COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS
N. P. STRICKLAND
Abstract.Let V0 and V1 be complex vector bundles over a space X. We
use the theory of divisors on formal groups to give obstructions in gene*
*ralised
cohomology that vanish when V0 and V1 can be embedded in a bundle U in
such a way that V0\ V1 has dimension at least k everywhere. We study var*
*i-
ous algebraic universal examples related to this question, and show that*
* they
arise from the generalised cohomology of corresponding topological unive*
*rsal
examples. This extends and reinterprets earlier work on degeneracy class*
*es in
ordinary cohomology or intersection theory.
1. Introduction
There are a number of different motivations for the theory developed here, b*
*ut
perhaps the most obvious is as follows. Suppose we have a space X with vector
bundles V0 and V1. (Throughout this paper, the term "vector space" refers to
finite-dimensional complex vector spaces equipped with Hermitian inner products,
and similarly for "vector bundle".) We define the intersection index of V0 and *
*V1 to
be the largest k such that V0 and V1 can be embedded isometrically in some bund*
*le
U in such a way that dim(V0x\ V1x) k for all x 2 X. We write int(V0, V1) for
this intersection index. Our aim is to use invariants from generalised cohomolo*
*gy
theory to estimate int(V0, V1), and to investigate the topology of certain univ*
*ersal
examples related to this question.
We will show in Proposition 5.3 that int(V0, V1) is also the largest k such *
*that
there is a linear map V0 -!V1 of rank at least k everywhere. This creates a lin*
*k with
the theory of degeneracy loci and the corresponding classes in the cohomology of
manifolds or Chow rings of varieties, which are given by the determinantal form*
*ula
of Thom and Porteous. The paper [9] by Pragacz is a convenient reference for co*
*m-
parison with the present work. The relevant theory is based strongly on Schubert
calculus, and could presumably be transferred to complex cobordism (and thus to
other complex-orientable theories) by the methods of Bressler and Evens [1].
However, our approach will be different in a number of ways. Firstly, we use
the language of formal groups, as discussed in [10] (for example). We fix an ev*
*en
periodic cohomology theory E with a complex orientation x 2 eE0CP 1. For any
space X we have a formal scheme XE = spf(E0X), the basic examples being
S := (point)E and G := CP 1E = spf(E0[[x]]), which is a formal group over S.
If V is a complex vector bundle over X, we write P V for the associated bun-
dle of projectivePspaces. It is standard that E0(P V ) = E0(X)[[x]]=fV (x), wh*
*ere
fV (x) = i+j=dim(Vc)ixj, where ci is the i'th Chern class of V . In geometric
____________
Date: November 5, 2002.
1991 Mathematics Subject Classification. 55N20,14L05,14M15.
Key words and phrases. vector bundle, divisor, formal group, degeneracy, Th*
*om-Porteous.
1
2 N. P. STRICKLAND
terms, this means that the formal scheme D(V ) := (P V )E is naturally embedded
as a divisor in G xS XE . Most of our algebraic constructions will have a very
natural interpretation`in terms of such divisors. We will also consider the bu*
*n-
dle U(V ) = x2X U(Vx) of unitary groups associated to V . A key point is that
E*U(V ) is the exterior algebra over E*X generated by E*-1P V . This provides a
very natural link with exterior algebra, and could be regarded as the "real rea*
*son"
for the appearance of determinantal formulae, which seem rather accidental in o*
*ther
approaches. Our divisorial approach also leads to descriptions of various cohom*
*ol-
ogy rings that are manifestly independent of the choice of complex orientation,*
* and
depend functorially on G. This functorality implicitly encodes the action of st*
*able
cohomology operations and thus gives a tighter link with the underlying homotopy
theory.
We were also influenced by work of Kitchloo [5], in which he investigates the
cohomological effect of Miller's stable splitting of U(n), and draws a link wit*
*h the
theory of Schur functions.
In Section 3 we use the theory of Fitting ideals to define an intersection in*
*dex
int(D0, D1), where D0 and D1 are divisors on G. In Section 4 we identify E*U(V )
with the exterior algebra generated by E*-1P V , and show that this identificat*
*ion
is an isomorphism of Hopf algebras. In Section 5 we use this to prove our first
main theorem, that int(V0, V1) int(D(V0), D(V1)); this implicitly gives all t*
*he
relations among Chern classes that are universally satisfied when int(V0, V1) *
* k
for some given integer k. Next, in Section 6 we study the universal examples of
our various algebraic questions, focusing on the scheme Intr(d0, d1) which clas*
*sifies
pairs (D0, D1) of divisors of degrees d0 and d1 such that int(D0, D1) k. Our *
*next
task is to construct spaces whose associated schemes are these algebraic univer*
*sal
examples. In Section 7 we warm up by giving a divisorial account of the general*
*ised
cohomology of Grassmannians and flag spaces, and then in Section 8 we show that
the space
I0r(d0, d1) := {(V0, V1) 2 Gd0(C1 ) x Gd1(C1 ) | dim(V0 \ V1) k}
satisfies I0r(d0, d1)E = Intr(d0, d1). (The origin of the present work is that *
*the au-
thor needed to compute the cohomology of certain spaces similar to I0r(d0, d1) *
*as
input to another project; it would take us too far afield to discuss the backgr*
*ound.)
This completes the main work of the paper, but we have added three more sec-
tions exploring the isomorphism E*U(V ) ' ~*E*-1P V in more detail. Section 9
treats some purely algebraic questions related to this situation, and in Sectio*
*ns 10
and 11 we translate all the algebra into homotopy theory. In particular, this g*
*ives
a divisorial interpretation of the work of Mitchell, Richter and others on filt*
*rations
of U(n): the scheme associated to the k'th stage in the filtration of X U(V )*
* is
D(V )k= k, and the scheme associated to X U(V ) is the free formal group over
XE generated by D(V ).
Appendix A gives a brief treatment of the functional calculus for normal oper*
*a-
tors, which is used in a number of places in the text.
Remark 1.1. There is a theory of degeneracy loci for morphisms with symmetries,
where the formulae involve Pfaffians instead of determinants. It would clearly *
*be
a natural project to reexamine this theory from the point of view of the present
paper, but so far we have nothing to say about this.
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 3
2.Notation and Conventions
2.1. Spheres. We take Rn [{1} as our definition of Sn, with 1 as the basepoint;
we distinguish S1 from the homeomorphic space U(1) := {z 2 C | |z| = 1}. Where
necessary, we use the homeomorphism fl :U(1) -!S1 given by
fl(z)= (z + 1)(z - 1)-1=i
fl-1(t)= (it + 1)=(it - 1).
One checks that fl(ei`) = cot(-`=2), which is a strictly increasing function of*
* ` for
0 < ` < 2ß.
2.2. Fibrewise spaces. We will use various elementary concepts from fibrewise
topology; the book of Crabb and James [2] is a convenient reference. Very few
topological technicalities arise, as our fibrewise spaces are all fibre bundles*
*, and the
fibres are usually finite complexes.
In particular, given spaces U and V over a space X, we write U xX V for the
fibre product, and UnXfor the fibre power U xX . .x.XU. If U is pointed (in oth*
*er
words, it has a specified section s: X -! U) and E is any cohomology theory we
write eE*XU = E*(U, sX). We also write X U for the fibrewise suspension of U,
which is the quotient of S1 x U in which {1} x U [ S1 x sX is collapsed to a
copy of X. This satisfies eE*X X U = eE*-1XU. We also write X U for the fibrew*
*ise
loop space of U, which is the space of maps ! :S1 -! U such that the composite
S1 -!U -! X is constant and !(1) 2 sX. If V is another pointed space over X,
we write U ^X V for the fibrewise smash product. If W is an unpointed space over
X then we write W+X = W q X, which is a pointed space over X in an obvious
way.
2.3. Tensor products over schemes. If T is a scheme and M, N are modules
over the ring OT, we will write M T N for M OT N. Similarly, we write ~kTM
for ~kOTM, the k'th exterior power of M over OT.
2.4. Free modules. Given a ring R and a set T , we write R{T } for the free
R-module generated by T .
3. Intersections of divisors
Let G be a commutative, one-dimensional formal group over a scheme S. Choose
a coordinate x so that OG = OS [[x]]. Let D0 and D1 be divisors on G defined
over S, with degrees d0 and d1 respectively. This means that ODi = OG =fi =
OS [[x]]=fi(x) for some monic polynomial fi(x) of degree di such that fi(x) = x*
*di
modulo nilpotents. It follows that ODi is a free module of rank di over OS , wi*
*th
basis {xj | 0 j < di}.
As D0 and D1 are closed subschemes of G we can form their intersection, so t*
*hat
OD0\D1 = OG=(f0, f1) = OS[[x]]=(f0(x), f1(x)).
Typically this will not be a projective module over OS, so some thought is requ*
*ired
to give a useful notion of its size. We will use a measure coming from the theo*
*ry
of Fitting ideals, which we now recall briefly.
Let R be a commutative Noetherian ring, and let M be a finitely generated R-
module. We can then find a presentation P1 OE1-!P0 OE0-!M, where P0 and P1 are
finitely generated projective modules of ranks p0 and p1 say, and M = cok(OE1).*
* The
4 N. P. STRICKLAND
exterior powers ~jPi are again finitely generated projective modules. We define
Ij(OE1) to be the smallest ideal in R modulo which we have ~j(OE1) = 0. More
concretely, if P0 and P1 are free then OE1 can be represented by a matrix A and
Ij(OE1) is generated by the determinants of all jxj submatrices of A. We then d*
*efine
Ij(M) = Ip0-j(OE1); this is called the j'th Fitting ideal of M. It is a fundame*
*ntal
fact that this is well-defined; this was already known to Fitting (see [8, Chap*
*ter 3],
for example), but we give a proof for the convenience of the reader.
Proposition 3.1. The ideal Ij(M) is independent of the choice of presentation of
M.
Proof.We temporarily write Ij(M, P*, OE*) for the ideal called Ij(M) above.
Put N = ker(OE0) and let fi :N -!P0 be the inclusion. Then OE1 factors as P1*
* ff-!
N -fi!P0, where ff is surjective. For any ideal J R we see that ~kff is surje*
*ctive
mod J, so ~kOE1 is zero mod J iff ~kfi is zero mod J. This condition depends on*
*ly on
the map OE0: P0 -!M, so we can legitimately define Ij(M, P0, OE0) := Ij(M, P*, *
*OE*).
Now suppose we have another presentation Q1 _1--!Q0 _0--!M, where Qi has
rank qi. Define Ø0: P0 Q0 -!M by (u, v) 7! OE0(u)+_0(v). It will suffice to pro*
*ve
that
Ij(M, P0, OE0) = Ij(M, P0 Q0, Ø0) = Ij(M, Q0, _0),
and by symmetry we need only check the first of these. By projectivity we can
choose a map ` :Q0 -!P0 with OE0` = _0, and define Ø1: P1 Q0 -!P0 Q0 by
(u, v) 7! (OE1(u) - `(v), v). It is easy to check that this gives another prese*
*ntation
P1 Q0 Ø1-!P0 Q0 Ø0-!M.
If k q0 then ~kØ1 is certainly nonzero, because the composite
kØ
~kQ0 -!~k(P1 Q0) ~--!~k(P0 Q0) -!~kQ0
is the identity, and ~kQ0 6= 0. If k > q0 and ~kØ1 = 0 then (by restricting to
~k-q0P1 ~q0Q0) we see that ~k-q0OE1 = 0.
For the converse, notice that ~*N is a graded ring for any module N, and that
~*ff is a ring map for any homomorphism ff of R-modules. One can check that
~j+q0(P1 Q0) is contained in the ideal in ~*(P1 Q0) generated by ~jP1. It
follows that if ~jOE1 = 0 then ~j+q0Ø1 = 0.
This shows that Ir(OE1) = Ir+q0(Ø1), and thus that Ir(M, P0, OE0) = Ir(M, P0
Q0, Ø0), as required.
It is clear that
I0(M) . . .Im (M) = R,
and we define
rank(M) = rankR(M) = min{r | Ir(M) 6= 0}.
We call rank(M) the Fitting rank of M. For example, if R is a principal ideal
domain with fraction field K, one can check that rank(M) = dimK (K R M) for
all M. However, we will mostly be interested in rings R with many nilpotents, f*
*or
which there is no such simple formula.
The following lemma is easily checked from the definitions.
Lemma 3.2. (a) The Fitting rank is the same as the ordinary rank for pro-
jective modules.
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 5
(b) If N is a quotient of M then rank(N) rank(M).
