Moduli Schemes

of Generically Simple Azumaya Modules

Norbert Hoffmann, Ulrich Stuhler

Received: December 16, 2004

Communicated by Ulf Rehmann

Abstract. Let A be an Azumaya algebra over a smooth projective variety X or more generally, a torsion free coherent sheaf of algebras over X whose generic fiber is a central simple algebra. We show that generically simple torsion free A-module sheaves have a projective coarse moduli scheme; it is smooth and even symplectic if X is an abelian or K3 surface and A is Azumaya. We explain a relation to the classical theory of the Brandt groupoid.

2000 Mathematics Subject Classification: 14J60 (Primary), 16H05 Keywords and Phrases: Moduli space, torsion free sheaf, Azumaya module, Brandt groupoid

Introduction

Let X be a smooth projective variety, e. g. a surface, over an algebraically closed field k. Let A be a sheaf of Azumaya algebras over X or more generally, a torsion free coherent sheaf of algebras over X whose generic fiber A_{η} is a central simple algebra over the function field of X. This paper is about moduli schemes of generically simple, locally projective A-module sheaves E. These moduli schemes are in close analogy to the Picard variety of X. In fact, our main result says that we do not need any stability condition for our sheaves E to construct coarse moduli schemes parameterizing them, say with fixed Hilbert polynomial or Chern classes. We find that these schemes are in general not proper over k, but they have natural compactifications: Working with torsion free sheaves E instead of only locally projective ones, we obtain projective moduli schemes. This gives lots of interesting moduli spaces, which certainly deserve further study. For example, we show in section 3 that they are smooth projective and even symplectic if X is an abelian or K3 surface and A is an Azumaya algebra. They are also related to classifying isomorphism types of Azumaya algebras A in a given central division algebra A_{η} = D, a topic already present in the classical literature on algebras; this relation is explained in section 4. We construct these moduli schemes in section 2. We use standard techniques from geometric invariant theory (GIT) and a boundedness result, which has been known for some time in the case of characteristic char(k) = 0, but is one of the deep results of A. Langer in [9] for char(k) = p > 0. Our construction works for any integral projective scheme X over k; the precise setup is formulated in section 1. As M. Lieblich has pointed out to us, this construction can also be seen as a special case of Simpson’s general result [16, Theorem 4.7], at least if char(k) = 0; for char(k) = p > 0 see also [10]. In section 3, we apply deformation theory to our A-modules, mainly in the case where X is a surface and A is an Azumaya algebra. Besides the smoothness mentioned above, we also show here that torsion free sheaves are really necessary to obtain projective moduli schemes, because locally projective sheaves of A-modules can degenerate to torsion free ones. During the final preparations of this paper, we were informed about the MITthesis of M. Lieblich [13, 12]. This thesis works much more systematically and abstractly and contains several results similar to ours in the language of algebraic stacks. We thank A. J. de Jong and M. Lieblich for informations concerning their work. Similar results have also been obtained independently by K. Yoshioka [17]; they have been used by D. Huybrechts and P. Stellari [8] to prove a conjecture of Caldararu. We thank F. Heinloth, J. Heinloth, Y. Holla and A. Langer for useful comments and discussions.

1 Families of A-modules

Let X be an integral projective scheme over the algebraically closed field k. Throughout this paper, A denotes a sheaf of associative O_{X}-algebras satisfying the following properties:

1. As a sheaf of O_{X}-modules, A is coherent and torsion free.

2. The stalk A_{η} of A over the generic point η ∈ X is a central simple algebra over the function field F = k(X) = O_{X,η}.

For example, X could be a smooth projective variety over k, and A could be a sheaf of Azumaya algebras over X.

Remark 1.1. If dim X = 1, then A_{η} is a matrix algebra over k(X) by Tsen’s theorem. So the first interesting case is dim X = 2.

Moduli Schemes of Generically Simple Azumaya Modules 371

Our main objects will be generically simple torsion free A-modules, i. e. sheaves E of left A-modules over X which are torsion free and coherent as O_{X}-modules and whose generic fiber E_{η} is a simple module over the central simple algebra A_{η}. By Wedderburn’s structure theorem, we have A_{η} ≅ Mat(n × n; D) for a division algebra D, say of dimension r^{2} over k(X); that E_{η} is simple means that it is Morita equivalent to a one-dimensional vector space over D. In particular, E has rank r^{2}n over O_{X}. Note that any such A-module E has only scalar endomorphisms: Indeed, End_{A}(E) is a finite-dimensional k-algebra; it has no zero-divisors because it embeds into the division algebra End_{A_{η}}(E_{η}) ≅ D^{op}. This implies End_{A}(E) = k because k is algebraically closed.

Lemma 1.2. Suppose that k ⊆ K is a field extension, and let K(X_{K}) be the function field of X_{K} := X ×_{k} Spec K. If D is a finite-dimensional division algebra over k(X), then D_{K} := D ⊗_{k(X)} K(X_{K}) is a division algebra, too.

Proof. Since k is algebraically closed, k(X) ⊗_{k} K is an integral domain; its quotient field is K(X_{K}). Suppose that D_{K} contains zero divisors. Clearing denominators, we can then construct zero divisors in D ⊗_{k(X)} (k(X)⊗_{k} K), which is clearly isomorphic to D ⊗_{k} K. Consequently, there is a finitely generated k-algebra A ⊆ K such that D ⊗_{k} A contains zero divisors. These zero divisors are automatically nonzero modulo some maximal ideal 𝔪 ⊂ A, so D ⊗_{k} A/𝔪 also contains zero divisors. But A/𝔪 ≅ k by Hilbert’s Nullstellensatz; hence D contains zero divisors. This contradiction shows that D_{K} has to be a division algebra if D is.

Corollary 1.3. If E is a generically simple torsion free A-module, then the pullback E_{K} of E to X_{K} is a generically simple torsion free module under the pullback A_{K} of A.

Proof. E_{K} is clearly torsion free and coherent over O_{X_{K}}. Since the generic fiber of E is Morita equivalent to a one-dimensional D-vector space, the generic fiber of E_{K} is Morita equivalent to a one-dimensional D_{K}-vector space; hence E_{K} is generically simple.

Definition 1.4. A family of generically simple torsion free A-modules over a k-scheme S is a sheaf E of left modules under the pullback A_{S} of A to X ×_{k} S with the following properties:

1. E is coherent over O_{X×_{k}S} and flat over S.

2. For every point s ∈ S, the fiber E_{s} is a generically simple torsion free A_{k(s)}-module.

Here k(s) is the residue field of S at s, and the fiber E_{s} is by definition the pullback of E to X ×_{k} Spec k(s).

