Descent for algebraic cobordism

0.17 MB·English

by  José Luis González

DESCENT FOR ALGEBRAIC COBORDISM
JOSE´ LUISGONZA´LEZANDKALLEKARU
3
ABSTRACT. Weprovetheexactnessofadescentsequencerelatingthealgebraiccobordism
1
0 groupsofaschemeanditsenvelopes.AnalogoussequencesforChowgroupsandK-theory
2 werepreviouslyprovedbyGillet.

n
a
J

5
1 1. INTRODUCTION

]
G We work in the category Schk of separated finite type schemes over a field k of charac-
A teristic zero. Recall that a proper morphism of schemes π : X˜ X is an envelope if for
. every subvariety V ⊂ X, there exists a subvariety V˜ ⊂ X˜, such t→hat π maps V˜ birationally
h
t onto V.
a
m
Gilletin[4]provedthatifπ : X˜ Xisanenvelope,withπprojective,thenthefollowing
[
sequenceis exact: →

1
v (1.1) A (X˜ × X˜) −p−1−∗−−p−2∗ A (X˜) −π∗ A (X) − 0.
2 ∗ X ∗ ∗
9 → → →
Here A is either the Chow theory or the K-theory of coherent sheaves, p : X˜ × X˜ X˜
2 ∗ i X
3 are the two projections and p ,π are the push-forward maps in either theory. The→goal
. i∗ ∗
1
ofthisarticleistoprove theexactnessoftheanalogoussequenceinthecobordism theory
0
3 Ω definedbyLevine andMorel [10].
∗
1
v: Theorem1.1. Letπ : X˜ X bean envelope,withπ projective. Thenthe sequence
i →
X Ω (X˜ × X˜) −p−1−∗−−p−2∗ Ω (X˜) −π∗ Ω (X) − 0
∗ X ∗ ∗
r
a → → →
isexact.
This theorem has several applications. We mention here two of them. The first appli-
cation is in the relationship between algebraic cobordism Ω (X) and algebraic K-theory
∗
G (X) (the Grothendieck group of the category of coherent sheaves on X). Levine and
0
Morel in [10] constructed anatural morphism
Ω (X)⊗ Z[β,β−1] − G (X)[β,β−1],
∗ L 0
→
andprovedittobeanisomorphismforsmoothX. Daiin[2]extendedthisisomorphismto
allschemesXthatcanbeembeddedinasmooth scheme;thisincludesallquasiprojective
schemes X. In [6] we build on Dai’s work to prove this isomorphism for all schemes X in
ThisresearchwasfundedbyNSERCDiscoveryandAcceleratorgrants.
1
2 JOSE´ LUISGONZA´LEZANDKALLEKARU
Sch . Since every scheme X admits a quasiprojective envelope, Theorem 1.1 implies that
k
the cobordism theory Ω as a functor on Sch is determined by its restriction to the full
∗ k
subcategory of quasiprojective schemes. A similar statement for G proved by Gillet [4]
0
then reducesthe problem to the case proved by Dai.
The second application of Theorem 1.1 is in the study of operational bivariant theories
of Fulton and MacPherson [3]. Kimura in [8] used the exact sequence (1.1) to give an
inductive method for findingthe operational Chowgroups ofsingular varieties from the
Chow groups of smooth varieties. His proof can be generalized to algebraic cobordism
andotherhomologytheories. In[5]westudytheoperationalbivarianttheoriesassociated
to certain homology theories that have the exact descent sequence. As a special case, we
describe the operational equivariant cobordism theory of toric varieties. This result is
based on previous work by Payne[12]and Krishna andUma[9].
2. AN OVERVIEW OF ALGEBRAIC COBORDISM THEORY
Algebraic cobordism theory was defined by Levine and Morel in [10]. Later, Levine
and Pandharipande [11] found a simpler presentation of the cobordism groups. We will
usetheconstruction ofLevine-Pandharipandeasthedefinition,butrefertoLevine-Morel
for itsproperties.
LetSm bethefullsubcategoryofSch whoseobjectsaresmoothquasiprojectiveschemes
k k
over Speck. By a smooth morphism we always mean a smooth and quasiprojective mor-
phism.
