If for xed rst Chern classc1 one can choose a minimal second Chern classc2, then all modules classied by MA/X;c1,c2 are actually locally projective A-modules of rank one.
Proof:
AssumeMhas Chern classes as described andMis not a locally projective, but just a torsion-free A-module of rank one. Then there is a canonical exact sequence
0 −−−−→ M −−−−→ M∗∗ −−−−→ T −−−−→ 0
withM∗∗ the bidual of M, hence c1(M) =c1(M∗∗), and the quotientT is an Artinian sheaf of nite length, since M is torsion-free. But thenc1(T) = 0 andc2(T)<0which implies that
c2(M∗∗) =c2(M) +c2(T)< c2(M).
But M∗∗ is also an A-module and the second Chern class of M is the minimal one, so the assumption was wrong andM ∼=M∗∗. SinceM∗∗ is reexive it is locally free onX, so by (1.56, still to come) it is a locally projective A-module and so is M.
Lemma 1.37 ([CC11, Proposition 4.2]):
Assume A is a terminal order on a smooth projective surfaceX. IfM is a locally projectiveA -module of rank one, then for every closed point p∈X we have an isomorphism ofAp-modules:
Mp ∼=Ap. 1.3 Noncommutative cyclic covers
In this section we want to describe a method that gives us explicit examples of orders on sur-faces with prescribed ramication data, the so-called noncommutative cyclic covering trick. For detailed information see [Cha05]. To do this we start with a smooth projective scheme X and dene the notion of an invertible OX-bimodule.
Denition 1.38:
An invertible bimodule onX is of the formLσ, whereL∈P ic(X)andσ∈Aut(X). The bimodule Lσ can be thought of as the OX-module L where the left action is the usual one OXL ∼=L and the right action is twisted by the automorphism σ, that isLOX ∼=σ∗L.
Using this denition one can compute the tensor product of invertible modules using the following formula:
Lσ⊗Mτ = (L⊗σ∗M)τ σ.
The bimoduleLσ denes an auto-equivalence of Coh(X) by Lσ⊗(−) :=L⊗σ∗(−). Using invertible bimodules one can dene so-called cyclic algebras on X.
Letσ∈Aut(X) be an automorphism of nite ordere, setG=hσ|σe= 1i and pickL∈P ic(X). AssumeDis an eective Cartier divisor onX and suppose there is an isomorphism of invertible bimodules
φ:Leσ −→ O∼ X(−D).
Denote byφalso the morphism φ:Leσ −→ O∼ X(−D),→ OX and consider it as a relation on the tensor algebra
T(X, Lσ) := L
n≥0
Lnσ.
We say that the relation φ satises the so-called overlap condition, if the following diagram commutes:
Lσ⊗Le−1σ ⊗Lσ −−−−→ Oφ⊗id X⊗Lσ
id⊗φ
y
y∼ Lσ⊗ OX −−−−→∼ Lσ
We dene the cyclic algebraA(X, Lσ, φ) by:
A(X, Lσ, φ) :=T(X, Lσ)/(φ). Ifφsatises the overlap condition, then one can show that:
A(X, Lσ, φ) =
e−1
L
n=0
Lnσ. The multiplication onA(X, Lσ, φ) is induced by
Lnσ⊗Lmσ −→
(Ln+mσ n+m < e Ln+mσ →φ Ln+m−eσ n+m≥e.
Example 1.39:
AssumeF is a eld and setX=Spec(F). Pick an automorphismσ∈Aut(F)of ordereand let Gbe the cyclic group generated by σ. Now if K denotes FG, thenF/K is a cyclic extension of degreee.
An invertible bimoduleLσ can be written as F z, such that we have za=σ(a)z for a∈F. The tensor powers are given byLnσ = (F z)n=F zn, where we havezna=σn(a)zn.
Now suppose there is a relation φ :F ze −→ F, then φ is dened by multiplication with some element b ∈ F such that ze = b. Now the overlap condition is equivalent to bz = zb, which impliesσ(b) =b or b∈FG =K.
The resulting cyclic algebra A(X, Lσ, φ) is the well known cyclic algebra F[z, σ]/(ze−b). Note that this algebra is a central simpleK-algebra of K-dimension e2.
Example 1.40:
We look at the previous example and takeF =C and σ = ( ) the complex conjugation on C, thusG=Z/2Z. As the bimodule we pickLσ =Cj withj2 =−1.
