The period-index problem for the Brauer group of an algebraic surface

by A.J. de Jong, MIT

Introduction

Let K be a field and let A be a central simple algebra over K, i.e., a finite dimensional algebra over K whose center is K. Such an algebra has the property that A ⊗_{K} K̄ ≅ M_{n}(K̄) for some integer n ∈ ℕ. The degree deg A of A is the integer n such that dim_{K}A = n^{2}. As is well known, we can write A ≅ M_{r}(D) where D is a central division algebra over K. The index ind A of A is the integer i such that dim_{K}D = i^{2}, i.e., it is the degree of D. Two central simple algebras A, A′ are called (Brauer) equivalent if there exist integers r, s ∈ ℕ such that A ⊗ M_{r}(K) ≅ A′ ⊗ M_{s}(K). This happens if and only if D ≅ D′. The set of equivalence classes of central simple algebras over K forms an abelian torsion group with addition given by [A] + [A′] = [A ⊗_{K} A′]. This group is the Brauer group Br(K) of K. Finally, the period or exponent exp A of A is the order of the class [A] of A in the Brauer group Br(K) of K. For this, and (a)–(d) below, see e.g. [9].

We recall that a splitting field for A is a field extension K ⊂ L such that A ⊗_{K} L ≅ M_{n}(L). The following properties are true in general: (a) the period divides the index, (b) any prime number dividing ind A also divides exp A, (c) for any splitting field L finite over K we have ind A divides [L : K], and in fact (d) ind A = min{[L : K] | L splits A}. It is clear that the values of exp A, ind A depend only on the class of A in the Brauer group, hence we can view them as functions on the Brauer group.

Theorem. Let k be a separably closed field and let k ⊂ K be a finitely generated field extension of transcendence degree 2. Let A as above have period prime to the characteristic of k. Then the period of A equals the index of A, in a formula: exp A = ind A.

We refer the reader to Section 1 for some further speculation, on the case of general transcendence degrees. Another question that seems to be still open is whether or not every A as in the theorem is cyclic.

The key idea of this paper is to produce a family of field extensions K ⊂ L_{t} that interpolates between a splitting field (at time t = 0) and the trivial extension (at time t = ∞). See for example Section 4. Deformation theory followed by specialization is used to transfer the equality exp A = ind A from time 0 to time ∞. See Section 6 for example.

The proof of the theorem follows from the following four steps.

Step 1. It suffices to prove the theorem in all cases where the period is a prime number ℓ not equal to the characteristic of k.

This is well known, but we include the argument for sake of completeness. Assume the theorem is known in the prime exponent case. We deduce the theorem by ascending induction on e = exp A. If e is not a prime number, then write e = ℓm. By assumption A^{⊗m} can be split by a field extension K ⊂ K′ of degree ℓ. This implies that A_{K′} has exponent dividing m. By the induction hypothesis applied to the pair (K′, A_{K′}), there exists a field extension K′ ⊂ L of degree dividing m splitting A_{K′}. Therefore A is split by the extension K ⊂ L of degree dividing ℓm and we conclude by the divisibility properties (a)-(d) mentioned just above the theorem.

Step 2. We may assume that k is algebraically closed.

Let A be a central simple algebra over K of period ℓ. Let K′ be the compositum of K and k̄ in an algbraic closure of K. If the result holds in case k is algebraically closed, i.e., if it holds for K′, then there exists a degree ℓ extension L′ of K′ splitting A. Hence there exists a finite field extension L of K of degree ℓp^{n} splitting A. By the divisibility properties (a)–(d) mentioned above we deduce that the index of A is ℓ. This proves step 2.

From now on in the rest of the article we assume that k is algebraically closed. By resolution of singularities for surfaces, for each K/k as in the theorem there exists a smooth projective surface X/k such that K ≅ k(X). Using Azumaya algebras one can define the Brauer group of any scheme. In particular, we have Br(X) and a map Br(X) → Br(K). For the definition of Azumaya algebras, the Brauer group of a scheme, and most of the standard results on these Brauer groups we refer to Grothendieck’s exposés [11]. We also refer to a joint work with M. Artin [3] where most of the results needed are reviewed and explained.

Step 3. The Theorem holds whenever the element of Br(K) comes from an element of the ℓ-torsion subgroup Br(X)[ℓ] of Br(X), where ℓ is as above.

This is proved in Section 6.

Step 4. Using the result of Step 2 we prove the result holds for any ℓ-torsion element of Br(K).

This is proved in Section 7.

A remark on the argument presented in this paper. The most technical argument of this paper is the proof of Step 3. Here we use the result, proved in Section 4, that any degree n finite field extension K ⊂ K′ of function fields of surfaces can be “deformed and specialized” to the trivial extension K ⊂ K × K × … × K. This is a trivial statement in itself, but it is complicated by our desire to keep track of various geometrical properties in order to prove Step 3 in Section 6. This necessitates a description of Azumaya algebras over reducible surfaces in Section 5. In Section 8 we briefly explain an alternative and simpler argument, due to M. Artin, which proves Step 3; this can be read instead of Sections 4, 5, and part of Section 6. The author has wanted to preserve Sections 4 and 5 since the arguments presented there may be of use in other situations; for example a similar argument occurs in [15].

A variety is an integral separated scheme of finite type over a field. A curve is a 1 dimensional variety, and a surface is a variety of dimension 2.

Acknowledgements. Thanks are due to M. Artin who made the author aware of the problem and pointed out some simplifications (as well as the example in Section 1). The author would also like to thank the referee for doing a very thorough job. This research was conducted by the author for the Clay Mathematics Institute. This material is based upon work supported by the National Science Foundation under Grant No. 9970049.

1. A lower bound on the index

In view of Tsen’s theorem for curves, and our theorem for surfaces, a natural guess for a general relation between the period and the index is that ind A divides (exp A)^{n−1} if A is defined over a function field in n variables (i.e., a finitely generated field extension of transcendence degree n of k). This question is raised in [6]. The power n − 1 cannot be improved. This was known in 1935, see [17] for examples; ours below is a special case. See [7] for some unramified examples.

Let u, v_{1}, ..., v_{n} be independent variables and let K = k(u; v_{1}, ..., v_{n}). Let f_{1}, ..., f_{n} ∈ k[u, v_{1}] be elements, with f_{1} = u. Assume that K[x_{1}, ..., x_{n}]/(x_{i}^{2} − f_{i}) is a domain (i.e. that the roots of the f_{i} are not related) and that f_{i}(0, 0) ≠ 0 for i > 1. Let B_{i} = K⟨x_{i}, y_{i}⟩ be the cyclic algebra over K defined by x_{i}^{2} = f_{i}, y_{i}^{2} = v_{i}, and y_{i}x_{i} = −x_{i}y_{i}.

1.1. Proposition. With the above notation, A := B_{1} ⊗_{K} B_{2} ⊗_{K} ⋯ ⊗_{K} B_{n} is a domain.

1.2. Corollary. In this case the transcendence degree of K over k is n + 1, the exponent of A is 2 and the index is 2^{n}.

Proof of the proposition. The algebra A is generated over k by x_{1}, ..., x_{n}, y_{1}, ..., y_{n}, with defining relations y_{i}x_{i} = −x_{i}y_{i}, all other pairs of variables commute, and relations x_{i}^{2} = f_{i}(x_{1}^{2}, y_{1}^{2}). In particular, for i > 1, y_{i} is a regular normal element, and A is graded by degree in y_{i}. To show that A is a domain, it suffices to show that the elements of degree zero in y_{i} are regular. This eliminates the variables y_{i}, i > 1, from consideration, and reduces us to the subring A_{0} generated by x_{1}, ..., x_{n}, y_{1}.