For X in Sch , let M(X) be the set of isomorphism classes of projective morphisms
k
f : Y X for Y ∈ Sm . This set is a monoid under disjoint union of the domains; let
k
M+(X→) be its group completion. The elements of M+(X) are called cycles. The class of
f : Y XinM+(X)isdenotedby[f : Y X]. ThegroupM+(X)isfreeabelian,generated
byth→e cycles [f : Y X] where Y isirred→ucible.
→
A double point degeneration is a morphism π : Y P1, with Y ∈ Sm of pure dimen-
k
sion, such that Y∞ = π−1( ) isa smooth divisor on Y→and Y0 = π−1(0)is a union A∪Bof
smooth divisors intersecti∞ng transversely along D = A∩B. Define P = P(O (A)⊕O ),
D D D
whereO (A)standsforO (A)| . (Notice thatP(O (A)⊕O ) =∼ P(O (B)⊕O )because
D Y D D D D D
∼
O (A+B) = O .)
D D
LetX ∈ Sch and letY ∈ Sm have pure dimension. Letp ,p be the two projections of
k k 1 2
X×P1. A double pointrelation isdefined bya projective morphism π : Y X×P1,such
that p ◦π : Y P1 isa double point degeneration. Let →
2
→
[Y X], [A X], [B X], [P X]
∞ D
→ → → →
be the cycles obtained bycomposing with p . The double pointrelation is
1
[Y X]−[A X]−[B X]+[P X] ∈ M+(X).
∞ D
→ → → →
DESCENTFORALGEBRAICCOBORDISM 3
Let R(X) be the subgroup of M+(X) generated by all the double point relations. The
cobordism group ofX isdefinedto be
Ω (X) = M+(X)/R(X).
∗
The group M+(X) is graded so that [f : Y X] lies in degree dimY when Y has pure
dimension. Since double point relations are→homogeneous, this grading gives a grading
on Ω (X). Wewrite Ω (X)for the degree npart ofΩ (X).
∗ n ∗
There is a functorial push-forward homomorphism f : Ω (X) Ω (Z) for f : X Z
∗ ∗ ∗
projective,andafunctorialpull-backhomomorphismg∗ : Ω (Z) →Ω (X)forg : X→ Z
∗ ∗+d
asmooth morphism ofrelativedimension d. Thesehomomorphi→sms are both defined→on
the cycle level. Levine and Morel also construct pull-backs along l.c.i. morphisms and,
more generally, refined l.c.i. pullbacks. Wewill not needthese pullbacks below.
Thecobordism theory hasexterior products
Ω (X)×Ω (W) − Ω (X×W),
∗ ∗ ∗
→
definedon the cycle level:
[Y X]×[Z W] = [Y ×Z X×W].
→ → →
TheseexteriorproductsturnΩ (Speck)intoagradedringandΩ (X)intoagradedmod-
∗ ∗
ule overΩ (Speck). When Xisin Sm , we denote by1 the class [id : X X].
∗ k X X
→
2.1. FirstChernClassOperators. AlgebraiccobordismisendowedwithfirstChernclass
operators
c˜ (L) : Ω (X) Ω (X),
1 ∗ ∗−1
→
associated to any line bundle L on X. This operator is also denoted by c˜ (L), where L is
1
the invertible sheaf of sections of L. We recall some properties of these operators that are
neededbelow.
A formal group law on a commutative ring R is a power series F (u,v) ∈ RJu,vK satis-
R
fying
(a) F (u,0) = F (0,u) = u,
R R
(b) F (u,v) = F (v,u),
R R
(c) F (F (u,v),w) = F (u,F (v,w)).
R R R R
Thus
FR(u,v) = u+v+Xai,juivj,
i,j>0
where a ∈ R satisfy a = a and some additional relations coming from property (c).
i,j i,j j,i
Wethink ofF asgivinga formal addition
R
u+ v = F (u,v).
