ALGEBRAS OF PRIME DEGREE ON

FUNCTION FIELDS OF SURFACES

Manuel Ojanguren and Raman Parimala

Abstract. Let X be a smooth projective surface over an algebraically closed field k of characteristic 0. Let D be a central division algebra of prime index over K = k(X). We prove that D is cyclic, thus proving, in this very special case, a conjecture that Albert probably never made [Sa].

0. Introduction

A central division algebra of degree n over a field K is said to be cyclic if it admits a cyclic splitting field of degree n over K. It is an open question whether central division algebras of prime degree l over any field are cyclic. It is a well-known result of Brauer-Hasse-Noether that all division algebras over number fields are cyclic. Let k be an algebraically closed field of characteristic zero, S a normal surface over k and R the completion or henselization of the local ring at a closed point of S. Then, over the quotient field K of R, Ford and Salman showed that all division algebras are cyclic. Over the quotient field of a 2-dimensional excellent henselian local domain with algebraically closed residue field, it was shown by Colliot-Thélène et al. [CTOP] that all division algebras of index coprime to residue field characteristic are cyclic, thus extending the result of Ford-Saltman. The method of approach here is to kill the ramification of the division algebra on a regular model over a henselian domain in a cyclic extension of K; one then uses the fact that the Brauer group of the model (or its l-primary part) is trivial.

In the case of the function field K of a surface X over an algebraically closed field k, to split a division algebra it is not sufficient to split its ramification, because the Brauer group of X may not be trivial. A theorem of de Jong [dJ] asserts that for every central division algebra D over K with index prime to the characteristic of K, the index coincides with the order of D in Br(K). In this paper we adapt the techniques of de Jong to prove that if the characteristic of K is zero, then every division algebra of prime degree over K is cyclic.

Given a smooth projective surface X over k and an Azumaya algebra A on X, we construct a finite flat morphism Y → X with Y smooth projective which splits A. Such a construction is announced by de Jong and Artin [dJ] under the more general setting in which the characteristic of k is coprime to index(A_{k(X)}). Following de Jong, we connect Y and X by a family W fibred over 𝔸^{1} and use this family to prove cyclicity when A is unramified over X. The ramified case reduces to the unramified case, following de Jong’s techniques closely.

Typeset by AMS-TEX

We thank Jean-Louis Colliot-Thélène for bringing to our attention the general position argument in Proposition 8.1 and for detecting a gap in a previous version of this article. We thank Shreeram S. Abhyankar for explaining to us the equivariant resolution of singularities in degree 2 extensions quoted in §9.

1. Representing Brauer classes

Let X be a noetherian regular integral 2-dimensional scheme, K its field of rational functions and A an Azumaya algebra over X.

Theorem 1.1. Suppose that the generic fiber A_{K} of A is of the form M_{n}(D), where D is a division algebra over K. Then the Brauer class of A can be represented by an Azumaya algebra Δ over X such that Δ_{K} = D.

We first recall the definition and some properties of maximal orders. Let R be an integral domain with field of fractions K and A an associative finite dimensional K-algebra. An R-order of A is an R-subalgebra Λ of A which is finitely generated and projective as an R-module and generates A over K: ΛK = A. It is easy to see that R-orders always exist. (A general reference for the theory of orders is [AG].)

If A is a central simple algebra over K and that R is integrally closed and noetherian, then every R-order is contained in a maximal one. If Λ is an R-Azumaya algebra such that A = Λ_{K}, then Λ is a maximal order of A.

Assume now that R is a discrete valuation ring. Then the following holds.

Theorem 1.2. Let R be a discrete valuation ring, K its field of fractions and A a central simple K-algebra. Any two maximal R-orders Λ and Γ in A are conjugate: Λ = uΓu^{−1} for some u ∈ A^{∗}. In particular, if one of the maximal orders is an Azumaya R-algebra, then so is the other one.

Proof. See for instance [AG], Proposition 3.5.

Proof of Theorem 1.1. Since the canonical map Br(X) → Br(K) is injective, any algebra Δ such that Δ_{K} = D will represent the Brauer class of A. Let x ∈ X be a point of codimension 1. Then O_{X,x} is a discrete valuation ring and A_{x}, being an Azumaya algebra over O_{X,x}, is a maximal order in A_{K} = M_{n}(D). Choose now a maximal order Δ(x) in D. It is easy to check that M_{n}(Δ(x)) is a maximal order in M_{n}(D). By Theorem 1.2, A_{x} and M_{n}(Δ(x)) are isomorphic, hence Δ(x) is Morita equivalent to A_{x} and is, therefore, an Azumaya algebra.

We have, for any codimension 1 point x ∈ X, an O_{X,x}-Azumaya algebra Δ(x) ⊂ D representing the class of A_{K} in Br(K). In other words, the K-algebra D is unramified over X. We patch all these algebras to get a global representative Δ of A_{K}, with generic fiber D.

To do this we can invoke (as in [CTS], Proposition 2.4, page 111 for the case of quadratic spaces) some general results on projective limits of schemes or prove a little lemma:

Lemma 1.3. Let X be a noetherian scheme of finite dimension, U a dense open subset of X, y ∈ X \U a codimension 1 point, V an open neighbourhood of y and W ⊂ U ∩V a dense open subset of X. There exists an open neighbourhood V′′ of y such that V′′ ∩ U ⊂ W.

Proof. The closed set Z = X \ W is of dimension n − 1 and contains y, hence Z has an irredundant decomposition

Z = \overline{\{y\}} ∪ \overline{\{y_{1}\}} ⋯ ∪ \overline{\{y_{r}\}} ∪ \overline{\{x_{1}\}} ∪ ⋯ ∪ \overline{\{x_{s}\}}

into closed irreducible sets, where y, y_{1}, …, y_{r} ∉ U and x_{1}, …, x_{s} ∈ U. Let F = \overline{\{x_{1}\}} ∪ ⋯ ∪ \overline{\{x_{s}\}} and V′ = X \ F ⊃ W. We have V′ ∩ U = ∅ and therefore U ∩ F = U ∩ Z. This shows that V′ ∩ U = U \ (U ∩ Z) = U ∩ (X \ Z) = U ∩ W = W. Thus we can take V′′ = V ∩ V′.

The next proposition shows that the local algebras Δ(x) can be patched, thus proving Theorem 1.1.

Proposition 1.4. Let X be a noetherian integral regular scheme of dimension 2, K its field of rational functions and A a central simple K-algebra. Suppose that A is unramified over X. There exists an Azumaya algebra Λ over X such that Λ_{K} = A.

Proof. The K-algebra A extends over some open set U as an Azumaya algebra Λ^{U} and we may assume that U is maximal with this property. Suppose that X \ U contains a point y of codimension 1. By assumption A is unramified at y, hence there exists an Azumaya O_{X,y}-algebra Γ^{y} ⊂ A such that Γ_{K}^{y} = A. We extend Γ^{y} to an Azumaya algebra Γ^{V} over some suitable open neighbourhood V of y. Since Γ_{K}^{V} = A = Λ_{K}^{U}, there exists an open set W ⊂ U ∩ V over which Γ_{W}^{V} ≃ Λ_{W}^{U} We now choose V′′ ⊂ W as in 1.3 and patch Λ^{U} with Γ_{V′′}^{V} over U ∩ V′′ to get an Azumaya algebra over U ∪ V′′. This shows that if U is maximal, then X \ U consists of finitely many closed points. By [CTS], Th. 6.13 applied to the group P GL_{n}, Λ^{U} extends to an Azumaya algebra on X.

2. Splitting extensions

We show how to split an Azumaya algebra over a surface X by a finite map Y → X.

Here and in the rest of the article we suppose that k is an algebraically closed field. We denote by Sing(X) the singular locus of a given scheme X.

Let

A_{n} = \frac{k[X_{11}, X_{12}, …, X_{nn}][T]}{(P (T))}

where P (T) is the characteristic polynomial of the generic matrix (X_{ij}) with 1 ≤ i, j ≤ n. Let Y_{n} = Spec(A_{n}). We study the singular locus of Y_{n}.

Lemma 2.1. Let β = diag(B_{1}, …, B_{m}) be a matrix consisting of m cyclic Jordan blocks

B_{i} = ( \begin{matrix}λ_{i} & 1 & 0 & · & · & · & 0 & 0 \\ 0 & λ_{i} & 1 & · & · & · & 0 & 0 \\ 0 & 0 & λ_{i} & · & · & · & 0 & 0 \\ · & · & · & · & · & · & · \\ 0 & 0 & 0 & · & · & · & λ_{i} & 1 \\ 0 & 0 & 0 & · & · & · & 0 & λ_{i} \end{matrix} )

with distinct eigenvalues λ_{i}. Then, for any i, the scheme Y_{n} is smooth at (β, λ_{i}).

Proof. We denote by I_{n} the identity matrix of size n. Developing the determinant of (X_{ij}) − T · I_{n} along the first column we get

±P (T) = (X_{11} − T)P_{1}(T) + X_{2,1}P_{2}(T) + ⋯ + X_{n,1}P_{n}(T)

where the polynomials P_{i} are the cofactors of the first column. Let k_{i} be the size of B_{i}. We see that P_{k_{1}}(T)(B, λ_{1}) is (up to sign) the determinant of a matrix of the form diag(I_{k_{1}−1}, B_{2} − λ_{1}I_{k_{2}}, …, B_{m} − λ_{1}I_{k_{m}}), it being understood that the first block is missing if k_{1} = 1. Since λ_{1} ≠ λ_{i}, this shows that ∂P (T)/∂X_{k_{1},1} = P_{k_{1}}(T) is not zero at (B, λ_{1}). Thus Y_{n} is smooth at (β, λ_{1}) and the same clearly holds for any other λ_{i}.

Lemma 2.2. Every neighbourhood of a matrix α with an eigenvalue λ ≠ 0 contains an invertible semisimple matrix with eigenvalue λ.

Proof. We may assume that α is in Jordan form. The given neighbourhood of α contains an open set defined by the non-vanishing of a polynomial g in the coordinates of the generic matrix (X_{ij}). We may assume that the diagonal entries of α are (λ, λ_{2}, …, λ_{n}). Since g(α) ≠ 0 we may find values λ′_{2}, …, λ′_{n} all distinct and different from λ and different from 0, such that when we replace λ_{i} by λ′_{i} in α we obtain an α′ for which g(α′) ≠ 0. This new α′ is in the given neighbourhood and is semisimple.

Let Y_{n} be as before and consider the finite map π : Y_{n} → 𝔸^{n^{2}} induced by the injection k[X_{11}, X_{12}, …, X_{nn}] → A_{n}. The projection C = π(Sing(Y_{n})) is a closed subscheme of 𝔸^{n^{2}} and is contained in the ramification locus of π, which is the closed subscheme of 𝔸^{n^{2}} whose closed points correspond to matrices with at least two equal eigenvalues.

Lemma 2.3. Let V ⊂ 𝔸^{n^{2}} be the set of semisimple invertible matrices with at least two coincident eigenvalues. Then V ⊆ C.

Proof. It suffices to check that any matrix of the form β = diag(μ_{1}, …, μ_{n−2}, λ, λ) is in C. We show that (β, λ) belongs to Sing(Y_{n}). Writing X_{ii} = μ_{i} +X_{i} for i ≤ n−2, X_{ii} = λ+X_{i} for i ≥ n − 1, T = λ + t and ν_{i} = μ_{i} − λ we see that P (T) is the determinant of the matrix

( \begin{matrix}ν_{1} + X_{1} & X_{12} & ⋯ & X_{1n} \\ X_{2,1} & ν_{2} + X_{2} & ⋯ & X_{2,n} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ ⋯ & ⋯ & X_{n−1} − t & X_{n−1,n} \\ ⋯ & ⋯ & X_{n,n−1} & X_{n} − t \end{matrix} )

and it is clear that it does not contain any linear term in X_{i},X_{ij} or t. Thus the variety it defines is singular at the origin, which corresponds to the point (β, λ) in the previous coordinates.

Lemma 2.4. Let W ⊂ M_{n}(k) be the set of all semisimple invertible matrices with at least n − 1 distinct eigenvalues. Then W is open and dense in M_{n}(k).

Proof. The set of all semisimple invertible matrices is open and dense in M_{n}(k). We claim that matrices having at least n − 1 distinct eigenvalues is open in M_{n}(k). In fact this set is the inverse image under the eigenvalue map M_{n} → 𝔸^{n}/S_{n} of the complement of the closed set of points with three equal coordinates. Hence W is open and clearly non empty.

By 2.4 the set U = W ∩ C of all semisimple invertible matrices with exactly two equal eigenvalues is open in C.

Lemma 2.5. The set U is dense in C.

Proof. Let (β, λ) be a point of Sing(Y_{n}). By 2.1, β, which we may assume to be in Jordan canonical form, contains at least two cyclic Jordan blocks with the same eigenvalue. We write β = diag(β_{1}, β_{2}, …, β_{r}) with the β_{i}’s cyclic Jordan blocks of size s_{i} and β_{1}, β_{2} having the same eigenvalue λ. Suppose that β is in the open set defined by f ≠ 0 for some polynomial function f in the entries X_{ij} of the generic n × n matrix. Let β̃ = diag(β̃_{1}, β̃_{2}, …, β̃_{r}) be a matrix where each β̃_{i} has the same size as β_{i} and the same offdiagonal entries. Suppose further that β̃ has n − 1 distinct eigenvalues, with β̃_{1} and β̃_{2} retaining the eigenvalue λ. Then β̃ is semisimple and, for a general β̃, f(β̃) ≠ 0. For example, if

β = ( \begin{matrix}λ & 1 & 0 & 0 & 0 \\ 0 & λ & 1 & 0 & 0 \\ 0 & 0 & λ & 0 & 0 \\ 0 & 0 & 0 & λ & 1 \\ 0 & 0 & 0 & 0 & λ \end{matrix} )

then

β̃ = ( \begin{matrix}λ_{1} & 1 & 0 & 0 & 0 \\ 0 & λ_{2} & 1 & 0 & 0 \\ 0 & 0 & λ & 0 & 0 \\ 0 & 0 & 0 & λ_{3} & 1 \\ 0 & 0 & 0 & 0 & λ \end{matrix} )

with λ, λ_{1}, λ_{2}, λ_{3} distinct.

Corollary 2.6. The dimension of C is equal to the dimension of U.

Lemma 2.7. The dimension of U is n^{2} − 3.

Proof. Let Σ_{n−1} ⊂ (k^{∗})^{n−1}/S_{n−1} be the set of all {λ, λ_{3}, …, λ_{n}} consisting of n−1 distinct elements of k^{∗}. Clearly Σ_{n−1} has dimension n − 1. Mapping each matrix in U to the set of its eigenvalues we obtain a surjective map p : U → Σ_{n−1}. The linear group GL_{n}(k) acts transitively on each fiber of p and the stabilizer of the matrix diag(λ, λ, λ_{3}, …, λ_{n}) is GL_{2}(k) × (k^{∗})^{n−2}. Hence the dimension of U is dim(GL_{n}(k)) − dim(GL_{2}(k) × (k^{∗})^{n−2}) + dim(Σ_{n−1}) = n^{2} − (4 + n − 2) + n − 1 = n^{2} − 3.

Corollary 2.8. The closed set Sing(Y_{n}) is of codimension 3.

Proof. The closure of U is C = π(Sing(Y_{n})) and π is a finite map.

