In this chapter we collect some formulas for the Chern classes of the bundles we are interested in.
We see these classes as elements in the cohomology with rational coecients. For a torsion-free moduleM of rankr on a smooth projective surfaceX we dene the associated determinant line bundle bydet(M) := (
r
VM)∗∗. Lemma 1.79:
Assume X is a smooth projective surface and let M and N be coherent OX-modules of rank r respectivelys. If one of these modules is locally free, then the Chern classes of the tensor product are given by:
• c1(M⊗OX N) =sc1(M) +rc1(N);
• c2(M⊗OX N) =sc2(M) + s2
c1(M)2+ (rs−1)c1(M)c1(N) + r2
c1(N)2+rc2(N). Lemma 1.80:
Let X be a smooth and projective surface and let T be a torsion sheaf on X, sitting in an exact sequence
0 −−−−→ M −−−−→φ N −−−−→ T −−−−→ 0 (4) whereM andN are torsion-free coherentOX-modules of the same rank. Then c1(T) is eective and one has:
c1(T) = P
codim(ξ)=1
lOX,ξ(Tξ){ξ}. Proof:
SinceM and N are torsion-free, the injection M ,→ N induces an injection det(M),→ det(N), see for example [Kob87, Proposition V.6.13]. We get an exact sequence
0 −−−−→ det(M) −−−−→det(φ) det(N) −−−−→ Q −−−−→ 0. (5) Tensoring this exact sequence with the line bundledet(N)−1 shows:
1. det(M)⊗ OX(D)∼=det(N) 2. Q ∼=OD⊗det(N)
for some eective Cartier divisorD. This impliesc1(det(M)⊗ OX(D)) =c1(det(N))or, by using c1(M) =c1(det(M)):
c1(N)−c1(M) =D.
With the exact sequence (4) and the properties of c1 we get: c1(T) =D.
Now eective Cartier divisors are the same as eective Weil divisors on X, and D corresponds to
P
codim(ξ)=1
lOX,ξ((OD)ξ){ξ}.
34 1.6 Chern class computations
Pick a prime divisor C of D with generic point ξ, then R := OX,ξ is a discrete valuation ring and the exact sequences (4) and (5) give us two exact sequences:
0 −−−−→ Mξ −−−−→φξ Nξ −−−−→ Tξ −−−−→ 0 0 −−−−→ det(M)ξ −−−−→det(φ)ξ det(N)ξ −−−−→ Qξ −−−−→ 0.
The rst sequence yieldsTξ=coker(φξ) while the second one gives (OD)ξ=R/det(φ)ξR. Since Mξ and Nξ are torsion-free, they are in fact free and of the same rank over the principal ideal domain R, hence coker(φξ) is of nite length. Using the structure theorem for modules over a principal ideal domain, one can see that lR(coker(φξ)) =lR(R/det(φξ)R). So
D= X
codim(ξ)=1
lR((OD)ξ){ξ}
= X
codim(ξ)=1
lR(R/det(φξ)R){ξ}
= X
codim(ξ)=1
lR(coker(φξ)){ξ}
= X
codim(ξ)=1
lR(Tξ){ξ}.
Corollary 1.81:
AssumeAis a maximal order on a smooth projective surfaceX and let M andN be torsion-free A-modules of rank one. If HomA(M, N)6= 0 then c1(N)−c1(M) is eective.
Proof:
By (1.47) a nontrivial element inHomA(M, N) gives rise to a short exact sequence:
0 −−−−→ M −−−−→ N −−−−→ T −−−−→ 0.
Now use (1.80).
Lemma 1.82 ([Fri98, Chapter 2]):
AssumeX is a smooth and projective surface andp∈X is a closed point. IfIp denotes the ideal sheaf of p andk(p) is the skyscraper sheaf at p, then we have an exact sequence
0 −−−−→ Ip −−−−→ OX −−−−→ k(p) −−−−→ 0 which shows that
c1(Ip) =c1(k(p)) = 0and c2(Ip) =−c2(k(p)) = 1. Theorem 1.83:
Assume A is a maximal order on a smooth projective surface X of rank r2, with ramication curves{Ci} and ramication indices {ei} for i= 1, . . . , l. Then we have:
c1(A) =−r22 Pl
We note that ifξis a point of codimension one inXwhich is not the generic point of a ramication curve, thenAξ is Azumaya and the trace gives an identication
(A∗)ξ = (Aξ)∗ ∼=Aξ
so that Qξ = 0 for these points. This is basically due to the fact that Aξ gets isomorphic to a matrix algebra over the completion of the local ringOX,ξ. A matrix algebra Mn(R) is self-dual with respect to the trace and since the trace is compatible with completion, we get the desired isomorphism.