FR R
4 JOSE´ LUISGONZA´LEZANDKALLEKARU
There exists a unique power series χ(u) ∈ RJuK such that F (u,χ(u)) = 0. Denote
R
[−1] u = χ(u). ComposingF andχ, we can form linearcombinations
FR R
[n ] u + [n ] u + ···+ [n ] u ∈ RJu ,...,u K
1 FR 1 FR 2 FR 2 FR FR r FR r 1 r
for n ∈ Zand u variables.
i i
There exists a universal formal group law FL, and its coefficient ring L is called the
Lazard ring. Thisring can beconstructed asthe quotientof the polynomial ring Z[A ]
i,j i,j>0
by the relations imposed by the three axioms above. The images of the variables A in
i,j
the quotient ring are the coefficients ai,j of the formal group lawFL. The ring Lis graded,
withAi,j havingdegreei+j−1. ThepowerseriesFL(u,v)isthenhomogeneousofdegree
−1 ifu and vboth havedegree −1.
It is shown in [10] that the graded group Ω (Speck) is isomorphic to L. The formal
∗
L
grouplawon describesthefirstChernclassoperatorsoftensorproductsoflinebundles
(property (FGL) below).
We list three properties satisfied by the first Chern class operators c˜ (L) : Ω (Y)
1 ∗
Ω (Y)for Y ∈ Sm and La linebundle on Y: →
∗−1 k
(Dim) For L ,...,L line bundleson Y,r > dimY,
1 r
c˜ (L )◦···◦c˜ (L )(1 ) = 0.
1 1 1 r Y
(Sect) IfLisalinebundleonY ands ∈ H0(Y,L)isasection such thatthezerosubscheme
i : Z ֒ Y of sissmooth, then
→
c˜ (L)(1 ) = i (1 ).
1 Y ∗ Z
(FGL) For two line bundlesL and Mon Y,
c˜1(L⊗M)(1Y) = FL(c˜1(L),c˜1(M))(1Y).
In the terminology of [10, 11] the three properties imply that Ω is an oriented Borel-
∗
Moore functor ofgeometric type.
The first Chern class operators of two line bundles commute: c˜ (L)◦ c˜ (M) = c˜ (M)◦
1 1 1
c˜ (L), and they are compatible with smooth (l.c.i.) pull-backs, projective push-forwards
1
and exterior products. The property (Sect) above implies that if L is a trivial line bundle
on X, then the first Chernclass operator of Liszero.
2.2. Divisor Classes. Recall that a divisor D on a smooth scheme Y ∈ Sch has strict
k
normalcrossings(s.n.c.) ifateverypointp ∈ Y thereexistsasystemofregularparameters
y ,...,y ,suchthatDisdefinedbytheequationym1···ymn = 0nearpforsomeintegers
1 n 1 n
m ,...,m .
1 n
LetD = r n D beanonzeros.n.c. divisoronaschemeY ∈ Sm ,withD irreducible.
Pi=1 i i k i
Letusrecall the construction by Levineand Morel [10]ofthe class [D |D|] ∈ Ω (|D|).
∗
→
DESCENTFORALGEBRAICCOBORDISM 5
Let
Fn1,...,nr(u ,...,u ) = [n ] u + [n ] u + ···+ [n ] u ∈ LJu ,...,u K.
1 r 1 FL 1 FL 2 FL 2 FL FL r FL r 1 r
Wedecompose this powerseries as
Fn1,...,nr(u1,...,ur) = XFJn1,...,nr(u1,...,ur)Yui,
J i∈J
where the sum runs over nonempty subsets J ⊂ {1,...,r}. The power series Fn1,...,nr are
J
such that u doesnot divide anynonzeroterm in Fn1,...,nr ifi ∈/ J.
i J
For i = 1,...,r, let L = O (D ). If J ⊂ {1,...,r}, let iJ : DJ = ∩ D ֒ |D|, and
i Y i i∈J i
LJ = L | . The class[D |D|] isdefinedin [10]as →
i i DJ
→
(2.1) [D |D|] = XiJ∗FnJ1,...,nr(LJ1,...,LJr)(1DJ),
→
J
where the sum runs over nonempty subsets J ⊂ {1,...,r} and Fn1,...,nr(LJ,...,LJ) is the
J 1 r
powerseries Fn1,...,nr evaluatedon the first Chernclasses ofLJ,...,LJ.