We now show how to use the reduced characteristic polynomial to split an Azumaya algebra over a surface. If L is a line bundle over some scheme and n a positive integer, we denote by L^{⊗n} the n-fold tensor product of L with itself and by L^{−n} the inverse of L^{⊗n}. Let X be a smooth projective surface over an algebraically closed field k and A an Azumaya algebra of rank n^{2} over X. Let L be a line bundle over X such that A ⊗_{O_{X}} L is generated by global sections s_{1}, …, s_{N} and let s be any global section of A ⊗_{O_{X}} L. Choose an arbitrary affine nonempty open set U ⊂ X over which L is principal: L_{|U} = O_{U}f for some f ∈ L(U). Then sf^{−1} ∈ A(U), which is an Azumaya algebra over O_{X}(U). Let

P_{f,U}(T) = T^{n} + b_{1}T^{n−1} + ⋯ + b_{n}

with b_{1}, …, b_{n} ∈ k[U] be the characteristic polynomial of sf^{−1}. We define J_{f,U} as the ideal of

Sym(L^{−1}|_{U}) = O_{U} ⊕ L^{−1}|_{U} ⊕ L^{−2}|_{U} ⊕ ⋯ = O_{U} ⊕ O_{U}f^{−1} ⊕ O_{U}f^{−2} ⊕ ⋯

generated by f^{−n} ⊕ b_{1}f^{−(n−1)} ⊕ ⋯ ⊕ b_{n}.

Lemma 2.9. Let Λ be an Azumaya algebra of rank n^{2} over a ring R. For any α ∈ Λ and any c ∈ R, the characteristic polynomial P_{α}(T) of α satisfies the relation c^{n}P_{α}(T) = P_{cα}(cT).

Proof. It immediately follows from the split case Λ = M_{n}(R).

Lemma 2.10. The ideal J_{f,U} does not depend on the choice of f.

Proof. We apply 2.9 with f = ug for some other generator g of L|_{U} and u invertible on U. (We note that the suffixes f or g stand for the elements s/f, s/g in the algebra). We have

P_{g,U}(T) = P_{u^{−1}f,U}(T) = u^{n}P_{f,U}(u^{−1}T) = T^{n} + ub_{1}T^{n−1} + ⋯ + u^{n}b_{n} .

Thus the ideal J_{g,U} is generated by

g^{−n} ⊕ b_{1}ug^{−(n−1)} ⊕ ⋯ ⊕ u^{n}b_{n} = u^{n}(f^{−n} ⊕ b_{1}f^{−(n−1)} ⊕ ⋯ ⊕ b_{n}) .

and coincides therefore with J_{f,U}. Patching the ideals J_{f,U} over a suitable affine covering of X yields a global ideal J_{s} of Sym(L^{−1}) that only depends on the section s. We call J_{s} the characteristic ideal of s.

We define a projective k-scheme Y_{s} with a finite map to X as the closed subscheme of Spec (Sym(L^{−1})) defined by the ideal J_{s}.

To simplify notation, if s = λ_{1}s_{1} + ⋯ + λ_{N}s_{N} we put λ = (λ_{1}, …, λ_{N}) ∈ k^{N}, J_{s} = J_{λ} and Y_{s} = Y_{λ} Theorem 2.11. Assume that k is of characteristic zero. There exists a nonempty open set U ⊂ k^{N} such that, for any λ ∈ U, Y_{λ} is a projective smooth surface.

Proof. We extend the base to X̃ = X × 𝔸^{N} where 𝔸^{N} = Spec (k[t_{1}, …, t_{N}]). Let Ã and L̃ be the inverse images of A and L under the projection π : X̃ → X. Put s̃ = t_{1}s_{1} + ⋯ + t_{n}s_{n} and let J̃_{t}(T) be the characteristic ideal of s̃ and Ỹ the closed subscheme of Spec(Sym(L̃^{−1})) defined by J̃_{t}(T). Look at the diagram

[figure: Ỹ; p; π; q; X; X × 𝔸^{N}; 𝔸^{N}]

The map π is clearly finite and flat and the two projections from X × 𝔸^{N} are flat, hence p is flat. We try to determine the singularities of Ỹ using the following lemma.

Lemma 2.12. Let f : Z → X be a flat map of schemes. Suppose that X is regular. If z ∈ Z is a singular point of Z, then z is a singularity of its fiber f^{−1}(f(z)).

Proof. Let C be the local ring of Z at z and A be the local ring of f(z). By assumption the maximal ideal of A is generated by a regular sequence (x_{1}, …, x_{m}). Since f is flat, C is faithfully flat over A and this sequence is still regular as a sequence in C. If z is not a singular point of its fiber, then C/(x_{1}, …, x_{m}) is regular and hence its maximal ideal is generated by a regular sequence (\overline{y}_{1}, …, \overline{y}_{r}). This implies that the maximal ideal of C is generated by the regular sequence (x_{1}, …, x_{m}, y_{1}, …, y_{r}), hence C is regular.

By 2.12 the singularities of Ỹ are contained in the union of the singularities of the fibers of p.

Lemma 2.13. The singular locus of every fiber p^{−1}(x) of p has codimension 3 in p^{−1}(x).

Proof. Let k(x) be the residue field of x ∈ X, Ω its algebraic closure and F_{x} the fiber of p at x. The geometric fibre A(\overline{x}) of A at x is a matrix algebra M_{n}(Ω) and

F\overline{x} = Spec (Ω[t_{1}, …, t_{N}][T]/(P_{x}(T))) ,

where P_{x}(T) is the characteristic polynomial of \overline{s} = (t_{1}s_{1}(x) + ⋯ + t_{N}s_{N}(x))/f(x) for some generator f of L|_{U}, U a neighbourhood of x. Since the sections s_{i}(x)/f(x) generate M_{n}(Ω) over Ω, by a linear change of coordinates we may assume that \overline{s} = t_{1}e_{1} +⋯+t_{m}e_{m} where m = n^{2} and {e_{1}, …, e_{m}} form a basis of M_{n}(Ω). Then

F\overline{x} = Y_{n} × Spec (Ω[t_{m+1}, …, t_{N}]) .

We proved that the singular locus of Y_{n} has codimension 3, hence the same holds for the singular locus of F\overline{x}. For every x ∈ X the fiber F_{x} is a finite cover of 𝔸^{N} and hence the dimension of F_{x} is N. Let Sing(Ỹ) be the singular locus of Ỹ. By 2.10, for every x ∈ X, the fiber at x of p|_{Sing(Ỹ)} : Sing(Ỹ) → X is contained in the singular locus of F_{x} and has therefore dimension at most N − 3. Since X is 2-dimensional, the dimension of Sing(Ỹ) is at most N − 1.

We now look at q : Ỹ → 𝔸^{N}. Since Sing(Ỹ) is at most (N − 1)-dimensional, its image q(Sing(Ỹ)) is contained in a proper closed subset of 𝔸^{N}. Choose an open set W ⊂ 𝔸^{N} which does not intersect q(Sing(Ỹ)) and let Ṽ = q^{−1}(W). We now have a map q : Ṽ → W of smooth varieties. This map is clearly flat and surjective and therefore, k being of characteristic zero, it is generically smooth (see [Ha_{1}], Ch. III, Corollary 10.7). This means that there exists a dense open set U ⊂ 𝔸^{N} such that q^{−1}(U) → U is smooth. For any λ ∈ U the fiber Y_{λ} = q^{−1}(λ) is smooth.

Let us denote by π_{λ} : Y_{λ} → X the map induced by π on the fibre Y_{λ}.

Proposition 2.14. There exists a dense open set U ⊂ k^{N} such that for any λ ∈ U and for all but finitely many closed points x ∈ X the fibre π_{λ}^{−1}(x) contains at least n−1 points.

Proof. For n = 2 we may take U = k^{N}. We now assume that n ≥ 3. Let J be a Jordan block of size m × m. A direct computation shows that its stabilizer in M_{m}(k) is the group G(m) that consists of all matrices of the form

( \begin{matrix}a_{1} & a_{2} & a_{3} & ⋯ & a_{m} \\ 0 & a_{1} & a_{2} & ⋯ & a_{m−1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & ⋯ & a_{1} \end{matrix} )

and is therefore of dimension m. Thus the stabilizer of diag(J_{1}, …, J_{r}) ∈ M_{n}(k) where J_{i} denotes a Jordan block of size m_{i}, contains a product of the form G(m_{1})×⋯×G(m_{r}) and is therefore of dimension at least n. It follows that the stabilizer of any matrix α ∈ M_{n}(k) is of dimension at least n. If α is a matrix with less than n − 1 eigenvalues, the eigenvalues may be chosen in at most n − 2 different ways. Thus the set of matrices with at most n − 2 distinct eigenvalues has dimension at most n^{2} − 2.

Consider now the closed set

V = {(λ, x) | P_{λ}(x, T) has at most n − 2 distinct roots} ⊂ 𝔸^{N} × X

and its projection V → X. It follows that the fiber of any x has dimension at most (N − n^{2}) + (n^{2} − 2) = N − 2 and hence dim(V) ≤ N. If the image of the projection p : V → 𝔸^{N} is not dense, then there is an open set U ⊂ 𝔸^{N} such that for λ ∈ U the characteristic polynomial P_{λ} has no triple root. If p(V) is dense in 𝔸^{N}, then for an open set U ⊂ 𝔸^{N} the fiber of λ ∈ U is finite ([Ha_{1}], Ch. II, Exercise 3.22, page 95). In this case, for each λ ∈ U, P_{λ}(x, T) has at most one double root except for a finite number of points x ∈ X.

Proposition 2.15. Suppose that the algebra A_{k(X)} is a division ring of index n. Then there exists an open set U of k^{N} such that, for each λ ∈ U, the variety Y_{λ} is integral. Proof. We extend the scalars to k(X). There exists an open set U of k^{N} such that for each λ ∈ U, s_{λ}/f generates a maximal subfield of degree n in A_{k(X)}. Since P (T) is of degree n and vanishes at s_{λ}/f, it must be irreducible over k(X), hence (Y_{λ})_{k(X)} is integral. This, together with the smoothness of Y, implies that Y is integral. Summing up, we have proved the following result. Theorem 2.16. Let X be a smooth projective surface over an algebraically closed field k of characteristic zero. Let A be an Azumaya algebra of rank n^{2} over X with A_{k(X)} a division ring. Then there exists an open set U of k^{N} such that for each λ ∈ U,

(1) Y_{λ} is a smooth, integral projective surface,

(2) the map π_{λ} : Y_{λ} → X is finite and flat,

(3) π_{λ}^{∗}(A) is trivial in Br(Y_{λ}),

(4) there are only finitely closed point on X which have fewer that n − 1 preimages in Y.

3. A splitting criterion

Proposition 3.1. Let X be a smooth projective surface over an algebraically closed field k and A an Azumaya algebra over X, of rank n^{2}. Assume that the characteristic of k is zero or a prime that does not divide n. Fix an element η ∈ H^{2}(X, μ_{n}) which maps to [A] ∈ _{n}Br(X) ⊂ H^{2}(X, 𝔾_{m}). Suppose that there exists a diagram

[figure: W; g; f; X; 𝔸^{1}]

with 𝔸^{1} = Spec(k[t]) and such that

(1) W is a 3-dimensional integral scheme with W_{\overline{k(t})} integral,

(2) the map f is proper,

(3) W_{1} = f^{−1}(1) has n irreducible components V_{i}, each with multiplicity 1 and such that g|_{V_{i}} : V_{i} → X is a birational isomorphism for every i,

(4) W is normal at the generic point of each V_{i},

(5) g^{∗}(η)|_{W_{0}} = 0 in H^{2}(W_{0}, μ_{n}). Then A_{k(X)} is a matrix algebra over k(X). Proof. Let R be the local ring of 𝔸^{1} at t = 0 and R^{h} its henselization. Let g_{h} : W ×_{𝔸^{1}} Spec(R^{h}) → X be the composite map W ×_{𝔸^{1}} Spec(R^{h}) → W ⟶^{g} X. The element g_{h}^{∗}(η) ∈ H^{2}(W ×_{𝔸^{1}} Spec(R^{h})) maps to zero in H^{2}(W_{0}, μ_{n}). By proper base change ([Mi], Ch. VI, 2.7), g_{h}^{∗}(η) = 0, hence there exists a finite étale map C_{0} ⟶^{α} 𝔸^{1} of a curve onto a neighbourhood of 0, such that if g_{C_{0}} : W ×_{𝔸^{1}} C_{0} → X denotes the restriction of g_{h}, then g_{C_{0}}^{∗}(η) = 0. We extend α : C_{0} → 𝔸^{1} to an α : C_{1} → 𝔸^{1} such that the point t = 1 is the image of a rational point of C_{1}. Such a point exists, since k is algebraically closed. Since W_{\overline{k(t})} is integral, the scheme W ×_{𝔸^{1}} C_{1} is integral, with generic point Spec(k(W ×_{𝔸^{1}} C_{0})). The class g_{C_{1}}^{∗}(η) ∈ H^{2}(W ×_{𝔸^{1}} C_{1}, μ_{n}) is generically zero. Since by (3) each V_{i} occurs with multiplicity 1 in the fibre of 1 and by (4) W is normal at the generic point of V_{i}, t − 1 generates the maximal ideal of the discrete valuation ring O_{W,V_{i}}. Let 1′ ∈ C_{1} be a rational point such that α(1′) = 1. Then V_{i} × 1′ ≃ V_{i} is an irreducible component of the fibre of 1′. Let S be its local ring in k(W ×_{𝔸^{1}} C_{1}). The maximal ideal of S is generated by a local parameter of C_{1} at 1′, hence S is a discrete valuation ring with quotient field k(W ×_{𝔸^{1}} C_{1}) and the map H^{2}(S, μ_{n}) → H^{2}(k(W ×_{𝔸^{1}} C_{1}), μ_{n}) is injective. Thus g_{C_{1}}^{∗}(η) restricts to zero in H^{2}(S, μ_{n}) and specializes to zero in

H^{2}(κ(V_{i} × {1′}), μ_{n}) = H^{2}(κ(V_{i}), μ_{n}) = H^{2}(k(X), μ_{n})

under the map g. The composite map k(X) → κ(V_{i}) ⟶^{g} k(X) being the identity, we have η_{k(X)} = 0.

4. Construction of families

We shall first construct a variety W satisfying the assumptions (1) to (4) of 3.1. Let X be a smooth projective surface over an algebraically closed field k and A an Azumaya algebra of dimension n^{2} on X. Let L be a line bundle on X such that A ⊗_{O_{X}} L is generated by global sections e_{1}, …, e_{N}. Let s = λ_{1}e_{1} + ⋯ + λ_{N}e_{N} with λ = (λ_{1}, …, λ_{N}) ∈ k^{N} and J_{s} ∈ Sym(L^{−1}) the characteristic ideal of s defined in §2. Recall that Y_{λ} is the subscheme of Spec(Sym(L^{−1})) defined by J_{s} and that locally on any affine open set U ⊂ X over which L|_{U} is generated by a section f, J_{s}|_{U} is generated by P_{f,U}(f^{−1}) = f^{−n}⊕b_{1}f^{−(n−1)}⊕⋯⊕b_{n} where P_{f,U}(T) = T^{n}+b_{1}T^{(n−1)}+⋯+b_{n} is the characteristic polynomial of s/f ∈ H^{0}(U, A). We choose λ such that Y_{λ} satisfies (1), (2) and (3) of Theorem 2.16. Let X̂ be the scheme X × 𝔸^{1}, p : X̂ → X its first projection and t the coordinate on 𝔸^{1}. We put L̂ = p^{∗}(L) and define an ideal in Sym(L̂^{−1}) as follows. Let w_{1}, …, w_{n} be n distinct global sections of L. We choose them in such a way that no function w_{i}/f over U is a zero of P_{f,U}(T). We denote by Û the inverse image of U. For simplicity, we still denote by the same letter a function (or a section of a bundle, or a polynomial, …) on an open set of X and its extension to X̂. Let Î_{f,U} be the ideal of Sym(L̂^{−1}|Û) generated by Q_{f,U}(t, f^{−1}) where

Q_{f,U}(t, T) = (1 − t)P_{f,U} + t(T − w_{1}/f) … (T − w_{n}/f) .

If we replace f by another generator g such that g = uf for some invertible function u on U, then, as in 2.10, we see that Î_{f,U} = Î_{g,U}. Therefore these ideals patch over X and give rise to an ideal Î_{s} of Sym(L̂^{−1}). We define W as the closed subscheme of Spec(Sym(L̂^{−1})) defined by Î_{s}. The composite

W → Spec(Sym(L̂^{−1})) → X

defines a map g : W → X and the second projection defines a map f : W → 𝔸^{1}.

