The structure of the cohomology ring of the filt schemes

26 Pages·0.35 MB·English

by  Takuro Mochizuki

The structure of the cohomology ring of the filt schemes
Takuro Mochizuki
3
0
Abstract
0
2 Let C be a smooth projective curve over the complex number field C. We investigate the structure of
thecohomology ringofthequotschemesQuot(r,n),i.e.,themodulischemeofthequotientsheavesofO⊕r
n C
with length n. We obtain a filtration on H∗(Quot(r,n)), whose associated graded ring has a quite simple
a
J structure. As a corollary, we obtain a small generator of the ring. We also obtain a precise combinatorial
7 description of H∗(Quot(r,n)) itself.
1 Forthatpurpose,weconsiderthecompletefiltschemesandweusea‘splittingprinciple’. Acompletefilt
schemehasaneasygeometricdescription: asequenceofbundlesofprojectivespaces. Thusthecohomology

] ringofthecompletefiltschemesareeasily determined. Inthecohomology ringofcompletefiltschemes, we
G
can do some calculations for the cohomology ring of quot schemes.
A Keywords: Quot scheme, cohomology ring, torus action, splitting principle, symmetric function.
. MSC: 14F45, 57R17.
h
t
a
1 Introduction
m
[ LetC beasmooth projectivecurveoverthecomplexnumberfieldC. LetE bealocally freecoherentsheaf

1 over C. Let n be a non-negative integer. Let Quot(E,n) denote the moduli scheme of quotient sheaves of
v E withlength n. Itis called thequotscheme. Inparticular, weusethenotation Quot(r,n) ifE =O⊕r. In
4 thepapers[7]and[8],westudiedthestructureofthecohomology ringofquotschemesQuot(r,l) (l=2,3).
8 We also studied the infinite quot schemes ∞ Quot(r,l) = Quot(∞,l). In particular, we determined the
r=1
1 generators and the relations among themSfor the cohomology rings H∗(Quot(∞,l)), H∗(Quot(r,l)) (l =
1 2,3). However, when l ≥ 4, the rings seem somewhat complicated for a direct calculation. Changing the
0
route for an attack, we introduce the ‘splitting principle’ for the quot schemes, which is an analogue of the
3
well known splitting principle in the theory of Grassmannian manifolds.
0
We introduce the filt schemes: Let E be a vector bundle over C, and l = (l ,...,l ) be a tuple of
/ 1 h
h non-negative integers. The filt scheme Filt(E,l) is the moduli of thesequence of quotients:
t
a
E −→F −→F −→···−→F −→F .
m h h−1 2 1
: It satisfies thefollowing condition:
v
i • Weput Gi:=Ker(Fi−→Fi−1). Then thelength of Gi is li.
X
r If l1 = n and li = 0 for any i > 1, the filt scheme Filt(E,(n,0,...,0)) is naturally identified with the
n
a
quot scheme Quot(E,n). On the other hand, if li =1 for any i, the variety Filt(E,(1,...,1)) is called the
complete filt scheme. We denote it by Filt(E,n) for simplicity of the notation. Inzpa}r|ticu{lar, we denote
Filt(O⊕r,n) by Filt(r,n). Weput Filt(∞,n):= ∞ Filt(r,n).
r=1
Thestructure of Filt(E,n) is quitesimple. AsSis described in thesubsection 2.4, it is a projective space
bundleoverFilt(E,n−1)×C. LetLndenotethecorrespondingtautologicallinebundle. Wedenotethefirst
Chern class of Ln by ωn. We have the natural morphism ηn,m : Filt(E,n) −→ Filt(E,m) for any m ≤ n,
by forgetting the parts Fj (m+1 ≤ j ≤ n). We denote the line bundle ηn∗,mLm and ηn∗,mωm by Lm and
ωm respectively. We have the naturally defined morphism Filt(E,n) −→ Cn by taking the supports of Gi.
Then the structure of the cohomology ring H∗(Filt(E,n)) over H∗(Cn) is easily and completely described
in termes of ωi (i= 1,...,n). (See the subsubsection 2.4.2.) When we take a limit, H∗(Filt(∞,n)) is the
polynomial ring over H∗(Cn) generated by ωi (i=1,...,n).
WehavethenaturalmorphismΨ:Filt(E,n)−→Quot(E,n),forgettingFi (i=1,...,n−1). Itinduces
the injection of the cohomology rings. Thus H∗(Quot(E,n)) is a subring of the ring H∗(Filt(E,n)). We
would like to do some calculation for H∗(Quot(E,n)) in H∗(Filt(E,n)).
1
Let consider the case E = rα−=10Lα. Let Grm denote the r-dimensional algebraic torus. We have the
decomposition of thefixed poinLt sets:
Quot(E,n)Grm = FQ(v), Filt(E,n)Grm = F(v).
v∈aB(n,r) v∈[0a,r−1]n
(See the subsubsection 2.3.3.) Due to the theories of Bianicki-Birula ([2]), Carrel and Sommes ([3]), or in
other words, dueto theMorse theoretic consideration, we obtain the morphisms:
ξQ(v;·):H∗(FQ(v))[−2co(v)]−→H∗(Quot(E,n)), (v∈B(n,r))
ξ(v;·):H∗(F(v))[−2co(v)]−→H∗(Filt(E,n)), (v∈[0,r−1]n).