J 1 r
When pushed forward to Y, the class [D |D|] becomes equal to c˜ (O(D))(1 ). To see
1 Y
this, compute, →
c˜ (O(D))(1 ) = Fn1,...,nr(L ,...,L )(1 )
1 Y 1 r Y
= XFJn1,...,nr(L1,...,Lr)Yc˜1(Li)(1Y).
J i∈J
Applyingthe property (Sect) repeatedly, weget
Yc˜1(Li)(1Y) = [DJ ֒ Y].
i∈J →
CompatibilityoffirstChernclassoperatorswithpull-backsoflinebundlesthengivesthe
desired divisor classformula pushed forward to Y.
We note that in the definition of divisor classes it is not necessary to assume that D
i
are irreducible. We may let them be smooth but possibly reducible divisors and then the
same formula holds.
2.3. APairofSmoothDivisors. LetAandBbesmoothdivisorsonY ∈ Sm thatintersect
k
transversely along D = A∩B. Consider
[A+B |A∪B|] = [A A∪B]+[B A∪B]+i F1,1 (O (A),O (B))(1 ),
∗ {1,2} D D D
→ → →
where i : D ֒ |A∪B| andO (A),O (B)stand forO (A)| ,O (B)| . Let
D D Y D Y D
→
P = P(O (A)⊕O ).
D D D
6 JOSE´ LUISGONZA´LEZANDKALLEKARU
Lemma2.1. With notation asabove, letO (A+B+C)| = O for somedivisorConY. Then
Y D D
F1,1 (O (A),O (B))(1 ) = −[P D]+β,
{1,2} D D D D
→
where
β = X bijlc˜1(OD(A))ic˜1(OD(B))jc˜1(OD(C))l(1D),
i,j≥0,l>0
for someuniversallydefinedb ∈ L thatdonot dependon Y,A,B,C.
ijl
Proof. Itis shown in the proof of [11,Lemma 9]that
−[P D] = F1,1 (O (A),O (−A))(1 ).
D {1,2} D D D
→
Substituting O (B+C) = O (−A),we get
D D
−[P D] = F1,1 (O (A),O (B+C))(1 )
D {1,2} D D D
→
= Xaijc˜1(OD(A))i−1c˜1(OD(B+C))j−1(1D),
i,j>0
witha thecoefficientsintheformalgrouplawofL. Tocomputec˜ (O (B+C)),weapply
ij 1 D
the formal group lawagain:
c˜1(OD(B+C)) = c˜1(OD(B))+c˜1(OD(C))+Xaijc˜1(OD(B))ic˜1(OD(C))j.
i,j>0
Now substituting this into the expression of −[P D], the terms that involve only
D
c˜ (O (A)) and c˜ (O (B)) give F1,1 (O (A),O (B))(→1 ), the remaining terms give the
1 D 1 D {1,2} D D D
class−β. (cid:3)
3. PROOF OF THE MAIN THEOREM
Wewill nowprove Theorem 1.1. Letus start with some notation. Recallthat
Ω (X) = M+(X)/R(X),
∗
where M+(X) is the group of cycles and R(X) is the subgroup of relations, generated by
double point relations. We identify Ω (Speck) with the Lazard ring L. Then Ω (X) is an
∗ ∗
L
-module.
Composition with π gives the push-forward map π : M+(X˜) M+(X), taking R(X˜)
∗
to R(X). Theinduced map Ω (X˜) Ω (X)is alsodenoted π . De→fine
∗ ∗ ∗
→
Ω (X) = π M+(X˜)/π R(X˜).
∗ π ∗ ∗
IfK = Ker(π : M+(X˜) M+(X)), then
π ∗
→
(3.1) Ω (X) =∼ M+(X˜)/(R(X˜)+K ).
∗ π π
There are natural mapsofL-modules that factor π as
∗
Ω (X˜) −φ Ω (X) −ψ Ω (X).