(c) If there is a presentation P -! Q -!M then rank(Q)-rank(P ) rank(M)
rank(Q).
(It is not true, however, that rank(M N) = rank(M) + rank(N); indeed, if
a 6= 0 and a2 = 0 then rank(R=a) = 0 but rank(R=a R=a) = 1.)
Definition 3.3. The intersection multiplicity of D0 and D1 is the integer
int(D0, D1) := rankOS(OD0\D1).
We also put
Intr(D0, D1) = spec(OS =Ir-1(OD0\D1)),
which is the largest subscheme of S over which we have int(D0, D1) r.
Remark 3.4. Let S0 be a scheme over S, so that G0:= G xS S0 is a formal group
over S0. We refer to divisors on G0as divisors on G over S0. Given two such div*
*isors
D0 and D1, we get a closed subscheme Intr(D0, D1) S0. We will use this kind of
base-change construction throughout the paper without explicit comment.
To make the above definitions more explicit, we will describe several differ*
*ent
presentations of OD0\D1 that can be used to determine its rank.
Construction 3.5. First, recall that we can form the divisor
D0 + D1 = spec(OG =f0f1) = spec(OS [[x]]=f0(x)f1(x)).
This contains D0 and D1, so we have a pullback square of closed inclusions as
shown on the left below. This gives a pushout square of OS -algebras as shown on
the right.
D0 \ D1 v______D0w OD0\D1 u u___OD0_
v| v| uu| uu|
| | | |
| | | |
| | | |
| | | |
|u |u | |
D1 v_____D0_+wD1 OD1 u u___OD0+D1_,
which gives a presentation
OD0+D1 -! OD0 OD1 -!OD0\D1.
Explicitly, this is just the presentation
OG =(f0f1) OE-!OG=f0 OG=f1 _-!OG=(f0, f1)
given by
OE(g mod f0f1)= (g mod f0, -g mod f1)
_(g0 mod f0, g1 mod f1)= g0 + g1 mod (f0, f1).
Although this is probably the most natural presentation, it is not easy to w*
*rite
down the effect of OE on the obvious bases of OG=(f0f1) and OG=fi. To remedy th*
*is,
we give an alternate presentation.
6 N. P. STRICKLAND
Construction 3.6. Let Ji be the ideal generated by fi and put J = J0J1. Then
Ji=J is free over OS with basis {xjfi(x) | 0 j < d1-i} and the inclusion maps
Ji-! OG give rise to a presentation
J0=J J1=J i-!OG=J = OD0+D1 -,!OG=(J0 + J1) = OD0\D1.
P
Let cijbe the coefficient of xdi-jin fi(x), so that ci0= 1 and fi(x) = di=j+k*
*cijxk.
Then
d0+jX
i(xjf0(x), 0)= c0,d0+j-kxk for0 j < d1
k=j
d1+jX
i(0, xjf1(x))= c1,d1+j-kxk for0 j < d0,
k=j
and this tells us the matrix for i in terms of the obvious bases of J0=J J1=J*
* and
OG =J. For example, if d0 = 2 and d1 = 3 the matrix is
0 1
c02 0 0 |c13 0
BB c01 c02 0 c12 c13 C
BB 1 c01 c02 ||c11c12 CCC
@ 0 1 c01 |1 c11 A
0 0 1 |0 1
In general, we have a square matrix with d0 + d1 rows and columns. The left
hand block consists of d1 columns, each of which contains d1 - 1 zeros. The rig*
*ht
hand block consists of d0 columns, each of which contains d0 - 1 zeros. Clearly
Intr(D0, D1) is the closed subscheme defined by the vanishing of the minors of *
*this
matrix of size d0+d1-r +1. In particular, Int1(D0, D1) is defined by the vanish*
*ing
of the determinant of theQwhole matrix, which is classicallyQknown as the resul*
*tant
of f0Qand f1. If f0(x) = i(x - ai) and f1(x) = j(x - bj) then the resultant*
* is
just i,j(ai- bj). We do not know of any similar formula for the other minors.
Construction 3.7. For a smaller but less symmetrical presentation, we can just *
*use
the sequence J1=J -!OG =J0 -!OG =(J0+ J1) induced by the inclusion of J1 in OG.
This is isomorphic to the presentation OG =J0 ~1-!OG=J0 -!OG =(J0 + J1), where
~1(g) = f1g. However, the isomorphism depends on a choice of coordinate on G
(because the element f1 does), so the previous presentation is sometimes prefer*
*able.
There is of course a similar presentation OG =J1 ~0-!OG=J1 -!OG =(J0 + J1).
Finally, we give a presentation that depends only on the formal Laurent seri*
*es
f0=f1 and thus makes direct contact with the classical Thom-Porteous formula.
Construction 3.8. Write MG = R((x)) = OG [x-1]. Note that f1(x)=xd1 is
a polynomial in x-1 whose constant term is 1 and whose other coefficients are
nilpotent, so it is a unit in R[x-1]. It follows that f1 is a unit in R((x)). *
* Put
Q = x-1R[x-1] R((x)), so that R((x)) = R[[x]] Q. Multiplication by the seri*
*es
xd1f0=f1 induces a map
P1 = _R[[x]]_fOE-!P0 = ___R((x))__d.
1R[[x]] x R[[x]] 1Q
We claim that the cokernel of OE is isomorphic to R[[x]]=(f0, f1) = OD0\D1, so *
*we
have yet another presentation of this ring. Indeed, the cokernel of OE is clear*
*ly given
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 7
by R((x))=(xd1f0f-11R[[x]]+xd1R[[x]]+Q). The element f1=xd1is invertible in R[x*
*-1]
so it is invertible in R((x)) and satisfies (f1=xd1)Q = Q. Thus, multiplication*
* by
this element gives an isomorphism
__________R((x))_________ ' _______R((x))______.
xd1f0f-11R[[x]] + xd1R[[x]] +fQ0R[[x]] + f1R[[x]] + Q
As R((x)) = R[[x]] Q, we see that the right hand side is just R[[x]]=(f0, f1)*
* as
claimed.
The elements {1, x, . .,.xd1-1} give a basis for both P0 and P1, and the matr*
*ix
elements of OE with respect to these bases are just the coefficients of f0=f1 (*
*suitably
indexed). More precisely, we have
X
f0=f1 = xd0-d1 cix-i,
i 0
where c0 = 1 and ci is nilpotent for i > 0. We take ci= 0 for i < 0 by conventi*
*on.
The matrix elements ijof OE are then given by ij= cd0+i-jfor 0 i, j < d1.
For example, if d0 = 3 and d1 = 5 then the matrix is
0 1
c3 c4 c5 c6 c7
BBc2 c3 c4 c5 c6 CC
= BBc1 c2 c3 c4 c5 CC.
@ 1 c1 c2 c3 c4 A
0 1 c1 c2 c3
Now suppose that our divisors Diarise in the usual way from vector bundles Viov*
*er
a stably complex manifold X, and we have a generic linear map g :V0 -!V1. Let Zr
be the locus where the rank of g is at most r, and let i: Zr -!X be the inclusi*
*on.
Generically, this will be a smooth stably complex submanifold of X, so we have
a class zr = i*[Zr] 2 E0X. The Thom-Porteous formula says that zr = det( r),
where r is the square block of size d1 - r taken from the bottom left of . Mo*
*re
explicitly, the matrix elements are ( r)ij= cd0-k+i-jfor 0 i, j < d1- r. Clea*
*rly
det( r) 2 Id1-r(OE) = Ir(OD0\D1). If Zr is empty then zr = 0. On the other
hand, Proposition 5.3 will tell us that int(D0, D1) > r and so Ir(OD0\D1) = 0, *
*so
det( r) = 0, which is consistent with the Thom-Porteous formula. It is doubtless
possible to prove the formula using the methods of this paper, but we have not *
*yet
done so.
Proposition 3.9. We always have int(D0, D1) min (d0, d1) (unless the base
scheme S is empty). If int(D0, D1) = d0 then D0 D1, and if int(D0, D1) = d1
then D1 D0.
Proof.The presentation OD1 ~0-!OD1 -!OD0\D1 shows that
int(D0, D1) = rank(OD0\D1) rank(OD1) = d1.
If this is actually an equality we must have ~d1-d1+1(~0) = 0 or in other words*
* ~0 =
0, so f0 = 0 (mod f1), so D1 D0. The remaining claims follow by symmetry.
Proposition 3.10. If there is a divisor D of degree k such that D D0 and
D D1, then int(D0, D1) k.
Proof.Clearly OD is a quotient of the ring OD0\D1, and it is free of rank k, so
int(D0, D1) = rank(OD0\D1) k.
8 N. P. STRICKLAND
Definition 3.11. Given two divisors D0, D1, we write Subr(D0, D1) for the scheme
of divisors D of degree r such that D D0 and D D1. The proposition shows
that the projection ß :Subr(D0, D1) -! S factors through the closed subscheme
Intr(D0, D1).
Remark 3.12. Proposition 3.9 implies that Intd0(D0, D1) is just the largest clo*
*sed
subscheme of S over which we have D0 D1. From this it is easy to see that
Subd0(D0, D1) = Intd0(D0, D1).
It is natural to expect that the map ß :Subr(D0, D1) -!Intr(D0, D1) should be
surjective in some suitable sense. Unfortunately this does not work as well as *
*one
might hope: the map ß is not faithfully flat or even dominant, so the correspon*
*ding
ring map ß* need not be injective. However, it is injective in a certain univer*
*sal
case, as we shall show in Section 6.
We conclude this section with an example where ß* is not injective. Let G be*
* the
additive formal group over the scheme S = spec(Z[a]=a2). Let D0 and D1 be the
divisors with equations x2 - a and x2, respectively. Then OD0\D1 = OS [x]=(x2 -
a, x2) = OS [x]=(a, x2), which is the cokernel`of'the map ~: OS[x]=x2 -!OS [x]=*
*x2
given by ~(t) = at. The matrix of ~ is a0 0a which is clearly nonzero, but
~2(~) = a2 = 0. It follows that int(D0, D1) = 1, so Int1(D0, D1) = S. However,
Sub1(D0, D1) is just the scheme D0 \ D1 = spec(OS [x]=(a, x2)), so ß*(a) = 0.
For a topological interpretation, let V0 be the tautological bundle over HP *
*1= S4,
and let V1 be the trivial rank two complex bundle. If we use the cohomology the*
*ory
E*Y = (H*Y )[u, u-1] (with |u| = 2) and let a be the second Chern class of V0 we
find that E0X = Z[a]=a2, and the equations of D(V0) and D(V1) are x2- a and x2.
Using the theory to be developed in Section 5 and the calculations of the previ*
*ous
paragraph, we deduce that V0 and V1 cannot have a common subbundle of rank one,
but there is no cohomological obstruction to finding a map f :V0 -!V1 with rank
at least 1 everywhere. To see that such a map does in fact exist, choose a subs*
*pace
W < H2 which is a complex vector space of dimension 2, but not an H-submodule.
We can then take the constant bundle with fibre H2=W as a model for V1. The
bundle V0 is by definition a subbundle of the constant bundle with fibre H2, so
there is an evident projection map f :V0 -!V1. As W is not an H-submodule, we
see that f is nowhere zero and thus has rank at least one everywhere, as claime*
*d.
4.Unitary bundles
In order to compare the constructions of the previous section with phenomena
in topology, we need a topological interpretation of the exterior powers ~kOD *
*when
D is the divisor associated to a vector bundle.
Let V be a complex vector bundle of dimension d over a space X. We can thus
form a bundle U(V ) of unitary groups in the evident way (so U(V ) = {(x, g) | *
*x 2
X and g 2 U(Vx)}). The key point is that E*U(V ) can be naturally identified wi*
*th
~*E*XE*-1P V (the exterior algebra over the ring E*X generated by the module
E*P V ). Moreover, we can use the group structure on U(V ) to make E*U(V ) into
a Hopf algebra over E*X, and we can make ~*E*XE*-1P V into a Hopf algebra
by declaring E*P V to be primitive. We will need to know that our isomorphism
respects these structures. All this is of course well-known when X is a point a*
*nd
E represents ordinary cohomology. Kitchloo [5] has shown that if one chooses the
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 9
right proof then the restriction on E can be removed. With just a few more word*
*s,
we will be able to remove the restriction on X as well.
We start by comparing U(V ) with a suitable classifying space. First let V be*
* a
vector space rather than a bundle. We let EU(V ) denote the geometric realisati*
*on
of the simplicial space {U(V )n+1}n 0 and we put BU(V ) = EU(V )=U(V ), which is
the usual simplicial model for the classifying space of U(V ). There is a well-*
*known
map j :U(V ) -! BU(V ), which is a weak equivalence of H-spaces. By adjunction
we have a map i : U(V ) -!BU(V ), which gives a map
i*: eE*BU(V ) -!Ee* U(V ) = eE*-1U(V ).