It follows from the hypothesis that the subring R of A_{0} generated by u, v_{1}, x_{2}, ..., x_{n} is a central commutative domain of dimension 2 over which A_{0} is defined by the relations x_{1}^{2} = u, y_{1}^{2} = v_{1}, and y_{1}x_{1} = −x_{1}y_{1}. Moreover, because f_{i}(0, 0) ≠ 0 for i > 1, the completion of R at the origin is the power series ring R̂ = k[[u, v_{1}]]. And the completion Â_{0} of A_{0} is the standard cyclic algebra over this power series ring, which we know to be a domain. □

2. Elementary transformations

In this section we describe a method to change a given Azumaya algebra along a divisor. This idea occurs in many contexts, in particular in the theory of moduli of vector bunldes. We refer to the beautiful book [13] by Huybrechts and Lehn on moduli of vector bundles for a discussion of this procedure in the context of vector bundles and many references.

Let X be a scheme and let A be an Azumaya algebra over X. Suppose that i : D → X is an effective Cartier divisor and suppose that we can write A|_{D} = End(\overline{V}) for some locally free sheaf \overline{V} over D. Given a locally direct summand \overline{F} ⊂ \overline{V} we can construct the elementary transform of A along D with respect to \overline{F}.

First let B ⊂ A be the subalgebra of sections of A which preserve the filtration \overline{F} ⊂ \overline{V}. So there is a short exact sequence

0 ⟶ B ⟶ A ⟶ i_{∗}Hom(\overline{F} , \overline{Q}) ⟶ 0

where \overline{Q} = \overline{V} /\overline{F}. The elementary transform is then an Azumaya algebra A′ containing B as a subalgebra and fitting into an exact sequence

0 ⟶ B ⟶ A′ ⟶ i_{∗}Hom(I_{D}/I_{D}^{2} ⊗ \overline{Q}, \overline{F}) ⟶ 0.

Étale locally we can describe the situation as follows. There exist (étale locally) a locally free sheaf V and compatible isomorphisms A ≅ End(V) and V |_{D} ≅ \overline{V}. Then we define a locally free sheaf V′ by the sequence

0 ⟶ V′ ⟶ V ⟶ i_{∗}\overline{Q} ⟶ 0,

and A′ ≅ End(V′). Moreover, upon choosing (étale locally) a splitting V = F ⊕ Q with \overline{F} = F |_{D} we see that

A = ( \begin{matrix}End(F) & Hom(Q, F) \\ Hom(F, Q) & End(Q) \end{matrix} ) and A′ = ( \begin{matrix}End(F) & I_{D}^{−1} ⊗ Hom(Q, F) \\ I_{D} ⊗ Hom(F, Q) & End(Q) \end{matrix} ) .

It is clear that these étale local constructions glue to give a global A′ as claimed above.

In the following lemma, we use the class of an Azumaya algebra A of rank n^{2}. This is the element cl(A) ∈ H^{2}(X, μ_{n}) which can be defined by the coboundary ∂ : H^{1}(X, ℙ𝔾𝕃_{n}) → H^{2}(X, μ_{n}) coming from the short exact sequence of sheaves

0 ⟶ μ_{n} ⟶ 𝕊𝕃_{n} ⟶ ℙ𝔾𝕃_{n} ⟶ 0

applied to the class of A in H^{1}(X, ℙ𝔾𝕃_{n}). We remark that if the Azumaya algebra has the form A = End(V) for some locally free sheaf V of rank n, then cl(A) = c_{1}(det V) in H^{2}(X, μ_{n}). In this case, we have det (V′) ≅ det (V) ⊗ I_{D}^{⊗(n−rank \overline{F})}. We generalize this as follows.

2.1. Lemma. Suppose that A has rank n^{2} over O_{X} and that n is invertible in O_{X}. In H^{2}(X, μ_{n}) we have

cl(A′) = cl(A) + (rank \overline{F}) · [D].

Proof. First we give a proof in case X is quasi-projective using Čech-cohomology. (This argument can be made to work on any scheme using a hypercovering instead of a Čech covering.) Below we indicate P. Gille’s suggestion for a proof in the general case. Choose an étale covering U = {U_{i} → X} such that there exist isomorphisms α_{i} : A|_{U_{i}} → End(V_{i}) and isomorphisms v_{i} : det V_{i} → O_{U_{i}}. Then there are canonical isomorphisms pr_{1}^{∗}(V_{i}) = pr_{2}^{∗}(V_{j}) ⊗ L_{ij} compatible with α_{i} for some L_{ij} on U_{ij} = U_{i} ×_{X} U_{j}. Really, L_{ij} ∈ H^{1}(U_{ij}, μ_{ℓ}) as v_{i} and v_{j} define an isomorphism L_{ij}^{⊗n} → O. In any case, our assumption on X assures that we can find a refinement of U such that L_{ij} is trivial in H^{1}(U_{ij}, μ_{n}), see [1]. Thus we may assume we have isomorphisms α_{ij} : pr_{1}^{∗}(V_{i}) → pr_{2}^{∗}(V_{j}) compatible with the α’s and v’s. Then

pr_{13}^{∗}(α_{ki}) ∘ pr_{23}^{∗}(α_{jk}) ∘ pr_{12}^{∗}(α_{ij})

is multiplication by an nth root of unity ζ_{ijk} ∈ μ_{n}(U_{ijk}). By definition, the class cl(A) of A is the class of the cocycle (ζ_{ijk}) ∈ C^{2}(U, μ_{n}).

We may also assume that there are functions g_{i} ∈ O(U_{i}) such that I_{D} is generated by g_{i} over U_{i}, and that there are invertible functions f_{ij} such that f_{ij}^{n}g_{i} = g_{j}. The cohomology class [D] is represented by the 2-cocycle (θ_{ijk}) defined by

θ_{ijk}^{−1} = pr_{13}^{∗}(f_{ki}) · pr_{23}^{∗}(f_{jk}) · pr_{12}^{∗}(f_{ij}).

The inverse comes in because [D] = −c_{1}(I_{D}).

At this point we simply observe how to change the choices made above in order to compute cl(A′). By the description of A′ above we see that there are subsheaves V′_{i} ⊂ V_{i} so that the quotient V_{i}/V′_{i} is a locally free sheaf of rank n − r = n − rank \overline{F} supported on U_{i} ×_{X} D and such that A′|_{U_{i}} = End(V′_{i}). Then we can define

v′_{i} : det V′_{i} = det V_{i} ⊗ I_{D}^{n−r} ⟶^{v_{i}g_{i}^{−n+r}} O_{U_{i}}.

Then we have to adjust α_{ij} accordingly, namely

α′_{ij} = α_{ij}f_{ij}^{n−r}.

This immediately implies that

cl(A′) = cl(A) − (n − r)[D]

as desired.

Here is a sketch of the proof in general (after a suggestion by P. Gille). Let f : Y → X be the Brauer-Severi variety associated to A. Then f^{∗}A ≅ End(V). Since the construction of the elementary transform commutes with flat pullback, we see the result in the cohomology of Y by the remark above the lemma. Note that R^{1}f_{∗}μ_{n} = (0). Thus H^{2}(X, μ_{n}) ↪ H^{2}(Y, μ_{n}), by the Leray spectral sequence for f. The result follows. □

The étale cohomology group H^{2}(X, 𝔾_{m}) is called the cohomological Brauer group of X. The class of A in the cohomological Brauer group of X is the image of cl(A) under the map on étale cohomology induced by the map of sheaves μ_{n} → 𝔾_{m}.

2.2. Proposition. Suppose that X is a quasiprojective scheme of dimension 2 over k and A is an Azumaya algebra over X of rank n^{2}, with n prime to the characteristic of k. Suppose that η ∈ H^{2}(X, μ_{n}) is such that cl(A) and η have the same image in H^{2}(X, 𝔾_{m}). Then there exists an elementary transformation A′ of A such that cl(A′) = η.

Proof. The Kummer sequence 0 → μ_{n} → 𝔾_{m} → 𝔾_{m} → 0 gives rise to the short exact sequence

0 → Pic(X)/n Pic(X) → H^{2}(X, μ_{n}) → H^{2}(X, 𝔾_{m})[n] → 0.

Write cl(A) − η = [D] with D an effective Cartier divisor on X. Now we use that Br(D) = (0), the Brauer group of a proper curve over a separably closed field is trivial, see [11, III Cor. 5.8]. Hence we can do an elementary transformation along D (using the lemma below) to obtain what we want by Lemma 2.1. □

2.3. Lemma. Let D be a 1-dimensional quasiprojective scheme over k with ample sheaf O_{D}(1), and let \overline{V} be a rank n locally free sheaf on D. For each r, 0 < r < n, there exists a locally direct summand \overline{F} ⊂ \overline{V} of rank r, namely, for m >> 0 the image of a general map

O_{D}(−m)^{r} ⟶ \overline{V}

is locally free.

Proof. This is a standard Bertini argument. First, for m >> 0 the sheaf \overline{V} (m) is generated by global sections. Set V = H^{0}(D, \overline{V} (m)). For each closed point d ∈ D we let T_{d} ⊂ V^{r} be the set of r-tuples of sections (s_{1}, …, s_{r}) which are linearly dependent in the fibre \overline{V} ⊗ κ(d) of \overline{V} at d. The codimension of T_{d} in V^{r} is n + 1 − r ≥ 2. There is a 1-dimensional family of subsets T_{d} as d varies over the points of D. More precisely, there is a closed subscheme T ⊂ D × V^{r} such that T_{d} = V^{r} × {d} ∩ T . The image of T under the projection map to V^{r} has codimension at least 1. Thus there is a point of V^{r} which is not in any of the T_{d}. Take such a point and think of it as a map O(−m)^{r} → \overline{V}. This gives the desired map. □

3. Obstructions

We recall that obstructions to deforming an Azumaya algebra A are given by a certain cohomology group in degree 2. There is a short exact sequence

0 ⟶ O_{X} ⟶ A ⟶ Der_{O}(A, A) ⟶ 0,

i.e., any O_{X}-derivation of A is locally innner. Hence A/O = Der_{O}(A, A). Using this the following lemma follows.

3.1. Lemma. Let X → X′ be a closed immersion defined by a sheaf of ideals I ⊂ O_{X′} of square zero. Let A be an Azumaya algebra over X. If H^{2}(X, I ⊗ (A/O)) = 0 then A lifts to X′.

Proof. Étale locally A lifts uniquely up to isomorphism. The sheaf of O′-isomorphisms of a given lift A′ reducing to the identity on A is canonically isomorphic to I ⊗ Der_{O}(A, A). As usual, the obstructions to glueing local lifts then lie in H^{2} of this sheaf. □

Set sA = Ker(A ⟶^{Trace} O). The trace pairing A ⊗ A → O, a ⊗ a′ ↦ Trace(aa′) induces a perfect duality between A/O and sA. Thus, if X is a 2 dimensional projective scheme over k, then the vanishing of H^{2}(X, A/O) is equivalent to the vanishing of H^{0}(X, sA ⊗ ω_{X/k}). Here ω_{X/k} is the dualizing sheaf on X as in [12, Chapter III]. If k has characteristic p > 0 and if p divides the rank of A over O then O ⊂ sA. In this case the nonvanishing of H^{0}(X, ω_{X/k}) implies the nonvanishing of H^{2}(X, A/O) for any A. Thus the following proposition is optimal in some sense.

3.2. Proposition. Suppose that X is a reduced connected projective scheme over k, equidimensional of dimension 2. Let A be an Azumaya algebra of rank n^{2} over O_{X}. Assume that n is not zero in k. Then there exists an elementary transform A′ of A such that cl(A′) = cl(A) in H^{2}(X, μ_{n}) and H^{0}(X, sA′ ⊗ ω_{X/k}) = 0.

Proof. We construct A′ as follows. First we pick a sufficiently ample divisor H on X and we pick a member D ∈ |nH|. Since Br(D) = (0) by [11, III Cor. 5.8], we may pick an isomorphism A|_{D} = End(\overline{V}) for some locally free sheaf \overline{V} on D. Next, we pick a sufficiently negative invertible sheaf L on D and a map ϕ : L → \overline{V}. Below we will indicate exactly what the conditions are under which the construction of the elementary transform A′ solves the problem posed by the proposition.

If \overline{V} ⊗ L^{−1} is globally generated, then \overline{F} = im (ϕ) is a locally direct summand of \overline{V} for a suitable choice of ϕ. See Lemma 2.3. We let A′ be the elementary transform of A along D with respect to \overline{F}. By Lemma 2.1 we have cl(A) = cl(A′) in H^{2}(X, μ_{n}). Thus it suffices to prove the vanishing result.

Recall that we have the following exact sequences:

0 ⟶ B ⟶ A ⟶ i_{∗}Hom(\overline{F} , \overline{Q}) ⟶ 0

and

0 ⟶ B ⟶ A′ ⟶ i_{∗}Hom(I_{D}/I_{D}^{2} ⊗ \overline{Q}, \overline{F}) ⟶ 0.

Note that the trace map Trace : A → O induces a surjective map B → O. Let us write sB for the kernel of this map. In the sequences above we can replace A, B, and A′ by sA, sB and sA′.

As a first step we claim that, if we choose H sufficiently ample, L sufficiently negative and suitable D and ϕ, the map

H^{0}(X, sA ⊗ ω_{X/k}) ⟶ H^{0}(X, i_{∗}Hom(\overline{F} , \overline{Q}) ⊗ ω_{X/k})

is injective.

To see this let T ⊂ X be a finite set of closed smooth points of X. For each t ∈ T let W_{t} = κ(t)^{n} denote the standard n-dimensional vector space over the residue field κ(t) ≅ k. Since A is an Azumaya algebra we can choose an isomorphism of κ(t)-algebras α_{t} : A ⊗ κ(t) ≅ End(W_{t}). This in turn induces an isomorphism A ⊗ ω_{X/k} ⊗ κ(t) = End(W_{t}) well determined up to nonzero scalars; the scalar depends on the choice of a basis of the 1-dimensional κ(t)-vector space ω_{X/k} ⊗ κ(t). Choose a line L_{t} ⊂ W_{t} for each t ∈ T. For any such set of choices we consider the subspace

U({α_{t}, L_{t}, t ∈ T }) ⊂ H^{0}(X, sA ⊗ ω_{X/k})

of sections whose values at the points t ∈ T correspond, via the isomorphisms α_{t} to endomorphisms of W_{t} which preserve the subspaces L_{t}. Pick a nonzero σ ∈ U({α_{t}, L_{t}, t ∈ T }) if there is one. Since ω_{X/k} is a torsion free sheaf (see Lemma 3.3 below), the nonzero element σ ∈ U({α_{t}, L_{t}, t ∈ T }) has a nonzero value σ_{x} at some smooth point x ∈ X(k). Choose an algebra isomorphism α_{x} : A ⊗ κ(x) = End(W_{x}) = M_{n}(k) and a compatible isomorphism A ⊗ ω_{X/k} ⊗ κ(x) = End(W_{x}) as above. Since n is not zero in k we see that σ_{x} is not a scalar endomorphism. Hence there is a line L_{x} in W_{x} which is not preserved by σ_{x}. Setting T′ = T ∪ {x}, we see that U({α_{t}, L_{t}, t ∈ T′}) does not contain σ. We conclude that there exist a choice of T, α_{t} and L_{t} such that U({α_{t}, L_{t}, t ∈ T }) = (0).

Let S be the finite set of closed points of X which were found in the lemma below. Let D ∈ |nH| be an effective Cartier divisor on X, which avoids S and contains T. It is straightforward to see that such a divisor exists if H is sufficiently ample. (In fact we may even choose D such that D is reduced but we will not use this.) As explained at the start of the proof there exists an isomorphism β : A|_{D} → End(\overline{V}) for some locally free sheaf \overline{V} on D. Since T ⊂ D the isomorphisms α_{t} and β_{t} combine to give algebra isomorphisms End(\overline{V}_{t}) = End(W_{t}). By Skolem-Noether this corresponds to an isomorphism of vector spaces \overline{V}_{t} ≅ W_{t} unique up to scalars. Using these isomorphisms we may think of L_{t} as a subspace of the fibre \overline{V}_{t} of \overline{V} at t.

According to the strategy explained at the beginning of the proof we now choose a map ϕ : L → \overline{V} for some very negative invertible sheaf L. Namely, we choose a section s of Hom(L, \overline{V}) = H^{0}(D, \overline{V} ⊗ L^{−1}) such that the value s_{t} is a nonzero element the subspace L_{t} ⊗ (L^{−1} ⊗ κ(t)) for each t ∈ T and otherwise general. Such a section will exist if L^{−1} is sufficiently ample. Exactly as in the proof of Lemma 2.3 one shows that the image \overline{F} of ϕ is a locally direct summand of rank 1 (again under the assumption that L^{−1} is sufficiently ample relative to the choices of T, D, and \overline{V}). The requirement on the values s_{t} garantees that \overline{F}_{t} is equal to L_{t} as subspaces of \overline{V}_{t}.

At this point it is clear that our claim holds true since the kernel of the displayed map above is contained in U({α_{t}, L_{t}, t ∈ T }) by construction!

To finish the proof it suffices to show that

H^{0}(X, i_{∗}Hom(I_{D}/I_{D}^{2} ⊗ \overline{Q}, \overline{F}) ⊗ ω_{X/k}) = 0.

This is the same space as the first of the following two spaces:

H^{0}(D, N_{D|X} ⊗ \overline{Q}^{∧} ⊗ L ⊗ ω_{X/k}) ↪ H^{0}(D, N_{D|X} ⊗ \overline{V}^{∧} ⊗ L ⊗ ω_{X/k}).

By the lemma below the scheme D is Cohen-Macaulay and the sheaf ω_{X/k} ⊗O_{D} is a Cohen-Macaulay module over O_{D}. Therefore the second space above vanishes when L is sufficiently negative, as D is equidimensional of dimension 1 (since X is equidimensional of dimension 2). Namely, if G is any Cohen-Macaulay module on D, then H^{0}(D, G⊗L) = Ext_{D}^{1}(G⊗L, ω_{D/k})^{∧} by Serre Duality on D. On the other hand, Ext_{D}^{1}(G⊗L, ω_{D/k}) = H^{1}(D, Hom(G, ω_{D/k})⊗L^{−1}) since the sheaf \underline{Ext}^{1}(G, ω_{D/k}) is zero by the Cohen-Macaulay assumptions. The vanishing then follows from standard vanishing for the higher cohomology groups of a coherent sheaf twisted by a sufficiently ample sheaf. □

A quasi-coherent O_{X}-module F is torsion free when the canonical map F → F ⊗ K_{X} is injective, where K_{X} is the sheaf of total fractions of O_{X}. Basically this means that no nonzero section of F is annihilated by a local section f of O_{X} which is a non-zero divisor on O_{X}.

3.3. Lemma. Let X be a reduced projective connected scheme over k, equidimensional of dimension 2. Then ω_{X/k} is torsion free. There exists a finite set of closed points S ⊂ X such that for any effective Cartier divisor D ⊂ X which avoids S the scheme D is Cohen-Macaulay and the sheaf ω_{X/k} ⊗ O_{D} is a Cohen-Macaulay module.

Proof. Since X is projective there is a finite morphism π : X → ℙ^{2}. The sheaf F = π_{∗}(O_{X}) is a coherent torsion free sheaf on ℙ^{2}. By duality for a finite morphism, we have π_{∗}ω_{X/k} = Hom(F, ω_{ℙ^{2}/k}) and hence ω_{X/k} is torsion free. Since ℙ^{2} is nonsingular we see that F is locally free outside of codimension 2. By [18, Corollary 18.17] we deduce that X is Cohen-Macaulay away from a finite set of points. Let S be the set of points where X is not Cohen-Macaulay. Then ω_{X/k} is a Cohen-Macaulay module over X − S, see [18, Theorem 21.8]. Thus the divisor D is Cohen-Macaulay as well, and ω_{X/k} ⊗ O_{D} is a Cohen-Macaulay module over D. □

4. Making families

Let X be a projective smooth surface over k and let k(X) ⊂ L be a finite separable field extension. In this section we construct a family of surfaces, which in some sense gives an interpolation between the field extension k(X) ⊂ L and the trivial field extension. The end result is the following lemma. (Presumably the result is valid for any finite field extension k(X) ⊂ L.)

4.1. Lemma. Given k(X) ⊂ L finite separable there exist: (a) a diagram

\begin{matrix}W & ⟶^{g} & X \\ f & ↓ \\ C \end{matrix}

of morphisms of projective varieties, (b) closed points 0, ∞ ∈ C(k), and (c) irreducible components Y ⊂ W_{0} = f^{−1}(0) and V ⊂ W_{∞} = f^{−1}(∞) such that

(i) C is a nonsingular projective curve,

(ii) f is flat,

(iii) W is normal of dimension 3,

(iv) Y is a smooth projective surface, g|_{Y} : Y → X is dominant and L ≅ k(Y) as k(X) extensions,

(v) W_{0} is reduced, connected and equidimensional of dimension 2,

(vi) g(W_{0} \ Y) is not dense in X, and

(vii) g|_{V} : V → X is birational and f is generically smooth along V.

Proof. Use the theorem of the primitive element to write

L = k(X)[θ]/(θ^{n} + a_{1}θ^{n−1} + … + a_{n})

with a_{i} ∈ k(X). Choose pairwise distinct b_{i} ∈ k and set

M = k(X)(t)[θ]/(θ^{n} + a_{1}θ^{n−1} + … + a_{n} + t(θ − b_{1})(θ − b_{2}) ⋯ (θ − b_{n})).

Let W → X × ℙ^{1} be the normalization of X × ℙ^{1} in M. Here t is the coordinate on ℙ^{1}. The morphisms f : W → ℙ^{1} and g : W → X are defined as the composition of the finite morphism W → X × ℙ^{1} with the two projections.

Of course, W does not yet satisfy all properties of the lemma but it is a good start. Remark that as W is normal, any fibre of f is S_{1} and hence reduced if and only if it is generically reduced. From our choice of M we see that W_{0} is integral, that g : W_{0} → X is dominant and that k(W_{0}) ≅ L as k(X) extensions. (This implies that the geometric generic fibre of f is integral.) Furthermore, W_{∞} = f^{−1}(∞) is reduced and has n irreducible components (corresponding one to one to the b_{i}) and each of these is mapped birationally to X by g. Pick one of these and call it V (in fact V is isomorphic to X, but we won’t use this). We also see that f is generically smooth along W_{0} and V (i.e., V occurs with multiplicity 1 in W_{∞}).

Let Y → W_{0} be a projective desingularization of W_{0}. By [12, II 7.17] there exists a sheaf of ideals I_{0} on W_{0} such that Y = Bl_{W_{0}}(I_{0}). Let I ⊂ O_{W} be the corresponding sheaf of ideals on W (i.e., defining the same closed subscheme). We let

b : W′ ⟶ W

be the blowing up of the sheaf of ideals I on W. The strict transform of W_{0} is isomorphic to Y; hence we may think of Y as a closed subscheme of W′. For reasons of dimension, Y ⊂ W′_{0} is an irreducible component of W′_{0} = (f′)^{−1}(0), where f′ = f ∘ b. (Similarly we write g′ = g ∘ b and later on we will encounter f′′, g′′ etc.)