And we havethedecompositions:
H∗(Quot(E,n))= Im(ξQ(v;·)), H∗(Filt(E,n))= Im(ξ(v;·))
v∈MB(n,r) v∈[0M,r−1]n
We havethefollowing relation (thesubsection 4.2):
1
Ψ∗(ξQ(v,a))= ξ(σv,σ(a)). (1)
|St(v)|
σX∈Sn
Thusthecalculation inH∗(Quot(r,n))isreducedtothecalculationinH∗(Filt(r,n)). Themaincalculation
will be done in Theorem 3.1. (See the formula (6). See also (15).) Although the equality (6) provides us
the method of the calculation, the explicit calculation seems rather complicated. Thus we introduce the
filtration definedas follows:
F H∗(Quot(E,n))= Im(ξQ(v;·)), F H∗(Filt(E,n))= Im(ξ(v;·)).
h h
v∈MB(n,r) v∈[0M,r−1]n
co(v)≤h co(v)≤h
Due to the equality (1) and Theorem 3.1, it is easy to derive that the filtrations are compatible with the
products. Moreoverthestructuresoftheassociated graded ringsarequitesimple(thesubsubsectoin4.2.1).
As a corollary, we obtain some sets of the generators of H∗(Quot(∞,n)). (Proposition 4.1, Proposition
4.2 and Theorem 4.1.)
Todeterminetheprecise structureof H∗(Quot(r,n)),we consider theSn-action ρon H∗(Filt(∞,n))=
H∗(Cn)[ω1,...,ωn], defined as follows (See subsection 4.3):
For σ∈Sn, we put ρ(σ)( ip∗i(ai)·ωili):= ip∗σ(i)(ai)·ωσli(i).
TheactionρpreservestheprodQucts. LetH∗(Cn)[Qω1,...,ωn]ρ,Sn denotetheSn-invariantpartwithrespect
to the action ρ. Then we haveΨ∗H∗(Quot(∞,n))=H∗(Cn)[ω1,...,ωn]ρ,Sn. (Theorem 4.2)
Remark 1.1 WealsousetheSn-actionκonH∗(Filt(r,n))definedbyκ(σ)ξ(v,a)=ξ(σ·v,σ(a)). (Seethe
subsection 3.1. Weomittodenoteκthere.) WealsohaveH∗(Filt(r,n))κ,Sn =Ψ∗H∗(Quot(r,n)). However
the action κ does not preserve the product.
Let J(r,n) denote the ideal of H∗(Quot(r,n)) generated by {ξQ(v)|v ∈ B(n,r+1)−B(n,r)}. Then
we haveH∗(Quot(r,n))=H∗(Quot(∞,n))/J(r,n). Thus, once we obtain thedescription of Ψ∗ξQ(v;a) in
terms of ωi, we can say that thestructureH∗(Quot(r,n)) is described.
The formulas (6) and (15) assures us that we have some algorithms to calculate Ψ∗ξQ(v;a). Each
algorithm will provide us a combinatorial decription. In the subsection 4.4, we give one of the description.
Although the result does not seem convenient for the practical calculation, we would like to remark the
following: Due to Theorem 4.2, the formula gives a symmetric function, that is, they are elements of
H∗(Cn)[ω1,...,ωn]ρ,Sn, which is not obvious at a sight.
Finally we see the structure of the cohomology ring of the infinite quot scheme of infinite length
Quot(∞,∞).
ThefiltschemesFilt(E,l)arethequitenaturalobjects: Theyaredefinedasthemodulischemeofsome
simple functors. They are smooth. They also have big symmetry. The author believes that it is always
significant for mathematics to investigate such objects in detail, and he expects that the they will provide
us very interesting examples.
2
This paper is a revision. The most main results are same as those of the original version. The original
version was written in May 2001, after the original versions of the papers [7] and [8]. The subsections 4.3
and 4.4 are added in January 2003.
Acknowledgement
The author is grateful to the colleagues in Osaka City University. In particular, he thanks Mikiya Masuda
for his encouragement and support.
TheauthorthanksthefinancialsupportsbyJapanSocietyforthePromotionofScienceandtheSumitomo
Foundation. The paper was revised during his stay at the Institute for Advanced Study. The author is
sincerely grateful to their excellent hospitality. He also acknowledges National Scientific Foundation for a
grant DMS 9729992, although anyopinions, findingsand conclusions orrecommendations expressed in this
material are those of theauthor.
2 Preliminary
2.1 Notation
We usethe following notation:
Z: The ring of integers, Z :={a∈Z|a≥0}, [0,n]:={a∈Z|0≤a≤n},
≥0
[1,n]:={a∈Z|1≤a≤n}, C: The complex numberfield, Q: The rational numberfield.