If ξ is the generic point of a ramication curveC then Aξ is a maximal R-order in Aη, where R =OX,ξ is a discrete valuation ring with maximal ideal m. It is known that maximal orders over discrete valuation rings are standard orders, see for example [Art86, Denition 2.13]. As the length of an R-module is preserved under an étale extension R → S of discrete valuation rings, we may assume, using Morita equivalence, thatAξ is of the form
We conclude thatQξ is given in this case by:
Qξ=
NowR/mis a simpleR-module and sinceRis a discrete valuation ring, we havem= (π)for some uniformizing element π ∈ R. This implies m−1 = (π−1) so that m−1/R =m−1/mm−1 ∼=R/m, hencem−1/Ris also simple. So we have:
36 1.6 Chern class computations
lR(R/m) =lR(m−1/R) = 1. Counting the entries above and under the diagonal gives
lR(Qξ) =e(e−1)2 lR(R/m) +e(e−1)2 lR(m−1/R) =e(e−1). Using Morita equivalence shows that, if rk(A) =r2 = (ef)2, we get
lR(Qξ) =f2e(e−1) =f2e2(1−1e) =r2(1−1e). So we nally getc1(A) =−r22
l
P
i=1
(1−e1
i)Ci. Lemma 1.84:
Assume A is a terminal order on a smooth projective surface X with ramication curves {Ci} and ramication indices {ei} for i= 1, . . . , l. If M is a torsion-free A-module of rank one and A∗ denotes the dual sheaf of A, then we have:
c1(A∗⊗AM) =c1(M)−2c1(A). Proof:
Since M is torsion-free, we have an exact sequence:
0 −−−−→ M −−−−→ M∗∗ −−−−→ Q −−−−→ 0 (6) whereM∗∗ is the bidual of M and codim(supp(Q)) = 2. SinceA∗ is locally free, it is a locally projective A-bimodule by (1.13), particularly it is a atA-module. So tensoring (6) we get:
0 −−−−→ A∗⊗AM −−−−→ A∗⊗AM∗∗ −−−−→ A∗⊗AQ −−−−→ 0 withcodim(supp(A∗⊗AQ)) = 2. We conclude:
c1(A∗⊗AM) =c1(A∗⊗AM∗∗).
This implies that it is enough to prove the lemma for locally projectiveA-modules.
The trace pairing
tr :A × A −−−−→ OX
gives us an embedding A ,→ A∗ (which is in fact an isomorphism away from the ramication locus), so there is an exact sequence:
0 −−−−→ A −−−−→ A∗ −−−−→ R −−−−→ 0 withsupp(R)⊂
l
S
i=
Ci. Using the atness of the locally projectiveA-module M we get an exact sequence:
0 −−−−→ M −−−−→ A∗⊗AM −−−−→ R ⊗AM −−−−→ 0.
But by (1.80), we see thatc1(A∗⊗AM)−c1(M) must be an eective divisorDand that it has the form
D=
l
P
i=1
lOX,ξi((R ⊗AM)ξi),
whereξi is the generic point of Ci.
We are now in the following situation: given a discrete valuation ring R = OX,ξ, with eld of fractions K, a maximal R-order A = Aξ with ramication index e and a projective A-module N =Mξ. We want to computelR((A∗⊗AN)/N).
SinceM is a torsion-freeA-module of rank one, we know thatMη =N⊗RKis a simpleA⊗RK -module. By (1.16) N is an indecomposable A-module. But then by (1.17) all possible modules N areA-isomorphic. So it is enough to compute the length for one A-module N and we choose N =A. We have to ndlR(A∗/A), which is r2(1−1e) by looking at the proof of (1.83). We get
D=r2
l
P
i=1
(1−e1
i)Ci.
So by comparisonD=−2c1(A), which proves thatc1(A∗⊗AM) =c1(M)−2c1(A). Lemma 1.85:
AssumeX and Y are smooth projective surfaces and A is an Azumaya algebra on X. We have the projectionsp and q from X×Y toX respectively Y. IfM is a coherentA-module andN a coherentp∗A-modules, then the class
ch(Ext0p∗A,q(p∗M, N)− Ext1p∗A,q(p∗M, N) +Ext2p∗A,q(p∗M, N)) inH∗(Y,Q) depends only on the classes of ch(M) and ch(N).
Proof:
SinceAis Azumaya onX, we have thatp∗Ais Azumaya onX×Y by (A.9). Thus for ally∈Y andi≥3we see:
Extip∗A,q(p∗M, N)⊗k(y)∼=ExtiA(M, Ny) = 0
due to the base change theorem and(p∗M)y ∼=M. SoExtip∗A,q(p∗M, N) = 0for all i≥3.
Therefore we may assume, like in the proof of (1.78) that there is a complex L0 −−−−→ L1 −−−−→ L2
of locally free sheaves onY withHi(L•) =Extip∗A,q(p∗M, N). We see that the class in question is
ch(H0(L•)− H1(L•) +H2(L•)).
UsingHi(L•) =ker(di)/im(di−1) and the exact sequence
0 −−−−→ ker(di) −−−−→ Li −−−−→ im(di) −−−−→ 0, we see that, by additivity ofch, the class is:
ch(L0−L1+L2).
NowLi =q∗Homp∗A(p∗Mi, N) whereM• →M is the locally projective resolution of M. Since pis atp∗M• →p∗M is a locally projective resolution ofp∗M. Now we use additivity ofchagain to get the following class: