SMOOTH FINITE SPLITTINGS OF

AZUMAYA ALGEBRAS OVER SURFACES

Manuel Ojanguren and Raman Parimala

Introduction

Let k be an algebraically closed field of characteristic zero, X a quasi-projective smooth surface over k and A an Azumaya algebra over X of rank n^{2}. We construct a smooth irreducible quasi-projective surface Y and a flat finite map π_{Y} : Y → X of degree n such that π^{∗}A is trivial in the Brauer group Br(Y). We further show that the Galois closure of Y over X is a smooth irreducible quasi-projective surface Z and that the Galois group of k(Z) over k(X) is the symmetric group S_{n}.

The smooth finite splitting Y → X was announced, for k of arbitrary characteristic, by Artin and de Jong [dJ], but no proof seems to have been published.

The splitting Y → X that we construct is locally of the form

Spec(O_{X,x}[T]/(P (T)))

where P (T) is the characterisitic polynomial of a section of A. This leads to a very easy construction of a deformation of Y into a union of copies of X, like the one in Lemma 5.1 of [dJ]. From this deformation, following the arguments in [dJ], we deduce a splitting criterionfor A. For the use of these results in the proof of de Jong’s theorem we refer to [CT].

We thank Jean-Louis Colliot-Thélène, Aise Johan de Jong, and David Saltman for several discussions.

1. The characteristic polynomial of the generic matrix

In this section we suppose that k is an algebraically closed field, of arbitrary charactersitic. 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}.

Typeset by AMS-TEX

Lemma 1.1. 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 1.2. 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. The injection k[X_{11}, X_{12}, …, X_{nn}] → A_{n} induces a finite map π : Y_{n} → 𝔸^{n^{2}}. 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 1.3. 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 1.4. 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 1.4 the set U = W ∩ C of all semisimple invertible matrices with exactly two equal eigenvalues is open in C.

Lemma 1.5. The set U is dense in C.

Proof. Let (β, λ) be a point of Sing(Y_{n}). By 1.1, 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 1.6. The dimension of C is equal to the dimension of U.

Lemma 1.7. 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 1.8. 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.

2. Finite smooth splittings

Let X be a smooth quasi-projective surface over an algebraically closed field k, and A an Azumaya algebra of degree n over X. Let K = k(X) be the field of rational functions of X and A_{K} the generic fibre of A. We do not assume that A_{K} is a division ring.

Lemma 2.1. There exists an element σ in A_{K} whose characteristic polynomial is irreducible, separable and has Galois group S_{n}.

Proof. Let σ_{1}, …, σ_{m} be a K-basis of A_{K} (m being equal to n^{2}). Let K ⊂ L be a separable finite extension of K such that A_{K} ⊗_{K} L = M_{n}(L). Let X_{1}, …, X_{m} be indeterminates and σ̃ = X_{1}σ_{1} + ⋯ + X_{m}σ_{m}. After an L-linear change of variables the characteristic polynomial Pσ̃(T) of σ̃ is the characteristic polynomial of the generic matrix, hence it is irreducible and separable over L(X_{1}, …, X_{m}), and has Galois group S_{n}. Since it is defined over K(X_{1}, …, X_{m}) it has the same properties over this smaller field. By Hilbert’s irreducibility theorem (see for instance [FJ], Prop. 16.1.5) there exist ξ_{1}, …, ξ_{m} in K such that the charactersitic polynomial of σ = ξ_{1}σ_{1} + ⋯ + ξ_{m}σ_{m} is irreducible, separable, with Galois group S_{n}.

We fix a smooth embedding of X in a projective space. If d is sufficiently large, the twisted sheaf A(d) is generated by global sections s_{1}, … s_{N−1} and, for some global section f of O_{X}(d) and σ as in Lemma 1, s_{N} = σf is a global section of A(d). We set L = O_{X}(d). Let s be any global section of A(d) = 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 2.2. 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 2.3. The ideal J_{f,U} does not depend on the choice of f.

Proof. We apply 2.2 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.

The ideal J_{s} defines a closed subscheme Y_{s} of Spec (Sym(L^{−1})) which is clearly finite and flat over X. 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}. We denote by π_{λ} : Y_{λ} → X the natural map.

Theorem 2.4. There exists a nonempty open set U ⊂ k^{N} such that, for any λ ∈ U, Y_{λ} is an irreducible quasi-projective surface.

Before proving this theorem we recall, without proof, two easy lemmas.

Lemma 2.5. Let π : Y → X be a flat dominant morphism, with X integral. Then Y is reduced if and only if the generic fibre of π is reduced.

Lemma 2.6. Let π : Y → X be a flat dominant morphism, with X integral. Then Y is irreducible if and only if the generic fibre of π is irreducible.

Proof of Theorem 2.4. We set 𝔸^{N} = Spec (k[t_{1}, …, t_{N}]) and extend the base to X̃ = X × 𝔸^{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; X̃; 𝔸^{N}]