The fact that j is an H-map means that i is primitive, or in other words that
i O ~ = i O (ß0 + ß1) 2 [ U(V )2, BU(V )].
We can also construct a tautological bundle T = EU(V ) xU(V )V over BU(V ).
We now revert to the case where V is a vector bundle over a space X, and
perform all the above constructions fibrewise. Firstly, we construct the bundle
BU(V ) = {(x, e) | x 2 X and e 2 BU(Vx)}. Note that each space BU(Vx) has a
canonical basepoint, and using these we get an inclusion X -!BU(V ).
A slightly surprising point is that there is a canonical homotopy equivalence
BU(V ) -!XxBU(d). Indeed, we can certainly perform the definition of T fibrewise
to get a tautological bundle over BU(V ), which is classified by a map q :BU(V *
*) -!
BU(d), which is unique up to homotopy. We can combine this with the projection
p: BU(V ) -!X to get a map f = (p, q): BU(V ) -!X x BU(d). The map p is a
fibre bundle projection, and the restriction of q to each fibre of p is easily *
*seen to be
an equivalence. It is now an easy exercise with the homotopy long exact sequence
of p to see that f is a weak equivalence. (Nothing untoward happens with ß0 and
ß1 because BU(d) is simply connected.)
Remark 4.1. Let q0: X -! BU(d) be the restriction of q. Then q0 classifies the
bundle T |X ' V , so in general it will be an essential map. Thus, if we just *
*use
the basepoint of BU(d) to make X x BU(d) into a based space over X, then our
equivalence f :BU(V ) ' X x BU(d) does not preserve basepoints, and cannot be
deformed to do so. If it did preserve basepoints we could apply the fibrewise l*
*oop
functor X and deduce that U(V ) ' X x U(d), but this is false in general.
It follows from the above that E*BU(V ) is a formal power series algebra over
E*X, generated by the Chern classes of T . It will be convenient for us to mod-
ify this description slightly by considering the virtual bundle T - V (where V *
*is
implicitlyPpulled back to BU(VP) by the map p: BU(V ) -! X). We have fT(t) =
td dk=0akt-k and fV (t) = td dk=0bkt-kPfor some coefficients ak 2 E0BU(V ) *
*and
bk 2 E0X so fT-V (t) = fT(t)=fV (t) = k 0ckt-k for some ck 2 E0BU(V ). For
k d we have ck = ak (mod b1, . .,.bd) and it follows easily that
E*BU(V ) = (E*X)[[c1, . .,.cd]].
Note that the restriction of T -V to X BU(V ) is trivial, so the classes ck r*
*estrict
to zero on X.
Next, consider the fibrewise suspension X U(V ). By dividing each fibre in*
*to
two cones we obtain a decomposition X U(V ) = C0[ C1 where the inclusion of X
in each Ciis a homotopy equivalence, and C0\C1 = U(V ). Using a Mayer-Vietoris
sequence we deduce that eE*X X U(V ) ' eE*-1U(V ) and that this can be regarded
10 N. P. STRICKLAND
as an ideal in E* X U(V ) whose square is zero. Moreover, the construction of i*
* can
be carried out fibrewise to get a map X U(V ) -!BU(V ) which is again primitiv*
*e.
It follows that i induces a map
i*: Ind(E*BU(V )) -!Prim(E*-1U(V )).
(Here Indand Prim denote indecomposables and primitives over E*X.) Note also
that Ind(E*BU(V )) is a free module over E*X generated by {c1, . .,.cd}.
To prove that i* is injective, we need to consider the complex reflection map
æ: X P V+X -! U(V ), which we define as follows. For t 2 S1 = R[{1} and x 2 X
and L 2 P Vx, the map æ(t, x, L) is the endomorphism of Vx that has eigenvalue
fl-1(t) on the line L, and eigenvalue 1 on L? . Here fl-1(t) = (it+1)=(it-1) 2 *
*U(1),
as in Section 2.1. Using this we obtain a map , = i O X æ: 2XP V+X -! BU(V ).
Our next problem is to identify the virtual bundle ,*(T -V ) over 2XP V+X . *
*For
this it is convenient to identify S2 with CP 1and thus 2P V+X with a quotient*
* of
CP 1x P V . We have tautological bundles H and L over CP 1and P V , whose Euler
classes we denote by y and x.
Lemma 4.2. We have ,*(T - V ) ' (H - 1) L. Moreover, there is a power series
g(s) 2 E0[[s]] with g(0) = 1 such that ,*ck = -yxk-1g(x) for k = 1, . .,.d. (If*
* E0
is torsion-free then g(s) = 1= log0F(x).)
Proof.In the proof it will be convenient to write TV and LV instead of T and L,
to display the dependence on V .
First consider the case where X is a point and V = C. Then æ: S1 -!U(1) =
U(C) is a homeomorphism and BU(C) ' CP 1. It is a standard fact that , :S2 -!
BU(C) can be identified with the inclusion CP 1-! CP 1, and thus that ,*TC = H.
In the general case, note that we have a map ,L :CP 1x P L -!BU(L) of spaces
over P V . The projection P L -! P V is a homeomorphism which we regard as
the identity. If we let ß :P V -! X be the projection, we have a splitting ß*V =
L (ß*V L). The inclusion L -! ß*V gives an inclusion U(L) -! ß*U(V ) and
thus an inclusion BU(L) -!ß*BU(V ), or equivalently a map OE: BU(L) -!BU(V )
covering ß. As TV = V xU(V )EU(V ) and U(L) acts trivially on ß*V L we see
that OE*TV = TL (ß*V L).
Next, we note that tensoring with L gives an isomorphism ø :U(C)xP V -! U(L)
and thus an isomorphism Bø :BU(C) x P V -! BU(L) with (Bø)*TL = TC L.
One can check that the following diagram commutes:
1 _____1
CP 1x P V _____wCP 1x P V wCP 1x P V
| | |
| | |
,Cx1| ,L | |,V
| | |
|u |u |u
BU(C) x P V ______wBU(L)'Bø______wBU(VO).E
It follows that ,*VTV ' (,C x 1)*(Bø)*OE*TV , and the previous discussion iden*
*tifies
this with (H L) (ß*V L). It follows that ,*V(TV -V ) ' (H L)-L = (H-1) L,
as claimed.
Now let g(s) be the partial derivative of t +F s with respect to t evaluated*
* at
t = 0. This is characterised by the equation t +F s = tg(s) + s (mod t2); it i*
*s clear
that g(0) = 1, and by applying logFwe see that g(s) = 1= log0F(s) in the torsio*
*n-free
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 11
case. As y2 = 0 we see that the Euler class of H L is x +F y = x + yg(x). Thu*
*s,
we have
fH L-L (t)= (t - x - yg(x))=(t - x)
= 1 - yg(x)t-1=(1 - x=t)
X
= 1 - yg(x)xk-1t-k.
k>0
The k'th Chern class of (H - 1) L is the coefficient of t-k in this series, w*
*hich is
-yg(x)xk-1 as claimed.
Corollary 4.3. The induced map ,*: Ind(E*BU(V )) -! E*( 2XP V+X , X) =
E*-2P V is an isomorphism.
Theorem 4.4. There is a natural isomorphism ~*E*-1P V -! E*U(V ) of Hopf
algebras over E*X.
Proof.Put ai = ,*ci 2 Prim(E*U(V )) for i = 1, . .,.d.Q Given a sequence I =
(i1, . .,.ir) with 1 i1 < . .<.ir d, put aI = jaij. We first claim that t*
*he
elements aI form a basis for E*U(V ) over E*X. This is very well-known in the c*
*ase
where X is a point (so U(V ) ' U(d)) and E represents ordinary cohomology; it c*
*an
proved using the Serre spectral sequence of the fibration U(d-1) -!U(d) -!S2d-1.
For a more general theory E we still have an Atiyah-Hirzebruch-Serre spectral
sequence Hp(S2d-1; EqU(d - 1)) =) Ep+qU(d). It follows easily that the elements
aI form a basis whenever X is a point. A standard argument now shows that they
form a basis for any X. Indeed, it follows easily from the above that they form*
* a
basis whenever V is trivialisable. We can give X a cell structure such that V *
*is
trivialisable over each cell, and then use Mayer-Vietoris sequences to check th*
*at the
elements aI form a basis whenever X is a finite complex. Finally, we can use the
Milnor exact sequence to show that the elements aI form a basis for all X.
The ring E*U(V ) is graded-commutative so we certainly have aiaj = -ajaiand
in particular 2a2i= 0 for all i. Suppose we can show that a2i= 0. Then i* exten*
*ds
to give a map ~*E*XInd(E*BU(V )) -!E*-1U(V ) of Hopf algebras, and from the
previous paragraph we see that this is an isomorphism. Combining this with the *
*iso-
morphism of Corollary 4.3 gives the required isomorphism ~*E*-1P V -! E*U(V ).
All that is left is to check that a2i= 0. For this we consider the case of *
*the
tautological bundle T over BU(d), and take E = MP = MU[u, u-1]. (We use this
2-periodic version of MU simply to comply with our standing assumptions on E;
we could equally well use MU itself.) Here it is standard that MP *BU(d) is a
formal power series algebra over MP *and thus is torsion-free. The ring MP *U(T*
* )
is a free module over MP *BU(d) and thus is also torsion-free. As 2a2i= 0 we mu*
*st
have a2i= 0 as required. More generally, for an arbitrary bundle V over a space
X we have a classifying map X -! BU(d) giving rise to a map U(V ) -! U(T ).
Moreover, for any E we can choose an orientation in degree zero and thus a ring
map MP -! E. Together these give a ring map MP *U(T ) -! E*U(V ), which
carries ai to ai. As a2i= 0 in MP *U(T ), the same must hold in E*U(V ).
We will need to extend the above result slightly to give a topological inter*
*preta-
tion of the quotient rings
~ rE*-1P V = ~*E*-1P V=~>rE*-1P V.
12 N. P. STRICKLAND
For this we recall Miller's filtration of U(V ):
FkU(V )= {g 2 U(V ) | codim(ker(g - 1)) k}
= {g 2 U(V ) | rank(g - 1) k}.
More precisely, this is supposed to be interpreted fibrewise, so
FkU(V ) = {(x, g) | x 2 X and g 2 U(Vx) and rank(g - 1) k}.
It is not hard to see that æ gives a homeomorphism X P V+X -! F1U(V ). It
is known from work of Miller [6] that when X is a point, the filtration is stab*
*ly
split. Crabb showed in [3] that the splitting works fibrewise; our outline of r*
*elated
material essentially follows his account.
We will need to recall the basic facts about the quotients in Miller's filtra*
*tion.
Consider the space
Gk(V ) = {(x, W ) | x 2 X , W Vx , dim(W ) = k}.
For each point (x, W ) 2 Gk(V ) we have a Lie group U(W ) and its associated Lie
algebra u(W ) = {ff 2 End(W ) | ff + ff* = 0}. These fit together to form a bun*
*dle
over Gk(V ) which we denote by u. Given a point (x, W, ff) in the total space o*
*f this
bundle one checks that ff-1 is invertible and that g := (ff+1)(ff-1)-1 is a uni*
*tary
automorphism of W without fixed points, so g 1W? 2 FkU(Vx) \ Fk-1U(Vx). It
is not hard to show that this construction gives a homeomorphism of the total
space of u with FkU(V )\Fk-1U(V ) and thus a homeomorphism of the Thom space
Gk(V )u with FkU(V )=Fk-1U(V ).
If g 2 FjU(Vx) and h 2 FkU(Vx) then ker(g - 1) \ ker(h - 1) has codimension
at most j + k, so gh 2 Fj+kU(V ), so the filtration is multiplicative. A less o*
*bvious
argument shows that it is also comultiplicative, up to homotopy:
Lemma 4.5. The diagonal map ffi :U(V ) -! U(V ) xX U(V ) is homotopic to a
filtration-preserving map.
Proof.For notational convenience, we will give the proof for a vector space; it*
* can
clearly be done fibrewise for vector bundles.
We regard U(1) as the set of unit complex numbers and define p0, p1: U(1) -!
U(1) as follows:
( 2
p0(z)= z ifIm(z) 0
1 otherwise
( 2
p1(z)= z ifIm(z) 0
1 otherwise.
Thus (p0, p1): U(1) -! U(1) x U(1) is just the usual pinch map U(1) -! U(1) _
U(1) U(1) x U(1).