Proposition 4.1. The triple (W, f, g) satisfies the conditions (2), (3) and (4) of Proposition 3.1. Furthermore W_{0} = f^{−1}(0) is a smooth projective surface.

Proof. Property (2) follows from the fact that W is finite, hence proper over X̂ which is proper over 𝔸^{1}. The fibre W_{1} is locally the spectrum of R[T]/((T − w_{1}/f) … (T − w_{n}/f)) whose irreducible components Spec(R[T]/(T − w_{i}/f)) have multiplicity 1 and map isomorphically onto Spec(R) under g. This proves (3). To show (4) let 𝔭_{i} be the generic point of V_{i} and U = Spec(R) a suitable affine open set such that its inverse image in W contains 𝔭_{i}. Then, locally at 𝔭_{i}, W is the spectrum of

S = (R[T, t]/((1 − t)P_{f,U}(T) + t(T − h_{1}) … (T − h_{n})))_{𝔭_{i}}

with h_{i} = w_{i}/f. Since T − h_{i} and 1 − t are in 𝔭_{i} we have 𝔭_{i} = (T − h_{i}, 1 − t). We assumed that P (h_{i}) ≠ 0 in K = S/𝔭_{i}, hence 𝔭_{i}S is generated by T − h_{i}. This proves that W is normal at the generic point of V_{i}. It is clear from the construction that W_{0} is smooth and projective. It remains to prove property (1).

The proof of irreduciblility will be completed in §6. We begin with the following lemma.

Lemma 4.2. Let

Φ_{f,U}(T) = T^{n}P_{f,U}(T^{−1}) = 1 + b_{1}T + ⋯ + b_{n}T^{n} ,

Ψ_{f,U}(T) = (1 − (w_{1}/f)T) ⋯ (1 − (w_{n}/f)T) = 1 + c_{1}T + ⋯ + c_{n}T

and

R_{f,U}(t, T) = (1 − t)Φ_{f,U}(T) + tΨ_{f,U}(T) = T^{n}Q_{f,U}(t, T^{−1}) .

Then Φ_{f,U}(f), Ψ_{f,U}(f) and R_{f,U}(t, f) do not depend on f and can be patched to yield global sections of

OX̂ ⊕ L̂ ⊕ ⋯ ⊕ L̂^{⊗n} .

Proof. Since these polynomials are determined by their restrictions to the generic fibre, to show that they patch it suffices to show that, for a fixed U, they do not depend on the choice of f. In fact we check that each b_{i}f^{i} patches to yield a sections of L̂^{⊗i}. Let f = ug for a section g of L|_{U}, u an invertible function on U. By 2.9 we have

P_{g,U}(T) = T^{n} + ub_{1}T^{n−1} + ⋯ + u^{n}b_{n}

and therefore

Φ_{g,U}(T) = 1 + b_{1}uT + ⋯ + b_{n}u^{n}T^{n} .

The assertion for Φ_{g,U} follows from ug = f and the same proof holds for Ψ_{g,U} and R_{g,U}.

Proposition 4.3. The varieties W, W ⊗_{k[t]} k(t) and W ⊗_{k[t]} \overline{k(t)} are integral.

We first recall some elementary well-known facts about integral schemes. A scheme is integral if it is irreducible and reduced. The condition that it is reduced is a local one and can be checked on each set of an affine open covering. The condition of irreducibility is a priori a global one. For quasi-projective schemes, though, it can be checked on affine open sets.

Lemma 4.4. Let X be a scheme such that any two points of X are contained in an irreducible open affine set. Then X is irredubible.

Proof. Suppose that X = Y ∪Z with Y and Z closed in X and both different fron X. Then there exist points y ∈ Y \Z and z ∈ Z\Y. Let U be an irreducible affine open set containing y and z. Then, since z ∉ U ∩ Y and y ∉ U ∩ Z, U = (U ∩ Y) ∪ (U ∩ Z) is a decomposition of U into two proper subset, which are closed in U, leading to a contradiction.

Lemma 4.5. Let f : Y → X be a flat morphism of quasi projective varieties, with X integral. Denote by k(X) the field of rational functions on X. The following two conditions are equivalent:

(1) Y is integral,

(2) Y ×_{X} Spec(k(X)) is integral.

Proof. Suppose that (2) holds. Choose any two points y and z in Y and an affine open set U ⊂ X that contains f(y) and f(z). The preimage W = f^{−1}(U) of U is a quasiprojective variety, hence it contains an affine open set V that contains y and z. We prove that V is integral. The map f : V → U is flat and since the coordinate ring k[U] of U injects into k(U), k[V] injects into k[V] ⊗_{k[U]} k(U) = k[V ×_{X} Spec(k(X))]. But if Y ×_{X} Spec(k(X)) is integral, then its open set V ×_{X} Spec(k(X)) is integral as well and therefore k[V] ⊂ Spec(k[V ×_{X} Spec(k(X))]) is integral. We have found an integral affine open set V containing y and z. By 4.4, Y is irreducible. It is obviously reduced because it can be covered by affine open sets like V. Suppose now that (1) holds. Then, for any open affine set V of Y, the ring k[V]⊗_{k[U]} k(X), as a localization of k[V], is integral. Hence every open affine set of Y ×_{X} Spec(k(X) is integral and, by 4.4, Y ×_{X} Spec(k(X)) is integral.

5. Irreducibility

For proving 4.3 we need a result of [AHS] in a slightly different form. For convenience of the readers we give the proof, even if it is the same as that of [AHS]. Let A be a noetherian normal affine domain over an algebraically closed field k of characterisitic zero, and K its field of fractions. Suppose that F (T) and G(T) are two polynomials in A[T] satisfying the following hypotheses:

(1) F (T) is of degree n ≥ 1 and generates a prime ideal of A[T],

(2) F (T) and G(T) are coprime in K[T] with deg(G(T)) ≤ deg(F (T)). We want to study the irreducibility of the polynomials

F (T) + λG(T) , λ ∈ k .

Note that, since A is normal, irreducibility over A and over K are the same by Gauss’ lemma.

We define the set Π ⊂ A[T] as the set of polynomials f(T) ∈ A[T] for which there exists a λ ∈ k and a proper factorization

F (T) + λG(T) = f(T)g(T) ,

i.e., f(T) and g(T) are not in A.

Let B be the affine domain

(A[T]/(F (T)))

and L its field of fractions. For any f ∈ A[T] we denote by \overline{f} its image in B.

Lemma 5.1. Let B be an affine domain over k and L its field of fractions. There exists a finite set of discrete valuations {v_{1}, …, v_{q}} such that the only elements b ∈ B for which v_{1}(b) = ⋯ = v_{m}(b) = 0 are the elements of k.

Proof. See [AHS], Lemma, page 55.

Let ϖ : L^{∗} → ℤ^{q} be defined by ϖ(x) = (v_{1}(x), …, v_{q}(x)). For elements h ∈ A[T] we write ϖ(h), v_{i}(h) for ϖ(\overline{h}), v_{i}(\overline{h}).

Lemma 5.2. Assume that F and G satisfy (1) and (2). If F + λG = fg is a proper factorization, then λ is uniquely determined by ϖ(f). Thus ϖ(Π) can be mapped bijectively onto the set of λ’s for which F + λG is reducible.

Proof. Suppose that

f_{1}g_{1} = F + λ_{1}G

and

f_{2}g_{2} = F + λ_{2}G ,

and that ϖ(f_{1}) = ϖ(f_{2}) and λ_{1} ≠ λ_{2}. Then \overline{f}_{1}/\overline{f}_{2} = μ for some μ ∈ k^{∗}. This means that f_{1}−μf_{2} ∈ (F), but F is of higher degree than f_{1}−μf_{2}, hence f_{1} = μf_{2}. Substituting in the two equations above and subtracting one from the other we get f_{2}(μg_{1} − g_{2}) = (λ_{1} − λ_{2})G. Then f_{2} divides G and since it divides F + λG it also divides F, which contradicts the assumption that F and G are coprime. Thus λ_{1} = λ_{2}.

Corollary 5.3. Under the assumptions (1) and (2), if (1 − λ)F + λG = fg is a proper factorization, then λ is uniquely determined by ϖ(g). Thus, if Λ is the set of proper factors of (1 − λ)F + λG, then ϖ(Λ) can be mapped bijectively onto the set of λ’s for which (1 − λ)F + λG is reducible.

Proof. This is just a restatement of Lemma 5.2.

6. Proof of Proposition 4.3

Consider the diagram

[figure: W; f; g; π; X; X ×_{k} 𝔸^{1}; 𝔸^{1} .]

The map π is finite and flat because for any f and U the polynomial

Q_{f,U}(t, T) = (1 − t)P_{f,U}(T) + t(T − w_{1}/f) ⋯ (T − w_{n}/f))

is monic in T. The horizontal maps are clearly flat, hence f and g are flat. We first check that W is integral. We denote by Q_{ξ}(t, T) the restriction of some Q_{f,U}(t, T) to the generic point ξ of X. By 4.5 it suffices to show that W ×_{X}Spec(k(X)) = Spec(k(X)[T, t]/(Q_{ξ}(t, T)) is integral. Since P_{f,U}(T) and (T −w_{1}/f) ⋯ (T −w_{n}/f)) are coprime in k(X)[T], Q_{ξ}(t, T) is irreducible.

Since W is flat over 𝔸^{1}, 4.3 implies that W ×_{𝔸^{1}} Spec(k(t)) is also integral.

Consider now W ⊗_{k[t]} \overline{k(t)} and suppose that it is not integral. This means that Q_{ξ}(t, T) factors in k(X) ⊗_{k} \overline{k(t)}[T]: Q_{ξ}(t, T) = Q_{1}(T) ⋯ Q_{m}(T) with Q_{i} defined and irreducible over k(X) ⊗_{k} \overline{k(t)}. In this case Q_{ξ}(t, T) already factors in the same way when we replace \overline{k(t)} by a finite extension k(t)[w]/(Φ(t, w)) of k(t), where Φ(t, w) = c_{0}(t)w^{m} + ⋯ + c_{m}(t) with c_{i}(t) ∈ k[t]. There exists g(t) ∈ k[t] such that Q_{i}(T) ∈ (k[t, w, 1/g]/(Φ(t, w)))[T]. If λ ∈ k is different from the roots of g(t)c_{0}(t), we can choose a μ ∈ k such that φ(λ, μ) = 0. Then, specializing w to μ and t to λ yields a map (k[t, w, 1/g]/(Φ(t, w))) → k. Denoting the images under specialization by “bar” we obtain Q_{ξ}(λ, T) = \overline{Q}_{1}(T) ⋯ \overline{Q}_{m}(T). This shows that Q_{ξ}(t, T) decomposes for all but a finite number of values of t in k. We want to show that this fact leads to a contradiction.

It suffices to show (by Gauss’ lemma) that, for some U and some f, Q_{f,U}(λ, T) is irreducible for almost all λ ∈ k. We now follow the notation of §4. The irreduciblility of Q_{f,U}(λ, T) is equivalent to the irreduciblility of R_{f,U}(λ, T), and the factorization of Q_{f,U}(λ, T) in irreducible monic factors yields a similar factorization R_{f,U}(λ, T) = R_{1}(T) ⋯ R_{m}(T) into irreducible factors with constant term 1.

We shall use the result of the preceding section with A = H^{0}(U, O_{X}), F = Φ_{f,U} and G = Ψ_{f,U}.

We denote by R_{ξ} the restriction of R_{f,U}(λ, T) to the generic point ξ of X and by R_{ξ,i} the restriction to ξ of R_{i}. It follows from 4.2 that R_{f,U}(λ, T) is the restriction of a global section R of O_{X} ⊕ L ⊕ ⋯ ⊕ L^{⊗n} and the sections of this bundle form a finite dimensional vector space over k. Since the factorization of R_{f,U}(λ, T) over k[U] into factors with constant term 1 is unique, each factor R_{ξ,i} of R_{ξ} extends to a unique factor of R_{f,U} for any U over which L|_{U} is generated by a global section f. By patching, we see that every R_{ξ,i} is the restriction of a global section of

O_{X} ⊕ L ⊕ ⋯ ⊕ L^{⊗n} .

This shows that the set Λ of proper factors φ of the polynomials R_{f,U}(λ, T) when λ varies over k, are contained in a finite dimensional vector space over k. In other words, we can write any φ ∈ Λ as φ = μ_{1}h_{1} + ⋯ + μ_{r}h_{r} for some fixed polynomials h_{1}, …, h_{r}. Let L be the field of fractions of B = k[U][T]/(Φ) where Φ = T^{n}P (λ, T^{−1}). The image of R_{f,U}(λ, T) in B is the class of λΨ_{f,U}(T) where Ψ is as defined in 4.2. For any discrete kvaluation v of L we have v(φ) ≥ min{v(h_{1}), …, v(h_{r})}, which shows that v(Λ) is bounded below. But if φψ = R_{f,U}(λ, T) (necessarily with λ ≠ 0 because Φ_{f,U} is irreducible) then −v(φ) = v(ψ) − v(R_{f,U}(λ, T)) = v(ψ) − v(Ψ_{f,U}(T)). Since ψ is also in Λ, the value of v(φ) is as well bounded above. This proves that if v_{1}, …, v_{q} are the valuations described in 5.1 (for B = k[U][T]/(Φ) and F (T) = Φ(T)), then the image of Λ in ℤ^{q} obtained by mapping φ ∈ Λ to (v_{1}(φ), …, v_{q}(φ)) is a finite set. We conclude, using 5.3, that R_{f,U}(λ, T) is reducible only for finitely many λ ∈ k. Thus Q_{ξ}(λ, T) is irreducible for almost all λ ∈ k, leading to a contradiction. This finishes the proof of Proposition 4.3.

7. Galois splittings

We now construct, for any λ ∈ k^{N}, a Galois covering Z_{λ} of X with group G, such that X = Z_{λ}/G. Notice that, in general, even if Y_{λ} is smooth and Y_{λ} → X is a projective map, the Galois closure of Y_{λ} is not smooth. Therefore, in order to have Y and Z smooth, we must construct both at the same time. We proceed as in the construction of Y_{λ}. Let U ⊂ X be an affine open set for which L|_{U} is isomorphic to O_{U}f for some section f on U . Let L, s_{1}, …, s_{N} ∈ H^{0}(X, A ⊗_{O_{X}} L) and s = λ_{1}s_{1}+⋯ λ_{n}s_{n} be as before. Let P_{f,U}(T) = T^{n}+b_{1}T^{n−1}+⋯+b_{n} be the characteristic polynomial of s/f ∈ A(U). We choose n isomorphic copies L_{1}, …, L_{n} of L and, for each i, f_{i} = f the generator of L_{i}|_{U}. Consider

T = Sym(L_{1}^{−1} ⊕ ⋯ ⊕ L_{n}^{−1}) .

Writing f_{i}^{−1}f_{j}^{−1} instead of f_{i}^{−1} ⊗_{O_{U}} f_{j}^{−1} we shall write the restriction of T to U simply as

⨁ O_{U}f_{1}^{−i_{1}} ⋯ f_{n}^{−i_{n}} .

Note that T |_{U} is isomorphic to O_{U}[T_{1}, …, T_{n}]. For 1 ≤ i ≤ n let σ_{i} be the i-th elementary symmetric function in the n variables T_{1}, …, T_{n}. We define J_{f,U} ⊂ T |_{U} as the ideal generated by

σ_{i}(f_{1}^{−1}, …, f_{n}^{−1}) − (−1)^{i}b_{i} 1 ≤ i ≤ n .

It corresponds in the polynomial algebra to the ideal generated by

σ_{i}(T_{1}, …, T_{n}) − (−1)^{i}b_{i} 1 ≤ i ≤ n .