∗ ∗ π ∗
→ →
DESCENTFORALGEBRAICCOBORDISM 7
The map φ is surjective by (3.1). We claim that ψ is also surjective. For every subvariety
Y ⊂ X, choose a resolution of singularities given by a projective birational morphism
Y˜ Y. Then the cycles [Y˜ X] generate Ω (X) as an L-module [10]. We can choose the
∗
res→olutions so thatY˜ Xfa→ctor through π : X˜ X, hence these classeslie in π M+(X˜).
∗
→ →
Thefollowing isthe mainingredient for proving Theorem 1.1.
Proposition 3.1. The mapψ : Ω (X) Ω (X)is anisomorphism.
∗ π ∗
→
Proof of Theorem 1.1. Let us assume Proposition 3.1 and prove Theorem 1.1. Clearly
the sequence is a complex by functoriality of push-forward. Moreover, π = ψ ◦ φ is
∗
surjective. We only need to prove exactness in the middle: Kerπ ⊂ Img(p − p ). By
∗ 1∗ 2∗
Proposition 3.1, Kerπ = Kerφ, which by (3.1) is the image of K in Ω (X˜). Now K is
∗ π ∗ π
generated bycycles
[f : Y X˜]−[g : Y X˜],
→ →
where π◦f = π◦g. These generators lift to cycles
[(f,g) : Y X˜ × X˜] ∈ Ω (X˜ × X˜). (cid:3)
X ∗ X
→
To prove Proposition 3.1, we follow the argument in [10, Chapter 6] showing that
Ω (X) Ω (X) is an isomorphism. Here Ω (X) is the group defined by cycles and
∗ D ∗ ∗ D
relations→transverse to a divisor D. The proof hastwo steps:
(1) Define a distinguished lifting M+(X) −d Ω (X) , such that the composition
∗ π
→
M+(X˜) −π∗ M+(X) −d Ω (X)
∗ π
→ →
isthe canonical homomorphism.
(2) Show that d maps R(X) to zero, hence it descends to d : Ω (X) Ω (X) , provid-
∗ ∗ π
inga leftinverse toψand proving that ψisinjective. →
3.1. Elimination of Indeterminacies. We will need to eliminate the indeterminacies of
rational mapsY 99K X˜. Weuse the following well-known theorem.
Theorem 3.2. (Hironaka [7]) Let Y be a smooth variety, D ⊂ Y a divisor with strict normal
crossings. Let f : Y˜ Y be a projective birational morphism, U ⊂ Y a nonempty open set such
thatf : f−1(U) U→isan isomorphism. Thenthere existsa sequenceof morphisms
→
Y = Y g1− Y g2− Y − ... gm−−1 Y ,
1 2 3 m
← ← ← ←
suchthat
(1) For each i the map g is the blowup of Y along a smooth center C , where C lies over
i i i i
YrU and intersects the union of the pull-back of D and the exceptional locus of Y Y
i
normally. →
(2) Therational mapY Y 99K Y˜ extendsto amorphismY Y˜.
m m
→ →
8 JOSE´ LUISGONZA´LEZANDKALLEKARU
RecallthatifYissmoothandDisans.n.c. divisoronY,thenasmoothsubschemeC ⊂ Y
is said to intersect D normally if at every point p ∈ Y we can choose a regular system of
parameters y ,...,y so that D is defined by yn1 ···ynr = 0 for some n ,...,n ∈ Z and
1 r 1 r 1 r
C is defined by vanishing of y ,...,y for some i ,...,i . If D is an s.n.c. divisor and
i1 ij 1 j
C intersects it normally, then the blowup of Y along C is smooth and the pull-back of D
together with the exceptional divisor is again ans.n.c. divisor.
The proof of the above theorem is reduced to the problem of principalizing an ideal
sheaf as follows. One may assume that f : Y˜ Y is the blowup of a coherent sheaf of
idealsI on Y with co-support in YrU. Ifthe seq→uence of blowups g : Y Y is such that
m
g∗(I) isprincipal, then the birational mapY Y˜ isa morphism. →
m
→
Wewill applyTheorem 3.2in the following situation.