Note that if g 2 U(V ) and r 2 {0, 1} then the eigenvalues of g lie in U(1) *
*so we
can interpret_pr(g)_as an endomorphism of V as in Appendix A. As pr(U(1)) U(1)
we see that pr(z)= pr(z)-1 for all z 2 U(1) and thus that pr(g)* = pr(g)-1, so *
*pr
gives a map from U(V ) to itself.
We now define ffi0:U(V ) -!U(V )xU(V ) by ffi0(g) = (p0(g), p1(g)). It is cl*
*ear that
the filtration of p0(g) is the number of eigenvalues of g (counted with multipl*
*icity)
lying in the open upper half-circle, and the filtration of p1(g) is the number *
*in the
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 13
open lower half-circle. Thus, the filtration of ffi0(g) is the number of eigenv*
*alues not
equal to 1, which is less than or equal to the filtration of g.
On the other hand, each map pr: U(1) -!U(1) has degree 1 and thus is homo-
topic to the identity, so ffi0 is homotopic to ffi.
Theorem 4.6. There is a natural isomorphism ~ = = = kvk2, so j is
an isometry. Conversely, if j is an isometry then it preserves inner products *
*so
= = for all v, v0 which means that j*jv = v.
Even if j is not an isometry we have = kjvk2 which implies that j*j
is injective. It is thus a strictly positive self-adjoint operator on V , so w*
*e can
define (j*j)-1=2 by functional calculus (as in Appendix A). We then define ^_=
j O (j*j)-1=2. This is the composite of j with an automorphism of V , so it has*
* the
same image as j. It also satisfies ^_*^_= 1, so it is an isometric embedding.
Proposition 5.3. Let V0 and V1 be bundles over a space X. Consider the following
statements:
(a)There exists a bundle V of dimension k and linear isometric embeddings
V0 i0-V -i1!V1.
(a0)There exists a bundle V of dimension k and linear embeddings V0 i0-V -i1!
V1.
(b) There exist a bundle W and isometric linear embeddings V0 j0-!W -j1V1
such that dim((j0V0x) \ (j1V1x)) k for all x 2 X.
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 15
(b0) There exist a bundle W and linear embeddings V0 j0-!W -j1V1 such that
dim((j0V0x) \ (j1V1x)) k for all x 2 X.
(c) There is a linear map f :V0 -!V1 such that rank(fx) k for all x 2 X.
Then (a),(a0))(b),(b0),(c).
Proof.The equivalences (a),(a0) and (b),(b0) follow immediately from Lemma 5.2.
(a))(b): Define W , j0 and j1 by the following pushout square:
i0
V v_____V0_w
v v
i1| |j0
| |
|u |u
V1 v_____W.wj1
Equivalently, we can write Vt0for the orthogonal complement of itV in Vtand then
W = V V00 V10.
(b))(c): Put f = j*1j0: V0 -!V1. By hypothesis, for each x we can choose an
orthonormal sequence u1, . .,.uk in (j0V0x) \ (j1V1x). We can then choose eleme*
*nts
vp 2 V0x and wp 2 V1x such that up = j0vp = j1wp. We find that =
= = ffipq. This implies that the elements fv1, . .,.fvk *
*are
linearly independent, so rank(f) k as required.
(c))(b): Note that f*xfx: V0x -!V0x is a nonnegative self-adjoint operator wi*
*th
the same kernel as fx, and thus the same rank as fx. Similarly, fxf*xis a nonne*
*gative
self-adjoint operator on V1x with the same rank as fx. More basic facts about t*
*hese
operators are recorded in Proposition A.2.
As in Definition A.3 we let ~j = ej(f*xfx) be the j'th eigenvalue of f*xfx (l*
*isted
in descending order and repeated according to multiplicity). We see from Propos*
*i-
tion A.4 that ~j is a continuous function of x. Moreover, as f*xfx has rank at *
*least
k we see that ~k > 0. Now define øx: [0, 1) -![0, 1) by øx(t) = max(~k, t), and
define ~x = øx(f*xfx) and x = ø(fxf*x). (Here we are using functional calculus*
* as
in Appendix A again.) One checks that fx~x = xfx and ~xf*x= f*x x. We now
have maps
~1=2:V0 -!V0
f :V0-!V1
(~ + f*f)-1=2:V0-!V0,
which we combine to get a map
j0 = (~1=2, f) O (~ + f*f)-1=2:V0 -!V0 V1.
Similarly, we define
j1 = (f*, 1=2) O ( + ff*)-1=2:V1 -!V0 V1.
It is easy to check that j*0j0 = 1 and j*1j1 = 1, so j0 and j1 are isometric em*
*beddings.
Now choose an orthonormal sequence v1, .p.,.vk_of eigenvectors of f*xfx, with
eigenvalues ~1, . .,.~k. Put v0i= fx(vi)= ~i2 V1; these vectors form an orthon*
*or-
mal sequence of eigenvectors of fxf*x, with the same eigenvalues. p ___
For i k we havep~i_ ~k > 0 so øx(~i) =p~i_so (~ + f*f)-1=2(vi) = vi= 2~i
and ~1=2(vi) = ~iviso j0(vi) = (vi, v0i)= 2. This is the same as j1(v0i), so *
*it lies in
(j0V0x) \ (j1V1x). Thus, this intersection has dimension at least k, as require*
*d.
16 N. P. STRICKLAND
We conclude this section with a topological interpretation of the scheme D(V0*
*)\
D(V1) itself.
Proposition 5.4. Let V0 and V1 be vector bundles over a space X, and let L0 and
L1 be the tautological bundles of the two factors in P V0 xX P V1. Then there i*
*s a
natural map S(Hom (L0, L1))E -! D(V0) \ D(V1), which is an isomorphism if the
map E*P (V0 V1) -!E*P V0 E*P V1 is injective.
Proof.We divide the sphere bundle S(V0 V1) into two pieces, which are preserved
by the evident action of U(1):
C0= {(v0, v1) 2 S(V0 V1) | kv0k kv1k}
C1= {(v0, v1) 2 S(V0 V1) | kv1k kv0k}.
The inclusions Vi-! V0 V1 give inclusions S(Vi) -!Ciwhich are easily seen to be
homotopy equivalences. It follows that Ci=U(1) ' P Vi. We also have
C0 \ C1 = {(v0, v1) | kv0k = kv1k = 2-1=2} ' S(V0) x S(V1).
Given a point in this space we have a map ff: Cv0 -!Cv1 sending v0 to v1. This
has norm 1 and is unchanged if we multiply (v0, v1) by an element of U(1). Using
this we see that (C0 \ C1)=U(1) = S(Hom (L0, L1)). Of course, we also have (C0 [
C1)=U(1) = P (V0 V1). We therefore have a homotopy pushout square as shown
on the left below, giving rise to a commutative square of formal schemes as sho*
*wn
on the right.
S(Hom (L0, L1))______wPiV00 S(Hom (L0, L1))E_____D(V0)w
| | | v|
| |j | |
i1| | 0 | |
| | | |
|u |u |u |u
P V1_________wPj(V01 V1) D(V1) v_______D(V0_w V1).
This evidently gives us a map S(Hom (L0, L1))E -! D(V0) \ D(V1).
To be more precise, we use the Mayer-Vietoris sequence associated to our pus*
*hout
square. This gives a short exact sequence
cok(f0) p-!E0S(Hom (L0, L1)) q-!ker(f-1 ),
where
fk = (j*0, j*1): EkP (V0 V1) -!EkP V0 EkP V1.
We have seen that cok(f0) = OD(V0)\D(V1), and the map p just corresponds to our
map
S(Hom (L0, L1))E -! D(V0) \ D(V1).
This map will thus be an isomorphism if f* is injective, as claimed.
6.Algebraic universal examples
Let G be a formal group over a formal scheme S. Later we will work with
bundles over a space X, and we will take S = XE and G = (CP 1 xX)E . We write
Div+d= Div+d(G) ' Gd= d, so ODiv+d= OS[[c1, . .,.cd]].
Fix integers d0, d1, r 0. We write Intr(d0, d1) for the scheme of pairs (D*
*0, D1)
where D0 and D1 are divisors of degrees d0 and d1 on G, and int(D0, D1) r.
In other words, if Di is the evident tautological divisor over Div+d0x Div+d1th*
*en
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 17
Intr(d0, d1) = Intr(D0, D1). We will assume that r min(d0, d1) (otherwise we
would have Intr(d0, d1) = ;.)
For a more concrete description, put
R = ODiv+dx Div+= OS[[c0j| j < d0]][[c1j| j < d1]].
0 d1
Let A be the matrix of i over R as in Section 3, and let I be the ideal in R ge*
*nerated
by the minors of A of size d0 + d1 - r + 1. Then Intr(d0, d1) = spf(R=I).
We will also consider a "semi-universal" case. Suppose we have a divisor D1 on
G over S, with degree d1. Let D0 be the tautological divisor over Div+d0. We can
regard D0 and D1 as divisors on G over Div+d0and thus form the closed subscheme
Intr(D0, D1) Div+d0. We denote this scheme by Intr(d0, D1).
We can also define schemes Subr(d0, d1) and Subr(d0, D1) in a parallel way.
Remark 6.1. Sub r(d0, d1) is just the scheme of triples (D, D0, D1) for which
D D0 and D D1. This is isomorphic to the scheme of triples (D, D00, D01) 2
Div+rx Div+d0-rx Div+d1-r, by the map (D, D00, D01) 7! (D, D + D00, D + D01).
Definition 6.2. We write Subr(D) for the scheme of divisors D0of degree r such
that D0 D. Using Remark 3.12 we see that Subr(D) = Subr(r, D) = Intr(r, D).
Theorem 6.3. The ring OIntr(d0,d1)is freely generated over
OS[[c0i| 0 < i d0 - r]][[c1j| 0 < j d1]]
by the monomials
Yd0
cff0:= cffi0i
i=d0-r+1
P
for which iffi d1- r. Moreover, if we let ß :Subr(d0, d1) -!Intr(d0, d1) be *
*the
usual projection, then the corresponding ring map ß* is a split monomorphism of
modules over ODiv+d(so ß itself is dominant).
1
The proof will be given after a number of intermediate results. It seems lik*
*ely
that the injectivity of ß* could be extracted from work of Pragacz [9, Section *
*3]. He
works with Chow groups of varieties rather than generalised cohomology rings of
spaces, and his methods and language are rather different; we have not attempted
a detailed comparison.
We start by setting up some streamlined notation. We put n = d0 - r and
m = d1-r. We use the following names for the coordinate rings of various schemes
of divisors, and the standard generators of these rings:
C0 = ODiv+d= OS[[u1, . .,.un+r]]
0
C1 = ODiv+d= OS[[v1, . .,.vm+r ]]
1
A = ODiv+n= OS[[a1, . .,.an]]
B = ODiv+m= OS[[b1, . .,.bm ]]
C = ODiv+r= OS[[c1, . .,.cr]].
(In particular, we have renamed c0iand c1ias ui and vi.) We put u0 = v0 = a0 =
b0 = c0 = 1. We define ui= 0 for i < 0 or i > n + r, and similarly for vi, ai, *
*biand
18 N. P. STRICKLAND
ci. The equations of the various tautological divisors are as follows:
X
f0(x)= uixn+r-i 2 C0[x]
Xi
f1(x)= vixm+r-i 2 C1[x]
Xi
f(x)= aixn-i 2 A[x]
Xi
g(x)= bixm-i 2 B[x]
Xi
h(x)= cixr-i2 C[x].
i
We write T0 for the set of monomials of weight at most m in un+1, . .,.un+r, and
T for the set of monomials of weight at most m in c1, . .,.cr. We also introduc*
*e the
subrings
C00= OS[[u1, . .,.un]] C0
C000= OS[[u1, . .,.un-1]] C00.
We note that the ring Q := OIntr(d0,d1)has the form (C0b C1)=I for a certain id*
*eal
I. The theorem claims that Q is freely generated as a module over C00bC1 by T0.
The map ß*: C0b C1 -!Ab B bC sends f0(x) to f(x)h(x) and f1(x) to g(x)h(x).
This induces a map ß*: Q -!Ab B bC, and the theorem also claims that this is a
split injection.
We will need to approximate certain determinants by calculating their lowest
terms with respectQto a certain ordering. More precisely, we consider monomials
of the form uff= n+ri=1uffii, and we order these by uff< ufiif there exists i*
* such
that ffi > fii and ffj = fij for j > i. The mnemonic is that u1 . . .un+r, so
any difference in the exponent of ui overwhelms any difference in the exponents*
* of
u1, . .,.ui-1.