We denote the corresponding moduli functor by

M = M_{A/X} : \underline{Sch}_{k} ⟶ \underline{Sets};

it sends a k-scheme S to the set of isomorphism classes of families E of generically simple torsion free A-modules over S. Our main goal is to construct and study coarse moduli schemes for this functor.

If E is a family of generically simple torsion free A-modules over S, then there is an open subset of S above which these A-modules are locally projective. However, we work with all torsion free A-modules because they satisfy the following valuative criterion for properness:

Proposition 1.5. Let V be a discrete valuation ring over k with quotient field K. Then the restriction map

M(Spec V) ⟶ M(Spec K)

is bijective.

Proof. Let π ∈ V be a uniformising element, and let l = V/(π) be the residue field of V. We denote by

X_{K} ⟶^{j} X_{V} ⟵^{i} X_{l}

the open embedding of the generic fiber and the closed embedding of the special fiber; here X_{A} := X ×_{k} Spec A for any k-algebra A. Let η (resp. ξ) be the generic point of X_{K} (resp. of X_{l}), and let

j_{η} : Spec O_{X_{V} ,η} ⟶ X_{V} (resp. j_{ξ} : Spec O_{X_{V} ,ξ} ⟶ X_{V})

be the ‘inclusion’ morphism of the subset {η} (resp. {ξ, η}) into X_{V}. Let E ∈ M(Spec K) be an A_{K}-module. Assume given an extension E ∈ M(Spec V) of E. Then E embeds canonically into j_{∗}E; in particular, the stalk E_{ξ} over the discrete valuation ring O_{X_{V} ,ξ} embeds into the generic fiber E_{η}. E is uniquely determined by E and E_{ξ} because

E = j_{∗}E ∩ j_{ξ,∗}E_{ξ} ⊆ j_{η,∗}E_{η};  (1)

this equation follows easily from the assumption that the special fiber i^{∗}E is torsion free, cf. the proof of [11, Proposition 6]. Moreover, the A-stable O_{X_{V} ,ξ}-lattice E_{ξ} ⊂ E_{η} is unique up to powers of π because its quotient modulo π is a simple module under the generic fiber of A_{l} by corollary 1.3. This implies that E is determined by E up to isomorphism, thereby proving injectivity. To prove surjectivity, we construct an extension E of E as follows: The simple A_{η}-module E_{η} is Morita equivalent to a one-dimensional vector space over the division algebra D_{K} = D ⊗_{k(X)} O_{X_{V} ,η}. Inside this vector space, we choose a free module of rank one over D_{V} := D⊗_{k(X)}O_{X_{V} ,ξ} and denote by E_{ξ} the Morita equivalent submodule of E_{η}. Then we define E by (1); this clearly defines a sheaf of A_{V} -modules over X_{V} which is flat over V, whose generic fiber j^{∗}E is E, and whose special fiber i^{∗}E is generically simple. According to the proof of [11, Proposition 6] again, E is coherent over O_{X_{V}}, and its special fiber i^{∗}E is torsion free. This shows E ∈ M(Spec V).

Remark 1.6. Suppose that X is smooth. In the trivial case A = O_{X}, generically simple locally projective A-modules are just line bundles, so they also satisfy the valuative criterion for properness; here locally projective and only torsion free modules lie in different connected components of the moduli space. This is no longer true for nontrivial A, even if A is a sheaf of Azumaya algebras over an abelian or K3 surface X: If A_{η} is not just a full matrix algebra over k(X), then every generically simple torsion free A-module is a degeneration of locally projective ones by theorem 3.6.iii below; in particular, the latter do not satisfy the valuative criterion for properness.

2 Construction of the coarse moduli schemes

We choose an ample line bundle O_{X}(1) on X and put d := dim(X). As usual, the Hilbert polynomial P (E) of a coherent sheaf E on X with respect to this choice is given by

P (E; m) := χ(E(m)) = ∑_{i=0}^{d}(−1)^{i} dim_{k} H^{i}(X; E(m))

where E(m) := E ⊗ O_{X}(1)^{⊗m}. See [7, Chapter 1.2] for details about P (E), in particular for the fact that it is a polynomial of degree d = dim(X) if E is torsion free. Recall that the Hilbert polynomial is locally constant in flat families. Keeping it fixed, we denote by

M_{A/X;P} : \underline{Sch}_{k} ⟶ \underline{Sets}

the subfunctor of M_{A/X} that parameterizes families E all of whose fibers E_{s} have Hilbert polynomial P. Our first aim is to prove that the class M_{A/X;P} (Spec k) of generically simple torsion free A-modules E with Hilbert polynomial P is bounded (in the sense of [7, Definition 1.7.5]); this will follow easily from the following apparently weaker statement:

Proposition 2.1. The class of coherent O_{X}-modules E with the following two properties is bounded:

1. E is torsion free and has Hilbert polynomial P.

2. E admits at least one A-module structure for which it is generically simple.

Proof. Suppose that E is such a sheaf on X. We recall a few concepts, which are all in [7, Chapter 1.2].

rk(E) := dim_{F} (E_{η}), deg(E) := c_{1}(E) · O_{X}(1)^{d−1}, μ(E) := \frac{deg(E)}{rk(E)} denote the rank, degree and slope of E. Writing the Hilbert polynomial as

P (E; m) = ∑_{i=0}^{d} α_{i}(E)\binom{m + i − 1}{i}

with integral coefficients α_{i} [7, Chapter 1.2], one has

deg(E) = α_{d−1}(E) − rk(E)α_{d−1}(O_{X}),

cf. [7, Definition 1.2.11]. As P (E) is fixed, it follows in particular that the slope μ(E) = μ is fixed. We denote

μ_{max}(E) := max{μ(E′)|0 ≠ E′ ⊆ E a coherent O_{X}-subsheaf of E}.

μ_{max}(E) is in fact the slope of the first subsheaf E_{max} ⊆ E in the μ-Harder-Narasimhan filtration of E [7, Section 1.6]. According to a deep result of A. Langer [9, Theorem 4.2], our class of O_{X}- modules E is bounded if the numbers μ_{max}(E) are bounded from above. To check the latter, we choose an integer m ∈ ℤ such that the coherent sheaf A(m) is generated by its global sections. Since E is generically simple, the multiplication map

A ⊗_{O_{X}} E_{max} ⟶ E

is generically surjective. Consequently, the induced map

H^{0}(A(m)) ⊗_{k} E_{max} ⟶ E(m)

is generically surjective, too. But E certainly has a μ-semistable torsion free quotient E′′ with μ(E′′) ≤ μ, e. g. the last quotient from the μ-Harder-Narasimhan filtration of E. It is easy to see that there is a nonzero map E_{max} → E′′(m), obtained by composing

E_{max} ⟶ E(m) ⟶ E′′(m).