As in the preceding section, it is easy to check that these ideals do not depend on the choice of f and can therefore be patched over the various U’s to obtain a global ideal J_{λ} ⊂ T . Let Z_{λ} be the closed subscheme of Spec(T ) defined by J_{λ}.

Proposition 7.1. The symmetric group S_{n} acts on Z_{λ} via its obvious action on T . The quotient Z_{λ}/S_{n} coincides with X and Y_{λ} coincides with the quotient Z_{λ}/S_{n−1}, where S_{n−1} is the isotropy group of 1.

Proof. It suffices to deal with the affine case. Let P (T) = T^{n} + b_{1}T^{n−1} + ⋯ + b_{n} be a monic polynomial with coefficients in a ring R and assume (it is the case we are ultimately interested in) that 2 is invertible in R. Let B be the quotient of R[T_{1}, …, T_{n}] by the ideal I generated by all polynomials σ_{i}(T_{1}, …, T_{n}) − (−1)^{i}b_{i}. Denote by σ′_{i}(T_{1}, …, T_{n−1}) the i-th elementary symmetric function in n − 1 indeterminates. To the quotient of Z_{λ} by S_{n} corresponds the ring of invariants B^{S_{n}}. Since S_{n} acts trivially on I and 2 is invertible in R we have

H^{1}(S_{n}, I) = Hom(S_{n}, I) = Hom(μ_{2}, I) = 0 .

Hence

(R[T_{1}, …, T_{n}]/I)^{S_{n}} = R[T_{1}, …, T_{n}]^{S_{n}}/I^{S_{n}} = R[σ_{1}, …, σ_{n}]/I = R .

Similarly we obtain

(R[T_{1}, …, T_{n}]/I)^{S_{n−1}} = R[σ′_{1}, …, σ′_{n−1}, T_{n}]/I .

The relations

σ_{1} = σ′_{1} + T_{n} , σ_{i} = σ′_{i} + T_{n}σ′_{i−1} for 2 ≤ i ≤ n − 1 and σ_{n} = T_{n}σ′_{n−1}

immediately give that R[σ′_{1}, …, σ′_{n−1}, T_{n}]/I = R[T_{n}]/(P (T_{n})).

Theorem 7.2. Assume that k is of characteristic zero. There exists a nonempty open set U ⊂ k^{N} such that, for any λ ∈ U, Z_{λ} is a projective smooth surface.

The proof requires some preliminaries. Let X_{ij} with i, j running from 1 to n be indeterminates and write P (T) = T^{n} + a_{1}T^{n−1} + ⋯ + a_{n} for the characteristic polynomial of the generic matrix (X_{ij}). Let A be the polynomial k-algebra in the X_{ij}. Consider another set T_{1}, …, T_{n} of indeterminates and put

B_{n} = A[T_{1}, …, T_{n}]/I

where I is the ideal generated by all the polynomials σ_{i}(T_{1}, …, T_{n})−(−1)^{i}a_{i} for 1 ≤ i ≤ n. Let Z_{n} = Spec(B_{n}). We want to determine Sing(Z_{n}).

A k-point of Z_{n} is a pair (α, t) with α ∈ M_{n}(k) and t = (t_{1}, …, t_{n}) ∈ k^{n} such that t_{1}, …, t_{n} are the eigenvalues of α, i.e. the roots of the characteristic polynomial of α, which we write as

P (α)(T) = T^{n} + a_{1}(α)T^{n−1} + ⋯ + a_{n}(α) .

Let π : Z_{n} → Spec(A) be the first projection and let S = π(Sing(Z_{n})). We want to compute the dimension of S.

Let (α, t) be a singularity of Z_{n}. Since no σ_{i}(T_{1}, …, T_{n}) involves the X_{ij} and no a_{j} involves the T_{i}, if we order the X_{ij} lexicographically, the Jacobian matrix of the equations σ_{i}(T_{1}, …, T_{n}) − (−1)^{i}a_{i} = 0 is of size (n^{2} + n) × n and looks as follows:

J = ( \begin{matrix}\frac{∂σ_{1}}{∂T_{1}} & ⋯ & \frac{∂σ_{n}}{∂T_{1}} \\ ⋮ & & ⋮ \\ \frac{∂σ_{1}}{∂T_{n}} & ⋯ & \frac{∂σ_{n}}{∂T_{n}} \\ \frac{∂a_{1}}{∂X_{11}} & ⋯ & \frac{∂a_{n}}{∂X_{11}} \\ ⋮ & & ⋮ \\ \frac{∂a_{1}}{∂X_{nn}} & ⋯ & \frac{∂a_{n}}{∂X_{nn}} \end{matrix} ) .

By 7.1, π is a finite map and the dimension of Z_{n} is n^{2}. The point (α, t) being a singularity of Z_{n}, the Jacobian criterion implies that the rank of J at (α, t) is at most n − 1. Thus, in particular, the determinant δ of the top n × n block of J must vanish at (α, t). It is well-known (and can be proved by an easy induction on n) that δ = ± ∏_{i<j}(T_{i} −T_{j}). This shows that α has at least two equal eigenvalues. In other words, denoting by V (–) the vanishing locus of a given set of polynomials, (α, t) belongs to the vanishing locus V (δ^{2}) of the discriminant δ^{2} of P (T). Consider now Sing(Z_{n}) ∩ V (a_{1}, …, a_{n}). Since Sing(Z_{n}) ⊂ V (δ^{2}) we have Sing(Z_{n} ∩ V (a_{1}, …, a_{n}) = Sing(Z_{n} ∩ V (δ^{2}, a_{1}, …, a_{n}). But the vanishing of a_{1}, …, a_{n−1} and δ^{2} already implies the vanishing of a_{n}; in fact, if T^{n} − a_{n} has a multiple root, then a_{n} = 0 (we are in characteristic 0). Thus

Sing(Z_{n}) ∩ V (a_{1}, …, a_{n−1}) = Sing(Z_{n}) ∩ V (a_{1}, …, a_{n})

and therefore dim(Sing(Z_{n})) ≤ dim(Sing(Z_{n})∩V (a_{1}, …, a_{n}))+n−1. The set V (a_{1}, …, a_{n}) is the set N of nilpotent matrices. On the other hand, the bottom block of the Jacobian matrix must have rank at most n − 1, which means that α is a singular point of N. This shows that Sing(Z_{n}) ∩ N ⊆ Sing(N) and from the previous inequality we obtain the next result.

Lemma 7.3. The dimension of Sing(Z_{n}) is at most dim(Sing(N)) + n − 1.

We now compute the dimension of Sing(N). We begin with the computation of the dimension of N.

Proposition 7.4. Let N ⊂ M_{n} denote the variety of nilpotent matrices. Then the dimension of N is n^{2} − n.

Proof. Since N is defined by the ideal (a_{1}, …, a_{n}) of A = k[X_{11}, X_{12}, …, X_{nn}], it suffices to show that this ideal has height n. Let I be the ideal generated by (a_{1}, … a_{n}, X_{ij} | i ≠ j). We claim that this ideal has height n^{2}. The ring A/I is isomorphic to k[X_{11}, X_{2,2}, …, X_{nn}]/J where J is the ideal generated by elementary symmetric functions σ_{1}, …, σ_{n} in X_{ii}. Since k[X_{11}, …, X_{nn}] is finite over k[σ_{1}, …, σ_{n}], the ideal J has height n in k[X_{11}, …, X_{nn}]. Hence I is supported only at closed points. Since the a_{i} are homogeneous, it follows that the ideal (a_{1}, …, a_{n}) has height n.

Lemma 7.5. A nilpotent matrix α whose Jordan form consists of only one cyclic block is not a singularity of N. More precisely, the determinant of (\frac{∂a_{i}}{∂X_{j1}} ) is not zero at α.

Proof. Let A be as before and P (T) = T^{n}+a_{1}T^{n−1}+⋯+a_{n} the characteristic polynomial of the generic matrix (X_{ij}). The variety of nilpotent matrices is N = V (a_{1}, …, a_{n}). We show that at

α = ( \begin{matrix}0 & 1 & 0 & · & · & · & 0 & 0 \\ 0 & 0 & 1 & · & · & · & 0 & 0 \\ 0 & 0 & 0 & · & · & · & 0 & 0 \\ · & · & · & · & · & · & · \\ 0 & 0 & 0 & · & · & · & 0 & 1 \\ 0 & 0 & 0 & · & · & · & 0 & 0 \end{matrix} )

the jacobian matrix (\frac{∂a_{i}}{∂X_{ij}} ) has rank n. We compute the n × n matrix (\frac{∂a_{i}}{∂X_{j1}} ). The

derivative of a_{i} by X_{j1} is the coefficient of T^{n−i} in \frac{∂P (T)}{∂X_{j1}}. Developing the determinant of (X_{ij}) − T I_{n} along the first column we find

±P (T) = (X_{11} − T)P_{1}(T) + X_{2,1}P_{2}(T) + ⋯ + X_{n,1}P_{n}(T)

where P_{i}(T) is the determinant of an (n − 1) × (n − 1) matrix M_{i}. At (X_{ij}) = α we find

M_{i}(α) = ( \begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix} )

with

B_{1} = ( \begin{matrix}1 & 0 & 0 & · & · & · & 0 & 0 \\ −T & 1 & 0 & · & · & · & 0 & 0 \\ 0 & −T & 1 & · & · & · & 0 & 0 \\ · & · & · & · & · & · & · \\ 0 & 0 & 0 & · & · & · & 1 & 0 \\ 0 & 0 & 0 & · & · & · & −T & 1 \end{matrix} )

of size j − 1 and

B_{2} = ( \begin{matrix}−T & 1 & 0 & · & · & · & 0 & 0 \\ 0 & −T & 1 & · & · & · & 0 & 0 \\ 0 & 0 & −T & · & · & · & 0 & 0 \\ · & · & · & · & · & · & · \\ 0 & 0 & 0 & · & · & · & −T & 1 \\ 0 & 0 & 0 & · & · & · & 0 & −T \end{matrix} )

of size n − j. Thus P_{j}(T) = ±T^{n−j} and \frac{∂a_{i}}{∂X_{j1}} (α) is ±1 for j = i and zero otherwise. This proves the lemma.

Lemma 7.6. The set N_{2} of nilpotent matrices whose Jordan form has exactly two cyclic blocks are dense in the set of nilpotent matrices whose Jordan form has two or more blocks.

Proof. Let α = diag(B_{1}, B_{2}, …, B_{m}) be a nilpotent matrix which we can assume to be in Jordan form with blocks B_{1}, …,B_{m}, m ≥ 3. Let g ≠ 0 with g ∈ A define a neighbourhood of α. We can find constants ϵ_{2}, …, ϵ_{m−1} such that replacing the zeros between the superdiagonals of B_{2} and B_{3}, between the superdiagonals B_{3} and B_{4} and so on, by the ϵ_{i} we obtain a matrix α′ such that g(α′) ≠ 0. Clearly α′ has two cyclic blocks.

Lemma 7.7. If α ∈ N has a Jordan form with two or more cyclic blocks, then α is a singularity of N.

Proof. We may assume that α is in Jordan form and can be written as α = diag(B_{1}, B_{2}, …, B_{m}) where m ≥ 2, each B_{i} is a cyclic Jordan block, B_{1} is of size p and B_{2} of size q. We can write the generic matrix as (X_{ij}) = (α + Y_{ij}). Then \frac{∂a_{i}}{∂X_{ij}} (α) = \frac{∂a_{i}}{∂Y_{ij}} (0). But in the matrix α+(Y_{ij}) the p-th line and the (p+q)-th line are linear homogeneous in the Y_{ij}, hence developing the determinant of α + (Y_{ij}) along these two lines we see that a_{n}(Y_{ij} | 1 ≤ i, j ≤ n) has no constant and no linear term. This shows that all the derivatives \frac{∂a_{n}}{∂Y_{ij}} vanish at the origin and therefore the Jacobian matrix \frac{∂a_{i}}{∂Y_{ij}} cannot be of rank n.

Corollary 7.8. The set N_{2} is dense in Sing(N)

The set N_{2} is the union of the GL_{n}(k)-orbits S_{p,q} of all the matrices of the form β = diag(B_{p}, B_{q}) where B_{p} is the nilpotent cyclic Jordan block of size p and B_{q} the nilpotent cyclic Jordan block of size q = n−p. In particular, it is the finite union of the constructible sets S_{p,q}. The dimension of S_{p,q} is n^{2} − s where s is the dimension of the isotropy group of β

Lemma 7.9. For n ≥ 3 the dimension of the isotropy group of diag(B_{p}, B_{q}) is n + 2 min(p, q). In particular it is always at least n + 2.

Proof. Let Γ ⊂ GL_{n}(K) be the isotropy group of β = diag(B_{p}, B_{q}). Let

γ = ( \begin{matrix}A & B \\ C & D \end{matrix} )

be an element of Γ, written with blocks A, B, C, D of suitable sizes. The condition γβγ^{−1} = β is equivalent to the conditions

AB_{p} = B_{p}A , DB_{q} = B_{q}D , BB_{q} = B_{p}B , CB_{p} = B_{q}C .

We compute the dimension of the linear subspace Γ_{0} of M_{n}(K) consisting of matrices that satisfy the four conditions above.

An explicit matrix computation shows that the first condition gives

A = ( \begin{matrix}a_{1} & a_{2} & a_{3} & · & · & · & a_{p−1} & a_{p} \\ 0 & a_{1} & a_{2} & · & · & · & a_{p−2} & a_{p−1} \\ 0 & 0 & a_{1} & · & · & · & a_{p−3} & a_{p−2} \\ · & · & · & · & · & · & · & · \\ 0 & 0 & 0 & · & · & · & a_{1} & a_{2} \\ 0 & 0 & 0 & · & · & · & 0 & a_{1} \end{matrix} )

A similar result holds for D, hence the matrices diag(A, D) in Γ_{0} span a linear space of dimension p + q = n.

Assume now that p ≤ q. An explicit computation shows that the third condition gives

B = ( \begin{matrix}0 & · & · & · & 0 & b_{1} & b_{2} & b_{3} & · & · & · & b_{p−1} & b_{p} \\ 0 & · & · & · & 0 & 0 & b_{1} & b_{2} & · & · & · & b_{p−2} & b_{p−1} \\ 0 & · & · & · & 0 & 0 & 0 & b_{1} & · & · & · & b_{p−3} & b_{p−2} \\ · & · & · & · & · & · & · & · & · & · & · & · & · \\ · & · & · & · & · & · & · & · & · & · & · & · & · \\ 0 & · & · & · & · & · & · & · & · & · & · & b_{1} & b_{2} \\ 0 & · & · & · & · & · & · & · & · & · & · & 0 & b_{1} \end{matrix} )

A similar result holds for C, hence, when p ≤ q the dimension of Γ_{0} is n + p + p = n + 2 min(p, q) and clearly this is also the dimension (as a variety) of Γ.

Proposition 7.10. For n ≥ 3 the dimension of Sing(N) is n^{2} − n − 2.

Proof. By 7.8 and 7.9, dim(Sing(N)) = dim(N_{2}) = n^{2} − min_{p,q}(dim(S_{p,q})). The isotropy group of minimal dimension is S_{1,n−1} which has dimension n + 2. Thus dim(N_{2}) = n^{2} − (n + 2).

Theorem 7.11. For n ≥ 3 the dimension of Sing(Z_{n}) is at most n^{2} − 3

Proof. This immediately follows from 7.3 and 7.10.