Sn: Then-th symmetric group.
• The order of a finite set S is denoted by|S|.
• Ingeneral, qj will denotetheprojection ontothej-thcomponent. Forexample,qj(l)denotesthej-th
component lj ∈Z≥0 of l=(l1,l2,...,ln)∈Zn≥0.
• For 1≤j,k ≤n, the transposition of j and k is denoted by τ . Namely τ denotes the element of
j,k j,k
Sn, satisfying τj,k(j)=k, τj,k(k)=j and τj,k(l)=l for any l6=j,k.
• Let n be a non-negative integer, and X be a variety. Then X(n) denote the n-th symmetric product
of X. For an element l=(l1,...,lh)∈Zh≥0, we put X(l) := hi=1X(li).
i−1 Q
• The element (0,...,0,1,0,...,0)∈Zn≥0 is denoted by ei. Weput cl := li=1ei.
• For any elemenzt}l|∈{Zn≥0, we put|l|:= ni=1qi(l). Here qi(l) denotesthPe i-th component of l.
• Ingeneral, theidentitymorphism ofXPis denotedbyidX. Weoften omit todenoteX,if thereareno
confusion.
• Let C be a smooth projective curve. We denote the diagonal of C×C by ∆. Let ∆ij (i6=j) denote
the divisor {(x1,...,xn) ∈Cn|xi =xj} of Cn. Let I be a subset of {1,...,n}, then ∆I denotes the
closed subset {(x1,...,xn)|xi=xj for any i,j ∈I}.
Wewill often usethe notation ∆I todenote thecorresponding cohomology class.
• Ingeneral,weusethenotationpttodenotethecohomologyclassofapointinatopologicalspaceX. Let
C bea smooth projective curve. Fora subset I ⊂{1,...,n}, we havethe projection i∈Iqi :Cn −→
C|I|. We have the cohomology class pt in H∗(C|I|) in the above sense. The pull bacQk ( i∈Iqi)∗pt is
denoted by ptI. Q
• Inthispaper,thecoefficientofthecohomologyringistherationalnumberfieldQ,ifwedonotmention.
• We will consider the polynomial ring H∗(X)[t0,...,tr−1] over H∗(X). Here X denotes a topological
space, and ti (i = 0,...,r−1) denote formal variables. For an element J = (j0,...,jr−1) ∈ Zr≥0,
we put tJ := ri=−01tjii. Then an element P ∈H∗(X)[t0,...,tr−1] has the description P = JaJ ·tJ
aJ ∈H∗(X)Q. Thetotal degreeof P with respect tothevariables t0,...,tr−1 isdenotedbyPdegt(P).
(cid:0)It is called tot(cid:1)al degree for simplicity. Wealso put as follows:
[P]:= aJ ·tJ.
|J|=Xdegt(P)
It is called thetop term of P.
3
• In principle, the polynomial ring H∗(X)[t0,...,tr−1] is obtained as the equivariant cohomology ring
HG∗rm(X) for the trivial action of Grm. For an element a ∈ HGjrm(X), the number j is called the
cohomological degree of a.
2.2 Some sets
2.2.1 B(l) and B(l,r)
Let l bea non-negative integer. Weput as follows:
B(l):= v =(v ,...,v)∈Zl v ≥v ≥···≥v ≥0 .
1 l ≥0 1 2 l
(cid:8) (cid:12) (cid:9)
Let r be a positive integer. Then we put as follows: (cid:12)
B(l,r):= v =(v ,...,v)∈B(l) v ≤r−1 .
1 l 1
(cid:8) (cid:12) (cid:9)
We havethenatural inclusion B(l)−→Zl . Then theset B(l,r(cid:12)) is contained in [0,r−1]l.
≥0
Letu=(u ,...,u)beanyelementofZl . Wehavetheuniqueelementv ofB(l),whichisareordering
1 l ≥0
of u. Thus we have the map Zl −→ B(l), which we denote by Nor, i.e. v := Nor(u). Clearly we have
≥0
Nor([0,r−1]l)⊂B(r,l). The map is called thenormalization.
Foran element v ∈B(l), we putco(v):= li=1vi ≥0. Thuswe obtain the map co:B(l)−→Z≥0.
Fora tuple l=(l1,...,lh)∈Zh≥0, we putPas follows:
h h
B(l):= B(li), B(l,r):= B(li,r).
iY=1 Yi=1
For an element v∗ = (v1,...,vh) ∈ B(l), we put co(v∗) := hi=1co(vi) ∈ Z≥0. For an element u ∈
h Zli , we put as follows: P
i=1 ≥0
Q
Nor(u):= Nor(u ),...,Nor(u ) ∈B(l).
1 h
(cid:0) (cid:1)
Thus we obtain themaps co:B(l)−→Z and Nor: h Zli −→B(l).
