fdet:eG0U !P U. We can inclu*
*de P U = G1U
in eG1U eGU in the usual way, then use the projection eGU !Ge0U from Proposi*
*tion 22.4. We
find that the resulting composite P U !P U is actually minus the identity, but*
* we can precompose
by minus the identity to get the required section.
Remark 22.8. Cartier duality identifies spec(E0P U) with Hom (C, Gm ), and the *
*proposition
suggests that spec(E0SGe0U) should be the quotient Map0(C, Gm )= Hom(C, Gm ). H*
*owever, there
are difficulties in interpreting this quotient, and it is in fact more useful t*
*o take a slightly different
approach as in [1, 2]. We will not give details here.
Next, recall that Greenlees has defined an equivariant analogue of connective*
* Ktheory (denoted
by kUA) by the homotopy pullback square
kUA ______Fw(EA+, kU)

 
 
 
u u
KUA _____Fw(EA+, KU).
If v 2 ß2kU is the Bott element then kU[v1] = KU and kU=v = H. It is not hard *
*to see that
there is a corresponding element in ßA2kUA with kUA[v1] = KUA and kUA=v = F (E*
*A+, H).
Proposition 22.9. The zeroth, second and fourth spaces of kUA are eGU, eG0U and*
* SGe0U respec
tively.
Proof.We take it as wellknown that the zeroth space of KUA is eGU, and KUA is *
*twoperiodic so
this is also the 2k'th space for all k. Let Xk denote the 2k'th space of kUA, s*
*o we have a homotopy
pullback square
Xk _____wF (EA+, BU<2k>)
 
 
i 
 
u u
GeU _____Fw(EA+,jZ x BU)
(where BU<0> is interpreted as Z x BU). In the case k = 0, the map i is the ide*
*ntity and so
X0 = eGU. In the case k = 1, the map i is just the inclusion
F (EA+, BU) !Z x F (EA+, BU) = F (EA+, Z x BU)
and the map j sends eGkU into {k} x F (EA+, BU). It follows easily that X1 = eG*
*0U. In the case
k = 2, we note that the cofibration 2kUA v!kUA !F (EA+, H) gives a fibration*
* X2 !X1 !
F (EA+, K(Z, 2)). We know that X1 = eG0U and Proposition 4.4 that F (EA+, K(Z, *
*2)) = P U.
One can check that the resulting map eG0U !P U is just dfet, and so X2 = SGe0*
*U as claimed.
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Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK
Email address: N.P.Strickland@sheffield.ac.uk