Since E_{max} and E′′(m) are μ-semistable, this implies

μ_{max}(E) = μ(E_{max}) ≤ μ(E′′(m)) = mμ(O_{X}(1)) + μ(E′′) ≤ mμ(O_{X}(1)) + μ.

This is the required bound for μ_{max}(E).

According to the proposition, there is an integer m = m_{A/X;P} with the following property: For every generically simple torsion free A-module E with Hilbert polynomial P, E(m) is generated by global sections, and H^{i}(E(m)) = 0 for all i > 0. We keep this m fixed in the sequel and denote by N := P (m) the common dimension of all the vector spaces H^{0}(E(m)).

Proposition 2.2. i) There is a fine moduli scheme R of finite type over k that parameterizes generically simple torsion free A-modules E with Hilbert polynomial P together with a basis of H^{0}(E(m)). ii) For l ≫ m, there is an ample line bundle L_{l} on R whose fibre at E is

canonically isomorphic to det H^{0}(E(l)).

iii) The algebraic group GL(N) over k acts on R by changing the chosen bases of the H^{0}(E(m)).

iv) There is a natural action of GL(N) on L_{l} that lifts the action in iii.

v) The scheme-theoretic stabilizer of every point in R(k) coincides with the scalars 𝔾_{m} ⊆ GL(N).

Proof. i) Let Quot_{P} (A(−m)^{N}) be Grothendieck’s Quot-scheme parameterizing coherent quotients E with Hilbert polynomial P of the O_{X}-module sheaf A(−m)^{N}. We can take for R the locally closed subscheme of Quot_{P} (A(−m)^{N}) defined by the following conditions:

1. The quotient sheaf E is torsion free.

2. E is an A-module, i. e. the kernel of A(−m)^{N} → E is an A-submodule.

3. As an A-module, E is generically simple.

4. The following composed map is an isomorphism:

k^{N} ⟶ H^{0}(A^{N}) ⟶ H^{0}(E(m)).

Here the left map is given by the unit of the algebra A.

(In particular, this proves that the class of A-modules E in question is bounded.) ii) By Grothendieck’s construction of Quot-schemes, the L_{l} are a fortiori ample line bundles on Quot_{P} (A(−m)^{N}). iii) and iv) also hold for the whole Quot-scheme, cf. [7, 4.3], and its subscheme R is clearly GL(N)-invariant.

v) Let E be a generically simple torsion free A-module with Hilbert polynomial P. Choose a basis of H^{0}(E(m)) and let G ⊆ GL(N) be the scheme-theoretic stabilizer of the corresponding point in R(k). It suffices to show that G and 𝔾_{m} have the same set of points with values in k and in k[ε] where ε^{2} = 0. Every point in G(k) corresponds to an automorphism of E; hence G(k) = k^{∗} because E has only scalar endomorphisms. Similarly, every point in G(k[ε]) corresponds to an automorphism of the constant family E[ε] over Spec k[ε]. Restricting from Spec k[ε] to Spec k, we get an exact sequence

0 ⟶ End_{A}(E) ⟶^{·ε} End_{A[ε]}(E[ε]) ⟶ End_{A}(E) ⟶ 0;

again because E has only scalar endomorphisms, it implies End(E[ε]) = k[ε] and hence G(k[ε]) = k[ε]^{∗}. This proves G = 𝔾_{m}.

Theorem 2.3. If l ≫ m, then every point of R is GIT-stable for the action of SL(N) ⊂ GL(N) with respect to the linearization L_{l}.

Proof. We carry the necessary parts of [7, Chapter 4.4] over to our situation. Put V := k^{N}; then the points of R correspond to quotients

ρ : V ⊗_{k} A(−m) → E.

We fix such a point. Let V′ ⊂ V be a proper vector subspace, and put

E′ := ρ(V′ ⊗_{k} A(−m)) ⊆ E.

Then E′ is an A-submodule of E with nonzero generic fibre; since E_{η} contains no proper A_{η}-submodules, this implies E′_{η} = E_{η}. So E and E′ have the same rank, i. e. their Hilbert polynomials have the same leading coefficient. Hence

dim(V) · χ(E′(l)) > dim(V′) · χ(E(l))  (2)

if l is sufficiently large. (We can find one l uniformly for all V′ because the family of vector subspaces V′ ⊂ V is bounded.) After these preliminaries, we can check that our point ρ in R satisfies the Hilbert-Mumford criterion for GIT-stability, cf.[7, Theorem 4.2.11]. So consider a nontrivial one-parameter subgroup λ : 𝔾_{m} → SL(V) = SL(N); we will rather work with the associated eigenspace decomposition V = ⨁_{n∈ℤ} V_{n} where 𝔾_{m} acts on V_{n} with weight n. Let R̄ be the closure of R in the projective embedding given by L_{l}; this is also the closure of R in Quot_{P} (V ⊗_{k} A(−m)) because the L_{l} are ample on the whole Quot-scheme. We have to look at the limit lim_{t→0} λ(t) · ρ in R̄. This is a fixed point for the 𝔾_{m}-action, so 𝔾_{m} acts on the fibre of L_{l} over it, necessarily with some weight −μ^{L_{l}}(ρ, λ) ∈ ℤ; what we have to show for stability is μ^{L_{l}}(ρ, λ) > 0. First we describe the limit point lim_{t→0} λ(t) · ρ as a point in the Quot-scheme. We put

V_{≤n} := ⨁_{ν≤n} V_{ν} ⊆ V and E_{≤n} := ρ(V_{≤n} ⊗_{k} A(−m)) ⊆ E.

Then V_{n} = V_{≤n}/V_{≤n−1}; we put E_{n} := E_{≤n}/E_{≤n−1}, thus obtaining surjections

ρ_{n} : V_{n} ⊗_{k} A(−m) ⟶ E_{n}.

Then

ρ̄:= ⨁_{n∈ℤ} ρ_{n} : V ⊗_{k} A(−m) ⟶ ⨁_{n∈ℤ} E_{n}

is also a point in Quot_{P} (V ⊗_{k} A(−m)); it is the limit we are looking for:

ρ̄ = lim_{t→0} λ(t) · ρ.