Proof of Theorem 7.2. If n = 2 then U = k^{N} and for any λ ∈ k^{N}, Z_{λ} = Y_{λ}. We therefore assume that n ≥ 3. In this case the proof is on similar lines as the proof of Theorem 2.11. We extend the base to X̃ = X×𝔸^{N} where 𝔸^{N} = Spec (k[t_{1}, …, t_{N}]) and define Ã, L̃ and L̃_{i} for 1 ≤ i ≤ n as the inverse images of A, L and the L_{i}’s under the projection π : X̃ → X. Repeating the construction of J_{λ} we obtain an ideal J_{t}, where t = (t_{1}, …, t_{N}), which specializes to J_{λ} when we specialize t to λ. The scheme Z̃ is the closed subscheme of

Spec(T̃) = Spec(Sym(L̃_{1}^{−1} ⊕ ⋯ ⊕ L̃_{n}^{−1}))

defined by J_{t}. Look at the diagram

[figure: Z̃; p; π; q; X; X × 𝔸^{N}; 𝔸^{N}]

The map π is clearly finite and flat and the two projections from X × 𝔸^{N} are flat, hence p is flat. By 2.12 the singularities of Z̃ are contained in the union of the singularities of the fibers of p. Since, by Theorem 7.11, the singularities of the closed fibres of p are at worst in codimension 3, we can argue exactly as in the proof of Theorem 2.11 and conclude that q is generically smooth, from which the assertion of Theorem 7.2 immediately follows.

In general the fibre Z_{λ} is not integral. This happens, for instance, if Y is already a Galois covering of X. But if we choose Z_{λ} smooth, then it will be a disjoint union of irreducible smooth varieties and picking one of them (which we call Z) we obtain a Galois covering π : Z → X. Let G be the Galois group of k(Z)/k(X), |G| its order and H ⊂ G the Galois group of k(Z)/k(Y). Clearly H is of index n in G.

Proposition 7.12. Suppose that n ≥ 3 and let x ∈ X be a closed point. Then

(a) if x has n preimages in Y, then π : Z → X is étale at x;

(b) if x has n−1 preimages y_{1}, …, y_{n−1} corresponding to the roots τ_{1}, …, τ_{n−1} of P (x, T) and τ_{1} is the unique double root of P (x, T), then π_{Z} : Z → Y is étale at y_{1};

(c) the fibre π^{−1}(x) in Z consists of |G|/2 points and each of them has multiplicity 2.

Proof. If the point x has n distinct preimages y_{1}, …, y_{n} in Y, then clearly π_{Z} is étale at y. We can identify each y_{i} with a root τ_{i} of P (x, T) and we can identify every point z of π^{−1}(x) with a permutation (σ(τ_{1}), …, σ(τ_{n})) where σ ∈ G and π_{Z} maps z to its first coordinate σ(τ_{1}). Since locally at y_{1} the map Z → Y is Galois with group H, the fibre of y_{1} consists of |H| = |G|/n points (τ_{1}, σ(τ_{2}), …, σ(τ_{n})) where σ ∈ H. When τ_{1} = τ_{2} and τ_{i} ≠ τ_{1} for i ≠ 1, 2, the fibre of y_{1} = y_{2} still consists of |H| distinct points (τ_{1}, σ(τ_{2}), …, σ(τ_{n})), hence π_{Z} is étale at y_{1}. This proves (b) and shows that every z ∈ π_{Z}^{−1}(y_{1}) has multiplicity 2 in π^{−1}(x). Since G operates transitively on π^{−1}(x), this is true of all the points in the fibre, whence (c).

Theorem 7.13. Let X be a projective smooth surface over an algebraically closed field k of characteristic zero and A an Azumaya algebra over X. Suppose that A_{k(X)} is a division algebra of prime degree l. Then there exist smooth integral projective surfaces Y and Z and finite flat maps

Z ⟶^{π_{Z}} Y ⟶^{π_{Y}} X

such that

(1) π_{Y}^{∗} A is trivial in the Brauer group of Y;

(2) the composite Z ⟶^{π_{Z}} Y ⟶^{π_{Y}} X is a (ramified) Galois covering;

(3) the degree of π_{Y} is l and the degree of π_{Z} is prime to l;

(4) there are only finitely closed point on X which have fewer that l − 1 preimages in Y;

(5) if π_{Y} is ramified at y ∈ Y and π_{Y} (y) has l − 1 preimages in Y, then π_{Z} is étale at y.

8. General sections of a very ample line bundle.

For a finite map f : V → W of varieties we shall always denote by Δ(V/W) the ramification locus of f in V and by δ(V/W) = f(Δ(Y/W)) the ramification locus of f in W, both understood with their reduced structure. Let Y ⟶^{π_{Y}} X be a map of smooth projective surfaces satisfying (1), (2), and (4) of 2.16. We put Δ = π_{Y}^{−1}(δ) the set-theoretical preimage of δ(Y/X) in Y. Note that Δ(Y/X) ⊂ Δ. We recall the local structure of π_{Y}. The construction of Y in §2 shows that, locally on X, Y is isomorphic to the spectrum of an extension S = R[T]/(P (T)), where R = O_{X,x} is the local ring of the closed point x = π_{Y} (y) of X and P (T) = T^{n} + b_{1}T^{n−1} + ⋯ + b_{n} is the characteristic polynomial of a generic element of an Azumaya algebra over R. We denote by m_{x} the maximal ideal of R and by m_{y} the maximal ideal of S corresponding to y. Let L be a very ample line bundle on Y, generated by N global sections s_{1}, …, s_{N}. We say that a general section s of L satisfies a certain property if there exists a Zariski open dense set U ⊂ k^{N} such that, for every λ = (λ_{1}, …, λ_{N}) ∈ U, s = ∑_{i} λ_{i}s_{i} satisfies the property.

We want to show that a general global section s of L has a smooth vanishing locus V (s) whose image π_{Y} (V (s)) has no singularity on δ(Y/X). We begin with a result of Colliot-Thélène ([CT], Lemma 3.1) . We set V (s) = D and π_{Y} (V (s)) = C.

Proposition 8.1. If s is a general section of L, then every point of C ∩ δ(Y/X) has at most one preimage on D

Proof. We essentially reproduce the proof given in [CT]. Consider the open set of Δ ×_{X} Δ

U = {(y_{1}, y_{2}) ∈ Δ ×_{k} Δ | y_{1} ≠ y_{2}, π_{Y} (y_{1}) = π_{Y} (y_{2})} .

Let H = H^{0}(Y, L) and Z a closed set in U ×_{k} H defined by

Z = {(y_{1}, y_{2}, s) | s(y_{1}) = s(y_{2}) = 0} .

Since U is a one-dimensional scheme and the fibres of the projection Z → U have dimension dim(H)−2, the dimension of Z is dim(H)−1. Hence the projection of Z into H is contained in a proper closed subset and any section in the complementary open set has the required property.

A general section s of L has smooth vanishing locus D which crosses Δ transversally. With such a choice of s, our next step is to insure that, if y is in D ∩ Δ(Y/X), then for some open neighbourhood U of y the restriction of π_{Y} to D ∩ U has a smooth image. Let φ = 0 be a local equation of D in a neighbourhood of y. The point y corresponds to a (multiple) root τ ∈ k of P (x, T) and the function φ can be lifted to a polynomial Φ ∈ R[T].

Proposition 8.2. Suppose that y is a closed point of Δ(Y/X) ∩ D, π_{Y} (y) = x and π_{Y}^{−1}(x) = {y, y_{2}, …, y_{r}} where y_{2}, …, y_{r} are not on D. Suppose further that

(⋆)  \frac{∂Φ(x, T)}{∂T}(τ) ≠ 0 .

Then π_{Y} induces an isomorphism O_{C,x} → O_{D,y}. Thus, D being smooth at y, C is smooth at x.

Proof. Note that by 8.1 every x ∈ δ(Y/X) ∩ C has exactly one preimage in D. Let R be the local ring O_{X,x} of X at x. Let y, y_{2}, …, y_{r} be the preimages of x in Y. Since x ∈ C ∩ δ(Y/X) has exactly one preimage y on D, the points y_{2}, …, y_{r} do not lie on D.The homomorphism π_{Y}^{∗} : O_{C,x} → O_{D,y} is the finite extension

χ : R/(P, Φ) ∩ R ⟶ R[T]/(P, Φ) .

We may assume, by a change of variable, that y ∈ Spec(R[T]) corresponds to the maximal ideal m_{y} = (m_{x}, T). Denoting by “bar” the reduction modulo m_{x}, we have \overline{P} (T) = T^{e}P_{1}(T) with e ≥ 2, P_{1}(0) ≠ 0, and \overline{Φ}(T) = T Φ_{1}(T). By (⋆) we have Φ_{1}(0) ≠ 0. Hence R[T]/(P, Φ, m_{x}) = k. By Nakayama’s lemma this implies that χ is an isomorphism.

It remains to show that the condition (⋆) can be realized at every point of Δ(Y/X) ∩ D. We can translate it in terms of sections in the following way. Let s ∈ H^{0}(Y, L). Let y ∈ Δ be a closed point at which the section s vanishes, x = π_{Y} (y) and L(y) the fibre of L at y. We define (∂s/∂T)(y) ∈ L(y) as follows. Let R = O_{X,x}, O_{Y,π_{Y}^{−1}(x)} = R[T]/(P (T)) and let e be a generator of L at O_{Y,π_{Y}^{−1}(x)}. Let s = φe with φ ∈ O_{Y,π_{Y}^{−1}(x)}. Then, φ(y) = 0. Set (∂s/∂T)(y) = (∂Φ/∂T)(y) · e(y) where Φ ∈ R[T] represents φ and e(y) the image of e in L(y). This definition is independent of the choice of e and Φ since P′(T) vanishes at y. The condition (⋆) is equivalent to (∂s/∂T)(y) ≠ 0.

Lemma 8.3. For every closed point y ∈ Δ(Y/X) there exists s ∈ H^{0}(Y, L) such that if s(y) = 0, then (∂s/∂T)(y) ≠ 0.

Proof. Let H^{0}(Y, L)_{y} be the vector space of all sections vanishing at y. By Bertini’s theorem ([Ha_{1}], Ch. II, 8.18) the map H^{0}(Y, L)_{y} → m_{y}/m_{y}^{2} sending s = φe to the class of φ is surjective. The maximal ideal m_{y} at y of R[T]/(P (T)) is generated by (m_{x}, (T − α)), where m_{x} is the maximal ideal of R and α a repeated root of \overline{P} (T) in k. By the surjectivity assertion in the theorem of Bertini stated above, there exists s ∈ H^{0}(Y, L)_{y} such that s = φe with φ − (T − α) ∈ m_{y}^{2}. For this choice of s, (∂s/∂T)(y) = e(y) ≠ 0.

Proposition 8.4. For a general s ∈ H^{0}(Y, L), V (s) is smooth, irreducible, intersects Δ transversally and avoids the ramification locus of π_{Y} |_{Δ} : Δ → δ(Y/X); further, for each intersection y of V (s) with Δ(Y/X), π_{Y} (V (s)) is smooth at π_{Y} (y).

Proof. The set of singular points of Δ and the ramification locus of π_{Y} |Δ : Δ → δ(Y/X) form a finite set Σ of closed points of Δ. By Bertini’s theorem, for a general s ∈ H^{0}(Y, L), s(y) ≠ 0 for y ∈ Σ and V (s) is smooth, irreducible and intersects Δ transversally. We show that for a general s ∈ H^{0}(Y, L), s(y) = 0 implies that (∂s/∂T)(y) ≠ 0. Let s_{1}, s_{2}, …, s_{N} be a set of generators of the k-vector space H^{0}(Y, L). Let s = ∑ X_{i}s_{i} where X_{i} are indeterminates and let \overline{s} be the restricition of s to Δ × 𝔸^{n}. We look at the closed set V (\overline{s}, ∂s/∂T) ⊂ Δ × A^{N}. We claim that the dimension of V (\overline{s}, ∂s/∂T) is at most N − 1. We check this at every smooth point y ∈ Δ. Let B be the local ring of Δ at y. Let e a local generator of s at y and s_{i} = f_{i}e, ∂f_{i}/∂T = g_{i}. We need to show that the dimension of B[X_{1}, …, X_{N}]/(∑ f_{i}X_{i}, ∑ g_{i}X_{i}) is at most N − 1. Suppose this dimension is greater than N − 1. In this case the height of the ideal (∑ f_{i}X_{i}, ∑ g_{i}X_{i}) is at most 1. Since B is a discrete valuation ring, there exists a, b, h ∈ B[X_{1}, …, X_{N}] such that ∑ f_{i}X_{i} = ah, ∑ g_{i}X_{i} = bh. Suppose that h does not belong to B. Degree considerations show that a belongs to B and, if a is not a unit in B, it divides each f_{i}. Thus the s_{i}’s vanish at y, contradicting the fact that they generate the sections of L globally. Therefore, a is a unit in B and ∑ g_{i}X_{i} = ba^{−1} ∑ f_{i}X_{i}. There exists by 8.3, λ = (λ_{1}, …, λ_{N}) such that s_{λ}(y) = 0 and ∂s_{λ}/∂T (y) ≠ 0 where s_{λ} = ∑ λ_{i}s_{i}. This contradicts the equality ∑ g_{i}X_{i} = ba^{−1} ∑ f_{i}X_{i}. The case h ∈ B is dealt with in the same way. Thus the dimension of V (s, ∂s/∂T) is at most N −1. Under the projection Δ×𝔸^{N} → 𝔸^{N}, the set V (s, ∂s/∂T) maps into a proper closed subset of 𝔸^{N}. Thus for an open set of λ ∈ 𝔸^{N}, s_{λ} has the property that if s_{λ}(y) = 0 for y ∈ Δ_{Y}, then \frac{∂s_{λ}}{∂T} ≠ 0. By 8.2 π_{Y} (V (s)) is smooth at π(y).

Proposition 8.5. For a general section s of L each point of π_{Y} (V (s)) has at least l − 1 distinct preimages in Y.

Proof. Since under π_{Y} : Y → X there only are finitely many points of X with fewer that l − 1 preimages (condition (4) of 2.16), if s is chosen such that V (s) avoids the preimages of these finitely many points, then s satisfies the condition of the proposition.

Theorem 8.6. Let Z ⟶^{π_{Z}} Y ⟶^{π_{Y}} X be maps of smooth projective surfaces as in Theorem 7.13. Let L be a very ample line bundle on Y, generated by global sections s_{1}, …, s_{N}. Assume that π_{Z}^{∗}L is very ample. For any λ = (λ_{1}, …, λ_{N}) set s_{λ} = λ_{1}s_{1} + ⋯ + λ_{N}s_{N}. Let D_{λ} be the vanishing locus of s_{λ} and C_{λ} = π_{Y} (D_{λ}) be the image of D_{λ} in X. Let D_{Z,λ} be the vanishing locus of π_{Z}^{∗}(s_{λ}) on Z. There exists a dense open subset U ⊂ k^{N} such that for every λ ∈ U the following five conditions are satisfied:

(1) D_{λ} is irreducible, smooth and intersects Δ transversally;

(2) D_{λ} avoids the ramification locus of π_{Y} |_{Δ} : Δ → δ(Y/X);

(3) every point of C_{λ} has at least n − 1 preimages in Y;

(4) no singularitiy of C_{λ} lies on δ(Y/X);

(5) D_{Z,λ} is smooth and irreducible.

Proof. Conditions (1) and (2) follow from 8.4. Condition (3) is a consequence of 8.5. Let x be in C_{λ} ∩ δ(Y/X) and let y be its unique preimage in D. If y ∈ Δ(Y/X), then C_{λ} is smooth at x by 8.4. If y belongs to Δ \ Δ(Y/X) then C_{λ} is smooth at x because D_{λ} is smooth, π_{Y} is étale at y and y is the unique preimage of x in D. Thus (4) is satisfied. To see condition (5) note that π_{Z}^{−1}(D_{λ}) is the vanishing locus of the section ∑_{i} λ_{i}π_{Z}^{∗}s_{i} of π_{Z}^{∗}L; since π_{Z}^{∗}L is very ample we can apply Bertini’s theorem ([Ha_{1}], Ch. II, 8.18).

9. Extracting a root from a bundle

The main result of this section is the following theorem.