≥0 i=1 ≥0
Q
Example
We havethenatural isomorphism B(cn)≃Zn≥0. Wehavethe natural isomorphism B(n·e1)≃B(n).
2.2.2 Dec(l,r)
Let l be an element of Zh≥0. A tuple l∗ = (l0,...,lr−1) ∈ Zh≥0 r is called an r-decomposition of l, if we
have rα−=10lα = l. Let Dec(l,r) denote the set of r-decom(cid:0)posit(cid:1)ions of l. We have the natural injection
Dec(lP,r)−→Dec(l,r+1) given by (l0,...,lr−1)7−→(l0,...,lr−1,0). Weput Dec(l):= r≥1Dec(l,r).
Foran element l∗ =(l0,l1,...,lr−1), we put as follows: S
r−1 r−1 h
co(l∗):= α·|lα|= α· qi(lα).
αX=0 αX=0 Xi=1
2.2.3 Isomorphism
Let l be an element of Zh . We have the natural bijection φ between B(l,r) and Dec(l,r). Let v =
≥0 ∗
(v1,...,vh) be an element of B(l,r). Then φ(v∗)=l∗ =(l0,...,lr−1) is determined as follows:
qj(lα):= i qi(vj)=α .
(cid:12)(cid:8) (cid:12) (cid:9)(cid:12)
(cid:12) (cid:12) (cid:12)
4
More visually, thecorrespondence is as follows:
lr−1,1 lr−2,1 l1,1 l0,1
(v ,v ,...,v )=(r−1,...,r−1,r−2,...,r−2,......,1,...,1,0,...,0,)
11 12 1l1
z }| { z }| { z }| { z }| {
lr−1,2 lr−2,2 l1,2 l0,2
(v ,v ,...,v )=(r−1,...,r−1,r−2,...,r−2,......,1,...,1,0,...,0,)
21 22 2l2
z }| { z }| { z }| { z }| {
.
.
.
lr−1,n lr−2,n l1,n l0,n
(vn,1,vn,2,...,vn,ln)=(r−1,...,r−1,r−2,...,r−2,......,1,...,1,0,...,0,).
z }| { z }| { z }| { z }| {
By theconstruction, we clearly haveco(φ(v ))=co(v ).
∗ ∗
2.3 Filt scheme
2.3.1 Definition
Let l = (l ,...,l ) be an element of Zh . Let C be a smooth projective curve over C. Let E be a locally
1 h ≥0
free coherent sheaf over C. The filt scheme Filt(E,l) is the moduli scheme of thesequence of quotients
E −→F −→F −→···−→F −→F , (2)
h h−1 2 1
satisfying thefollowing conditions:
• The length of thesheaf Gi=Ker(Fi −→Fi−1) is li for each i.
In particular, Filt(E,cn) is denoted by Filt(E,n). It is called the complete filt scheme. On the other
hand, Filt(E,n·e ) is denoted by Quot(E,n). It is theusual quot scheme.
1
2.3.2 Deformation
A sequenceas in (2) is denoted byE →F∗. For such sequence, we putKj :=Ker(E −→Fj) and K0 :=E.
Then we havethenatural morphisms:
Hom(Ki,Kj)−→Hom(Ki,Kj−1), Hom(Ki,Kj)−→Hom(Ki+1,Kj).
In all, we obtain the following morphism f:
hi=1Hom(Ki,Ki) −−−−f−→ hi=1Hom(Ki,Ki−1).
L L
Here the first term stands at degree −1. It is easily checked that f is injective, and the support of the
coherent sheaf Cok(f) is 0-dimensional. By using the obstruction theory given in [7], we can conclude that
the filt scheme Filt(E,l) is smooth, and the tangent space is given by H0(Cok(f)). It is easy to see that
dim(H0(Cok(f)))=rank(E)·|l|. In all, we havethefollowing.
Proposition 2.1 The variety Filt(E,l) is smooth and of rank(E)·|l|-dimension.
The projection of of Filt(E,l)×C onto the j-th component by qj. We have the universal sequence of
the quotients:
q∗(E)−→Fu −→···−→Fu.
2 h 1
We put Ku :=Ker(q∗(E)−→Fu). Then we obtain the following complex overFilt(E,l)×C:
j 2 j
h Hom(Ku,Ku) −−−f−u−→ h Hom(Ku,Ku ).
i=1 i i i=1 i i−1
L L
Then tangent bundleof Filt(E,l)×C is given by thevector bundle q (Cok(fu)).
1∗
5
2.3.3 Torus action and the fixed point sets
Let consider the case E = rα−=10Lα, for line bundles Lα (α = 0,...,r−1) over C. Let Grm denotes the
r-dimensionaltorus. ThenwLehavethenaturalGrm-actiononE,whichinducestheGrm-action onFilt(E,l).
Foran l ∈Dec(l,r), we put as follows:
∗
r−1 r−1
Fl(l∗):= Filt(Lα,lα)≃ C(lα).
αY=0 αY=0
In particular, we use thenotation F(l∗) and FQ(l∗) instead of Fcl(l∗) and Fl·e1(l∗) for any l.