To prove this, just copy the proof of [7, Lemma 4.4.3], replacing O_{X}(−m) by A(−m) everywhere. The second step is to consider the fibre of L_{l} over ρ̄. It is by definition

det H^{0}(⨁_{n∈ℤ} E_{n}(l));

this is canonically isomorphic to the tensor product of the determinant of cohomology of the E_{n}(l). Now 𝔾_{m} acts on V_{n} with weight n, so it acts on the fiber in question with weight

−μ^{L_{l}}(ρ, λ) = ∑_{n∈ℤ} n · χ(E_{n}(l)),

cf. [7, Lemma 4.4.4]. Finally, we use the preliminaries above to estimate this sum. If we apply (2) to V′ = V_{≤n}, E′ = E_{≤n} and sum up, we get

∑_{n∈ℤ} ( dim(V) · χ(E_{≤n}(l)) − dim(V_{≤n}) · χ(E(l))) > 0;  (3)

note that only finitely many summands are nonzero because V_{≤n} is zero or V for almost all n since almost all V_{n} are zero. Put

a_{n} := dim(V) · χ(E_{n}(l)) − dim(V_{n}) · χ(E(l));

again, all but finitely many of these integers are nonzero, and

∑_{n∈ℤ} a_{n} = dim(V) · χ(E(l)) − dim(V) · χ(E(l)) = 0.

If we write a_{≤n} := ∑_{ν≤n} a_{ν}, then

∑_{n} a_{≤n} + ∑_{n} na_{n} = 0

because the sum of the a_{n} is zero. Hence (3) is equivalent to

∑_{n∈ℤ} n( dim(V) · χ(E_{n}(l)) − dim(V_{n}) · χ(E(l))) < 0.

But ∑_{n} n dim V_{n} = 0 because 𝔾_{m} acts on V with determinant 1. Thus we obtain

∑_{n∈ℤ} n( dim(V) · χ(E_{n}(l))) < 0,

i. e. −μ^{L_{l}}(ρ, λ) · dim(V) < 0. This proves that the Hilbert-Mumford criterion for GIT-stability is satisfied here.

Now we can state our main result. See [15] for the concepts ‘geometric quotient’ and ‘coarse moduli scheme’.

Theorem 2.4. i) The action of GL(N) on R described in proposition 2.2.iii above admits a geometric quotient

M_{A/X;P} := GL(N)\R

which is a separated scheme of finite type over k.

ii) The quotient morphism

R ⟶ M_{A/X;P}

is a principal PGL(N)-bundle (locally trivial in the fppf-topology).

iii) M_{A/X;P} is a coarse moduli scheme for the moduli functor M_{A/X;P}.

iv) M_{A/X;P} is projective over k.

Proof. i) According to geometric invariant theory [15, Theorem 1.10 and Appendix 1.C] and theorem 2.3 above, the action of SL(N) on R admits a geometric quotient M_{A/X;P} which is quasiprojective over k. Then M_{A/X;P} is also a geometric quotient for the action of GL(N) on R because both groups act via PGL(N) = PSL(N). ii) This morphism is affine by its GIT-construction. According to [15, Proposition 0.9], it suffices to show that the action of PGL(N) on R is free, i. e. that

ψ : PGL(N) × R ⟶ R × R, (g, r) ↦ (g · r, r)

is a closed immersion. The fibers of ψ over k-points of R × R are either empty or isomorphic to Spec k by proposition 2.2.v above; furthermore, ψ is proper due to [15, Proposition 0.8]. Hence ψ is indeed a closed immersion. iii) Part i implies that M_{A/X;P} is a coarse moduli scheme for the functor

\underline{GL(N)\R} : \underline{Sch}_{k} ⟶ \underline{Sets}

that sends a k-scheme S to the set of GL(N)(S)-orbits in R(S). However, this functor is very close to the moduli functor M = M_{A/X;P}: We have a morphism from the functor represented by R to M; it simply forgets the extra structure. If S is a scheme over k, then two S-valued points of R have the same image in M(S) if and only if they are in the same GL(N)(S)-orbit. Thus we get a morphism of functors

ϕ : \underline{GL(N)\R} ⟶ M

which is injective for every scheme S over k. The image of ϕ consists of all sheaves E ∈ M(S) for which the vector bundle pr_{∗}E(m) over S is trivial, where pr : X ×_{k} S → S is the canonical projection.

In particular, ϕ is bijective whenever S is the spectrum of a field, and it induces an isomorphism between the Zariski sheafifications of both functors. It follows that M_{A/X;P} is also a coarse moduli scheme for the functor M. iv) We already know that M_{A/X;P} is quasiprojective. Furthermore, it satisfies the valuative criterion for properness by proposition 1.5.

In particular, the moduli functor M_{A/X} of all generically simple torsion free A-modules has a coarse moduli scheme

M_{A/X} = ∐_{P} M_{A/X;P}

which is a disjoint sum of projective schemes over k. If X is smooth of dimension d, then we have another such decomposition

M_{A/X} = ∐_{c_{1},...,c_{d}} M_{A/X;c_{1},...,c_{d}}

given by fixing the Chern classes c_{i} ∈ CH^{i}(X), the Chow group of cycles modulo algebraic equivalence. Indeed, each M_{A/X;c_{1},...,c_{d}} is open and closed in some M_{A/X;P} where P is given by Hirzebruch-Riemann-Roch. If X is a smooth projective surface, then this decomposition reads

M_{A/X} = ∐_{{c_{1}∈NS(X) \atop c_{2}∈ℤ}} M_{A/X;c_{1},c_{2}}.

3 Deformations and smoothness

We introduce the usual cohomology classes that describe deformations of a coherent A-module E, following Artamkin [1]. By definition, a deformation E of E over a local artinian k-algebra (A, 𝔪) with residue field k is a (flat) family E of coherent A-modules parameterized by Spec A together with an isomorphism k ⊗_{A} E ≅ E. Consider first the special case A = k[ε] with ε^{2} = 0. Then we have an exact sequence of A-modules

0 ⟶ k ⟶^{·ε} k[ε] ⟶ k ⟶ 0.  (4)

By definition, the Kodaira-Spencer class of the deformation E over k[ε] is the Yoneda extension class

ks(E) := [0 ⟶ E ⟶^{·ε} E ⟶ E ⟶ 0] ∈ Ext_{A}^{1}(E, E)

obtained by tensoring (4) over A with E.

Lemma 3.1. The Kodaira-Spencer map ks is a bijection between isomorphism classes of deformations of E over k[ε] and elements of Ext_{A}^{1}(E, E).

Proof. Let E be an A-module extension of E by E. Then E becomes an A[ε]- module if we let ε act via the composition E ↠ E ↪ E. According to the local criterion for flatness [4, Theorem 6.8], E is flat over k[ε] and hence a deformation of E. This defines the required inverse map.