Lemma 6.4. Suppose we have integers fli satisfying 0 fl0 < . .<.flm < m + r,
and we put Mij= un+r+i-fljfor 0 i, j m, where uk is interpreted as 0 if k <*
* 0
or k > n+r. Then the lowest term in det(M) is the product of the diagonal entri*
*es,
so m
Y
det(M) = un+r+i-fli+ higher terms.
i=0
Remark 6.5. Determinants of this type are known as Schur functions.
Q m
Proof.Put ffi = i=0un+r+i-fli. Let M0ibe obtained from M by removing the 0'th
row and i'th column. The matrixQM00has the same general form as M so by in-
duction we have det(M00) = mi=1un+r+i-fli+ higher terms. If we expand det(M)
along the top row then the 0'th term is un+r-fl0det(M00) = ffi + higher terms. *
*As
0 fl0 < . .<.flm we have fli i + fl0 and so ffi only involves variables uj *
*with
j n + r - fl0. The remaining terms in the row expansion of det(M) have the
form (-1)iun+r-fl0+idet(M0i) for i > 0, and un+r-fl0+iis either zero (if i > fl*
*0) or a
variable strictly higher than all those appearing in ffi. The lemma follows eas*
*ily.
Lemma 6.6. The ring Q is generated by T0 as a module over C00bC1.
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 19
Proof.Let J be the ideal in C00bC1 generated by u1, . .,.un and v1, . .,.vm+r ,*
* so
(C00bC1)=J = OS . We also put C000= (C0b C1)=J = OS [[un+1, . .,.un+r]]. As J is
topologically nilpotent, it will suffice to prove the result modulo J. We will *
*thus
work modulo J throughout the proof, so that f1 = xm+r , and we must show that
Q=J is generated over OS by T0.
Let ~: C000[[x]]=xm+r -!C000[[x]]=xm+r be defined by ~(t) = f0t, and let M *
*be
the matrix of ~ with respect to the obvious bases. It is then easy to see that
Q=J = C000=I, where I is generated by the minors of M of size m + 1. The entries
in M are Mij= un+r+i-j.
We next claim that all the generatorspuk_are nilpotent mod I, or equivalently
that uk = 0 in the ring R = C000= I for all k. By downward induction we may
assume that ul= 0 in R for k < l n + r. We consider the submatrix M0 of M
given by M0ij= Mi,n+r-k+j= ui+k-j for 0 i, j m. By the definition of I we
have det(M0) 2 I and thus det(M0) = 0 in R. OnQthe other hand, we have ul= 0 for
l > k so M0 is lower triangular so det(M0) = iM0ii= um+1k. Thus uk is nilpote*
*nt
in R but clearly Nil(R) = 0 so uk = 0 in R as required. It follows that Q=J is a
quotient of the polynomial ring OS [un+1, . .,.un+r] OS[[un+1, . .,.un+r]].
Now let W be the submodule of Q=J spanned over OS by T0; we must prove
that this is all of Q=J. As 1 2 W , it will suffice to show that W is an ideal.*
* In
the light of the previous paragraph, it will suffice to show that W is closed u*
*nder
multiplication by the elements un+1, . .,.un+r, or equivalently that W contains*
* all
monomials of weight m + 1.
We thus let ff = (ffn+1, . .,.ffn+r) be a multiindex of weight m+1. There isQ*
*then
a unique sequence (fi0, . .,.fim ) with n + r fi0 . . .fim > n and uff= i*
*ufii.
Put fli= n+r+i-fii, so that 0 fl0 < . .<.flm < m+r. Let Mffbe the submatrix
of M consisting of the first m + 1 columns of the rows of indices fl0, . .,.flm*
* , so the
(i, j)'th entry of Mffis un+r+i-flj. Note that the elementsQrff:= det(Mff)Qlie *
*in I.
Lemma 6.4 tells us that the lowest term in rffis iun+r+i-fli= iufii= uff.
It is clear that the weight of the remaining terms is at most the size of Mff, *
*which
is m + 1. By an evident induction, we may assume that their images in C000=I li*
*e in
W . As rff2 I we deduce that uff2 W as well.
Corollary 6.7. Let D1 be a divisor of degree d1 on G over S0, for some scheme S0
over S. Then OIntr(d0,D1)is generated over OS0[[c01, . .,.c0,d0-r]] by the mono*
*mials
Q d0 ff
cff0= i=d0-r+1c0iifor which |ff| d1 - r.
Proof.The previous lemma is the universal case.
We next treat the special case of Theorem 6.3 where n = 0 and so r = d0. As
remarked in Definition 6.2, the map ß :Subr(d1) = Subr(r, d1) -!Intr(r, d1) is *
*an
isomorphism in this case.
Lemma 6.8. Let D be a divisor of degree d on G over S. For any r Pd we
let Pr(D) denote the scheme of tuples (u1, . .,.ur) 2 Gr such that ri=1[ur] *
* D.
Then OPr(D)is free of rank d!=(d - r)! over OS.
Proof.There is an evident projection Pr(D) -! Pr-1(D), which identifies Pr(D)
with the divisor D - [u1] - . .-.[ur-1] on G over Pr-1(D). This divisor has deg*
*ree
d - r + 1, so OPr(D)is free of rank d - r + 1 over OPr-1(D). It follows by an
evident induction that OPr(D)is free over OS , with rank d(d - 1) . .(.d - r + *
*1) =
d!=(d - r)!.
20 N. P. STRICKLAND
Lemma 6.9. Let D be a divisor of degree d on G overPS, let D0be the tautological
divisor of degree r over Subr(D), and let f(x) = ri=0cixr-i be the equation of
D. Then the set T of monomials of degree at most d - r in c1, . .,.cr is a basi*
*s for
OSubr(D)over OS .
` '
Proof.Put K = |T |; by elementary combinatorics we find that K = dr . Put
R = OSubr(D). Using T we obtain an OS -linear map fi :OKS-! R, which is surjec-
tive by Lemma 6.6; we must prove that it is actually an isomorphism.
Now consider the scheme Pr(D); Lemma 6.8 tells us that the ring R0:= OPr(D)
is a free module over OS of rank d!=(d - r)! = r!K. On the other hand, Pr(D) can
be identified with the scheme of tuples (D0, u1, . .,.ur) where D0 2 Subr(D) and
D0= [u1] + . .+.[ur]. In other words, if we change base to Subr(D) we can regard
Pr(D) as Pr(D0), and now Lemma 6.8 tells us that R0is free of rank r! over R.
Now choose a basis e1, . .,.er!for R0over R. We can combine this with fi to *
*get
a map fl :OSr!K -!R0. This is a direct sum of copies of fi, so it is surjective*
*. Both
source and target of fl are free of rank r!K over OS. Any epimorphism between f*
*ree
modules of the same finite rank is an isomorphism (choose a splitting and then *
*take
determinants). Thus fl is an isomorphism, and it follows that fi is an isomorph*
*ism
as required.
Corollary 6.10. The set T is a basis for B bC over C1.
Proof.This is the universal case of the lemma.
Corollary 6.11. The set T is a basis for Ab B bC over C00bC1.
Proof.Note that Ab B bC = (B bC)[[a1, . .,.an]]. For 0 < i n we have
X
ß*ui= ajck = ai+ ci mod decomposables,
i=j+k
where cimay be zero, but aiis definitely nonzero. It follows that our ring Ab B*
* bC
can also be described as (B bC)[[u1, . .,.un]], or equivalently as C00bB bC. T*
*he
claim now follows easily from the previous corollary.
Now let T1 be the set of monomials of the form uincff11.c.f.fnnfor which 0 *
* i <
|ff| m. These monomials can be regarded as elements of Ab B bC, giving a map
(C000bC1){T1} -!Ab B bC. The map ß*: C0b C1 -!Ab B bC also gives us a map
(C00bC1){T0} -!Ab B bC, and by combining these we get a map
OE: (C00bC1){T0} (C000bC1){T1} -!Ab B bC ' (C00bC1){T }
of modules over C000bC1. Our main task will be to prove that this is an isomorp*
*hism.
The proof will use the following lemma.
Lemma 6.12. Let R be a ring, and let ff: M -! N be a homomorphism of mod-
ules over R[[x]]. Suppose that M can be written as a product of copies of R[[x*
*]],
and similarly for N. Suppose also that the induced map M=xM -! N=xN is an
isomorphism. Then ff is also an isomorphism.
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 21
Proof.We have diagrams as shown below, in which the rows are easily seen to be
exact:
M=xkM v_____M=xk+1Mwx ______wM=xMw
| | |
ff| ff| |ff
| | |
|u |u |u
N=xkN v______N=xk+1Nwx______wN=xNw
We see by induction that the maps M=xkM -!N=xkN are all isomorphisms, and
the claim follows by taking inverse limits.
Our map OE is a map of modules over the ring
C000bC1 = OS[[u1, . .,.un-1, v1, . .,.vm+r ]].
Q 1
Moreover, we have C00bC1 = (C000bC1)[[un]] ' k=0C000bC1. Now let J be the ide*
*al
in C000bC1 generated by {u1, . .,.un-1, v1, . .,.vm+r }, so (C000bC1)=J = OS an*
*d OE
induces a map __
OE:OS[[un]]{T0} OS{T1} -!OS [[un]]{T }.
Note also that OS {T } is the image of C in (Ab B bC)=J and is thus a subring of
OS[[un]]{T_}. By an evident inductive extension of the lemma, it will suffice t*
*o show
that OEis an isomorphism.
Lemma 6.13. We have un+j = uncj + wj (mod J) for some polynomial wj in
c1, . .,.cr.
Proof.ForPany monic polynomialPp(x) of degree d we write ^p(y) = ydp(1=y). If
p(x) = irixd-ithen ^p(y) = iriyi. Note that bpq= ^p^q, and that ^p(0) = 1. *
*As we
work mod (ui| i < n) we have ^f0= 1 (mod yn). As we work mod (vj | j m + r)
we have ^f1= 1. We also have fh = f0 and gh = f1, so ^f^h= ^f0= 1 (mod yn) and
^g^h= ^f1= 1. It follows easily that ^f= ^g(mod yn), so ai= bi for i < n.
We now have to distinguish between the case m < n and the case m n. First
suppose that m < n. Then for i > n we have ai = bi = 0, and also bn = 0, and
ai = bi for i < n by the previous paragraph. This implies that f^- ^g= anyn.
We also have (f^- ^g)^h= f^0- 1, and by comparing coefficients we deduce that
anci = un+i for i = 0, . .,.r. The case i = 0 gives un = an, so un+i = unci for
i = 1, . .,.r, so the lemma is true with wi= 0.
Now suppose instead that m n. As ai= 0 for i > n we have
Xm
f^- ^g- (an - bn)yn = - biyi2 C[y].
i=n+1
We now multiply this by ^hand use the fact that (f^- ^g)^h= ^f0- 1. By comparing
coefficients of yn we find that un = an - bn. In view of this, our equation rea*
*ds
Xm
^f0- 1 - unyn^h= -( biyi)^h2 C[y].
i=n+1
P
The right hand side has the form j>0wjyn+j with wj 2 C, and by comparing
coefficients we see that un+j = uncj+ wj as claimed.
22 N. P. STRICKLAND
__ |ff|
Proof of Theorem 6.3.Lemma 6.13 tells us that OE(uff) is un cffplus terms invol*
*ving_
lower powers of un. It follows easily that if we filter the source and target o*
*f OEby
powers of un, then_the resulting map of associated graded modules is a isomorph*
*ism.
It follows that OEis an isomorphism, and thus that OE is an isomorphism. It fol*
*lows
that the map (C00bC1){T0} -!Ab B bC is a split monomorphism of modules over
C000bC1 (and thus certainly of modules over C1). We have seen that this map fac*
*tors
as *
(C00bC1){T0} _-!Q ß-!Ab B bC,
where _ is surjective by Lemma 6.6. It follows that _ is an isomorphism and that
ß* is a split monomorphism, as required.
7.Flag spaces
In the next section we will (in good cases) construct spaces whose associated
formal schemes are the schemes Subr(D(V0), D(V1)) and Intr(D(V0), D(V1)) con-
sidered previously. As a warm-up, and also as technical input, we will first co*
*nsider
the schemes associated to Grassmannian bundles and flag bundles. The results di*
*s-
cussed are essentially due to Grothendieck [4]; we have merely adjusted the lan*
*guage
and technical framework.