Next, we let

ν : W′′ ⟶ W′

be the normalization morphism. Note that ν is an isomorphism away from W′′_{0}, that ν is finite and that ν is an isomorphism generically along Y. (Use that b is an isomorphism generically along Y, so W′ is normal at the generic point of Y.) Thus ν^{−1}(Y)_{red} → Y is a birational finite morphism, so ν^{−1}(Y)_{red} ≅ Y as Y is nonsingular. By abuse of notation we think of Y as a closed subscheme of W′′ as well, and as before f′′ is generically smooth along Y.

The blow up b may have introduced “new” irreducible components in W′_{0} and hence it may no longer be the case that W′′_{0} is reduced. For this reason we apply one last operation which we describe now. Pick a finite (ramified and possibly inseparable) morphism of nonsingular projective curves π : C → ℙ^{1}. Let 0 ∈ C(k) be a point with π(0) = 0, also let ∞ ∈ C(k) be a point with π(∞) = ∞. Set

W′′′ = (W′′ ×_{ℙ^{1}} C)^{normalized} ⟶^{π′} W′′.

For a suitable choice of C, π, 0 the scheme W′′′_{0} will be (generically) reduced. References are [14, Lemma 2.3] or [15, Proposition 2.1] (argument based on [5]) or [16]. As before we see that (π′)^{−1}(Y)_{red} ≅ Y. Also, there is a unique irreducible component V′′′ of W′′′_{∞} mapping to V ×_{∞} ∞ in W ×_{ℙ^{1}} C.

It is fairly easy to see that W′′′, C, f′′′, g′′′, Y, V′′′ is a solution to the problem posed in the lemma. Perhaps one word on the validity of property (vi). This follows from the observation that under the sequence of maps

W′′′ ⟶^{π′} W′′ ⟶^{ν} W′ ⟶^{b} W ⟶ X × ℙ^{1} ⟶ X

every irreducible component of W′′′_{0} with the exception of Y gets blown down by the morphism b. □

5. Brauer groups of reducible surfaces

Let X be a scheme and suppose that X_{1}, X_{2} are two closed subschemes. Let Z = X_{1} ∩ X_{2} be the scheme theoretic intersection and assume that the sequence

0 ⟶ O_{X} ⟶ O_{X_{1}} ⊕ O_{X_{2}} ⟶ O_{Z} ⟶ 0

is a short exact sequence. In other words, the scheme X is obtained by gluing X_{1} and X_{2} along Z. Locally free sheaves on X correspond to triples (F_{1}, F_{2}, ϕ) consisting of a locally free sheaf on X_{1}, a locally free sheaf on X_{2} and a gluing map of the restrictions to Z.

The short exact sequence above induces a short exact sequence of sheaves on the small étale site of X:

0 ⟶ O_{X}^{∗} ⟶ O_{X_{1}}^{∗} ⊕ O_{X_{2}}^{∗} ⟶ O_{Z}^{∗} ⟶ 0.

This in turn gives a long exact sequence of étale cohomology groups of which we will only need the following part:

Pic(X_{1}) ⊕ Pic(X_{2}) ⟶^{ρ} Pic(Z) ⟶^{∂} H^{2}(X, 𝔾_{m}) ⟶ H^{2}(X_{1}, 𝔾_{m}) ⊕ H^{2}(X_{2}, 𝔾_{m}).

The map ρ maps (L_{1}, L_{2}) to L_{1}|_{Z} ⊗ L_{2}|_{Z}^{−1}.

5.1. Proposition. Assume that X is a reduced projective scheme over k, equidimensional of dimension 2. Suppose that X_{1}, X_{2} are (scheme theoretic) unions of irreducible components of X, that Z = X_{1} ∩ X_{2} has dimension ≤ 1, and that X is obtained by gluing X_{1}, X_{2} along Z as above. Let n be prime to the characteristic of k. Any element of order n in ker (H^{2}(X, 𝔾_{m}) → H^{2}(X_{1}, 𝔾_{m}) ⊕ H^{2}(X_{2}, 𝔾_{m})) is the class of an Azumaya algebra A of rank n^{2} over X.

No doubt the proposition is true also in case n is divisible by the characteristic of k. The strategy of the proof is as follows. We will construct a list of candidates for the Azumaya algebras A and then we will show that these fill out ker (H^{2}(X, 𝔾_{m})[n] → H^{2}(X_{1}, 𝔾_{m}) ⊕ H^{2}(X_{2}, 𝔾_{m})).

As expected the construction depends on a choice of L_{Z} ∈ Pic(Z) such that L_{Z}^{⊗n} is in the image of ρ. Let us choose a pair of invertible sheaves L_{i} ∈ Pic(X_{i}) and an isomorphism L_{Z}^{⊗n} ⊗ L_{1}|_{Z} → L_{2}|_{Z}. Then we can find a pair of locally free sheaves \overline{V}_{i}, i = 1, 2 of rank n on Z such that det \overline{V}_{i} ≅ L_{i}|_{Z} and an isomorphism α : \overline{V}_{1} ⊗ L_{Z} → \overline{V}_{2}. (For example we can take \overline{V}_{1} = O_{Z}^{n−1} ⊕ L_{1}|_{Z} and \overline{V}_{2} = \overline{V}_{1} ⊗ L_{Z}.)

The lemma below applies to the quadruples (X_{i}, Z, \overline{V}_{i}, L_{i}) and we obtain locally free sheaves V_{i} on X_{i} such that det V_{i} ≅ L_{i} and such that V_{i}|_{Z} ≅ \overline{V}_{i}. Let us define the Azumaya algebra A by gluing A_{i} = End(V_{i}) using the isomorphism over Z induced by α : \overline{V}_{1} ⊗ L_{Z} → \overline{V}_{2}.

Now we would be done if we could show that the class [A] of the Azumaya algebra A constructed above is equal to ∂[L_{Z}]. Since we do not know how to perform this calculation, we argue instead as follows. The cardinality of the set ker (H^{2}(X, 𝔾_{m})[n] → H^{2}(X_{1}, 𝔾_{m}) ⊕ H^{2}(X_{2}, 𝔾_{m})) is the cardinality of the set

\frac{\{L_{Z} ∈ Pic(Z) | L_{Z}^{n} ∈ im (ρ)\}}{im (ρ)}.

This cardinality is finite, at least if n is prime to the characteristic of k, since then Pic(X_{i})/n Pic(X_{i}) and Pic(Z)[n] are finite. Therefore it suffices to show that if A and A′ are constructed as above starting with L_{Z} and L′_{Z}, then [A] = [A′] in H^{2}(X, 𝔾_{m}) only if L′_{Z} ⊗ L_{Z}^{−1} ∈ im (ρ).

To see this we work out what it means that [A] = [A′]. Since the map Br(X) → H^{2}(X, 𝔾_{m}) is injective [11, I nr. 2], this means that the Azumaya algebra A^{opp} ⊗ A′ is isomorphic to End(V) for some locally free sheaf V on X. By our construction of A and A′ we get isomorphisms

V |_{X_{i}} ⟶ V′_{i} ⊗ V_{i}^{∧} ⊗ G_{i}

for certain invertible O_{X_{i}}-modules G_{i}. Over Z these isomorphism have to be compatible with the isomorphisms α : \overline{V}_{1} ⊗ L_{Z} → \overline{V}_{2} and α′ : \overline{V} ′_{1} ⊗ L′_{Z} → \overline{V} ′_{2}. It is easy to see that this induces an isomorphism L_{Z} ⊗ (L′_{Z})^{−1} → G_{1}|_{Z} ⊗ G_{2}^{−1}|_{Z}.

This finishes the proof of the proposition, except for the lemma just below. To state it, let X be a projective scheme equidimensional of dimension 2 over k. We will assume that the smooth locus of X → Spec k is dense in X. (This is slightly weaker then assuming that X is reduced, which is actually sufficient for our purposes.) Further, let Z ⊂ X be a closed subscheme of dimension ≤ 1.

5.2. Lemma. Let notation be as explained above. Suppose that \overline{V} is a locally free sheaf of constant rank r on Z and L an invertible O_{X}-module such that det \overline{V} ≅ L|_{Z}. Then there exists a locally free sheaf V on X such that det V ≅ L and V |_{Z} ≅ \overline{V}.

Proof. Choose a very ample sheaf O_{X}(1) on X. Pick m large enough. Then there exists a short exact sequence

0 ⟶ O_{Z}(−m)^{r−1} ⟶ \overline{V} ⟶ \overline{Q} ⟶ 0

for some invertible O_{Z}-module \overline{Q}. See Lemma 2.3. Our assumptions imply that there exists Q ∈ Pic(X) such that Q|_{Z} ≅ \overline{Q}. Think of \overline{V} as an element of Ext_{Z}^{1}(\overline{Q}, O_{Z}(−m)^{r−1}).

There are maps

Ext_{X}^{1}(Q, O_{X}(−m)^{r−1}) ⟶^{α} Ext_{Z}^{1}(\overline{Q}, O_{Z}(−m)^{r−1}) ⟶^{β} Ext_{X}^{1}(Q, O_{Z}(−m)^{r−1}).

It is elementary to check that the map β is injective. The cokernel of the map β ∘α is a subspace of the group Ext_{X}^{2}(Q, I_{Z}(−m)^{r−1}). Thus if this group vanishes then there exists an extension F of Q by O_{X}(−m)^{r−1} lifting \overline{V}.

To get around the possibility that the obstruction group is nonzero we argue as follows. Let T be a finite set of smooth closed points of X, disjoint from Z, having at least one in each of the irreducible components of X of dimension 2. Then X \ T has cohomological dimension 1 for quasi-coherent sheaves (for example if T is the intersection of two effective ample divisors, then X \ T is the union of two affines). Hence Ext_{X\T}^{2} (Q, I_{Z}(−m)^{r−1}) = H^{2}(X \ T, Q^{∗} ⊗ I_{Z}(−m)^{r−1}) = (0). Thus we can find an extension V′ of \overline{V} to X \ T with det (V′) ≅ L. Then j_{∗}(V′) = V works, where j : X \ T → X is the inclusion morphism. Here we use that a finitely generated reflexive module over a 2 dimensional regular local ring is free. □

6. Unramified classes

In this section we prove the assertion of step 3, using the results above. Let X be a projective nonsingular surface over k and let η ∈ Br(X) have prime order ℓ, ℓ ≠ char(k). (The argument in this section works just as well when the order of η is not prime as long as it is not divisible by the characteristic of k.) To start off, we choose η̃ ∈ H^{2}(X, μ_{ℓ}) lifting the image of η in H^{2}(X, 𝔾_{m}). This is possible since H^{2}(X, μ_{ℓ}) surjects onto H^{2}(X, 𝔾_{m})[ℓ], see the proof of Proposition 2.2.

Let k(X) ⊂ L be a separable splitting field for η, and let

\begin{matrix}W & ⟶^{g} & X \\ f & ↓ \\ C \end{matrix}

and 0, ∞, Y, V be as in Lemma 4.1. The strategy of the proof is as follows. We are going to construct an Azumaya algebra A on W_{0} of rank ℓ^{2} having cl(A) = η̃|_{W_{0}}. Then we are going to deform it to get an Azumaya algebra on an étale neighbourhood of W_{0} in W. Finally we specialize along the divisor V to get the desired division algebra over k(X).

Let R ⊂ W_{0} be the (scheme theoretic) union of the other irreducible components of W_{0}. Let Z ⊂ W_{0} be the scheme theoretic intersection of Y and R. Then dim Z ≤ 1 and since W_{0} is reduced there is a short exact sequence

0 ⟶ O_{W_{0}} ⟶ O_{Y} ⊕ O_{R} ⟶ O_{Z} ⟶ 0.

In other words, we are in the situation as described in Proposition 5.1.

By construction η maps to zero in Br(k(Y)) = Br(L). For the nonsingular surface Y we have Br(Y) ⊂ Br(k(Y)) (see [11, II Cor. 1.10]), hence η maps to zero in Br(Y) and to zero in H^{2}(Y, 𝔾_{m}) also. Let Γ ⊂ X be the reduced closed image of the map g|_{R} : R → X. By construction, Γ is a curve. Then η|_{Γ} = 0 since Br(Γ) = 0 by [11, III Cor. 5.8]. Note that g|_{R} : R → X factors through Γ and hence (g|_{R})^{∗}(η) = 0 in H^{2}(R, 𝔾_{m}) as well.

In other words we can apply Proposition 5.1 and get an Azumaya algebra of rank ℓ^{2} over W_{0} whose class in H^{2}(W_{0}, 𝔾_{m}) is equal to (g|_{W_{0}})^{∗}(η). By Proposition 2.2 there exists another Azumaya algebra A of rank ℓ^{2} over W_{0} such that cl(A) = (g|_{W_{0}})^{∗}η̃.

Now that we have our desired Azumaya algebra A, we may replace it by another which is unobstructed and has class cl(A) = η̃|_{W_{0}}, see Proposition 3.2. The proof of the following lemma is completely standard; we do not claim any originality in the arguments.

6.1. Lemma. There exist an étale neighbourhood (C′, 0) → (C, 0) and an Azumaya algebra B on W ×_{C} C′ such that B_{0} ≅ A.

Sketch of proof. Deformation: Since A is unobstructed, there exists an Azumaya algebra A_{1} on W_{1} = W ×_{C} Spec O_{C,0}/𝔪^{2} lifting A, see Lemma 3.1. (Note that the ideal defining the closed immersion W_{0} ↪ W_{1} is isomorphic to the structure sheaf of W_{0}.) It is easy to show that H^{2}(W_{1}, A_{1}/O) = (0) as well. By the same reasoning we can lift A_{1} to an Azumaya algebra A_{2} on W_{2} = W ×_{C} Spec O_{C,0}/𝔪^{4}. And so on.

Algebraization: By Grothendieck’s theorem on the existence of coherent sheaves [EGA III, 5.1.4] there exists an Azumaya algebra B̂ on W ×_{C} Spec Ô_{C,0} such that

A_{i} ≅ B̂ ⊗ O_{C,0}/𝔪^{2^{i}}.

Approximation: We refer the reader to the works of M. Artin; we give references to [4] where this kind of argument is explained as well. Note that the Azumaya algebra B̂ can be defined over a finite type O_{C,0}- subalgebra of Ô_{C,0}. Now we use that Ô_{C,0} can be written as a direct limit of smooth O_{C,0}-algebras [4, 3.6 Proposition 8]. Thus there exists a smooth O_{C,0}-algebra P, together with a map σ : P → Ô_{C,0} and an Azumaya algebra B_{P} over W ×_{C} Spec P wich gives back B̂ upon base change by σ. Finally, as P is smooth over O_{C,0}, we can choose a quotient P → O′ such that the induced ring map O_{C,0} → O′ is an étale extension of local rings, and such that the closed point of Spec O′ ⊂ Spec P corresponds to the closed point defined by σ [4, 2.3 Proposition 5]. The resulting étale neighbourhood C′ = Spec O′ with induced Azumaya algebra are the solution to our problem. □

We claim that after replacing (C′, 0) by a further étale neighbourhood we have cl(B) = η̃|_{W ×_{C}C′}. Namely, the proper base change theorem for the morphism p : W ×_{C} C′ → C′ and the sheaf μ_{ℓ} says that the stalk (R^{2}p_{∗}μ_{ℓ})_{0} is equal to H^{2}(W_{0}, μ_{ℓ}). The class cl(B) and the class η̃ give two sections of the sheaf R^{2}p_{∗}(μ_{ℓ}) whose stalks agree by construction. The claim follows.