Now let (A, 𝔪) be arbitrary again, and let Ã be a minimal extension of A. In other words, (Ã, 𝔪̃) is another local artinian k-algebra with residue field k, and A ≅ Ã/(ν) where ν ∈ Ã is annihilated by 𝔪̃. Then we have an exact sequence of A-modules

0 ⟶ k ⟶^{·ν} 𝔪̃ ⟶ A ⟶ k ⟶ 0.  (5)

By definition, the obstruction class of the deformation E over A is the Yoneda extension class

ob(E; k ↪^{·ν} Ã ↠ A) := [0 ⟶ E ⟶^{·ν} 𝔪̃ ⊗_{A} E ⟶ E ⟶ E ⟶ 0] ∈ Ext_{A}^{2}(E, E)

obtained by tensoring (5) over A with E. Whenever we want to mention A, we write ob_{A} instead of ob; on the other hand, we may omit k ↪^{·ν} Ã ↠ A if they are clear from the context.

Lemma 3.2. The obstruction class ob(E; k ↪^{·ν} Ã ↠ A) vanishes if and only if E can be extended to a deformation Ẽ over Ã.

Proof. If Ẽ is a deformation over Ã extending E, then we can tensor it over Ã with the diagram

[figure: 0; k; ·ν; 𝔪̃; 𝔪; 0; 0; k; ·ν; Ã; A; 0;]

this gives us a morphism of short exact sequences of A-modules

[figure: 0; E; ·ν; 𝔪̃ ⊗_{A} E; 𝔪 ⊗_{A} E; 0; 0; E; Ẽ; E; 0.]

The existence of such a morphism implies ob(E; k ↪^{·ν} Ã ↠ A) = 0 due to the standard exact sequence

… Ext_{A}^{1}(E, E) ⟶ Ext_{A}^{1}(𝔪 ⊗_{A} E, E) ⟶ Ext_{A}^{2}(E, E) …

Conversely, ob(E; k ↪^{·ν} Ã ↠ A) = 0 implies the existence of such a morphism of short exact sequences of A-modules. Then Ẽ becomes an Ã ⊗_{k} A-module if we let any a ∈ 𝔪̃ ⊂ Ã act via the composition Ẽ ↠ E ⟶^{a⊗} 𝔪̃ ⊗_{A} E ↪ Ẽ. According to the local criterion for flatness [4, Theorem 6.8], Ẽ is flat over Ã and hence a deformation of E extending E.

In the special case A = k[ε] and Ã = k[δ] with δ^{3} = 0, we note that (5) is the Yoneda product of (4) with itself and hence

ob(E; k ↪^{·δ^{2}} k[δ] ↠ k[ε]) = ks(E) × ks(E).  (6)

From now on, we assume that X is smooth. Then we have a trace map

tr_{O_{X}} : Ext_{O_{X}}^{i} (E, E) ⟶ H^{i}(X, O_{X})

for every coherent O_{X}-module E: it is defined using a finite locally free resolution of E, cf. [7, 10.1.2]. If E is a coherent A-module, then we define the trace map tr = tr_{A/O_{X}} as the composition

tr = tr_{A/O_{X}} : Ext_{A}^{i}(E, E) ⟶ Ext_{O_{X}}^{i} (E, E) ⟶^{tr_{O_{X}}} H^{i}(X, O_{X})

where the first map is induced by the forgetful functor from A-modules to O_{X}-modules. Similarly, one can define a trace map

tr_{A/O_{X}}^{ω_{X}} : Ext_{A}^{i}(E, E ⊗_{O_{X}} ω_{X}) ⟶ H^{i}(X, ω_{X})

where ω_{X} is the canonical line bundle on X; cf. [7, p. 218]. Now suppose that E is a deformation of E over A. Then we have a line bundle det E over X_{A} = X ×_{k} Spec A: it is defined using a finite locally free resolution of E as an O_{X_{A}}-module, cf. [7, 1.1.17 and Proposition 2.1.10].

Proposition 3.3. If Ã is a minimal extension of A, then

tr(ob_{A}(E; k ↪ Ã ↠ A)) = ob_{O_{X}} (det E; k ↪ Ã ↠ A) ∈ H^{2}(X, O_{X}).

In particular, tr(ob_{A}(E)) = 0 if the Picard variety Pic(X) is smooth.

Proof. The forgetful map Ext_{A}^{2}(E, E) → Ext_{O_{X}}^{2} (E, E) maps ob_{A}(E) to ob_{O_{X}} (E) by definition. It is known that tr_{O_{X}} maps the latter to ob_{O_{X}} (det E); cf. Artamkin’s paper [1] for the computation.

For the rest of this section, we assume that A is even a sheaf of Azumaya algebras over the smooth projective variety X of dimension d.

Proposition 3.4. Every coherent sheaf E of A-modules has a resolution of length ≤ d = dim(X) by locally projective sheaves of A-modules.

Proof. If m is sufficiently large, then the twist E(m) is generated by its global sections; this gives us a surjection ∂_{0} of E_{0} := A(−m)^{N} onto E for some N. Applying the same procedure to the kernel of ∂_{0} and iterating, we obtain an infinite resolution by locally free A-modules

… E_{d} ⟶^{∂_{d}} E_{d−1} … E_{1} ⟶^{∂_{1}} E_{0} ⟶^{∂_{0}} E.

We claim that the image of ∂_{d} is locally projective over A; then we can truncate there, and the proposition follows. It suffices to check this claim over the complete local rings Ô_{X,x} at the closed points x of X; there A becomes a matrix algebra Â_{x}, so the resulting Â_{x}- modules Ê_{i,x} are Morita equivalent to Ô_{X,x}-modules. Since Ô_{X,x} has homological dimension d, the image of ∂_{d,x} : Ê_{d,x} → Ê_{d−1,x} is projective over Â_{x}. Hence the image of ∂_{d} is indeed locally projective over A.

Our main tool to control the extension classes introduced above will be the following variant of Serre duality. To state it, we fix an isomorphism H^{d}(X, ω_{X}) ≅ k.

Proposition 3.5. We still assume that X is smooth of dimension d and that A is a sheaf of Azumaya algebras. If E and E′ are coherent A-modules, then the Yoneda product

Ext_{A}^{i}(E, E′) ⊗ Ext_{A}^{d−i}(E′, E ⊗ ω_{X}) ⟶ Ext_{A}^{d}(E, E ⊗ ω_{X})

followed by the trace map

tr_{A/O}^{ω_{X}} : Ext_{A}^{d}(E, E ⊗ ω_{X}) ⟶ H^{d}(X, ω_{X}) ≅ k

defines a perfect pairing of finite-dimensional vector spaces over k.