Theorem 9.1. Let Z ⟶^{π_{Z}} Y ⟶^{π_{Y}} X be maps of smooth projective surfaces satisfying (2) to (5) of Theorem 7.13. Suppose that n is equal to a prime l. Given a line bundle L_{Y} on Y with its class in

Pic(Y)/l · Pic(Y) ⊂ H^{2}(Y, μ_{l})

a pull-back of a class ζ ∈ H^{2}(X, μ_{l}), there exists a map h_{X} : X̃ → X which is proper, generically cyclic of degree l and such that

(1) the normalization Ỹ of Y ×_{X} X̃ is smooth,

(2) if h_{Y} : Ỹ → Y is the natural map, then h_{Y}^{∗} (L_{Y}) vanishes in

Pic(Ỹ)/lPic(Ỹ) + π∗ ; Ỹ (Pic(X̃)) .

This section is entirely dedicated to the proof of this theorem.

For any pair of morphisms W → V and Spec(A) → V we denote the fibre product W ×_{V} Spec(A) by W_{A}. Let Z^{1}(V) denote the free abelian group on codimension 1 cycles of any variety V. For a codimension 1 closed set W ⊂ V we denote by {W } the cycle in Z^{1}(V) corresponding to W and by [W] the class of W in the divisor class group of V.

By modifying L_{Y} by the l^{n}-th power of an ample line bundle L_{0} we may assume that L_{Y} is very ample. Since the pull-back of an ample line bundle by the finite map π_{Z} is ample ([Ha_{2}], I, 4.4) we may further assume that π_{z}^{∗}L is very ample. By the previous section there exists a smooth connencted curve D_{Y} on Y such that L_{Y} = O_{Y} (D_{Y}) and D_{Y} satisfies conditions (1) to (5) of Theorem 8.6. Let π_{Y} (D_{Y}) = D_{X}. We have (π_{Y})_{∗}([D_{Y}]) = r[D_{X}]. Note that r is the degree of k(D_{Y}) over k(D_{X}) and thus r ≤ l. If r = l, D_{Y} is the only divisor on Y lying over D_{X}, it is not ramified at its generic point, and π_{Y}^{∗} [D_{X}] is [D_{Y}]. Therefore L_{Y} = (π_{Y})^{∗}(O_{X}(D_{X})). In this case we can take X̃ = X and Ỹ = Y. Assume now that r < l. By assumption we have [L_{Y}] = (π_{Y})^{∗}(ζ) ∈ Pic(Y)/l ⊆ H^{2}(Y, μ_{l}). Then r[D_{X}] = (π_{Y})_{∗}(π_{Y})^{∗}(ζ) = lζ = 0 in H^{2}(X, μ_{l}). Since r < l this implies [D_{X}] = 0 in Pic(X)/l. Let L_{1} be a line bundle on X for which there is an isomorphism φ : O_{X}(D_{X}) → L_{1}^{⊗l}. Choosing a section s : O_{X} → L_{1}^{⊗l} vanishing precisely along D_{X} we can define, using the dual map (L_{1}^{−1})^{⊗l} → O_{X} (see [BPV], I, §17), an O_{X}-algebra structure on

S = O_{X} ⊕ L_{1}^{−1} ⊕ ⋯ ⊕ (L_{1}^{−1})^{⊗(l−1)} .

Let X′ = Spec(S) and g_{X} : X′ → X the map giving the O_{X}-algebra structure on S. Note that locally on X the scheme X′ looks like the spectrum of R[t]/(t^{l} − f), f a local equation of D_{X}. This map is a cyclic cover of degree l, ramified precisely along D_{X}. In particular (g_{X})^{∗}{D_{X}} = l{D′}, where D′ is an effective Cartier divisor on X′. We put Y′ = Y ×_{X} X′, Z′ = Z ×_{X} X′ and denote by π_{Y′} : Y′ → X′, π_{Z′} : Z′ → Y′ the obvious projections. We have a commutative diagram

[figure: Z′; π_{Z′}; g_{Z}; Y′; g_{Y}; π_{Y′}; X′; g_{X}; Z; π_{Z}; Y; π_{Y}; X]

Let π = π_{Y} ∘ π_{Z} and π′ = π_{Y′} ∘ π_{Z′}.

Proposition 9.2. The schemes Z′, Y′ and X′ are irreducible and normal. In particular, they have a finite number of singular points. Further, Sing(X′) ∩ δ(Y′/X′) = ∅ and Sing(Y′) ∩ Δ(Y′/X′) maps under π_{Y} g_{Y} to δ(Y/X) ∩ D_{X}.

Proof. Since the ramifications δ(Y/X) and δ(X′/X) = D_{X} have no common component, k(Z) and k(Y) are disjoint from k(X′) over k(X). The maps π and π_{Y} are flat, hence by 4.5, Z′ and Y′ are integral. By construction X′, Y′ and Z′ are locally of the form Spec(S) where S = R[t]/(t^{l} − f) and R is a regular ring. Thus X′, Y′ and Z′ are locally complete intersections and to show that they are normal it suffices to show that they are nonsingular in codimension 1 ([Ha_{1}], Ch. II, 8.23). The singularities of X′ are at most over the singularities of D_{X} (see [BPV], Ch. I, §17), which by condition (4) of 8.6 are away from δ(Y/X). Thus Sing(X′) is finite and Sing(X′) ∩ δ(Y′/X′) = ∅. We now prove that Sing(Y′) is finite. The proof for Z′ is similar. Let (a, b) ∈ Sing(Y′) ⊂ Y × X′ with c = π_{Y} (a) = g_{X}(b) ∈ X. If c ∉ δ(Y/X), then π_{Y} is étale at a and therefore b ∈ Sing(X′) which is finite. If c ∉ D_{X}, then g_{X} is étale at b and hence g_{Y} is étale at (a, b). Since Y is smooth, Y′ is smooth at (a, b) and therefore the remaining singularities can only be among the finitely many points of Y′ that lie over δ(Y/X) ∩ D_{X}. Thus Sing(Y′) is the disjoint union of Sing(Y′) ∩ Δ(Y′/X′) and Sing(Y′) ∩ π_{Y′}^{−1}(Sing(X′)) and Sing(Y′) ∩ Δ(Y′/X′) maps to δ(Y/X) ∩ D_{X}.

We have

g_{X}^{∗}{D_{X}} = l{D′}

for some curve D′ and therefore π_{Y′}^{∗}g_{X}^{∗}{D_{X}} = l{D_{1}} for some {D_{1}} ∈ Z^{1}(Y′). On the other hand, π_{Y}^{∗} {D_{X}} = {D_{Y} } + {D_{2}} for some {D_{2}} ∈ Z^{1}(Y). Since D_{Y} has no common component with the discriminant, {D_{2}} has no component equal to {D_{Y} } and therefore the same is true of g_{Y}^{∗} {D_{Y} } and g_{Y}^{∗} {D_{2}}. Thus

(†)  π_{Y′}^{∗}g_{X}^{∗}{D_{X}} = l{D_{1}} = g_{Y}^{∗} π_{Y}^{∗} {D_{X}} = g_{Y}^{∗} {D_{Y} } + g_{Y}^{∗} {D_{2}} .

We conclude that g_{Y}^{∗} {D_{Y} } = l{D_{Y′}} for some {D_{Y′}} ∈ Z^{1}(Y′). The difficulty is that {D_{Y′}} is a Weil divisor but not necessarily a Cartier divisor. The rest of the proof consists in trying to replace it by a Cartier divisor.

We start by replacing X′ by a smooth surface X̂ and discuss the singularities of Ŷ = Y ×_{X} X̂. Let ψ_{X} : X̂ → X′ be a resolution of singularities of X′; ψ_{X} is a proper birational morphism which is an isomorphism outside Sing(X′) and with X̂ smooth. We set Ŷ = Y ×_{X} X̂, Ẑ = Z ×_{X} X̂ and consider the commutative diagram

[figure: Ẑ; πẐ; ψ_{Z}; Ŷ; ψ_{Y}; πŶ; X̂; ψ_{X}; Z′; π_{Z′}; Y′; π_{Y′}; X′ .]

We set π̂ = πŶ πẐ. Since Ŷ is étale over X̂ away from δ(Ŷ/X̂) and X̂ is smooth, Ŷ is smooth away from δ(Ŷ/X̂). Since ψ_{X} is an isomorphism outside Sing(X′), and Sing(X′)∩ δ(Y′/X′) = ∅, ψ_{X} and ψ_{Y} are isomorhisms along δ(Y′/X′) and Δ(Y′/X′) respectively. Thus ψ_{Y} maps Σ = Sing(Ŷ) bijectively onto Sing(Y′) ∩ Δ(Y′/X′) which is finite and lies over to δ(Y/X) ∩ D_{X} in X by 9.2. Similar arguments lead to the fact that Sing(Ẑ) lies over δ(Y/X) ∩ D_{X}. We set Σ_{0} = πŶ (Σ) and Σ_{1} = π^{−1}(Σ_{0}). We denote by A = O_{X̂,Σ_{0}} the semilocal ring at Σ_{0}, A^{h} the henselization of A at its radical, and by A_{x}^{h} = O_{X̂,x}^{h} the factor of A^{h} corresponding to a point x ∈ Σ_{0}. Similarly, we define the rings B, C, and so on, by Spec(B) = Ŷ ×X̂ Spec(A), Spec(C) = Ẑ ×X̂ Spec(A), Spec(B^{h}) = Ŷ ×X̂ Spec(A^{h}), Spec(C^{h}) = Ẑ ×X̂ Spec(A^{h}) and further, for any x ∈ Σ_{0}, B_{x} = B ⊗_{A} A_{x}, C_{x} = C ⊗_{A} A_{x}, B_{x}^{h} = B ⊗_{A} A_{x}^{h}, C_{x}^{h} = C ⊗_{A} A_{x}^{h}, and similarly B_{y}, C_{z} and B_{y}^{h}, C_{z}^{h} when y ∈ Σ and z ∈ Σ_{1}. We further denote by R the semilocal ring of X at the image of Σ_{0}, by R_{x} the local ring of X at the image of an x ∈ Σ_{0}, and by R^{h} and R_{x}^{h} their respective henselizations.

Lemma 9.3. Let z be a point of Σ_{1}, y = πẐ(z) and x = π̂(z).

(a) The singular locus of Ẑ is Σ_{1}.

(b) If y is in Σ then A_{x}^{h} → B_{y}^{h} is a quadratic extension, B_{y}^{h} → C_{z}^{h} is an isomorphism and B_{x}^{h} ≃ B_{y}^{h} × (A_{x}^{h})^{l−2}.

(c) If y is not in Σ then A_{x}^{h} → B_{y}^{h} is an isomorphism, B_{y}^{h} → C_{z}^{h} is a quadratic extension and B_{x}^{h} ≃ (A_{x}^{h})^{l}.

(d) The A_{x}^{h}-algebra C_{x}^{h} is a product of |G|/2 quadratic extensions of A_{x}^{h}.

(e) Denoting by Cl(−) the class group, if y ∈ Σ we have

Cl(B) = Cl(B_{x}) = Cl(B_{y}) = Cl(C_{z}) ⊆ Cl(C_{z}^{h}) = ℤ/l .

In particular Cl(B) is either zero or ℤ/l.

Proof. In a suitable neighbourhood of Σ the map πẐ : Ẑ → Ŷ coincides with π_{Z′} : Z′ → Y′ which, by 7.12, is étale at every y ∈ Σ. Hence over every x ∈ Σ_{0} there is a singularity z of Ẑ which clearly is in Σ_{1}. Since G acts transitively over Σ_{1}, we have Sing(Ẑ) = Σ_{1}. This proves (a). Let u be a semilocal equation of D_{X} in R. Since ψ_{X} and ψ_{Y} are isomorphisms along Σ_{0} we have

B = \frac{R[T, t]}{(P (T), t^{l} − u)} = \frac{A[T]}{(P (T))} .

We know by 8.6 that P [T] has one double root and l − 2 simple roots at every x ∈ Σ_{0}, hence we may assume that

\frac{R^{h}[T]}{(P (T))} ≃ \frac{R^{h}[T]}{(T^{2} − θ)} × ∏_{x∈Σ_{0}} (R_{x}^{h})^{l−2}

where θ is a semilocal equation of δ(Y/X) at the image of Σ_{0}. Note that locally θ must be a regular parameter because R^{h}[T]/(P (T)) is regular. Extending scalars to A_{x}^{h} we find

B_{x}^{h} = \frac{A_{x}^{h}[T]}{(T^{2} − θ)} × ∏_{x∈Σ_{0}} (A_{x}^{h})^{l−2}

from which we see that if y corresponds to the double root, then B_{y}^{h} = A_{x}^{h}[T]/(T^{2} − θ) is a quadratic extension of A_{x}^{h}. In this case we know that π_{Z} is étale at z, hence the extension B_{y}^{h} → C_{z}^{h} is an isomorphism. This proves (b). If y ∉ Σ then π_{Y} is étale at y and therefore the extension A_{x}^{h} → B_{y}^{h} is an isomorphism. On the other hand, because of the Galois action, all the extensions A_{x}^{h} → C_{σ(z)}^{h} are isomorphic to each other, therefore they are all quadratic. This proves (c). It follows from (b) and (c) that the Galois extension A_{x}^{h} → C_{x}^{h} = ∏_{z} C_{z}^{h} is a product of quadratic extensions and since C_{x}^{h} is of rank |G| over A_{x}^{h}, the assertion (d) follows. Let f ∈ B be a semilocal equation of D_{Y} at Σ. By the construction of D_{Y} made in §8 (see Lemma 8.3), in A^{h}[T]/(T^{2} − θ) we can write f = aT − π where a is a unit of A^{h} and π ∈ A^{h} is a semilocal parameter (that is a parameter of A_{x}^{h} for every x ∈ Σ_{0}). Further, u = N_{Y/X}(f) = π^{2} − a^{2}T^{2} = π^{2} − a^{2}θ = fg for g = −aT − π. Then, at any y ∈ Σ, we have

B_{y}^{h} ≃ \frac{R_{x}^{h}[T]}{(T^{2} − θ, t^{l} − a^{2}T^{2} + π^{2})} = \frac{S[t]}{(t^{l} − fg)}

where x is the image of y in Σ_{0} and S = A_{x}[T]/(T^{2}−θ) is a regular 2-dimensional henselian ring. Its maximal ideal is (f, g) because D_{Y} and Δ(Y/X) meet transversally and they are defined, respectively, by f and T. The class group of B_{y}^{h} is ℤ/l generated by the prime 𝔭(y) = (t, aT − π) = (t, f) or by the prime 𝔮(y) = (t, g) whose class is the inverse of the class of 𝔭(y) . Since 𝔭(y) comes from B_{y}, it follows that Cl(B_{y}) is generated by the ideal 𝔭(y) ∩ B_{y}, which we still denote by 𝔭(y)). Since Cl(B) → ∏_{y∈Σ} Cl(B_{y}) is injective, inverting t we obtain a factorial ring B[1/t] and therefore Cl(B) is generated by those prime ideals of height 1 that contain t. Two of them are 𝔭 = (t, f) and 𝔮 = (t, g). Any other one must coincide with 𝔭 or 𝔮 at a given y ∈ Σ and therefore it must coincide with 𝔭 or 𝔮 because B is a domain. At every y ∈ Σ the class of 𝔭 is the inverse of the class of 𝔮, the divisor class group of B is generated by each of them and is, therefore, zero or ℤ/l. The rest of the proposition follows from the fact that the maps induced on divisor class groups are surjective for localizations and injective for henselizations of local rings.

We now look at the pull-back under ψ_{Y} and ψ_{X} of the Cartier divisors occurring in (†). We have

ψ_{Y}^{∗} g_{Y}^{∗} {D_{Y} } + ψ_{Y}^{∗} g_{Y}^{∗} {D_{2}} = l · ψ_{Y}^{∗} π_{Y′}^{∗}{D′} .