Finally, we specialize. Let \overline{C}′ be a projective nonsingular model of the curve C′ and let ∞′ ∈ \overline{C}′(k) be a point lying over ∞ ∈ C(k). Let ζ be the generic point of V ×_{∞} {∞′}. By construction the local ring

R = O_{V ×_{C} \overline{C}′,ζ}

is a discrete valuation ring with residue field k(X) and field of fractions k(W ×_{C} C′). In fact, R is a k(X)- algebra and the class of B ⊗ f.f.(R) is the image of η under the map Br(k(X)) → Br(R) → Br(f.f.(R)). By Lemma 6.2 below we obtain the result.

6.2. Lemma. Suppose that R is a discrete valuation ring with fraction field K and residue field κ. Suppose that η is a class in Br(R) whose restriction to Br(K) has index i. Then the image of η in Br(κ) has index dividing i.

Proof. The completion R̂ of R is a discrete valuation ring with the same residue field. The index of î of η over K̂ divides i. Hence we may assume that R is complete.

Let D be a division algebra of degree i over K representing η. We claim that the maximal R-order B in D is an Azumaya algebra over R whose class is η. From this the result follows immediately. We will now prove the claim.

The order B is the set of elements of D whose determinant lie in R, and B is a (noncommutative) discrete valuation ring with valuation ν(b) = ord_{R}(det (b)). The assumption of the lemma implies that there exists an s ∈ ℕ and an R-order B′ in M_{s}(D) which is Azumaya. The R-submodule B′ · B^{s} of the D vector space D^{s} is a (B′, B)-bimodule. As a right B-module it is torsion free, hence free of rank s (as B is a dvr). Since B′ acts on the left on this module we get an injection B′ → M_{s}(B). Any Azumaya algebra over R is a maximal order hence we get B′ = M_{s}(B). Thus M_{s}(B) is Azumaya over R and it follows easily that B is Azumaya over R. □

7. The ramified case

In this section we prove the assertion of Step 4. Let X be a projective nonsingular surface over k and let η ∈ Br(k(X)) be an element of order (exponent) ℓ. We assume that ℓ is a prime number not equal to the characteristic of k.

Let D ⊂ X be the ramification divisor of the class η, in other words we assume that η comes from an element of Br(X \ D) which we also denote η. After blowing up X we may assume that D is a normal crossings divisor on X. Pick effective divisors E, E′ on X such that (1) D ∪ E has normal crossings, (2) E is smooth, and (3) D+E ≡ ℓ(D+E′) (rational equivalence). We leave it to the reader to show such a pair (E, E′) exists. Let L = O_{X}(−D − E′) and let s_{1} ∈ Hom(L^{⊗ℓ}, O_{X}) be a nonzero section which is zero along D + E. Let s_{2} ∈ Hom(L, O_{X}) be a nonzero section which vanishes along D + E′. Then s_{2}^{ℓ} ∈ Hom(L^{ℓ}, O_{X}) vanishes to order ℓ along D+E′. In other words, the rational function h = s_{1}/s_{2}^{ℓ} has divisor div(h) = D+E −ℓ(D+E′).

Using these choices we construct a sheaf of algebras R over X × ℙ^{1}. Set N = pr_{1}^{∗}(L) ⊗ pr_{2}^{∗}(O_{ℙ^{1}}(−1)) and

R = ⨁_{i=0}^{ℓ−1} N^{⊗i}.

The multiplication maps N^{i} ⊗ N^{j} → N^{i+j} are the obvious ones for i + j < ℓ, and for i + j ≥ ℓ we use a map ϕ : N^{ℓ} → O_{X×ℙ^{1}} to get N^{i} ⊗ N^{j} → N^{i+j} → N^{i+j−ℓ}. Let (λ, μ) denote homogenous coordinates on ℙ^{1}. We take (using the obvious notational convention)

ϕ = λ^{ℓ}s_{1} + μ^{ℓ}s_{2}^{ℓ}.

Finally, we denote Y = \underline{Spec} R → X × ℙ^{1}.

Let t = μ/λ be the coordinate on ℙ^{1}. The reader may verify that Y is an integral scheme with function field a cyclic extension of k(X × ℙ^{1}) given by the equation u^{ℓ} = t^{−ℓ}h + 1, where h = s_{1}/s_{2}^{ℓ} as above. There are morphisms f : Y → ℙ^{1} and g : Y → X. Set Y_{t} = f^{−1}(t) for t ∈ ℙ^{1}(k). The surface Y_{t} is a cyclic covering of X (via the morphism g) which ramifies over the divisor D + E_{t} = V (s_{1} + t^{ℓ}s_{2}^{ℓ}). This formula is used to define E_{t}. Note that E_{0} = E.

Let S ⊂ X be the set of nodes of the curve D ∪ E ⊂ X. Note that every curve in the pencil ℙ(ks_{1} + ks_{2}^{ℓ}) ⊂ |D + E| has singularities at every s ∈ S. Furthermore, the divisor D + E = D + E_{0} has only ordinary double points (at the points of S). Thus there is a nonempty Zariski open U ⊂ ℙ^{1} such that for t ∈ U(k) the divisor D + E_{t} has a node at every s ∈ S and no other singularities. We conclude that Y_{t} for t ∈ U has an A_{ℓ−1}-singularity above each s ∈ S and no other singularities.

A word about the fibre Y_{∞}. This has ℓ irreducible components. Pick one of these, say V ⊂ Y_{∞}. The morphism g|_{V} : V → X is an isomorphism (left to the reader; later we only need that it is birational which is clear). The morphism f is generically smooth along V (as V occurs with multiplicity 1 in Y_{∞}).

Let Y_{ξ} be the generic fibre of the morphism f. By the results above this is a normal projective surface over κ(ξ) = k(t). In fact it has A_{ℓ−1} singularities, one for each s ∈ S and no other singularities. This implies the minimal resolution W_{ξ} → Y_{ξ} is not only regular, but in fact W_{ξ} → Spec k(t) is smooth. There exists a sheaf of ideals I_{ξ} on Y_{ξ} whose support is the singular locus and such that the corresponding blowing up is the minimal resolution of Y_{ξ}:

b_{ξ} : W_{ξ} ⟶ Y_{ξ}.

Let I ⊂ O_{Y} be a sheaf of ideals, whose zero set has dimension ≤ 1 and whose restriction to Y_{ξ} gives back I_{ξ}. (For example we can take I such that V (I) is the scheme theoretic closure of V (I_{ξ}).) We let

b : W ⟶ Y

be the blowing up of Y in the sheaf of ideals I. We write f′ : W → ℙ^{1} for the composition f ∘ b and g′ : W → X for the composition g ∘ b. We now have the following diagram:

D  ⟶ X ⟶ X ⟶ X
↑  ^{g′}↑  ^{g}↑  ↑
(g′)^{−1}(D) ⟶ W ⟶^{b} Y ⟶ X × ℙ^{1}
↓  ^{f′}↓  ^{f}↓  ↓
ℙ^{1}  ⟶ ℙ^{1} ⟶ ℙ^{1} ⟶ ℙ^{1}

We also introduce the notation X̆ = X \ D and W̆ = W \ (g′)^{−1}(D) ; so that there is an induced morphism g′ : W̆ → X̆.

Since b_{ξ} is the minimal resolution of Y_{ξ}, and since W_{ξ} is smooth over κ(ξ) we see that, after possibly shrinking the open U of ℙ^{1}, the morphism W_{U} → U is smooth and for t ∈ U(k) the morphism W_{t} → Y_{t} is the minimal resolution of Y_{t}. It follows that the divisor (g′)^{−1}(D) ⊂ W is a relative normal crossings divisor over U.