Corollary 3.3. Let Y be a smooth variety and D an s.n.c. divisor on Y. Let φ : Y X be a
proper morphism. Since π : X˜ X is an envelope, there exists a subvariety Z ⊂ X˜→mapping
birationally onto φ(Y) ⊂ X. Let→V ⊂ φ(Y) be anonempty open subsetsuch that π|−1(V) V is
Z
an isomorphism,and letU = φ−1(V) ⊂ Y. →
Then there exists a sequence of blowups g : Y Y of smooth centers that lie over YrU
m
and intersect the inverse image of D together with th→e exceptional locus normally, such that the
compositionY Y X factorsthrough X˜.
m
→ →
Proof. Let Y˜ be the component of Y × Z that dominates Y. Since Z φ(Y) is projective
X
and birational, the projection f : Y˜ Y isalso projective andbiration→al. Moreover, f isan
isomorphism over U. Wemaynow→applyTheorem 3.2. (cid:3)
Thefollowing result can beviewed asanembeddedelimination of indeterminacies.
Corollary 3.4. Let W be a smooth variety, D,E be effective divisors on W such that D + E
has s.n.c., and φ : W X be a proper morphism. Then there exists a birational morphism
g : W˜ W, obtained b→y a sequence of blowups of smooth centers that lie over |D| and intersect
thepull→-backofD+Etogetherwiththeexceptionallocusnormally,suchthatforeverycomponent
D˜ of the pull-backD˜ = g∗(D),the compositionD˜ W˜ W Xfactorsthrough X˜ X.
i i
→ → → →
Proof. Let D be a component of D. Let D′ be the divisor D′ = (D+E−qD )| , where q
i i Di
is the coefficient of D in D+E. We apply Corollary 3.3 to the map D X and the s.n.c.
i i
divisor D′ on D . The result is a sequence of blowups D˜ D so tha→t the composition
i i i
D˜ D X factors through X˜. The centers of the blow→ups all lie over D rU, where
i i i
U =→φ−1(V→) for some nonempty open V ⊂ φ(D ).
i
Let us now perform the same sequence of blowups on W (blow up W along the same
centers lying in D and in the strict transforms of D ), to get g : W˜ W. Then D˜ is iso-
i i i
morphic to the strict transform of D in W˜ . Such blowups introduce→new components to
i
DESCENTFORALGEBRAICCOBORDISM 9
thedivisorg∗(D). However,weclaimthatallthesenewexceptionalcomponentshaveim-
ageinXofsmallerdimensionthantheimageofD . Indeed,bythechoiceofU,thecenters
i
ofblowupslieovertheclosedsetφ(D )rV ⊂ X,andsodotheexceptionaldivisors. Thus,
i
by induction on the dimension of the image in X, we can resolve the indeterminacies of
all components of(the pull-backof) D. (cid:3)
3.2. Distinguished Liftings. Let [Y X] be an elementof M(X), with Y irreducible. We
construct alifting of thiscycle to Ω (→X) .
∗ π
Let W = Y ×P1, D = Y ×{0} ⊂ W, and f : W X be the composition of the projection
to Y and Y X. We apply Corollary 3.4 to this→situation (with E = 0) to find a blowup
g : W˜ W.→LetD˜ = g∗(D)be the pull-backof D,a possibly nonreduced s.n.c. divisor.
→
Consider the class [D˜ |D˜|] ∈ Ω (|D˜|). Note that any map Z |D˜| from an irre-
∗
ducible variety Z has im→age in a component D˜ of D˜. Since D˜ →X factors through
i i
X˜ X, the composition Z D˜ X also factors. Similarly, an irre→ducible double point
i
de→generation in |D˜|, when→pushe→d forward to X, factors through X˜. It follows that the
push-forward map Ω (|D˜|) Ω (X) factors through Ω (X) . We define a distinguished
∗ ∗ ∗ π
liftingof[Y X]tobethe im→ageof[D˜ |D˜|]inΩ (X) . Notethatadistinguished lifting
∗ π
dependson→the choice ofthe blowup W˜→ W,but not on the liftings D˜ X˜ of D˜ X.
i i
→ → →
3.3. ProductofDivisor Classes. LetDandEbeeffectivedivisorsonaschemeW ∈ Sm ,
k
such that D+Ehass.n.c. We definethe class
[D•E |D|∩|E|] ∈ Ω (|D|∩|E|)
∗
→
with the property that, when pushed forward to W,it becomesequalto
c˜ (O (D))◦c˜ (O (E))(1 ).