Since the centres of the blow-up for ψ_{Y} are away from Δ(Y′/X′) and since g_{Y}^{∗} {D_{Y} } and g_{Y}^{∗} {D_{2}} have no common composnents in Y′ and meet only along Δ(Y′/X′), it follows that ψ_{Y}^{∗} g_{Y}^{∗} {D_{Y} } and ψ_{Y}^{∗} g_{Y}^{∗} {D_{2}} have no common component in Ŷ. Therefore

ψ_{Y}^{∗} g_{Y}^{∗} {D_{Y} } = l{DŶ }

for some DŶ on Ŷ and

ψ_{Z}^{∗}g_{Z}^{∗}{D_{Z}} = l{DẐ}

where DẐ = π∗ ; Ẑ(DŶ). We shall now construct maps X̃ → X̂ and Ỹ → Ŷ satisfying the conditions of Theorem 9.1.

We recall a result of Abhyankar and Manish Kumar:

Lemma 9.4. Let R be a 2-dimensional complete local domain which is regular, with quotient field K. Let L be a quadratic extension of K and S the normalization of R in L. Suppose that the extension L/K is defined by z^{2} = ux^{i}y^{j} where (x, y) = m is the maximal ideal of R, u is a unit and 0 ≤ i, j ≤ 1. If (i, j) ≠ (1, 1) then S is regular. Suppose (i, j) = (1, 1) and let X → Spec(R) be the blow-up of Spec(R) at the closed point. Then the normalization of X in L is regular.

Proof. See [AMK], §2.

Let y be a singular point of Ŷ mapping to x ∈ X̂, to x_{0} in X and to y_{0} in Y. We note that x_{0} ∈ δ(Y/X) ∩ D_{X}. Let Ô_{Ŷ ,y} and Ô_{X̂,x} be the completions of O_{Ŷ ,y} and O_{X̂,x} respectively. By condition (3) of Theorem 8.6 the image x_{0} of y in X has l−1 preimages in Y, hence Y ×_{X} Spec(Ô_{X,x_{0}}) splits as a disjoint union of l − 2 copies of Spec(Ô_{X,x_{0}}) and one degree 2 extension Spec(Ô_{Y,y_{0}}) of Spec(Ô_{X,x_{0}}). It follows by base change that Ô_{Ŷ ,y} is a degree 2 extension of Ô_{X̂,x}.

Let Σ_{0} be the image of Σ in X̂. Clearly Σ_{0} ⊂ δ(Ŷ/X̂). Let φ_{1} : X̃_{1} → X̂ be a proper birational morphism which is an isomorphism outside Σ_{0} and such that the total transform of δ(Ŷ/X̂) has normal crossings over Σ_{0}. Let Σ̃_{0} ⊂ X̃_{1} be the image of the set of singular points of the normalization of Y × X̃_{1}. Let φ_{2} : X̃ → X̃_{1} be the blow-up at Σ̃_{0} and φ_{X} = φ_{2}φ_{1}. We have a commutative diagram

[figure: Z ×_{X} X̃; φ_{Z}; Y ×_{X} X̃; φ_{Y}; X̃; φ_{X}; Ẑ; π_{Z}; Ŷ; π_{Y}; X̂]

Let n_{Y} : Ỹ → Y ×_{X} X̃, n_{Z} : Z̃ → Z×_{X} X̃ be the normalization maps. We claim that Ỹ and Z̃ are smooth. Outside Σ_{0}, φ_{X} is an isomorphism and hence φ_{Y} is also an isomorphism outside π−1 ; Ŷ(Σ_{0}) ⊃ Σ = Sing(Ŷ). Let y be a singular point of Y ×_{X} X̃. The image of y in

Ŷ is contained in π−1 ; Ŷ(Σ_{0}) which maps to δ(Y/X) ∩ D_{X} in X. Thus the image x_{0} of y in X belongs to δ(Y/X) ∩ D_{X}. The point x_{0} has l − 1 preimages in Y in view of condition (3) of 8.6. Let x be the image of y in X̃. Let O_{Ỹ ,x} be the semilocal ring of Ỹ at π−1 ; Ỹ(x) and \overline{O}_{Ỹ ,x} be its integral closure in k(Ỹ). Let

Ô_{X̃,x} and Ô_{Ỹ ,y} denote respectively the completion of O_{X̃,x} and O_{Ỹ ,y}. Then O_{Ỹ ,x} ⊗_{O_{_{X},x}}

Ô_{X̃,x} splits as a product

Ô_{Ỹ ,y} × Ô_{X̃,x} × ⋯ × Ô_{X̃,x} .

Since \overline{O}_{Ỹ ,x} is finite over O_{X̃,x}, the tensor product \overline{O}_{Ỹ ,x} ⊗_{O_{_{X},x}} Ô_{X̃,x} is the completion of \overline{O}_{Ỹ ,x} with respect to its radical and hence, by [ZS], Ch. VIII, §30, Theorem 32, it is normal. Clearly it is the normalization of

O_{Ỹ ,x} ⊗_{O_{_{X},x}} Ô_{X̃,x} = Ô_{Ỹ ,y} × Ô_{X̃,x} × ⋯ × Ô_{X̃,x} .

Let \overline{Ô}_{Ỹ ,y} denote the normalization of Ô_{Ỹ ,y}. By construction the ramification divisor on

Ô_{X̃,x} for the extension Ô_{X̃,x} → Ô_{Ỹ ,y} has normal crossings and further \overline{Ô}_{Ỹ ,y} is regular by 9.4. It follows that \overline{O}_{Ỹ ,x} ⊗_{O_{_{X},x}} Ô_{X̃,x} is regular and hence \overline{O}_{Ỹ ,x} itself is regular. This

shows that Ỹ is smooth. Exactly the same argument shows that Z̃ is smooth because, by 8.3, Z → Y is étale at the image of y in Y.

We thus have a commutative diagram

[figure: Z̃; πZ̃; f_{Z}; π̃; h_{Z}; Ỹ; f_{Y}; πỸ; X̃; f_{X}; h_{X}; Ẑ; π_{Z}; g_{Z}ψ_{Z}; Ŷ; g_{Y} ψ_{Y}; π_{Y}; X̂; g_{X}ψ_{X}; Z; π_{Z}; π; Y; π_{Y}; X]

where the undefined maps are the obvious ones, f_{Z} = φ_{Z}n_{Z}, f_{Y} = φ_{Y} n_{Y} and f_{X} = φ_{X}. The surfaces X̃, Ỹ and Z̃ are smooth, π̃ is finite, and Z̃ is Galois over X̃ with group G = Gal(k(Z)/k(X)).

Proposition 9.5. The line bundle (h_{Z})^{∗}(O_{Z}(D_{Z})) belongs to

l · Pic(Z̃) + π^{∗}(Pic(X̃))

To prove 9.5 we need several preliminaries. By construction φ_{X} is an isomorphism outside Σ_{0}, hence φ_{Z} is an isomorphism outside Z ×_{X} Σ_{0}. It follows that Z ×_{X} X̃ is smooth —in particular normal— outside Z ×_{X} Σ_{0}, hence the normalization map n_{Z} is an isomorphism outside Σ_{1}. It follows that f_{z} = φ_{Z}n_{Z} is an isomorphism outside Σ_{1}. We denote by F_{1} = f_{Z}^{−1}(Σ_{1}) the exceptional fibre for f_{Z}, F = f_{Y}^{−1}(π−1 ; Ŷ(Σ_{0})) the exceptional fibre for f_{Y} F_{0} = f_{X}^{−1}(Σ_{0}) the exceptional fibre for f_{X}. To simplify notation we write F, F_{1} instead of F_{B}, (F_{1})_{C} etc. Let L̃ be the line bundle h_{Z}^{∗}(O_{Z}(D_{Z})). Since Ẑ \ Σ_{1} is smooth, the restriction DẐ|_{Ẑ\Σ_{1}}

is a Cartier divisor and defines a line bundle L_{0} on Z̃ \ F_{1} ≃ Ẑ \ Σ_{1}. We have an exact sequence

0 → Pic_{F_{1}}(Z̃) → Pic(Z̃) → Pic(Z̃ \ F_{1}) → 0

where Pic_{F_{1}}(Z̃) is the subgroup generated by the irreducible components of F_{1}. Thus there exists a line bundle L̃_{0} on Z̃ whose restriction to Z̃ \ F_{1} is isomorphic to f_{Z}^{∗}(L_{0}). The line bundle L̃(L̃_{0}^{−1})^{⊗l} restricted to Z̃ \ F_{1} is trivial and hence its class belongs to Pic_{F_{1}}(Z̃). We fix a point x ∈ Σ_{0} and let z_{1}, …, z_{m} ∈ Σ_{1} be the preimages of x in Ẑ. Let F_{x} be the fibre of x in X̃ and, for 1 ≤ j ≤ m, let F_{j} be the fibre of z_{j} in Z̃. By Zariski’s main theorem ([Ha_{1}], Ch. III, Cor. 11.4) the curves F_{x} and F_{1}, …, F_{m} are connected. Let E be an irreducible component of F_{x} and D_{j}^{k}, k = 1…. , m_{j} the irreducible components of

F_{j} mapping to E. The class [L̃(L̃_{0}^{−1})^{⊗l}] is the class of a divisor

∑_{x∈Σ_{0}} ∑_{E} ∑_{j} r_{j}^{k}D_{j}^{k}

as E runs over the irreducible components of F_{x}. Denoting C_{z_{j}}^{h} by C_{j}^{h} and, as usual, Z̃ ×Ẑ C_{j}^{h} by Z̃_{C_{j}^{h}} and so on, we have a commutative diagram with exact rows

\xymatrix{0 \ar[r] & Pic(Z̃_{C_{j}^{h}})/l \ar[r] & H^{2}(Z̃_{C_{j}^{h}}, μ_{l}) \\  & α_{Z} \ar[d] & β \ar[d] \\ 0 \ar[r] & Pic(F_{j})/l \ar[r] & H^{2}(F_{j}, μ_{l}).}

By the proper base change theorem ([Mi], Ch. VI, 2.7) β is an isomorphism and by [CTOP], Th. 1.7, (c) α_{Z} is surjective, hence α_{Z} is an isomorphism. Further, for any connected curve C the inclusion C_{red} ↪ C induces an isomorphism Pic(C) → Pic(C_{red}). We thus have isomorphisms

α_{Z} : Pic(Z̃_{C_{j}^{h}})/l ≃ Pic((F_{j})_{red}))/l

and, similarly,

α_{X} : Pic(X̃_{A_{x}^{h}})/l ≃ Pic((F_{x})_{red}))/l .

By a result of Artin ([Ar], §1) the degree map yields identifications

Pic((F_{x})_{red}))/l = ⨁_{E} ℤ/l and Pic((F_{j})_{red}))/l = ⨁_{E} ⨁_{D_{j}^{k}} ℤ/l

where E runs over the irreducible components of F_{x} and D_{j}^{k} runs over the irreducible components of F_{j} mapping to E. We have a commutative diagram

[figure: Pic(X̃_{A_{x}^{h}})/l; π^{∗}; α_{X}; ⨁_{j}Pic(Z̃_{C_{j}^{h}})/l; α_{Z}; ⨁_{E}ℤ/l; θ; ⨁_{E} ⨁_{D_{j}^{k}}ℤ/l]

where the vertical isomorphisms are obtained by associating to a divisor its intersection multiplicity with each E and D_{j}^{k} respectively and θ = (θ_{E,D_{j}^{k}}) where θ_{E,D_{j}^{k}} is the degree of k(D_{j}^{k}) over k(E). Since the Galois action transitively permutes the D_{j}^{k} ’s lying over E, these degrees are all equal to some f_{E}. Since, for x ∈ Σ_{0}, g_{X}(x) is in D_{X} ⊂ X, by 8.6 x has at least l − 1 ≥ 2 preimages in Ŷ and therefore the preimage of E in Ỹ has at least two components. Since the degree of Ỹ → X̃ is l and the degree of Z̃ → Ỹ is prime to l, this implies that f_{E} is prime to l. The Galois action on Pic(Z̃) transitively permutes the components in ⨁_{D_{j}^{k}} ℤ/l. Hence θ maps ⨁_{E} ℤ/l isomorphically onto the subgroup of ⨁_{E} ⨁_{D_{j}^{k}} ℤ/l consisting of all the elements fixed by the Galois action and the same is true of π^{∗}. Since [D_{Z}] ∈ Pic(Z)/l is the pull-back of an element in H^{2}(X, μ_{l}) under π, it is Galois invariant; its pull-back in Pic(Z̃)/l and its restriction to Pic(Z̃_{A^{h}})/l are again Galois invariant. Thus [ ∑_{E} ∑_{j} r_{j}^{k}D_{j}^{k}] belongs to the image of π^{∗}. We claim that [ ∑_{E} ∑_{j} r_{j}^{k}D_{j}^{k}] is of the form π^{∗}(∑ r_{E}[E]), where E varies in the exceptional fibre of X̃ → X̂. Since X̂ is smooth at x, A_{x}^{h} is regular and we have an exact sequence

0 → Pic_{F_{x}}(X̃_{A_{x}^{h}}) → Pic(X̃_{A_{x}^{h}}) → Pic(X̃_{A_{x}^{h}} \ F_{x}) → 0

with Pic(X̃_{A_{x}^{h}} \ F_{x}) = Pic(Spec(A_{x}^{h}) \ {x}) = 0. The claim follows.

Let M be the line bundle on X̃ representing the class of ∑_{x∈Σ_{0}} ∑_{E} r_{E}[E]. We claim that N = L̃(L̃_{0}^{−1})^{⊗l}π^{∗}(M^{−1}), which belongs to Pic_{F_{1}}(Z̃), is an l-th power in Pic(Z̃). We note that, by construction, it maps to zero in Pic(Z̃_{A_{x}^{h}})/l for all x ∈ Σ_{0}.

Lemma 9.6. The map

Pic_{F_{1}}(Z̃_{A}) → Pic_{F_{1}}(Z̃_{A^{h}})

is an isomorphism.

Proof. We have a commutative diagram

[figure: H^{0}(Z̃_{A}, 𝔾_{m}); γ_{0}; H^{0}(Z̃_{A} \ F_{A}, 𝔾_{m}); δ_{0}; H_{F_{1}}^{1}(Z̃_{A}, 𝔾_{m}); φ_{0}; Pic_{F_{1}}(Z̃_{A}); H^{0}(Z̃_{A^{h}}, 𝔾_{m}); γ; H^{0}(Z̃_{A^{h}} \ F_{A^{h}}, 𝔾_{m}); δ; H_{F_{1}}^{1}(Z̃_{A^{h}}, 𝔾_{m}); φ; Pic_{F_{1}}(Z̃_{A^{h}})]

with exact rows (see [Mi], III, 1.27 and the correction in [CTO], Proposition 4.4) and where φ_{0} and φ are surjective. Since Z̃_{A} → Ẑ_{A} is proper and birational we have 𝔾_{m}(Z̃_{A}) = 𝔾_{m}(Ẑ_{A}). For the same reason, 𝔾_{m}(Z̃_{A^{h}}) = 𝔾_{m}(Ẑ_{A^{h}}). Since Z̃_{A} \ F_{1} ≃ Ẑ_{A} \ Σ_{1} and Z̃_{A^{h}} \ F_{1} ≃ Ẑ_{A^{h}} \ Σ_{1} we have

𝔾_{m}(Z̃_{A} \ F_{1}) = 𝔾_{m}(Ẑ_{A} \ Σ_{1}) and 𝔾_{m}(Z̃_{A^{h}} \ F_{1}) = 𝔾_{m}(Ẑ_{A^{h}} \ Σ_{1}).

Since Ẑ_{A} is a 2-dimensional connected scheme and Σ_{1} a finite set of closed points of Ẑ,

𝔾_{m}(Ẑ_{A} \ Σ_{1}) = 𝔾_{m}(Ẑ_{A}) and 𝔾_{m}(Ẑ_{A^{h}} \ Σ_{1}) = 𝔾_{m}(Ẑ_{A^{h}}).