Let V be a bundle of dimension d over a space X. We write Pr(V ) for the spa*
*ce
of tuples (x, L1, . .,.Lr) where x 2 X and L1, . .,.Lk 2 P Vx and Li is orthogo*
*nal
to Lj for i 6= j. Recall also that in Lemma 6.8 we defined Pr(D(V )) to be the
scheme over XE of tuples (u1, . .,.ur) 2 Gr for which [u1] + . .+.[ur] D(V ).
Proposition 7.1. There is a natural isomorphism Pr(V )E = Pr(D(V )).
Proof.For each i we have a line bundle over Pr(V ) whose fibre over (x, L1, . .*
*,.Lr)
is Li. This is classified by a map Pr(V ) -! CP 1 , which gives rise to a map
ui:Pr(V )E -! G. The direct sum of these line bundles corresponds to the divisor
[u1] + . .+.[ur]. This direct sum is a subbundle of V , so [u1] + . .+.[ur] D*
*(V ).
This construction therefore gives us a map Pr(V )E -! Pr(D(V )).
In the case r = 1 we have P1(V ) = P V and P1(D(V )) = D(V ) so the claim
is that (P V )E = D(V ), which is true by definition. In general, suppose we kn*
*ow
that Pr-1(V )E = Pr-1(D(V )). We can regard Pr(V ) as the projective space of t*
*he
bundle over Pr-1(V ) whose fibre over a point (x, L1, . .,.Lr-1) is the space Vx
(L1 . . .Lr-1). It follows that Pr(V )E is just the divisor D(V )-([u1]+. .+.[u*
*r-1])
over Pr-1(D(V )), which is easily identified with Pr(D(V )). The proposition fo*
*llows
by induction.
Remark 7.2. One can easily recover the following more concrete statement. The
ring E0Pr(V ) = OPr(D(V ))is the largest quotient ring of (E0X)[[x1, . .,.xr]] *
*in which
Q k
the polynomial fV (t) is divisible by i=1(t - xi). It is a free module over *
*E0X
with rank d!=(d - r)!, and the monomials xffwith 0 ffi d - i (for i = 1, . .*
*,.r)
form a basis. More details about the multiplicative relations are given in Sect*
*ion 9.
We next consider the Grassmannian bundle
Gr(V ) = {(x, W ) | x 2 X , W Vx and dim(W ) = r}.
Proposition 7.3. There is a natural isomorphism Gr(V )E = Subr(D(V )).
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 23
Proof.Let T denote the tautological bundle over Gr(V ). This is a rank r subbun*
*dle
of the pullback of V so we have a degree r subdivisor D(T ) of the pullback of *
*D(V )
over Gr(V )E . This gives rise to a map Gr(V ) -!Subr(D(V )).
Next, consider the space Pr(V ). There is a map Pr(V ) -! Gr(V ) given by
(x, L1, . .,.Lr) 7! (x, L1 . . .Lr). This lifts in an evident way to give a h*
*ome-
omorphism Pr(V ) ' Pr(T ). Of course, this is exactly parallel to the proof of
Lemma 6.9. Over Pr(D(T )) we have points a1, . .,.ar of G with coordinate values
x1, . .,.xr 2 OPr(D(T))say. Let B be the set of monomials xffwith 0 ffi r - i
for i = 1, . .,.r. From our earlier analysis of Subr(D(V )) and Pr(D(T )) we s*
*ee
that B is a basis for OPr(D(T))over OSubr(D(V )). We also see from Remark 7.2
(applied to the bundle T ) that B is a basis for E0Pr(T ) over E0Gr(V ). This m*
*eans
that our isomorphism f :OPr(D(V ))-!E0Pr(V ) is a direct sum (indexed by B) of
copies of our map g :OSubr(D(V ))-!E0Gr(V ). It follows that g must also be an
isomorphism.
Remark 7.4. Lemma 6.9 now gives us an explicit basis for E0Gr(V ) over E0X,
consisting of monomials in the Chern classes of the tautological bundle T .
8.Topological universal examples
In this section we construct spaces whose associated formal schemes are the
algebraic universal examples considered in Section 6.
We first consider the easy case of the schemes Subr(D0, D1).
Definition 8.1. Given vector bundles V0 and V1 over X, we define Gr(V0, V1) to
be the space of quadruples (x, W0, W1, g) such that
(a)x 2 X;
(b) Wi is an r-dimensional subspace of Vixfor i = 0, 1; and
(c)g is an isometric isomorphism W0 -!W1.
(We would obtain a homotopy equivalent space if we dropped the requirement that
g be an isometry.)
If Vi is the evident tautological bundle over BU(di) we write Gr(d0, d1) for
Gr(V0, V1). More generally, if V is a bundle over X and d0 0 we can let V1 be
the pullback of V to BU(d0) x X, and let V0 be the pullback of of the tautologi*
*cal
bundle over BU(d0); in this context we write Gr(d0, V ) for Gr(V0, V1).
Theorem 8.2. There is a natural map p: Gr(V0, V1)E -! Subr(D(V0), D(V1)). In
the universal case this is an isomorphism, so
Gr(d0, d1)E = Subr(d0, d1).
More generally, there is a spectral sequence
Tor**E*BU(d0)xBU(d1)(E*X, E*Gr(d0, d1)) =) E*Gr(V0, V1),
whose edge map in degree zero is the map
p*: OSubr(D(V0),D(V1))-!E0Gr(V0, V1).
The spectral sequence collapses in the universal case. (We do not address the q*
*ues-
tion of convergence in the general case.)
24 N. P. STRICKLAND
Proof.First, we can pull back the bundles Vi from X to Gr(V0, V1) (without
change of notation). We also have a bundle over Gr(V0, V1) whose fibre over a
point (x, W0, W1, g) is W0; we denote this bundle by W , and note that there
are natural inclusions V0- 1 W -g!V1. We then have divisors D(W ) and D(Vi)
on G over Gr(V0, V1)E with D(W ) D(V0) and D(W ) D(V1), so the triple
(D(V0), D(V1), D(W )) is classified by a map Gr(V0, V1)E -! Subr(D(V0), D(V1)).
We next consider the universal case. As our model of EU(d) we use the space
of orthonormal d-frames in C1 , so BU(d) is just the Grassmannian of d-planes in
C1 . Given a point
(u_, v_) = (u1, . .,.ud0, v1, . .,.vd1) 2 EU(d0) x EU(d1)
we construct a point ((V0, V1), W0, W1, g) 2 Gr(d0, d1) as follows:
(a) V0 is the span of u1, . .,.ud0
(b) V1 is the span of v1, . .,.vd1
(c) W0 is the span of u1, . .,.ur
(d) W1 is the span of v1, . .,.vr
(e) g is the map W0 -!W1 that sends ui to vi.
This gives a map f :EU(d0)xEU(d1) -!Gr(d0, d1). Next, the group U(d0)xU(d1)
has a subgroup U(r) x U(d0 - r) x U(r) x U(d1 - r), inside which we have the
smaller subgroup consisting of elements of the form (h, k0, h, k1). It is not*
* hard
to see that ' U(r) x U(d0 - r) x U(d1 - r), and that f gives a homeomorphism
(EU(d0) x EU(d1))= -! Gr(d0, d1). Moreover, EU(d0) x EU(d1) is contractible
and acts freely so Gr(d0, d1) ' B = BU(r) x BU(d0 - r) x BU(d1 - r), so
Gr(d0, d1)E = Div+rx Div+d0-rx Div+d1-r= Subr(d0, d1) as claimed.
In the general case we can choose maps fi:X -!BU(di) classifying Vi, and this
gives rise to a pullback square as follows:
Gr(V0, V1)_____Gr(d0,wd1)
| |
| |
| |
| |
|u |u
X _____wBU(d0) x BU(d1).
The vertical maps are fibre bundle projections so this is actually a homotopy p*
*ull-
back square. This give an Eilenberg-Moore spectral sequence as in the statement
of the theorem. On the edge we have
E*X E*BU(d0)xBU(d1)E*Gr(d0, d1),
which is the same as
E*X E0BU(d0)xBU(d1)E*Gr(d0, d1).
We can now identify this as the tensor product of E*X with OSubr(d0,d1)over
ODiv+dx Div+. The part in degree zero is easily seen to be OSubr(D(V0),D(V1))as
0 d1
claimed.
We next show that our map Gr(V0, V1)E -! Subr(D(V0), D(V1)) is an isomor-
phism in the semiuniversal case as well as the universal case. We start by anal*
*ysing
the semiuniversal spaces Gr(d0, V ) in more familiar terms.
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 25
Proposition 8.3. There are natural homotopy equivalences Gr(d0, V ) ' Gr(V ) x
BU(d0 - r) (and in particular Gr(r, V ) ' Gr(V )).
Proof.A point of Gr(d0, V ) is a tuple (V0, x, W0, W1, g) where V0 2 Gd0(C1 ),
x 2 X, W0 2 Gr(V0), W1 2 Gr(Vx) and g :W0 -! W1. We can define a map
f :Gr(d0, V ) -!Gr(V ) x BU(d0 - r) by f(V0, x, W0, W1, g) = (x, W1, V0 W0). *
*It
is not hard to see that this is a fibre bundle projection, and that the fibre o*
*ver a
point (x, W, V 0) is the space of linear isometric embeddings from W to C1 V *
*0.
This space is homeomorphic to the space of linear isometric embeddings of Cr in
C1 , which is well-known to be contractible. Thus f is a fibration with contrac*
*tible
fibres and thus is a weak equivalence.
Corollary 8.4. The map Gr(d0, V )E -! Subr(d0, D(V )) is an isomorphism.
Proof.Recall that Subr(d0, D(V )) is the scheme of pairs (D1, D) where D1 is a
divisor of degree d1, D is a divisor of degree r and D D1 \ D(V ). There is an
evident isomorphism Subr(D(V )) xS Div+d0-r-!Subr(d0, D(V )) sending (D0, D)
to (D + D0, D). The proposition tells us that Gr(d0, V ) = Gr(V ) x BU(d0 -
r). We already know that BU(d0 - r)E = Div+d0-r, and Proposition 7.3 tells us
that Gr(V )E = Subr(D(V )). We therefore have an isomorphism Gr(d0, V )E =
Subr(D(V )) xS Div+d0-r= Subr(d0, D(V )). (This involves an implicit Künneth
isomorphism, which is valid because BU(d0 - r) has only even-dimensional cells.)
We leave it to the reader to check that this isomorphism is the same as the map
considered previously.
We now turn to parallel results for the schemes Intr(D(V0), D(V1)).
Definition 8.5. Given vector bundles V0 and V1 over a space X, we define Ir(V0,*
* V1)
to be the space of pairs (x, f) where f :V0x -!V1x is a linear map of rank at l*
*east
r. We define the universal and semiuniversal spaces Ir(d0, d1) and Ir(d0, V ) b*
*y the
evident analogue of Definition 8.1.
Remark 8.6. There is a natural map
Gr(V0, V1) -!Ir(V0, V1),
sending (x, W0, W1, g) to (x, f), where f is the composite
V0 proj--!W0 g-!W1 inc--!V1.
This gives a homeomorphism of Gr(V0, V1) with the subspace of Ir(V0, V1) consis*
*ting
of pairs (x, f) for which f*f and ff* are idempotent.
Definition 8.7. We define a natural map q :Ir(V0, V1) -! Intr(D(V0), D(V1)) as
follows. If we let ß denote the projection Ir(V0, V1) -! X then we have a tau-
tological map f :ß*V0 -! ß*V1 which has rank at least r everywhere. Propo-
sition 5.3 now tells us that int(ß*V0, ß*V1) r. We can therefore apply The-
orem 5.1 and deduce that the map Ir(V0, V1)E -! XE factors through a map
q :Ir(V0, V1) -!Intr(D(V0), D(V1)) XE as required.
Later we will show that the map q is an isomorphism in the universal case.
For this, it will be convenient to have an alternative model for the universal *
*space
Ir(d0, d1).
26 N. P. STRICKLAND
Proposition 8.8. Put
I0r(d0, d1) = {(V0, V1) 2 Gd0(C1 ) x Gd1(C1 ) | dim(V0 \ V1) k}.
Then I0r(d0, d1) is homotopy equivalent to Ir(d0, d1).
Proof.The basic idea is to refine the proof of Proposition 5.3. We will take Gd*
*(C1 )
as our model for BU(d). We write I = Ir(d0, d1) and I0= I0r(d0, d1) for brevity.
We will need various isometries between infinite-dimensionalpvector_spaces. We
define ffi :C1 -!C1 C1 by ffi(v) = (v, v)= 2, and we define ` :C1 C1 -!