1 W 1 W W
Let D = n D and E = p D , where D , i = 1,...,r are irreducible divisors. For
Pi i i Pi i i i
i = 1,...,r, let L = O (D ). If J ⊂ {1,...,r} is such that n 6= 0 and p 6= 0 for some
i W i j i
i,j ∈ J, letiJ : DJ = ∩ D ֒ |D|∩|E|, and LJ = L | .
i∈J i i i DJ
→
Letthe class[D•E |D|∩|E|] bedefined bythe formula
→
Xi∗I∪JFJn1,...,nr(L1I∪J,...,LrI∪J)FIp1,...,pr(L1I∪J,...,LrI∪J) Y c˜1(LiI∪J)(1DI∪J).
I,J i∈I∩J
Here the sum runs over pairs of nonempty subsets I,J ⊂ {1,...,r}, such that n 6= 0 and
j
p 6= 0 for all j ∈ J and i ∈ I. As in the case of divisor classes, it is enough to assume that
i
the divisors D are smooth but not necessarilyirreducible.
i
We claim that [D • E |D| ∩ |E|], when pushed forward to |D|, becomes equal to
c˜ (O (E)| )[D |D|]. To→seethis, weapplyc˜ (O (E)| )(whichweshorten toc˜ (O(E)))
1 W |D| 1 W |D| 1
→
10 JOSE´ LUISGONZA´LEZANDKALLEKARU
to the definition of [D |D|]:
→
c˜1(O(E))[D |D|] = XiJ∗FnJ1,...,nr(LJ1,...,LJr)c˜1(O(E))(1DJ).
→
J
Wecan nowuse the same divisor class formula to compute c˜ (O(E))(1 ):
1 DJ
c˜ (O(E))(1 ) = Fp1,...,pr(LJ,...,LJ)(1 )
1 DJ 1 r DJ
= XFIp1,...,pr(LJ1,...,LJr)Yc˜1(LJi)(1DJ).
I i∈I
Applyingthe property (Sec), we get
Yc˜1(LJi)(1DJ) = Y c˜1(LJi)([DI∪J DJ]).
i∈I i∈I∩J →
Now putting the formulas back together and using the compatibility of the first Chern
classoperators with pull-backs proves the claim.
Note that the above definition is symmetric in EandD,
[D•E |D|∩|E|] = [E•D |D|∩|E|].
→ →
This implies that, when pushed forward to |D| ∪ |E|, the classes c˜ (O(E))[D |D|] and
1
c˜ (O(D))[E |E|] become equal. →
1
→
When D is a smooth divisor that does not have common components with E, then
E′ = E| isan s.n.c. divisor on |D|and one hasthat
|D|
[D•E |D|∩|E|] = [E′ |E′|].
→ →
To prove this, let D = D , E = p D , and let the sum in the definition run over
1 Pi>1 i i
nonempty subsets J ⊂ {1} and I ⊂ {2,...,r}. Since F1,0,...,0 = 1 for J = {1}, the sum
J
simplifiesto the expression definingthe divisor class [E′ |E′|].
→
3.4. Double Cobordisms. Let W be a scheme in Sm , f : W P1 × P1 a morphism,
k
D = f∗(P1 ×{0}),E = f∗({0}×P1). Assume that D+Eisan s.n.c.→divisor on W. Write
D =XaiDi +XαiFi,
i i
E =XbiEi +XβiFi,
i i
where F are the components of D+E lying over (0,0) ∈ P1 ×P1 and D ,E are the other
i i i
components of Dand E. Wemayassume that α ,β > 0.
i i
Lemma3.5. With notation asabove, letD′ = a D . Thenthe classes
Pi i i
[E•D |E|∩|D|] and [E•D′ |E|∩|D′|]
→ →
becomeequal whenpushedforwardto |E|.