Then we havejr=rj for r∈R andji =−ij for i∈C. So we see that
A(X, Lσ, φ) =C[j,( )]/(j2+ 1) =C⊕Cj=R⊕Ri⊕Rj⊕Rij. But the last algebra is known as the Hamiltonian quaternionsH.
We will be most interested in such examples whereD= 0, that is the relationφ:Leσ −→ O∼ X is an isomorphism. In this cases there are some lemmas which are of interest to us:
18 1.3 Noncommutative cyclic covers Lemma 1.41 ([BA12, Theorem 2.4]):
Assume X and Y are smooth projective surfaces, such that there is a cyclic coverπ :X→Y of degreeewith Galois groupG=< σ >. LetA(X, Lσ, φ)be a cyclic algebra coming from a relation of the form φ:Leσ −→ O∼ X. Ifφsaties the overlap condition then A(X, Lσ, φ) denes an order A on Y (via π) and its ramication over C ⊂Y is exactly the ramication ofπ above C. Lemma 1.42 ([BA12, Theorem 2.5]):
Assume X and Y are smooth projective surfaces, such that there is a cyclic cover π : X → Y of degree e, with Galois group G =< σ > and totally ramied at D ⊂X. Consider the cyclic algebra A(X, Lσ, φ) coming from a relation of the formφ:Leσ −→ O∼ X. Then the ramication of A(X, Lσ, φ) along π(D) is the cyclic cover of D given by the e-torsion line bundle L|D.
Lemma 1.43 ([BA12, Lemma 2.8]):
A cyclic algebra A(X, Lσ, φ) is a maximal order onY if for all irreducible components Ci of the ramication divisor, the cover C˜i is irreducible.
We are interested in relations of the formLeσ −→ O∼ X. Using the denition of the tensor product for bimodules, we see thatLeσ =L⊗OXσ∗L⊗OX. . .⊗OX(σe−1)∗Lsinceσe=idby denition. So if we consider P ic(X) as a G-Set for G=< σ >, then we are looking for L∈P ic(X) such that L∈ker(1 +σ+. . .+σe−1). So these line bundles can be classied by using group cohomology.
Since Gis cyclic, the cohomology of anyG-SetM can be read o the sequence . . . −−−−→N M −−−−→D M −−−−→N M −−−−→D . . . .
where D = (1−σ) and N = (1 +σ +. . .+σe−1). Now 1-cocylces of the G-set P ic(X) are exactly the line bundles with the desired relations. We will also write L for the class of the line bundle L in H1(G, P ic(X)). Here we have H0(G, P ic(X)) = ker(D) = P ic(X)G and H1(G, P ic(X)) = ker(N)/im(D). Using the group cohomology we can now see when a certain relation satises the overlap condition.
Lemma 1.44 ([BA12, Proposition 2.10]):
Assume X andY are smooth projective surfaces such that there is a cyclic coverπ :X →Y of degreeeand the lowest common multiple of the ramication indices of π ise. Then all relations created from elements of H1(G, P ic(X)) satisfy the overlap condition.
Finally, we would like to know if the orders constructed via the noncommutative cyclic covering trick are generically nontrivial, meaning we want to know if their Brauer classes are nontrivial inBr(k(Y)). Again using group cohomology this can be checked:
Lemma 1.45 ([Cha05, Corollary 4.4]):
AssumeXandY are smooth projective surfaces and suppose that there is cyclic coverπ :X→Y of degree e, with Galois group G=< σ > and totally ramied at one irreducible divisor D⊂X. Suppose further thatD is not torsion in P ic(X). Then there is a group monomorphism
Ψ :H1(G, P ic(X))→Br(k(X)/k(Y))
given as follows: if L ∈ P ic(X) represents a 1-cocycle in H1(G, P ic(X)) then any relation φ:Leσ −→ O∼ X satises the overlap condition and Ψ(L) =k(Y)⊗OY A(X, Lσ, φ) in Br(k(Y)).
HereBr(k(X)/k(Y)) =ker(f :Br(k(Y))→Br(k(X)), wheref(A) =A⊗k(Y)k(X). Lemma 1.46 ([CK11, Proposition 2.6]):
Assume A is a cyclic algebra constructed via the noncommutative cyclic covering trick, that is A=A(X, Lσ, φ), then there is a natural isomorphism:
ExtpA(A⊗X N,−)∼=ExtpO
X(N,−) for all p≥0 and all coherent OX-modules N.
1.4 Hom and Ext for modules over orders