Let Filt(E,l)Grm denote the fixed point set of Filt(E,l∗) with respect to the torus action above. Then
we havethenatural isomorphism:
Filt(E,l)Grm ≃ Fl(l∗).
l∗∈Daec(l,r)
On Filt(E,l)Grm ×C, theuniversal sequenceis decomposed as follows:
r−1
q2∗Lα−→Fαu,h −→Fαu,h−1 −→···−→Fαu,1 .
Mα=0(cid:16) (cid:17)
We put Kαu,j :=Ker(q2∗Lα −→Fαu,j) and Kαu,0 :=q2∗Lα. Weput as follows:
Nl(l ):=Cok h Hom(Ku ,Ku )−→ h Hom(Ku ,Ku ) ,
+ ∗ α<β j=1 α,j β,j α<β j=1 α,j β,j−1
(cid:16) (cid:17)
L L L L
Nl(l ):=Cok h Hom(Ku ,Ku )−→ h Hom(Ku ,Ku ) .
− ∗ α>β j=1 α,j β,j α>β j=1 α,j β,j−1
(cid:16) (cid:17)
L L L L
Then we put Nl(l ) = Nl(l ) Nl(l ). Then q (Nl(l )) is the normal bundle of Fl(l ). We denote it
∗ + ∗ − ∗ 2∗ ∗ ∗
by Nl(l∗). L
2.3.4 The decompositions due to Bianicki-Birula, Carrel and Sommes
Letconsiderthesub-torusGm ⊂Grm givenbyt7−→(t,t2,...,tr). ThenweobtaintheGm-actiononNl(l∗).
The negative part Nl(l ) is given byq (Nl(l )). It is easy to see theequality rank(Nl(l ))=co(l ).
− ∗ 2∗ − ∗ − ∗ ∗
Dueto thetheory of Bianicki-Birula [2] or Morse-theoretic consideration, we have the decomposition of
Filt(E,l ) into theunstable manifolds:
∗
Filt(E,l )= Fl(l )+.
∗ ∗
l∗∈Daec(l,r)
It satisfies thefollowing:
• The complex codimension of Fl(l )+ is rank(Nl(l ))=co(l ).
∗ − ∗ ∗
• Wehavethe Gm-equivariantmorphism πl,l∗ :Fl(l∗)+ −→Fl(l∗). It is an affine bundle.
• Let x bea point of Fl(l∗)+. Then we have lim t·x=πl,l∗(x)∈Fl(l∗).
Gm∋t→0
When l = l·e , the moduli theoretic meaning of Fl·e1(l )+ is given in [7]. Following Carrel and Sommes
1 ∗
[3], we consider the natural embedding Fl(l )+ −→ Fl(l )×Filt(E,l). The closure Fl(l )+ induces the
∗ ∗ ∗
correspondence map:
ξl(l ;·):H∗(Fl(l ))[−2co(l )]−→H∗(Filt(E,l)).
∗ ∗ ∗
Due tothe result of Carrel and Sommes, we havethedecomposition:
H∗(Filt(E,l))≃ H∗(Fl(l ))[−2co(l )].
∗ ∗
l∗∈DMec(l,r)
Wehave already given the bijection Dec(l,r)≃B(l,r). We identify them. Then we havethe morphism
ξl(v :·):H∗(Fl(v ))[−2co(v )]−→H∗(Filt(E,l)), and thedecomposition:
∗ ∗ ∗
H∗(Filt(E,l))= Im(ξl(v ;·))≃ H∗(Fl(v ))[−2co(v )].
∗ ∗ ∗
l∗∈MB(l,r) v∗∈MB(l,r)
6
By using thedecomposition, we introducethefollowing filtration:
F H∗(Filt(E,l))= Im(ξl(v ;·))
h ∗
v∗∈MB(l,r)
co(v∗)≤h
The filtration is dependingon a decomposition E = rα−=10Lα.
L
2.4 Complete filt scheme
2.4.1 An inductive description of complete filt schemes
Let qj denotethe projection of Filt(E,n)×C ontothe j-th component. Wehavethe universal sequenceof
quotients q∗(E)−→Fu −→Fu −→···−→Fu. We put Ku :=Ker(q∗(E)−→Fu), which is locally free
2 n n−1 1 n 2 n
onFilt(E,n)×C. ThusweobtaintheassociatedprojectivespacebundleP(Ku∨). Wedenotetheassociated
n
tautological line bundleby Ln+1. Wesee the following rather obviouslemma.
Lemma 2.1 The variety Filt(E,n+1) is naturally isomorphic to P(Ku∨).
n
Proof Let U be a scheme. We denote the projection of U ×C onto the j-th component by qj. Let
q2∗(E) −→ Fn+1 −→ Fn −→ ··· −→ F1 be a sequence of quotients such that such that the length of
Ker(Fj −→ Fj−1) is 1. We naturally obtain the sequence, q2∗(E) −→ Fn −→ ··· −→ F1. It gives the
morphism g :U −→Filt(E,n) such that (g×idC)∗Fju =Fj. Here g×idC is naturally induced morphism
U ×C −→Filt(E,n)×C. Weput as follows:
Kn :=Ker(q2∗(E)−→Fn)=(g×idC)∗Knu, Gn+1 :=Ker(Fn+1 −→Fn).
Then we have the naturally induced morphism Kn −→ Gn+1 defined over U ×C. It induces the section
U −→P(K∨). As a result, we obtain the morphism U −→P(Ku∨). In particular, we obtain the morphism
n n
Φn+1 :Filt(E,n+1)−→P(Knu∨).
LetπndenotetheprojectionofP(Knu∨)−→Filt(E,n)×C. Weputχn :=q1◦πnandρn:=q2◦πn. Then
weobtainthemorphismχn×idC :P(Knu∨)×C −→Filt(E,n)×C. Thenwehavethefollowing morphisms:
(χn×idC)∗Knu −→q2∗E −→(χn×idC)∗Fnu −→···−→(χn×idC)∗F1u.
OnP(Knu∨),wehavethenaturallydefinedmorphismψn :χ∗nKnu −→Ln+1. Wehavetheclosedembedding
(id×ρn) : P(Knu∨) −→ P(Knu∨)×C. Let Xn denote the image. Then we obtain the morphism λn :
(χn×idC)∗Knu⊗OXn −→q1∗Ln+1⊗OXn,which isinducedbyψn andtheisomorphism Xn ≃P(Knu∨). We
putKnu+1 :=Ker(λn),andFnu+1 :=Cok(Knu+1 −→q2∗E). Wedenote(χn+1×id)∗Fju byFju forj =1,...,n.
Then we obtain the sequence q∗(E) −→ Fu −→ Fu −→ ··· −→ Fu defined over P(Ku∨)×C. In
2 n+1 n 1 n
particular, we obtain the morphism Ψn+1 :P(Knu∨)−→Filt(E,n+1).
It is easy to see that Φn+1 and Ψn+1 are inverses each other. Thus Filt(E,n+1) and P(Knu∨) are
isomorphic via themorphisms.
Wewill not distinguish them in the following. By our construction, we also knowthefollowing:
Corollary 2.1 The sheaf Gnu+1 := Ker(Fnu+1 −→ Fnu) is isomorphic to q1∗Ln+1⊗OXn. In particular, we
have q1∗ Gnu+1 ≃Ln+1.
(cid:0) (cid:1)
2.4.2 The structure of H∗(Filt(E,n),Z)
In the rest of the subsection 2.3, we consider the cohomology ring with Z-coefficient. For any m ≤ n, we
have thenatural morphism ηn,m :Filt(E,n)−→Filt(E,m) defined by thefollowing functor:
E →Fn →···→Fm →···→F1 7−→ E →Fm →···→F1 .
(cid:16) (cid:17) (cid:16) (cid:17)
We clearly haveηn,m :=χm◦χm+1◦···◦χn−1. Wedenote ηn∗,mLm by Lm for simplicity of notation.
For the universal filtration q∗(E) −→ Fu −→ ··· −→ Fu, we put Gu := Ker(Fu −→ Fu ). Then the
2 n 1 j j j−1
support of Gju induces the morphism pj : Filt(E,n) −→ C. Note that pn is same as ρn−1 in the previous
subsection. In all, we havethe morphism p= nj=1pj :Filt(E,n)−→Cn. We have theinduced morphism
p∗ : H∗(Cn) −→ H∗(Filt(E,n)). For any elQement a ∈ H∗(Cn), the element p∗(a) is denoted by a for
simplicity of notation.
7
Wehave themorphism ρn−1×idC :Filt(E,n)×C −→C×C. Then Xn−1 is same as (ρn−1×idC)∗∆.
It is reworded as follows: Wehave themorphism p×idC :Filt(E,n)×C −→Cn×C. Then Xn−1 is same
as (p×idC)∗∆n,n+1. In particular, thesheaf Gnu overFilt(E,n)×C is isomorphic tothe following:
q1∗Ln⊗Cok O(−∆n,n+1)−→O .
(cid:16) (cid:17)
Herewedenotethepullbackofthedivisor∆n,n+1 viathemorphism p×idC bythesamenotation ∆n,n+1.
Hence thesheaf Gu over Filt(E,n)×C is isomorphic to thefollowing:
j
q1∗Lj⊗Cok O(−∆j,n+1)−→O .
(cid:16) (cid:17)
In the K-theory of thecoherent sheaves on Filt(E,n), we havetheequality Ku =q∗(E)− n Gu. Hence
n 2 j=1 j
we obtain the following equality for thetotal Chern classes: P
n
c Ku =c q∗(E) × c Gu −1.
∗ n ∗ 2 ∗ j
(cid:0) (cid:1) (cid:0) (cid:1) jY=1 (cid:0) (cid:1)
We denote the first Chern class of Lj by ωj. Note that the composition Filt(E,n+1)−→Filt(E,n)×
C −→ C is same as pn+1 :Filt(E,n+1) −→ C. Thus we obtain the following relation in the cohomology
ring H∗(Filt(E,n+1)):
r r n
Xj=0aj·ωnr−+j1=0, Xj=0aj =p∗n+1(cid:0)c∗(E)(cid:1)·jY=11+ω1j+−ω∆jjn+1.
Duetothegeneraltheoryfortheprojectivespacebundles,therelationdeterminesthestructureofH∗(Filt(E,n+
1)) over H∗(Filt(E,n)) with thegenerator ωn+1.
By an inductiveargument, we obtain thefollowing proposition.
Proposition 2.2 TheringH∗(Filt(E,n))isthequotientringofH∗(Cn)[ω1,...,ωn]dividedbythefollowing
relations for 1≤h≤n:
r
a(h)·ωr−j.
j h
Xj=0
Here a(jh) is described in terms of ∆i,h and ωj (i,j ≤h−1) as follows:
r h−1
a(h) = 1+ωi−∆ih.
Xj=0 j iY=1 1+ωi
2.4.3 Some easy corollaries
As acorollary derived from thedescription above, weknow thestructureof thecohomology ringof infinite
complete filt schemes. Let Lα (α = 0,1,2...) be line bundles over C. We put Eβ := α≤βLα, and
E∞ := αLα. Then we have the naturally defined inclusions Filt(Eβ,n) −→Filt(Eβ+1,n)L. Then we can
considerLthelimit Filt(E∞,n).
Corollary 2.2 H∗(Filt(E∞,n)) isthe polynomial ringover H∗(Cn) generated bythe first Chernclasses ωi
for i=1,...,n
Ingeneral,wehavetheclassifyingmapsµi :Filt(E,n)−→P∞ correspondingtothelinebundlesLi (i=
1,...,n), for any locally free sheaf E over C. In all we have the morphism µ = ni=1µi : Filt(E,n) −→
(P∞)n. Thuswe obtain the morphism p×µ:Filt(E,n)−→Cn×(P∞)n. Q
Corollary 2.3 Let consider thecase C =P1. Thenthe morphismFilt(E ,n)−→ P1×P∞ n ishomotopy
∞
equivalent. (cid:0) (cid:1)
8
Let E be a locally free coherent sheaf over C. Recall that the sheaf Gu defined over Filt(E,n)×C is
i
isomorphic to q1∗Li⊗Cok O(−∆i,n+1) → O . Here ∆i,n+1 denotes the pull back of the divisor ∆i,n+1 of
Cn×C. Hencethetotal C(cid:16)hern class of Gi is(cid:17)described as follows:
c∗(Gi)= 1+ω1i+−ω∆ii,n+1 =Xj∞=0(cid:18)∆1i+,nω+i1(cid:19)j =1+∆i,n+1hX∞=0(−ωi)h+∆2i,n+1(1+ωi)−2. (3)
In particular, we havec1(Gi)=∆i,n+1 and c2(Gi)=−∆i,n+1·ωi+(2−2g(C))·pti,l+1. Hereg(C) denotes
the genusof C.
Thej-thChernclasscj(Gi)inducesthecorrespondencemapψj(i) :H∗(C)[−2(j−1)]−→H∗(Filt(E,n)).
The following equalities are easily checked:
ψ1(i)(a)=p∗i(a), ψ2(i)(1)=−ωi+(2−2g(C))·p∗i(pt). (4)
Proposition 2.3 Thus the following set generates the ring H∗(Filtcomp(E,l)) over Q:
n
Imψ(i)∪{ψ(i)(1)} .
1 2
i[=1(cid:16) (cid:17)
Proof The claim follows from (4).
The proposition will be generalized in thecase of theother filt schemes.
3 Some products in H∗(Filt( r−1 L ,n))
α=0 α
L
3.1 Preliminary
WOnehthaveeoatlhreeradhyankdn,owwnethhaevreintghestnruacttuurraellyofdHefi∗(cid:0)neFdiltc(oLhorαm−=1o0lLogαi,cna)l(cid:1)c,lwashsiecshisξcdne(svcr∗i;bae)dbvy∗ ω∈i (Bi(=cn1,r.).,.,an)∈.
H∗(Fcn(v∗)) . Wecalculate the action of ωn on ξcn(v∗;a). In thefollowin(cid:8)g, we put E(cid:12)(cid:12)= rα−=10Lα.
(cid:9) L
3.1.1 Replacement of notation
RecallthatB(cn,r)isnaturallyisomorphicto[0,r−1]n. Thusweusethenotationv =(v1,...,vn)instead
of v∗ =(v1,...,vn). Moreover we use F(v) instead of Fcn(v). Similarly we use ξ(v) instead of ξcn(v;1).