Proof. We start with the special case that E and E′ are locally projective over A. Then the Ext-groups in question are Zariski cohomology groups of the locally free O_{X}-module sheaves Hom_{A}(E, E′) and Hom_{A}(E′, E) ⊗ ω_{X}. But Hom_{A}(E, E′) and Hom_{A}(E′, E) are dual to each other by means of an appropriate local trace map, using the fact that the trace map A⊗_{O_{X}} A → O_{X} is nowhere degenerate because A is Azumaya. Hence this special case follows from the usual Serre duality theorem for locally free O_{X}-modules. If E and E′ are not necessarily locally projective over A, then we choose finite locally projective resolutions, using proposition 3.4. Induction on their length reduces us to the case where E and E′ have resolutions of length one by A-modules for which the duality in question holds. Now Ext_{A}^{i}(E, E′) and Ext_{A}^{d−i}(E′, E ⊗ ω)^{dual} are δ-functors in both variables E and E′, and the pairing defines a morphism between them. An application of the five lemma to the resulting morphisms of long exact sequences proves the required induction step.

Theorem 3.6. Let X be an abelian or K3 surface over k, and let A be a sheaf of Azumaya algebras over X. Suppose A_{η} ≅ Mat(n × n; D) for a central division algebra D of dimension r^{2} over the function field k(X).

i) The moduli space M_{A/X} of generically simple torsion free A-modules E is smooth. ii) There is a nowhere degenerate alternating 2-form on the tangent bundle

of M_{A/X}.

iii) If r ≥ 2, then the open locus M_{A/X}^{lp} of locally projective A-modules E is dense in M_{A/X}.

iv) If we fix the Chern classes c_{1} ∈ NS(X) and c_{2} ∈ ℤ of E, then

dim M_{A/X;c_{1},c_{2}} = Δ/(nr)^{2} − c_{2}(A)/n^{2} − r^{2}χ(O_{X}) + 2

where Δ = 2r^{2}nc_{2} − (r^{2}n − 1)c_{1}^{2} is the discriminant of E.

Proof. i) We have to check that all obstruction classes

ob_{A}(E; k ↪ Ã ↠ A) ∈ Ext_{A}^{2}(E, E)

vanish. Pic(X) is known to be smooth; using proposition 3.3, it suffices to show that the trace map

tr_{A/O_{X}} : Ext_{A}^{2}(E, E) ⟶ H^{2}(X, O_{X})

is injective. But it is straightforward to check that this map is Serre-dual to the natural map

H^{0}(X, ω_{X}) ⟶ Hom_{A}(E, E ⊗ ω_{X}),

which is an isomorphism because ω_{X} is trivial and E has only scalar endomorphisms. ii) Mukai’s argument in [14] carries over to our situation as follows. We fix an isomorphism ω_{X} ≅ O_{X}. The Kodaira-Spencer map identifies the tangent space T_{[E]}M_{A/X} with Ext_{A}^{1}(E, E). On this vector space, the Serre duality 3.5 defines a nondegenerate bilinear form. Indeed, this form is just the Yoneda product

Ext_{A}^{1}(E, E) ⊗ Ext_{A}^{1}(E, E) ⟶ Ext_{A}^{2}(E, E);

the right hand side is isomorphic to k by Serre duality again. Equation (6) implies that this bilinear form is alternating because all obstruction classes vanish here. iii) Let E be a generically simple torsion free A-module, and let Quot_{l}(E/A) be the moduli scheme of quotients E ↠ T where T is a coherent A-module of finite length l. This is a closed subscheme of Grothendieck’s Quot-scheme Quot_{lnr}(E) parameterizing those exact sequences of coherent sheaves 0 → E′ → E → T → 0 for which E′ is an A-submodule, i. e. the composition

A ⊗ E′ ↪ A ⊗ E ⟶^{·} E ↠ T

vanishes; here (nr)^{2} = rk(A). In particular, Quot_{l}(E/A) is projective over k. We show by induction that Quot_{l}(E/A) is connected; cf. [7, 6.A.1]. Let Drap_{l_{1},l_{2}}(E/A) be the moduli scheme of iterated quotients E ↠ T_{1} ↠ T_{2} where T_{i} is a coherent A-module of finite length l_{i} for i = 1, 2; this is again a closed subscheme of some Flag-scheme [7, 2.A.1] and hence projective over k. Sending such an iterated quotient to T_{1} and to the pair (T_{2}, supp(T_{1}/T_{2})) defines two morphisms

Quot_{l+1}(E/A) ⟵^{θ_{1}} Drap_{l+1,l}(E/A) ⟶^{θ_{2}} Quot_{l}(E/A) × X.

Using Morita equivalence over the complete local rings at the support of torsion sheaves, it is easy to see that θ_{1} and θ_{2} are both surjective; moreover, the fibers of θ_{2} are projective spaces and hence connected. This shows that Quot_{l+1}(E/A) is connected if Quot_{l}(E/A) is; thus they are all connected. Let E still be a generically simple torsion free A-module; we have to show that its connected component in M_{A/X} contains a locally projective A-module. Let

E^{∗} := Hom_{O_{X}} (E, O_{X})

be the dual of E; this is a sheaf of right A-modules. The double dual E^{∗∗} is a sheaf of left A-modules again; it is locally free over O_{X} and hence locally projective over A. We have an exact sequence

0 ⟶ E ⟶^{ι} E^{∗∗} ⟶^{π} T ⟶ 0

where T is a coherent A-module of finite length l. There is a natural map

Quot_{l}(E^{∗∗}/A) ⟶ M_{A/X}

that sends a quotient to its kernel; since Quot_{l}(E^{∗∗}/A) is connected, we may assume that T is as simple as possible, i. e. that its support consists of l distinct points x_{1}, …, x_{l} ∈ X where the stalks T_{x_{i}} are Morita-equivalent to coherent skyscraper sheaves of length one. In this situation, we adapt an argument of Artamkin [2] to show that E can be deformed to a locally projective A-module if r ≥ 2. We consider the diagram

[figure: Ext_{A}^{1}(E, E); δ; Ext_{A}^{2}(T, E); π^{∗}; ι_{∗}; Ext_{A}^{2}(E^{∗∗}, E); Ext_{A}^{2}(T, E^{∗∗}); ⨁_{i=1}^{l} Ext_{A}^{2}(T_{x_{i}}, E^{∗∗}).]