The scheme b^{−1}(V) has a unique irreducible component V′ which dominates V and is birational to it. (Because we are blowing up Y in a 1-dimensional closed subscheme.) Note that V′ ⊂ W_{∞} is an irreducible component and that f′ : W → ℙ^{1} is generically smooth along V′.

Let ξ̄ be a geometric point lying over ξ; in fact let κ(ξ̄) be an algebraic closure of κ(ξ) = k(t). Write Xξ̄ = X ⊗_{k} κ(ξ̄). The restriction of g′ to the generic fibre W_{ξ} induces a morphism g′ ; ξ̄: Wξ̄ → Xξ̄ of degree ℓ that ramifies over Dξ̄ = D ⊗ κ(ξ̄).

Claim. The restriction (g′ ; ξ̄)^{∗}(η) is an element of Br(Wξ̄).

To see this let C ⊂ Wξ̄ be any irreducible divisor. We will show that the pullback (g′ ; ξ̄)^{∗}(η) does not ramify along C. Let us explain what this means. Let O_{W ⊗κ(ξ̄),C}^{h} be the henselization of the local ring O_{W ⊗κ(ξ̄),C} of the generic point of C on Wξ̄. We want to show that η comes from an element of Br(O_{W ⊗κ(ξ̄),C}) and we will show that η restricts to the zero element of the Brauer group of the fraction field of O_{W ⊗κ(ξ̄),C}^{h}. It is enough to show this, since η ramifies at the closed point of Spec O_{W ⊗κ(ξ̄),C} if and only if it ramifies at the closed point of Spec O_{W ⊗κ(ξ̄),C}^{h}. This follows upon inspecting the exact sequences of [11, III Proposition 2.1] for the schemes Spec O_{W ⊗κ(ξ̄),C}^{h} and Spec O_{W ⊗κ(ξ̄),C}.

After we have shown that (g′ ; ξ̄)^{∗}(η) does not ramify along any curve of the nonsingular surface Wξ̄ we deduce the claim from [11, II Proposition 2.3].

There are several cases to distinguish. If C maps onto a divisor \overline{C} on Xξ̄ which is not contained in Dξ̄ then η does not ramify along \overline{C} so the result is obvious. If C maps onto an irreducible component of Dξ̄ then the result is clear by construction. Thus we may assume g′ ; ξ̄(C) is a closed point x ∈ Xξ̄. Let O = κ(ξ̄){u, v} be the (strict) henselization of O_{X⊗κ(ξ̄),x}. If x is not a node of Dξ̄, then η maps to the trivial element of Br(f.f.(O)). This is clear if x ∉ Dξ̄, and if x ∈ Dξ̄, then there exists a choice of coordinates (u, v) such that Dξ̄ : u = 0, and we can use that Br(O[1/u]) = (0). Hence in these cases η certainly does not ramify along C. If x is a node of Dξ̄ (so x ∈ S in particular), then there exists a choice of the coordinate system (u, v) such that Dξ̄: uv = 0, and such that the corresponding henselian local ring of the singular point of Yξ̄ is O′ = κ(ξ̄){u, v}[z]/(z^{ℓ} − uv). Now η restricts to an element of Br(O[1/uv])[ℓ]. This group is generated by the Azumaya algebra O[1/uv]⟨s, t⟩/(s^{ℓ} − u, t^{ℓ} − v, st − ζ_{ℓ}ts). The reader checks easily that this element maps to the trivial class in Br(O′[1/uv]). Hence certainly η does not ramify along C.

Such local considerations are not new, see e.g. [2], [10], and [8].

If (g′ ; ξ̄)^{∗}(η) is zero then k(Wξ̄) is a splitting field of degree ℓ over κ(ξ̄)(X) and so there exists a rank ℓ^{2} division algebra Dξ̄ over κ(ξ̄)(X) whose Brauer class is ηξ̄. In this case the result follows from the following standard lemma whose proof we leave to the reader.

7.1. Lemma. Let k ⊂ K be a finitely generated field extension, and let η ∈ Br(K). Supppose that k ⊂ Ω is an extension of algebraically closed fields, and let K_{Ω} be the compositum of K and Ω over k. Then the period and index of η are the same as the period and index of the restriction η_{Ω} ∈ Br(K_{Ω}).

We continue our argument and we may suppose that (g′ ; ξ̄)^{∗}(η) is not trivial. By the result of in Step 3, there exists a rank ℓ^{2} Azumaya algebra A on Wξ̄ whose Brauer class is (g′ ; ξ̄)^{∗}(η).

Recall that κ(ξ̄) is the algebraic closure of the field κ(ξ) = k(t). Thus κ(ξ̄) is the direct limit of the rings Γ(C, O_{C}) where ξ̄ → C → ℙ^{1} and C is a smooth curve. Similarly Wξ̄ is the limit of the schemes W_{C} = W ×_{ℙ^{1}} C. Since Brauer groups and moduli of Azumaya algebras commute with direct limits, there exists an Azumaya algebra B over W_{C} = W ×_{ℙ^{1}} C for some such C such that the class of B equals the restriction of η to W_{C} and such that B restricts back to A on Wξ̄. (In particular rank B = ℓ^{2}.) This means the class of B in Br(k(W_{C})) is equal to the image of η in this Brauer group.

Let \overline{C} be the nonsingular projective model of C and let ∞′ ∈ \overline{C}(k) be a point mapping to ∞ in ℙ^{1}. Let ζ be the generic point of the irreducible closed subscheme V′ ×_{∞} ∞′ of W ×_{ℙ^{1}} \overline{C}. At this point we use that the morphism f′ is generically smooth along the irreducible component V′ (birational to X) of the fibre W_{∞}. Thus the local ring

R = O_{W ×_{ℙ^{1}} \overline{C},ζ}

is a discrete valuation ring with residue field k(X) and field of fractions k(W_{C}). Clearly, B ⊗ k(W_{C}) is unramified over R since its class comes from k(X) via the map Br(k(X)) → Br(R). The desired conclusion follows directly from Lemma 6.2.

8. M. Artin’s simplification

In this last section we present another proof of Step 3; we adopt the notations of the introduction. We give a sketch only as a full account of this will appear in [3].

Let η ∈ Br(X)[ℓ], and suppose that A is an Azumaya algebra of degree n over X representing η. We may and do assume that n is prime to the characteristic of k. Let L be an invertible sheaf on X and let L → X be the corresponding line bundle. For any section σ ∈ H^{0}(X, A ⊗ L) we have its characteristic polynomial P (T) = T^{n} + a_{1}T^{n−1} + … + a_{n} with a_{i} ∈ H^{0}(X, L^{i}). Let Y ⊂ L be the zero set of this polynomial. Thus Y → X is a finite flat degree n morphism. We omit the proof of the following claim.

8.1. Claim. (a) For sufficiently ample L and a general section σ the surface Y is nonsingular.

(b) For such a σ the class η|_{Y} is trivial in Br(Y). □

The great advantage of this is that there is an obvious family of surfaces Y_{t} that Y fits into. Namely, choose auxiliary pairwise distinct sections s_{1}, …, s_{n} of L and define Y_{t} to be the zero set of

P_{t} = P (T) + t(T − s_{1}) ⋯ (T − s_{n}).

For example, it is completely trivial to verify the conditions of Lemma 4.1 for this family.

Choose a lift η̃ ∈ H^{2}(X, μ_{ℓ}) of the image of η in H^{2}(X, 𝔾_{m}). It is clear that η̃|_{Y} is the Chern class of some invertible sheaf N on Y. Hence the Azumaya algebra A = End(O_{Y}^{ℓ−1} ⊕ N) has rank ℓ^{2} and its class in H^{2}(Y, μ_{ℓ}) is equal to η̃|_{Y}. The rest of the argument is exactly the same as the argument in Section 6 starting from the paragraph just above Lemma 6.1.

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