Note that we haveξcn(v;a)=a·ξ(v). Then we havethedecomposition:
H∗(Filt(E,n))= H∗(Cn)·ξ(v).
v∈[0M,r−1]n
We will use the equivariant cohomology group HG∗rm(Filt(E,n)). Since the action of Grm preserves the
subvarieties F(v)+, the class ξ(v) is naturally lifted to the equivariant cohomology class. We denote them
by thesame notation. Moreover, theygive the base of HG∗rm(Filt(E,n)) over thering H∗(Cn)[t0,...,tr−1].
Wealso havetheequivariant first Chern class of Li, which is denoted also by ωi.
The inclusion F(v)⊂Filt(E,n) induces thenaturally defined ring morphism:
res(v):HG∗rm(Filt(E,n))−→HG∗rm(F(v))≃H∗(F(v))[t0,...,tr−1].
For an element f of HG∗rm(Filt(E,n)), theelement res(v)(f)∈H∗(F(v))[t0,...,tr−1]is denoted by fv.
3.2 Some lemmas
We preparesome lemmas which are obtained by some geometric considerations.
Lemma 3.1 We have the following equality:
ωiv =tvi− ∆k,i+p∗ic1(Lvi).
Xk<i
vk=vi
9
Proof We only have to show the equality in the case i = n. Recall that Filt(E,n) is isomorphic to the
associated projectivespacebundlewithKnu−1 overFilt(E,n−1)×C,andLn isthetautological linebundle
under the identification. Then the restriction Ln|F(v) is isomorphic to Ker(q2∗Lvn −→ Fvn,n−1). It is
isomorphic to thefollowing line bundle:
q2∗Lvn ⊗O − ∆in .
(cid:16) i<nX,vi=vn (cid:17)
Then we obtain theresult. Note thecomposition Filt(E,n)−→Filt(E,n−1)×C −q→2 C is same as pn.
Definition 3.1 We introduce the order ≤ on [0,r−1]n defined as follows:
0
Let v =(v1,...,vn) and v′ =(v1′,...,vn′) be elements of [0,r−1]n. Then v ≤0 v′ if and only if
vi ≤vi′ for any i=1,...,n.
By the inclusion B(n,r) ⊂ [0,r−1]n, we have the induced order on B(n,r). We denote the order by the
same notation.
Recall that [f] denotes thetop term for f ∈H∗(F(v))[t0,...,tr−1] (thesubsection 2.1).
Lemma 3.2 Assume that w∈[0,r−1]n satisfies w≥ v. Then we have the following equality:
0
n vj−1
w
[ξ(v) ]= (twj −ti).
jY=1 iY=1
In particular, we have deg (ξ(v)w)=co(v). Reversely, deg (ξ(v)w)=co(v) implies w ≥ v.
t t 0
Proof LetP =(x1,...,xn)beapointofCnsuchthatxi6=xj (i6=j). Thenwehavetheclosedembedding
Filt(E,n)×Cn{P}−→Filt(E,n). NotethatFilt(E,n)×Cn{P}isisomorphicto(Pr−1)n. Weonlyhaveto
consider theimage of ξ(v) to H∗(Filt(E,n)×Cn {P}).
Lemma 3.3 If ξ(v)w 6=0, then we have an element σ∈Sn such that σ(v)≤w.
Proof LetconsiderthemorphismΨ:Filt(E,n)−→Quot(E,n). ThenΨ(F(v))iscontainedinFQ(Nor(v)),
and Ψ(F(v)+)⊂FQ(Nor(v))+. In [7], we obtained that FQ(u)+∩FQ(u′)6=∅⇐⇒u ≤ u′. Thus we are
0
done.
Lemma 3.4 Let w be an element of [0,r−1]n, such that there exists σ∈Sn satisfying σ(v)=w. We put
I :={i|vi 6=wi}. Then we have the following inequality:
w
deg ξ(v) ≤co(v)−|I|+1.
t
Proof Let consider therestriction of ξ(v) to Filt(E,n)×Cn (Cn−∆I). It is easy to see the following:
F(v)+∩F(w)∩ Filt(E,n)×Cn (Cn−∆I) =∅.
(cid:2) (cid:3)
Thus we are done.
Letw beanelementof[0,r−1]n. WeputJ(v,w):={σ∈Sn|σ(v)≤w}. Wedenotetheprojectionof
[0,r−1]nontothei-thcomponentbyqi. Foranyelementσ∈J(v,w),weputI(σ):={i|qi(v)6=qi(σ(v))}.
Then we obtain thenumber d(v,w):=min |I(σ)| σ∈J(v,w) .
(cid:8) (cid:12) (cid:9)
Lemma 3.5 We have the following inequality: (cid:12)
w
deg (ξ(v) )≤co(v)−d(v,w)+1.
t
Proof Similar to Lemma 3.4.
Corollary 3.1 Let w be an element of [0,1] satisfying co(w)=co(v) and deg (ξ(v)w)=co(v)−1. Then
t
there exists a transposition τ such that τ(v)=w. Namely, there exist i6=j satisfying vi =wj, vj =wi and
v =w (k6=i,j).
k k
10