Here π^{∗} is Serre-dual to π_{∗} : Hom_{A}(E, E^{∗∗}) → Hom_{A}(E, T) because ω_{X} ≅ O_{X}. But the only morphisms from E to E^{∗∗} are the multiples of ι; hence π^{∗} = 0, and the connecting homomorphism δ from the long exact sequence is surjective. ι_{∗} corresponds under Serre duality and Morita equivalence to the direct sum of the restriction maps

Hom_{O_{x_{i}}} (O_{x_{i}}^{r}, k_{x_{i}}) ⟶ Hom_{O_{x_{i}}} (𝔪_{x_{i}} ⊕ O_{x_{i}}^{r−1}, k_{x_{i}}) where O_{x} = Ô_{X,x} is the complete local ring of X at x, 𝔪_{x} ⊆ O_{x} is its maximal ideal, and k_{x} = O_{x}/𝔪_{x} is the residue field. Assuming r ≥ 2, these restriction maps are obviously nonzero. Hence there is a class ξ ∈ Ext_{A}^{1}(E, E) whose image in Ext_{A}^{2}(T_{x_{i}}, E^{∗∗}) is nonzero for all i. Since all obstruction classes vanish, we can find a deformation E of E over a smooth connected curve whose Kodaira-Spencer class is ξ; it remains to show that a general fiber E′ of E is locally projective over A. Forming the double dual, we get an exact sequence

0 ⟶ E′ ⟶ (E′)^{∗∗} ⟶ T′ ⟶ 0.

An explicit computation using Morita equivalence shows that the forgetful map

k^{r} ≅ Ext_{A}^{2}(T_{x_{i}}, E^{∗∗}) ⟶ Ext_{O_{X}}^{2} (T_{x_{i}}, E^{∗∗}) ≅ k^{n^{2}r^{3}}

is injective. Hence the Kodaira-Spencer class of E in Ext_{O_{X}}^{1} (E, E) also has nonzero image in Ext_{O_{X}}^{2} (T_{x_{i}}, E^{∗∗}) for all i. According to [2, Corollary 1.3 and the proof of Lemma 6.2], this implies that E′ is less singular than E, i. e. the length of T′ as an O_{X}-module is strictly less than nr at every point of its support. But T′ is an A-module, so these lengths are all divisible by nr; hence T′ = 0, and E′ is locally projective over A. iv) Using i and iii, it suffices to compute the dimension of

T_{[E]}M_{A/X} ≅ Ext_{A}^{1}(E, E) ≅ H^{1}(End_{A}(E))

for a generically simple locally projective A-module E. Note that H^{0} and H^{2} are Serre-dual to each other and hence both one-dimensional here. The endomorphism sheaf End_{A}(E) is an Azumaya algebra of rank r^{2} over X, and the natural map

A ⊗_{O_{X}} End_{A}(E) ⟶ End_{O_{X}} (E)

is an isomorphism; this is easily checked by reducing to the case that A is a matrix algebra. Furthermore, c_{1}(A) is numerically equivalent to zero because A ≅ A^{dual} using the trace over O_{X}; similarly, c_{1}(End_{A}(E)) and c_{1}(End_{O_{X}} (E)) are also numerically equivalent to zero. Using this, the formalism of Chern classes yields

r^{2}c_{2}(A) + (nr)^{2}c_{2}(End_{A}(E)) = Δ  (7)

where Δ is the discriminant of E as above. Hence

χ(End_{A}(E)) = −Δ/(nr)^{2} + c_{2}(A)/n^{2} + r^{2}χ(O_{X})

by Hirzebruch-Riemann-Roch.

4 Left and right orders

We still assume that X is smooth projective of dimension d and that A is a sheaf of Azumaya algebras over X; furthermore, we suppose that the generic fiber A_{η} is a division algebra D of dimension r^{2} over k(X). In this case, generically simple locally projective A-modules are just locally free A-modules of rank one; of course every such A-module E can be embedded into D. The endomorphism sheaf of such a left A-module E is then just an order in D acting by right multiplication on E, exactly as in the classical picture of the Brandt groupoid [3, VI, §2, Satz 14]. The Picard group Pic(X) acts on the moduli scheme M_{A/X} by tensor product: a line bundle L ∈ Pic(X) acts as E ↦ E ⊗_{O_{X}} L. The projective group scheme Pic^{0}(X) of line bundles L algebraically equivalent to zero acts on the individual pieces M_{A/X;c_{1},...,c_{d}} because c_{i}(E ⊗ L) = c_{i}(E) for all i. The same remarks hold for

M_{A/X}^{lp} = ∐_{c_{1},...,c_{d}} M_{A/X;c_{1},...,c_{d}}^{lp}

where the superscript lp denotes the open locus of locally projective (and hence locally free) A-modules.

Proposition 4.1. There is a geometric quotient of M_{A/X}^{lp} by the action of Pic(X); it is a disjoint sum of separated schemes of finite type over k. Its closed points correspond bijectively to isomorphism classes of Azumaya algebras over X with generic fiber D.

Proof. Pic(X) acts with finite stabilizers; this follows from

det(E ⊗ L) ≅ det(E) ⊗ L^{⊗r^{2}}.  (8)

For fixed Chern classes c_{1}, …, c_{d}, let G ⊆ Pic(X) be the subgroup of all line bundles L that map M_{A/X;c_{1},...,c_{d}}^{lp} to M_{A/X;c_{1},...,c_{d}}^{lp}. Then G contains Pic^{0}(X), and its image in NS(X) = Pic(X)/Pic^{0}(X) is contained in the r^{2}-torsion and hence finite, so G is a projective group scheme. Therefore a geometric quotient of M_{A/X;c_{1},...,c_{d}}^{lp} by G exists and is separated and of finite type over k, according to [6, Exposé V, Théorème 7.1]. It remains to construct the announced bijection. As

End_{A}(E)^{op} ≅ End_{A}(E ⊗ L)^{op} =: A′,

which is again an Azumaya algebra with generic fiber D, we obtain a well defined map from closed points of the quotient to isomorphism classes of such A′. Conversely, given A, A′, the possible locally free A-modules E of rank one with End_{A}(E)^{op} ≅ A′ all differ only by tensoring with line bundles. This can be seen as follows: Suppose that E and E′ are locally free A-modules of rank one with

End_{A}(E)^{op} ≅ End_{A}(E′)^{op} ≅ A′.