C1 by `(v, w) = (v0, w0, v1, w1, . .).. Next, it is well-known that the space*
* of
linear isometric embeddings of C1 in itself is contractible, so we can choose a
continuous family of isometries OEt with OE0 = `ffi and OE1 = 1. Similarly, we*
* can
choose continuous families of isometric embeddings _t0, _t1:C1 -! C1 C1 with
_00(v) = `(v, 0) and _01(v) = `(0, v) and _10(v) = _11(v) = v.
We now define a map ff: I0 -! I by ff(V0, V1) = (V0, V1, f), where f is the
orthogonal projection map from V0 to V1. This acts as the identity on V0 \ V1 a*
*nd
thus has rank at least k. If we choose n large enough that V0 + V1 Cn and let
V0 i0-!Cn -i1V1 be the inclusions, then f = i*1i0.
Next, we need to define a map fi :I -!I0. Given (V0, V1, f) 2 I we can constr*
*uct
maps
~: V0-! V0
:V1-! V1
j0: V0-!V0 V1 < C1 C1
j1: V1-!V0 V1 < C1 C1
as in the proof of the implication (c))(b) in Proposition 5.3, so dim(j0V0\j1V1)
k. We can thus define fi :I -!I0 by fi(V0, V1, f) = (`j0V0, `j1V1).
Suppose we start with (V0, V1) 2 I0, define f :V0 -! V1 to be the orthogonal
projection, and then define j0, j1 as above so that fiff(V0, V1) = (`j0V0, `j1V*
*1).
Observe that f*f :V0 -!V0 decreases distances, and acts as the identity on V :=
V0\ V1. If we let ~1, . .,.~d0 be the eigenvalues of f*f (listed in the usual w*
*ay) we
deduce that ~1 = . .=.~k = 1 and that 0 ~i 1 for all i. It follows from this
that ~ and are the respective identity maps, so
j0= (1, f) O (1 + f*f)-1=2
j1= (f*, 1) O (1 + ff*)-1=2.
p _
In particular, we have j0(v) = j1(v) = (v, v)= 2for v 2 V , so j0|V = j1|V = f*
*fi|V .
Next, for 0 t 1 we define jt0:V0 -!C1 C1 by
jt0= (i0, ti0 + (1 - t)f) O (1 + t2 + (1 - t2)f*f)-1=2.
One can check that this is an isometric embedding, with j00= j0 and j10= ffi|V0*
* and
jt0|V = ffi|V for all t. Similarly, if we put
jt1= (i1, ti0 + (1 - t)f*) O (1 + t2 + (1 - t2)ff*)-1=2,
we find that this is an isometric embedding of V1 in C1 C1 with j01= j1 and
j11= ffi|V1 and jt1|V = ffi|V for all t. It follows that (`jt0V0, `jt1V1) 2 I0*
* for all t,
and this gives a path from fiff(V0, V1) = (`j0V0, `j1V1) to (`ffiV0, `ffiV1). R*
*ecall that
we chose a path {OEt} from `ffi to 1. The pairs (OEtV0, OEtV1) now give a path *
*from
(`ffiV0, `ffiV1) to (V0, V1) in I0. Both of the paths considered above are easi*
*ly seen
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 27
to depend continuously on the point (V0, V1) 2 I0 that we started with, so we h*
*ave
constructed a homotopy fiff ' 1.
Now suppose instead that we start with a point (V0, V1, f) 2 I; we need a
path from fffi(V0, V1, f) to (V0, V1, f). We have fi(V0, V1, f) = (`j0V0, `j1V*
*1), so
fffi(V0, V1, f) = (`j0V0, `j1V1, f0), where f0:`j0V0 -!`j1V1 is the orthogonal *
*pro-
jection. One can check that this is characterised by f0(`j0(v)) = `j1(j*1j0(v))*
*. Next,
for 0 t 1 we define kt0:V0 -!V0 V1 by
p __________
kt0= ( 1 - t2 + t2~, tf) O (1 - t2 + t2~ + t2f*f)-1=2.
This is an isometric embedding with k10= j0 and k00(v) = (v, 0). Similarly, we
define kt1:V1 -!V0 V1 by
p__________
kt1= (tf*, 1 - t2 + t2) O (1 - t2 + t2 + t2ff*)-1=2,
and we define f0t:`kt0V0 -!`kt1V1 by
f0t(`kt0(v)) = `kt1(j*1j0(v)),
so f01= f0. The points (kt0V0, kt1V1, f0t) give a path from fffi(V0, V1, f) to *
*(`(V0
0), `(0 V1), f00) in I.
Next, we define f00t:_t0V0 -! _t1V1 by f00t(_t0(v)) = _t1(j*1j0(v)). The poi*
*nts
(_t0V0, _t1V1, f00t) give a path from (`(V0 0), `(0 V1), f00) to (V0, V1, j*
**1j0) in I.
Using Proposition A.2 one can check that
p__ *
j*1j0= ( + ff*) O (fp __~+ f) O (~ + f f)
= f O (2~1=2(~ + f*f)-1).
The map i := 2~1=2(~ + f*f)-1 is a strictly positive self-adjoint automorphism *
*of
V0, so the same is true of t+(1-t)i for 0 t 1. The points (V0, V1, f O(t+(1*
*-t)i))
form a path from (V0, V1, j*1j0) to (V0, V1, f). All the paths considered depe*
*nd
continuously on the point (V0, V1, f) that we started with, so we have defined a
homotopy fffi ' 1.
Theorem 8.9. The map q :Ir(d0, d1)E -! Intr(d0, d1) is an isomorphism.
Proof.We first replace Ir(d0, d1) by the homotopy-equivalent space I0r(d0, d1).*
* We
write Ir = I0r(d0, d1) and Gr = Gr(d0, d1) for brevity, and similarly for Intra*
*nd
Subr. We first claim that there is a commutative diagram as follows.
OIntrv_____OSubrw
| |
q| p|
| |'
| |
|u |u
E0Ir _____wE0Gr.
Indeed, the isomorphism
p: OSubr(d0,d1)-!E0Gr(d0, d1)
comes from Theorem 8.2, and the map q comes from Definition 8.7. It was proved *
*in
Theorem 6.3 that the top horizontal map is a split monomorphism of OS -modules,
and it follows that the same is true of the map q :OIntr-!E0Ir.
We now specialise to the case where E is H[u, u-1], the two-periodicQversion
of the integer Eilenberg-MacLane spectrum. We then have E0X = kH2kX for
28 N. P. STRICKLAND
all spaces X. This splits each of the rings on the bottom row of our diagram
as a product of homogeneous pieces, and it is not hard to check that there is a
unique compatible way to split the rings on the top row. We know that q is a sp*
*lit
monomorphism; if we can show that the source and target have the same Poincar'e
series, it will follow that q is an isomorphism. If r = min(d0, d1) then Intr= *
*Subr
so the claim is certainly true. To work downwards from here by induction, it wi*
*ll
suffice to show that
P S(H*Ir+1) - P S(H*Ir) = P S(OIntr+1) - P S(OIntr)
for all r.
To evaluate the left hand side, we consider the space
Ir \ Ir+1 = {(V0, V1) 2 Gd0(C1 ) x Gd1(C1 ) | dim(V0 \ V1) = k}.
Let G0rbe the space of triples (V, V00, V10) of mutually orthogonal subspaces o*
*f C1
such that dim(V ) = r and dim(Vi) = di- r. This is well-known to be a model
of BU(r) x BU(d0 - r) x BU(d1 - r) and thus homotopy-equivalent to Gr; the
argument uses frames much as in the proof of Theorem 8.2. Let W be the bundle
over G0rwhose fibre over (V, V00, V10) is Hom (V10, V00). If ff 2 Hom (V00, V1*
*0) and we
put V0 = V V10and V1 = V graph(ff) then V0 \ V1 = V and so (V0, V1) 2 Ir.
It is not hard to see that this construction gives a homeomorphism of the total
space of W with Ir \ Ir+1. This in turn gives a homeomorphism of the Thom
space (G0r)W with the quotient space Ir=Ir+1. By induction we may assume that
H*Ir+1 is concentrated in even degrees, and it is clear from the Thom isomorphi*
*sm
theorem that the same is true of He*(G0r)W . This implies that H*Ir is in even
degrees and that P S(H*Ir) - P S(H*Ir+1) = P S(He*(G0r)W ). As W has dimension
(d0 - r)(d1 - r), we see that P S(He*(G0r)W ) = t2(d0-r)(d1-r)P S(H*G0r). We al*
*so
know that H*G0r' OSubr. The conclusion is that
P S(H*Ir) - P S(H*Ir+1) = t2(d0-r)(d1-r)P S(OSubr).
We next evaluate P S(OIntr+1) - P S(OIntr). Put
R*r= Z[[c01, . .,.c0,d0-r, c11, . .,.c1,d1]].
We knowQfrom Theorem 6.3 that OIntrisPfreely generated over R*rby the mono-
mialsQ ri=1cffi0,d0-r+ifor whichP ri=1ffi d1 - r. It follows that the mono*
*mi-
als ri=0cffi0,d0-r+ifor whichP r1ffi d1 - r form a basis for OIntrover R*r*
*+1.
Similarly, those for which ri=0ffi < d1 - r form a basis for OIntr+1over R*r+*
*1.
Thus,Pif we let M* be the module generated over R*r+1by the monomials with
r P r *
1ffi d1 - r 0ffi, we find that P S(OIntr+1) - P S(OIntr)P=rP S(M ).
It is not hard to check that the monomials for which 0ffi= d1-rQform a basis
for M*Pover R*r. Next, let N* be generated over Z by the monomials ri=0cffiif*
*or
which r0ffi= d1-r; note that this involves the variables 1 = c0, . .,.cr rath*
*er than
the variables cd0-r, . .,.cd0used in M*. Because deg(cd0-r+i) = deg(ci) + 2(d0-*
* r)
we have Y Y X
deg( cffid0-r+i) = deg( cffii) + 2(d0 - r) ffi.
i i i
Using this, we see that P S(M*) = t2(d0-r)(d1-r)P S(N*)P S(R*r). However, Corol-
lary 6.11 essentially says that OSubr ' R*r N* as graded Abelian groups, so
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 29
P S(N*)P S(R*r) = P S(OSubr), so
P S(OIntr+1) - P S(OIntr) = t2(d0-r)(d1-r)P S(OSubr) = P S(H*Ir) - P S(H*Ir+1).
As explained previously, this implies that q is an isomorphism in the case E =
H[u, u-1]. We next consider the case E = MU[u, u-1]. Let I be the kernel of the
usual map MU* -!Z. Because H*Ir is free of finite type and concentrated in even
degrees, we see that the Atiyah-Hirzebruch spectral sequence collapses and that
the associated graded ring grIMU*Ir is isomorphic to grI(MU*)b H*Ir. Using
this it is not hard to check that q is an isomorphism in the case E = MU[u, u-1]
also. Finally, given an arbitrary even periodic ring spectrum E we can choose a
complex orientation in eE0CP 1and thus a ring map MU[u, u-1] -!E. Using this,
we deduce that q is an isomorphism for all E.
Corollary 8.10. Let V0 and V1 be bundles of dimensions d0 and d1 over a space
X. Then there is a spectral sequence
Tor**E*BU(d0)xBU(d1)(E*X, E*Ir(d0, d1)) =) E*Ir(V0, V1),
whose edge map in degree zero is the map
q*: OIntr(D(V0),D(V1))-!E0Ir(V0, V1).
The spectral sequence collapses in the semiuniversal and universal cases. (We do
not address the question of convergence in the general case.)
Proof.This is another Eilenberg-Moore spectral sequence.
9. The schemes PkD
P Let D be a divisor of degree d on G over S, with equation f(t) = fD (t) =
d d-i
i=0cix 2 OS [t], say. In this section we assemble some useful facts about *
*the
scheme PkD. This is a closed subscheme of Gk, so OPkD = OS [[x0, . .,.xk-1]]=Jk
for some idealQJk; our main task will be to find systems of generators for Jk. *
*We
put pi(t) = j* . .>.ffk-1. Put xff:= xff0
. . .xffk-12 ODk. Then {xff| ff 2 A} is a basis for ODk, and {~k(xff) | ff 2 A0}
isPa basis for ~kOD . Moreover, if ff 2 A0 and we write _k~k(xff) = altk(xff) =
fi2Acfffixfiwe see that cffff= 1 and cfffi= 0 if fi 2 A0 and fi 6= ff. It fo*
*llows
that ~k is surjective and _k is a split injection of OS -modules, as indicated *
*in the
diagram.
Lemma 9.4 tells us that ~k factor as ~0kp* for some ~0k:OPkD -! ~kOD , and a
diagram chase shows that ~0kis surjective. This gives the right hand triangle *
*of
the diagram. We simply define _0k= p*_k to get the left hand triangle. As p* is
k-equivariant we have
altkp* = p*altk= p*_k~k = _0k~0kp*.