Thus γ and γ_{0} are isomorphisms. and therefore φ and φ_{0} are isomorphisms. Let h : Z̃_{A^{h}} → Z̃_{A} be the projection. We note that h^{∗}𝔾_{m} = 𝔾_{m}, hence by étale excision using a limit argument [Mi] the map H_{F_{1}}^{1}(Z̃_{A}, 𝔾_{m}) → H_{F_{1}}^{1}(Z̃_{A^{h}}, 𝔾_{m}) is an isomorphism and the lemma is proved.

To complete the proof of Theorem 9.1 we may assume that Cl(B) = ℤ/l, because if B is factorial then Weil divisors coincide with Cartier divisor and there is nothing to prove.

Proposition 9.7. Identifying Cl(B) with a subgroup of Cl(C) under the pull-back map, Cl(C) is generated by the Galois conjugates of π∗ ; Ẑ(Cl(B)) in Cl(C).

Proof. We know by 8.6 (5) that the inverse image of D_{Y} in Z is an irreducible smooth curve D_{Z}. Let h be a local equation of D_{Z} on Σ_{1}. Then for every z ∈ Σ_{1} we can choose a σ ∈ G for which σ(z) lies over a singular y ∈ Σ. From C_{z}^{h} ≃ B_{y}^{h} it follows that Cl(C_{z}^{h}) is generated by the ideal (t, σ(h)). This shows that Cl(C) is generated by the primes containing some σ(h) and the argument used in the proof of 9.3 shows that Cl(C) is generated by the Galois conjugates of π∗ ; Ẑ(Cl(B)).

Lemma 9.8. The group Pic(Z̃_{C^{h}}) has no l-torsion.

Proof. Using the fact that Pic(Z̃_{C^{h}} \ F_{1}) = Cl(C^{h}), we have the exact sequence

(†)  0 → Pic_{F_{1}}(Z̃_{C^{h}}) → Pic(Z̃_{C^{h}}) → Cl(C^{h}) → 0

where Pic_{F_{1}}(Z̃_{C^{h}}) is the subgroup of Pic(Z̃_{C^{h}}) generated by the irreducible components of F_{1}. Suppose that their number is r. Then they generate a free abelian group isomorphic to ℤ^{r} because Pic_{F_{1}}(Z̃_{C^{h}}) carries a nondegenerate quadratic form given by intersection multiplicities ([Mu], §1). It follows that Pic(Z̃_{C^{h}}) is a finitely generated group of rank r, of the form ℤ^{r} ⊕ T, where T is its torsion subgroup. Since Cl(C^{h}) is l-torsion, T can only be l-torsion, but we have seen that

Pic_{F_{1}}(Z̃_{C^{h}})/l ≃ (ℤ/l)^{r}

hence T must be zero.

Corollary 9.9. The group Pic(Z̃_{C}) has no l-torsion.

Proof. The corollary follows from the commutative diagram with exact rows

[figure: 0; Pic_{F_{1}}(Z̃_{C}); ≃; α; Pic(Z̃_{C}); Cl(C); β; 0; 0; _{Pic_{F_{1}}(Z̃_{C^{h}})}; _{Pic(Z̃_{C^{h}})}; Cl(C^{h}); 0]

using that α is an isomorphism by 9.6 and that β is injective.

Lemma 9.10. We have

(Cl(C))^{G} = (Cl(C^{h}))^{G} = 0.

Proof. It suffices to show that (Cl(C^{h}))^{G} = 0. We have

C^{h} = ∏_{z∈Σ_{1}} C_{z}^{h}

and each factor on the right is isomorphic to

\frac{R_{x}^{h}[T, t]}{(T^{2} − θ, t^{l} − a^{2}T^{2} + π^{2})} .

The isotropy group G_{z} ⊂ G of z acts by changing the sign of T and therefore maps the class of the prime t, aT − π to the class of (t, −at − π) which is its inverse and is distinct from the class of (t, at − π) because l is at least 3. Hence (Cl(S_{z}^{h}))^{G_{z}} = 0. From this the claim follows.

Proposition 9.11. The map

Pic_{F_{1}}(Z̃_{C^{h}}) ↪ Pic(Z̃_{C^{h}})

induces an injection

(Pic_{F_{1}}(Z̃_{C^{h}})/l)^{G} ↪ (Pic(Z̃_{C^{h}})/l)^{G}

Proof. From the exact sequence (†) used for proving 9.8, using multiplication by l and the snake lemma we get a long exact sequence

lPic(Z̃_{C^{h}}) → Cl(C^{h}) → Pic_{F_{1}}(Z̃_{C^{h}})/l → Pic(Z̃_{C^{h}})/l.

By Lemma 9.8, Pic(Z̃_{C^{h}}) has no l-torsion, hence we have an exact sequence

0 → Cl(C^{h}) → Pic_{F_{1}}(Z̃_{C^{h}})/l → Pic(Z̃_{C^{h}})/l

from which the assertion immediately follows, since, by Lemma 9.10, (Cl(C^{h}))^{G} = 0.

Proof of Proposition 9.5. Since Z̃_{C} \(Z̃_{C} \F_{1}) → F_{1} is an isomorphism, we have a patching diagram [Jo]

\xymatrix{Z̃_{C} \ F_{1} \ar[r] \ar[d] & Z̃_{C} \ar[d] \\ Z̃ \ F_{1} \ar[r] & Z̃}

and an exact sequence

(⋆⋆)  0 → Pic(Z̃) → Pic(Z̃_{C}) ⊕ Pic(Z̃ \ F_{1}) → Cl(C) → 0.

Surjectivity on the right is due to the the fact that Pic(Z̃ \ F_{1}) surjects onto Cl(C). From this sequence, multiplying by l, using the snake lemma and remembering that_{l}Pic(Z̃_{C}) = 0 and that Cl(C) is l-torsion, we obtain a long exact sequence

(‡) 0 → _{l}Pic(Z̃) → _{l}Pic(Z̃ \ F_{1}) → Cl(C) →

→ Pic(Z̃)/l → Pic(Z̃_{C})/l → Pic(Z̃ \ F_{1})/l

We now distinguish two cases:

I. Suppose that _{l}Pic(Ỹ \ F) → Cl(B) is surjective. Consider the commutative diagram

[figure: Pic(Ỹ \ F); (f_{Y})_{∗}; i; 0; Pic(Ŷ); Cl(Ŷ); j; Cl(B); 0]

Recall that g_{Y}^{∗} {D_{Y} } = l{DŶ } for some Weil divisor DŶ. By assumption j{DŶ } = i(γ) for some γ ∈ _{l}Pic(Ỹ \ F). Then, since j({DŶ } − (f_{Y})_{∗}(γ)) = 0, there exists a γ′ ∈ Pic(Ŷ) such that {DŶ } = (f_{Y})_{∗}(γ)) + γ′. We conclude that ψ_{Y}^{∗} g_{Y}^{∗} {D_{Y} } = l{DŶ } = lγ′ because lγ = 0. From this it follows that h_{Z}^{∗}{D_{Z}} is divisible by l in Pic(Z̃).

II. Suppose that _{l}Pic(Ỹ \ F) → Cl(B) is not surjective, in which case, since Cl(B) = ℤ/l, it is the zero map. By Proposition 9.7, _{l}Pic(Z̃ \ F_{1}) → Cl(C) is also zero, hence from the exact sequence (‡) we conclude that the map _{l}Pic(Z̃) → _{l}Pic(Z̃ \ F_{1}) is an isomorphism. In this case the exact sequence

0 → Pic_{F_{1}}(Z̃) → Pic(Z̃) → Pic(Z̃ \ F_{1}) → 0

yields that Pic_{F_{1}}(Z̃)/l ↪ Pic(Z̃)/l is injective. The element [N] ∈ Pic_{F_{1}}(Z̃) is Galois invariant in Pic(Z̃)/l and hence, because of this injection, it is in (Pic_{F_{1}}(Z̃)/l)^{G}. From the exact sequence

0 → Pic_{F_{1}}(Z̃_{C}) → Pic(Z̃_{C}) → Cl(C) → 0

multiplying by l, using the snake lemma and remembering that _{l}Pic(Z̃_{C}) = 0 we get the exact sequence

0 → Cl(C) → Pic_{F_{1}}(Z̃)/l → Pic(Z̃_{C})/l.

Taking invariants under G and using the fact that Cl(C)^{G} = 0 (Proposition 9.10) we see that [N] ∈ (Pic_{F_{1}}(Z̃)/l)^{G}. On the other hand, we know that [N] is zero in Pic(Z̃_{C^{h}})/l, hence by Proposition 9.11 it is zero in Pic_{F_{1}}(Z̃_{C^{h}})/l. Noting that Z̃_{C} = Z̃ ×Ẑ Spec(C) = Z̃ ×_{hx} Spec(A) = Z̃_{A}, 9.6 implies that [N] vanishes in Pic_{F_{1}}(Z̃_{C})/l. Since the map Pic_{F_{1}}(Z̃) → Pic_{F_{1}}(Z̃_{C}) is surjective, there exists N_{1} ∈ Pic_{F_{1}}(Z̃) such that NN_{1}^{⊗l} maps to zero in Pic(Z̃_{C}). The element NN_{1}^{⊗l} also maps to zero in Pic(Z̃ \ F_{1}). By the exact sequence (⋆⋆), NN_{1}^{⊗l} is zero in Pic(Z̃), completing the proof of the proposition.

We now complete the proof of Theorem 9.1 which says that the class of h_{Y}^{∗} (O_{Y} (D_{Y})) in Pic(Ỹ) belongs to

l · Pic(Ỹ) + π∗ ; Ỹ Pic(X̃).

This immediately follows from 9.4 and the fact that πZ̃ is a finite map of degree dividing (l − 1)! which is coprime to l.

10. Proof of the main theorem

Theorem 10.1. Let X be a smooth projective surface over an algebraically closed field k of characteristic zero. Let D be a central division algebra of prime degree l over k(X). Then D is cyclic.

Proof. If l = 2 then D is a quaternion algebra and every quadratic extension in D is cyclic. If l = 3 then D is cyclic by a result of Wedderburn [W]. However we show for all primes l ≥ 3 that the cyclic extension k(X̃) of k(X) of degree l splits D. Suppose first that D is unramified over X. By §1 the class of D is equivalent to an Azumaya algebra of rank l^{2} over X. Let η ∈ H^{2}(X, μ_{l}) be an element mapping to the class of D in H^{2}(X, 𝔾_{m}). Let π_{Y} : Y → X be a generic splitting of D as in Theorem 7.13. The class π^{∗}(η) ∈ H^{2}(Y, μ_{l}) maps to zero in H^{2}(Y, 𝔾_{m}) since the map H^{2}(Y, 𝔾_{m}) → H^{2}(k(Y), 𝔾_{m}) is injective, Y being smooth. In view of the Kummer exact sequence

1 → μ_{l} → 𝔾_{m} → 𝔾_{m} → 1

we have the exact sequence

0 → Pic(Y)/l → H^{2}(Y, μ_{l}) → H^{2}(k(Y), 𝔾_{m})

from which we see that there exists a line bundle L on Y such that π^{∗}(η) = [L] in Pic(Y)/l. By Theorem 9.1 there exists a map h_{X} : X̃ → X which is proper, generically cyclic of degree l and such that the normalization Ỹ of Y ×_{X} X̃ is smooth; further if h_{Y} : Ỹ → Y is the natural map, then h_{Y}^{∗} (L_{Y}) vanishes in

Pic(Ỹ)/lPic(Ỹ) + π∗ ; Ỹ (Pic(X̃)) .

We show that k(X̃) splits D. Let h_{Y}^{∗} (L) = L_{1}^{⊗l} ⊗_{O_{^{Ỹ}}} π∗ ; Ỹ (M) where M is a line bundle on

X̃. The element ξ = η +[M^{−1}] ∈ H^{2}(X̃, μ_{l}) maps to the class of A in H^{2}(X̃, 𝔾_{m}). Further π∗ ; Ỹ (ζ) is zero in H^{2}(Ỹ, μ_{l}).

Let W and the maps f : W → 𝔸^{1}, g : W → X be as constructed in §4. Let P (T) be the characteristic polynomial of a general element of D_{k(X)} defining k(Y). We have already seen in 9.2 that P (T) is irreducible over k(X̃) = k(X′). Repeating the proof of the irreducibility of W over \overline{k(t)} we see that W′ = W ×_{X} X̃ is integral and remains irreducible over \overline{k(t)}. Let f′, g′ denote the base changes of f, g to X̃. Then W′_{0} = (f′)^{−1}(0) = Y ×_{X} X̃ and g′|_{W′} : W′_{0} → X̃ is finite. Further, W′_{1} = (f′)^{−1}(1) has l irreducible components V′_{i}, and g′|_{V′_{i}} : V′_{i} → X̃ is a birational isomorphism (in fact an isomorphism). It is also clear that W′ is normal at the generic point of V′_{i}. Let I be the ideal of O_{Y ×_{X} X̃} such that the normalization map Ỹ → Y ×_{X} X̃ is the blow up at I ([Ha_{1}], Ch. II, Th. 7.17 ). Let W̃ be the blow up of W′ at the ideal (I, t) and f̃, g̃ the obvious maps from W̃ to 𝔸^{1}, X̃. The fibre W̃_{0} = Ỹ and W̃ is birational to W′ away from W̃_{0}. We have g^{∗}(ζ)|W̃_{0} = 0 in

H^{2}(W̃_{0}, μ_{l}). Thus conditions (1) to (5) of 3.1 are satisfied by W̃, f̃, g̃. Hence the class of B in Br(k(X̃)), which is the same as the class of D, is zero.

Suppose now that D is ramified on X. The proof in this case consists in a reduction to the unramified case, which is described in de Jong’s paper [dJ], §1. We follow de Jong’s notation.

Let A be an Azumaya algebra over k(X). Let h = (g, f) : W → X × ℙ^{1} be the map constructed in [dJ], satisfying

(1) W\overline{ξ} is smooth, where \overline{ξ} is the geometric generic point of ℙ^{1}.

(2) The extension of A to the function field L\overline{ξ} of W\overline{ξ} is unramified on W\overline{ξ}.

(3) The fiber of f at infinity has l components of multiplicity 1, each birationally isomorphic to X under g. By the unramified case, there exists an element a ∈ L\overline{ξ} such that D ⊗ L\overline{ξ}(a^{1/l}) is split. There is a finite map p : C → ℙ^{1} with C a smooth affine curve such that a belongs to the fraction field L_{C} of W ×_{ℙ^{1}} C and D ⊗ L_{C} is split by L_{C}(a^{1/l}) and hence is cyclic. The map p extends to p̃ : C̃ → ℙ^{1} where C̃ is a smooth projective curve containing C as an open set. Let ∞′ be a point of C̃ mapping to the point ∞ of ℙ^{1}. The discrete valuation of k(X × ℙ^{1}) corresponding to a the generic point of X × ∞ extends to a discrete valuation of M = k(W ×_{ℙ^{1}} C̃), with the same residue field. Let R be the corresponding discrete valuation ring. The algebra D extended to M being cyclic, is given by a symbol (u, v)_{ζ}, where ζ is a primitive l-th root of 1 in k. Since it is unramified at R, we may assume, by the following lemma that u and v are units of R̂, the completion of R. Its specialization to the residue field of R is therefore cyclic and coincides with the class of D.

Lemma 10.2. Let R̂ be a complete discrete valuation ring with quotient field K. Assume that K contains a primitive 2l-th root of 1. Then a cyclic algebra of index l over K and unramified at R̂ is represented by a symbol (u, v)_{ζ} with u and v units of R̂.

Proof. We fix ζ and write (−, −) for (−, −)_{ζ}. Let the κ be the residue field of R. Let A be represented by the symbol (ut^{m}, vt^{n}) ∈ H^{2}(K, μ_{l}). Since A is unramified the residue of (ut^{m}, vt^{n}) is zero. Since κ contains μ_{l} this residue is 1 = (u^{n}/v^{m}) ∈ κ^{∗}/(κ^{∗})^{l}. Thus we have

(ut^{m}, vt^{n}) = (u, v) + (u, t^{n}) + (t^{m}, v) + (t^{m}, t^{n}) .