We choose embeddings of E and E′ into A_{η} = D; this also embeds End_{A}(E)^{op} and End_{A}(E′)^{op} into D. The given isomorphism between them induces an automorphism of D, i. e. conjugation with an element of D; altering the embedding E′ ↪ D by a right multiplication with this element, we may assume that End_{A}(E)^{op} = End_{A}(E′)^{op} =: A′ as subalgebras of D. There is an open subscheme U ⊆ X such that A|_{U} = E|_{U} = E′|_{U} = A′|_{U}. Furthermore, X \U is a finite union of divisors D_{1}, … D_{l}; it is enough to study the question at the generic points x_{i} of the D_{i}. There the local ring O_{X,x_{i}} is a discrete valuation ring; over its completion Ô_{X,x_{i}}, A becomes isomorphic to a matrix algebra, so we can describe the situation using Morita equivalence as follows: Ê_{x_{i}}, Ê′_{x_{i}} correspond to lattices over Ô_{X,x_{i}} in F_{x_{i}}^{r} (F_{x_{i}} the completion of F at x_{i}) such that EndÔ_{x_{i}} (Ê_{x_{i}})^{op} = EndÔ_{x_{i}} (Ê′_{x_{i}})^{op} = Â′_{x_{i}}. But then it is an

easy exercise to see that Ê′_{x_{i}} = π_{i}^{N_{i}} Ê_{x_{i}} for some N_{i} ∈ ℤ where π_{i} ∈ Ô_{x_{i}} is a uniformising element. From this our claim follows. This shows that the map above is injective. For the surjectivity, we consider two Azumaya algebras A, A′ ⊆ D and define E(U) := {f ∈ D : A|_{U} ·f ⊆ A′|_{U}}. Using Morita equivalence as above, it is easy to check that E is a locally free A-module of rank one with End_{A}(E)^{op} = A′; this proves the surjectivity.

Remark 4.2. If A′ is another sheaf of Azumaya algebras over X with generic fiber A′_{η} ≅ D = A_{η}, then the moduli spaces M_{A/X} and M_{A′/X} are isomorphic. Indeed, the preceeding proof shows that there is a locally free A-module E of rank one with End_{A}(E)^{op} ≅ A′; then E is a right A′-module, and one checks easily that the functor E ⊗_{A′} defines an equivalence from left A′-modules to left A-modules.

Remark 4.3. If X is a surface, then this quotient can be decomposed explicitly into pieces of finite type as follows: The action of Pic(X) preserves the discriminant Δ(E) ∈ ℤ, so we get a decomposition

M_{A/X}^{lp}/Pic(X) = ∐_{Δ∈ℤ} M_{A/X;Δ}^{lp}/Pic(X).

Now the first Chern class c_{1}(E) ∈ NS(X) decomposes M_{A/X;Δ}^{lp} into pieces of finite type. But c_{1}(E ⊗ L) = c_{1}(E) + r^{2}c_{1}(L), and r^{2}NS(X) has finite index in NS(X), so M_{A/X;Δ}^{lp}/Pic(X) is indeed of finite type over k. According to equation (7), fixing Δ(E) corresponds to fixing c_{2}(A′) ∈ ℤ where A′ = End_{A}(E)^{op}. If X is an abelian or K3 surface, then theorem 3.6.iv yields

dim M_{A/X;Δ}^{lp}/Pic(X) = c_{2} − (r^{2} − 1)χ(O_{X})

where c_{2} = c_{2}(A′) = Δ/r^{2} − c_{2}(A) ∈ ℤ is the second Chern class of the Azumaya algebras A′ that this quotient parameterizes.

Remark 4.4. M. Lieblich [13, 12] has compactified such moduli spaces of Azumaya algebras using his generalized Azumaya algebras, i. e. algebra objects in a derived category corresponding to endomorphism algebras of torsion free rank one A-modules.

Remark 4.5. Since the automorphism group of the matrix algebra Mat(r×r) is PGL(r), Azumaya algebras of rank r^{2} correspond to principal PGL(r)-bundles. Moduli spaces for the latter have recently been constructed and compactified by T. Gomez, A. Langer, A. Schmitt and I. Sols [5].

References

[1] I. V. Artamkin. On the deformation of sheaves. Izv. Akad. Nauk SSSR, 52(3):660–665, 1988. Translation: Math. USSR Izv., 32(3):663–668, 1989.

[2] I. V. Artamkin. Deformation of torsion-free sheaves on an algebraic surface. Izv. Akad. Nauk SSSR, 54(3):435–468, 1990. Translation: Math. USSR Izv., 36(3):449–485, 1991.

[3] M. Deuring. Algebren. Zweite, korrigierte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 41. Springer-Verlag, Berlin, 1968.

[4] D. Eisenbud. Commutative algebra. With a view toward algebraic geometry. Springer-Verlag, New York, 1995.

[5] T. Gomez, A. Langer, A. Schmitt and I. Sols. Moduli Spaces for Principal Bundles in Arbitrary Characteristic. preprint math.AG/0506511.

[6] A. Grothendieck and M. Demazure. SGA 3: Schémas en groupes. I: Propriétés générales des schémas en groupes. Springer-Verlag, Berlin, 1970.

[7] D. Huybrechts and M. Lehn. The geometry of moduli spaces of sheaves. Vieweg-Verlag, Braunschweig, 1997.

[8] D. Huybrechts and P. Stellari. Proof of Caldararu’s conjecture. An appendix to a paper by Yoshioka. preprint math.AG/0411541.

[9] A. Langer. Semistable sheaves in positive characteristic. Ann. of Math. (2), 159(1):251–276, 2004.

[10] A. Langer. Moduli spaces of sheaves in mixed characteristic. Duke Math. J., 124(3):571–586, 2004.

[11] S. G. Langton. Valuative criteria for families of vector bundles on algebraic varieties. Ann. of Math. (2), 101:88–110, 1975.

[12] M. Lieblich. Moduli of twisted sheaves. preprint math.AG/0411337.

[13] M. Lieblich. Moduli of Twisted Sheaves and Generalized Azumaya Algebras. PhD thesis, Massachusetts Institute of Technology, 2004.

[14] S. Mukai. Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math., 77(1):101–116, 1984.

[15] D. Mumford and J. Fogarty. Geometric invariant theory. Second Edition. Springer-Verlag, Berlin, 1982.

[16] C. T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. I. Publ. Math. IHES, 79:47–129, 1994.

[17] K. Yoshioka. Moduli spaces of twisted sheaves on a projective variety. preprint math.AG/0411538.

Norbert Hoffmann Mathematisches Institut Bunsenstraße 3–5 D-37073 Göttingen Germany hoffmann@uni-math.gwdg.de Ulrich Stuhler Mathematisches Institut Bunsenstraße 3–5 D-37073 Göttingen Germany stuhler@uni-math.gwdg.de 390

Documenta Mathematica 10 (2005)