As p* is surjective, this proves that _0k~0k= altk, so the top triangle commute*
*s.
32 N. P. STRICKLAND
We next study certain orbit schemes for actions of k. Recall that OGk =
OS[[xi | i < k]] has a topological basis consisting of monomials in the variabl*
*es
xi. This basis is permuted by k, and the sums of the orbits form a topological
basis for the invariant subring OGkk= OGk= k = ODiv+k. It is clear from this
analysis that our quotient construction commutes with base change, in other wor*
*ds
(S0xS Gk)= k = S0xS (Gk= k) for any scheme S0 over S. Similarly, the set
{xff| ffi< d for alli} is a basis for ODk that is permuted by k, so the orbit *
*sums
give a basis for ODkkand we have a quotient scheme Dk= k = spf(ODkk) whose
formation commutes with base change. By comparing our bases we see that the
projection OG -! OD = OG =fD induces a surjective map OGk= k -!ODk= k. In
other words, we have a commutative square of schemes as shown, in which j and
j0 are closed inclusions, and q2 is a faithfully flat map of degree d!.
j
Dk v____________Gkw
| |
| |
q1| |q2
| |
|u |uu
Dk= k v_____Gk=wkj=0Div+k
One might hope to show that Pk(D)= k = Subk(D) in a similar sense, but this
is not quite correct. For example if D = 3[0] (so fD (t) = t3) and k = 2 then
OP2D = OS[[x, y]]=(x3, x2 + xy + y2). If we define a basis of this ring by
{e0, . .,.e5} = {1, x, y, x2, -x2 - xy, x2y},
we find that the generator of 2 has the effect
e0 $ e0 , e1 $ e2 , e3 $ e4 , e5 $ -e5.
If OS has no 2-torsion we find that OP22Dis spanned by {1, x + y, xy} and thus *
*is
equal to OSub2(D). However, if 2 = 0 in OS we have an additional generator x2y,*
* so
OP22Dis strictly larger than OSub2(D). This example also shows that the formati*
*on
of OP22Dis not compatible with base change.
The following proposition provides a substitute for the hope described above.
Proposition 9.6. There is a commutative diagram as follows, in which i, i0, j a*
*nd
j0 are closed inclusions, and q0 and q2 are faithfully flat of degree k!. Moreo*
*ver, the
outer rectangle is a pullback, and if Jk := ker(i*) then ker((i0)*) = Jkk.
j
PkD v___________Dkwiv___________Gk_w
| | |
| | |
q0| q1| |q2
| | |u
|uu |u |u
Subk(D) v________Dk=wki0v____Gk=_kw=jDiv+k0
Proof.We have already produced the right hand square. The map i is just the
obvious inclusion. The map q0 sends (a0, . .,.ak-1) 2 PkD to [a0] + . .+.[ak-1]*
* 2
Subk(D); it was observed in the proof of Lemma 6.9 that this makes OPkD into a
free module of rank k! over OSubk(D), so q0 is faithfully flat of degree k!.
The points of Subk(D) are the divisors of degree k contained in D, so Subk(D*
*) is
a closed subscheme of Div+k; we write m0: Subk(D) -!Div+kfor the inclusion, and
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 33
note that m0q0 = q2ji. As q0 is faithfully flat and m0q0 factors through Dk= k *
*we
see that m0factors through Dk= k, so there is a unique map i0: Subk(D) -!Dk= k
such that m0= j0i0. As m0is a closed inclusion, the same is true of i0. A point*
* of
the pullback of mPand q2 is a list a = (a0, . .,.ak-1) of pointsPof G such that*
* the
divisor q2(a) = r[ar] lies in Subk(D), and thus satisfies r[ar] D. It fol*
*lows
from the definitions that this pullback is just PkD as claimed.
As q0 is faithfully flat we have ker((i0)*) = ker(q*0(i0)*) = ker(i*q*1). By *
*construc-
tion, q*1is just the inclusion of the k-invariants in ODk, so ker(i*q*1) = ker*
*(i*) k =
Jkk as claimed.
Corollary 9.7. ~kOD is naturally a module over OSubk(D).
Proof.We can certainly regard ODk as a module over the subring ODk= k = ODkk,
and the map altk:ODk -!ODk respects this structure. This makes ~kOD =
image(altk) into a module over ODkk. If a 2 Jkk and b 2 ODk then a altk(b) =
altk(ab) but ab 2 Jk so altk(ab) = 0. This shows that ~kOD is annihilated by Jk*
*k,
so it is a module over ODkk=Jkk = OSubk(D), as claimed.
We next identify ~kOD as a module over OSubk(D). Let D0 be the tautological
divisor of degree k over Subk(D). Then OD0 is naturally a quotient of the ring
OD S OSubk(D), which contains the subring OD = OD 1. This gives us a map
OD -!OD0, which extends to give a map OE: ~kSOD -!~kSubk(D)OD0.
Proposition 9.8. The map OE: ~kSOD -! ~kSubk(D)OD0 is an isomorphism of free
rank-one modules over OSubk(D).
Proof.Put T = Subk(D) for brevity. Note that
OD = OS{xi| i < d}
~kSOD= OS{xi0^ . .^.xik-1| d > i0 > . .>.ik-1}
OD0 = OT {xi| i < k}
~kTOD0= OT {xk-1 ^ . .^.x0}.
` " particular, ~ksod free rank over , ~ktod0 one ot know 6.9 ra* *nk suppose moment oe homomorphism -modules. oe(xk-1 ^ .^.x0)="xk-1" .^.x0, element generates ~ktod0, surjective. as source targ* *et are same finite an isomorphism claimed. still need prove linear theqargument *9.4 reduce case where d divisor equation d-1i="0(t" yi) def* *ined overqthe ring r [[y0, .,.yd-1]], can invert discrimina* *nt w="i6=j(yi-yj)." reuse notation proof lemma, w-1od="F" (n, w-1r) w-1odk="F" (nk, w-1opkd="F" w-1r). 9.6 w-1ot image w-1odkkin w-1opkd, 34 symmetric functions nk w-1r. write n+k="{n_2" n0> . .>.nk-1} then Nk = k x N+kas k-sets so w-1OPkD =
F (N+k, w-1R). On the other hand, OT is also a quotient of Rb ODiv+k, which is *
*the
ring of symmetric power series in k variables over R; a symmetric power series p
corresponds to the function n_7! p(yn_) := p(yn0, . .,.ynk-1).
If n_2 Nk we put en_= en0 . . .enk-1, so these elements form a basis for
w-1ODk over w-1R. Similarly, the set {~k(en_) | n_2 N+k} is a basis for w-1~kOD*
* .
Using the previous paragraph we see that p.~k(en_) = p(yn_)~k(en_), which tells*
* us
the OT -module structure on w-1~kOD .
We next analyse w-1OD0. This is a quotient of the ring
Y
w-1OT OD = F (N+k, w-1OD ) = w-1R{ei| i < d}.
n_2N+k
It is not hard to check that the relevant ideal is a product of terms In_, wher*
*e In_is
spanned by the elements ei that do not lie in the list en0, . .,.enk-1. Thus
Y
w-1OD0 = w-1R{enj | j < k}
n_
Y
w-1~kTOD0 = w-1R.en0^ . .^.enk-1.
n_
Let e0n_be the element of this module whose n_'th component is en0^. .^.enk-1, *
*and
whose other components are zero. Clearly {e0n_| n_2 N+k} is a basis for w-1~kTO*
*D0
over w-1R. As a symmetric power series p corresponds to the function n_7! p(yn_)
and e0n_is concentrated in the n_'th factor we have p.e0n_= p(yn_)e0n_. It is a*
*lso easy to
see that OE(~k(en_)) = e0n_, and it follows that OE is OT -linear as claimed.
We next give a formula for OE in terms of suitable bases of ~kSOD and ~kSubrO*
*D0.
(This could be used to give an alternative proof that OE is an isomorphism.)
Proposition 9.9. Suppose we have an element xfi0^ . .^.xfik-12 ~kSOD, where
0 fi0 < . . .< fid-k-1. Let fl0, . .,.flk-1 be the elements of {0, . .,.d - *
*1} \
{fi0, . .,.fid-k-1}, listed in increasing order. Then
OE(xfi0^ . .^.xfik-1) = x0 ^ . .^.xk-1. det(ck+i-flj)0 i,j 0. Put Y = {x 2 X | |x| kffk + 1}, which is compact. *
*As
A is dense we can choose p 2 A with |f - p| < ffl=4 on Y . As p 2 A can choose *
*ffi
such that kp(fi) - p(ff)k < ffl=4 whenever kfi - ffk < ffi. We may also assume *
*that
ffi < 1, which means that when kfi - ffk < ffi we have fi 2 Y . Now if |f - g| *
*< ffl=4
on Y and kff - fik < ffi then
kf(ff) - g(fi)k kf(ff) - p(ff)k + kp(ff) - p(fi)k + kp(fi) - f(fi)k + kf(fi) *
*- g(fi)k
< ffl=4 + ffl=4 + ffl=4 + ffl=4 = ffl,
as required.
The following proposition is an elementary exercise in linear algebra.
Proposition A.2. Let ff: V -! W be a linear map. Then ff*ff and ffff* are
self-adjoint endomorphisms of V and W with nonnegative eigenvalues. For each
t > 0 the map ff gives an isomorphism of ker(ff*ff - t) with ker(ffff* - t), so*
* the
nonzero eigenvalues of ff*ff and their multiplicities are the same as those of *
*ffff*.
If f :[0, 1) -!R then ff O f(ff*ff) = f(ffff*) O ff.
Definition A.3. We write w(V ) = {ff 2 End (V ) | ff* = ff} (the space of self-
adjoint endomorphisms of V ). If ff 2 w(V ) then the eigenvalues of ff are real*
*, so
we can list them in descending order, repeated according to multiplicity. We wr*
*ite
ek(ff)Qfor the k'th element in this list, so e1(ff) . . .en(ff) and det(t - f*
*f) =
k(t - ek(ff)).
We will need the following standard result:
COMMON SUBBUNDLES AND INTERSECTIONS OF DIVISORS 43
Proposition A.4. The functions ek: w(V ) -!R are continuous.
Proof.Let fl be a simple closed curve in C and let m be an integer. Let U be the
set of endomorphisms of V that have precisely m eigenvalues (counted according *
*to
multiplicity) inside fl, and no eigenvalues on fl. A standard argument with Rou*
*ch'e's
theorem shows that U is open in End(V ).
Given real numbers r R, consider the rectangular contour flr,Rwith corners
at r i and R i. Clearly ek(ff) > r iff ff has at least k eigenvalues inside*
* flr,Rfor
some R. It follows that {ff | ek(ff) > r} is open, as is {ff | ek(ff) < r} by a*
* similar
argument. This implies that ek is continuous.
References
[1]P. Bressler and S. Evens. Schubert calculus in complex cobordism. Trans. Am*
*er. Math. Soc.,
331(2):799-813, 1992.
[2]M. Crabb and I. James. Fibrewise homotopy theory. Springer-Verlag London Lt*
*d., London,
1998.
[3]M. C. Crabb. On the stable splitting of U(n) and U(n). In Algebraic topolo*
*gy, Barcelona,
1986, pages 35-53. Springer, Berlin, 1987.
[4]A. Grothendieck. Sur quelques propri'et'es fondamentales en th'eorie des in*
*tersections.
S'eminaire C. Chevalley, Ecole Normale Sup'erieure, 2, 1958.
[5]N. Kitchloo. Cohomology splittings of Stiefel manifolds. Preprint, 1999.
[6]H. Miller. Stable splittings of Stiefel manifolds. Topology, 24(4):411-419,*
* 1985.
[7]S. A. Mitchell. A filtration of the loops on su(n) by Schubert varieties. M*
*ath. Z., 193(3):347-
362, 1986.
[8]D. G. Northcott. Finite free resolutions. Cambridge University Press, Cambr*
*idge, 1976. Cam-
bridge Tracts in Mathematics, No. 71.
[9]P. Pragacz. Enumerative geometry of degeneracy loci. Ann. Sci. 'Ecole Norm.*
* Sup. (4),
21(3):413-454, 1988.
[10]N. P. Strickland. Formal schemes and formal groups. In J. Meyer, J. Morava,*
* and W. Wilson,
editors, Homotopy-invariant algebraic structures: in honor of J.M. Boardman*
*, volume 239
of Contemporary Mathematics. American Mathematical Society, 1999.
*