The map π is clearly finite and flat and the two projections from X × 𝔸^{N} are flat, hence p and q are flat. We set Ỹ_{K} = Ỹ ×_{X} Spec(K) and q_{K} : Ỹ_{K} → 𝔸_{K}^{N} the restriction of q to Ỹ_{K}. We first note that, by the choice of s_{N} made above, the fibre q_{K}^{−1}(0, …, 0, 1) is integral. By Theorem 9.7.7 of [Gr], to prove the theorem it suffices to show that the geometric generic fibre of q is integral. Let Ω be an algebraic closure of k(t_{1}, …, t_{N}), Ỹ_{Ω} = Ỹ ×_{𝔸^{N}} Spec(Ω) the generic fibre of q, X̃_{Ω} = X ×_{k} Ω and π_{Ω} : Ỹ_{Ω} → X̃_{Ω} the extension of π. Let S be the integral closure of k[t_{1}, …, t_{N}] in Ω and Λ = K ⊗_{k} S. We set Ỹ_{Λ} = Ỹ ×X̃ Spec(Λ), X̃_{Λ} = Spec(Λ) and π_{Λ} : Ỹ_{Λ} → X̃_{Λ} the extension of π. Assume that Ỹ_{Ω} is not integral. Since π_{Ω} is flat, by 2.5 and 2.6 the generic fibre of π_{Ω} is not integral. But π_{Λ} is also flat and has the same generic fibre as π_{Ω}, hence, again by 2.5 and 2.5, Ỹ_{Λ} is not integral. The characterstic polynomial P_{s̃/f}(T) ∈ K[t_{1}, …, t_{N}] that generates J̃_{t}(T) over a suitable open set of X is clearly separable over K(t_{1}, …, t_{N}), hence Ỹ_{Λ} is reduced by Lemma 2.5. If Ỹ_{Λ} is not integral, being reduced it has more than one component and since π_{Λ} is finite and flat, each component maps surjectively onto X̃_{Λ} and hence no fibre is integral. Let z be a point of X̃_{Λ} over the point (0, …, 0, 1) of 𝔸_{K}^{N}. Specializing at z we get a contradiction with the irreducibility of π_{Λ}^{−1}(0, …, 0, 1) = Spec(K) ×_{X} Y_{(0,...,0,1)}.

Corollary 2.7. Let U be as in 2.4. For any λ ∈ W the field k(Y_{λ}) splits A_{K}.

Proof. By construction the field k(Y_{λ}) is a maximal subfield of A_{K}.

We now show that, assuming that k is of characteristic zero, a general fibre is smooth.

Proposition 2.8. The dimension of Sing(Ỹ) is at most N − 1.

Proof. We try to determine the singularities of Ỹ using the following lemma.

Lemma 2.9. 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 2.9 the singularities of Ỹ are contained in the union of the singularities of the fibers of p.

Lemma 2.10. 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 2.9, 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.

Theorem 2.11. There exists a nonempty open set V ⊂ k^{N} such that, for any λ ∈ V, Y_{λ} is a smooth integral quasi-projective surface. Further, the pull-back π_{λ}^{∗}A is trivial in Br(Y_{λ}).

Proof. 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 W̃ = q^{−1}(W). We now have a map q : W̃ → W of smooth varieties. This map is clearly flat and surjective and therefore, if k is of characteristic zero, it is generically smooth (see [Ha], Ch. III, Corollary 10.7). By definition of generic smoothness there exists a dense open set U′ ⊂ 𝔸^{N} such that q^{−1}(U′) → U′ is smooth. Thus for any λ ∈ U′ the fiber Y_{λ} = q^{−1}(λ) is smooth. By 2.4, if λ ∈ U then Y_{λ} is integral, hence for any λ ∈ V = U ∩ U′ the surface Y_{λ} is smooth and integral. By 2.7 the field k(Y_{λ}) splits A_{K}. But Y_{λ} being smooth, the canonical map Br(Y_{λ}) → Br(k(Y_{λ})) is injective and thus π_{λ}^{∗}A is trivial in Br(Y_{λ}).

Scholium 2.12 (suggested by D. Saltman). For any λ ∈ V the O_{X}-algebra (π_{λ})_{∗}O_{Y_{λ}} embeds into A as a smooth locally free maximal commutative subalgebra.

Proof. Write Y, s and π instead of Y_{λ}, s_{λ} and π_{λ}. Over any sufficiently small affine open set U the line bundle L is generated by a local section f and π_{∗}O_{Y} (U) = k[U][T]/(P_{s/f}(T) maps isomorphically onto k[U][s/f], which is a commutative subalgebra of A(U). It is easy to see that these local isomorphisms patch to give an isomorphism of (π_{λ})_{∗}O_{Y_{λ}} onto a subsheaf S of subalgebras of A locally generated by sections of the form s/f. The generic fibre of S is a maximal subfield K(s/f) ≃ k(Y) of A_{K}. Since S(U) = k[U][s/f] is smooth, it is integrally closed and therefore it is a maximal k[U]-order of K(s/f). This shows that it is a maximal commutative subalgebra of A(U).

Remark. Theorem 2.11 is not true in positive characteristic. Let for instance X be the affine plane X = Spec(k[u, v]) over a field of characteristic p ≠ 0 and A the trivial Azumaya algebra M_{2}(O_{X}) over X. Then A is generated by its global sections

s_{1} = ( \begin{matrix}1 & u^{p} \\ 0 & 0 \end{matrix} ) , s_{2} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , s_{3} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) , s_{4} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ,

and the generic splitting that we denoted Ỹ is the spectrum of

S = k[u, v, t_{1}, t_{2}, t_{3}, t_{4}][T]/(P (T))

where P is the determinant of T − (t_{1}s_{1} + t_{2}s_{2} + t_{3}s_{3} + t_{4}s_{4}). We find

P (T) = T^{2} − (t_{1} + t_{4})T + t_{1}t_{4} − t_{2}t_{3} − u^{p}t_{1}t_{3} .

The algebra S is smooth over k if and only if P, P′, ∂P/∂u and ∂P/∂v have no common zero over the algebraic closure of k(t_{1}, t_{2}, t_{3}, t_{4}). But in fact, eliminating T, u and v we find the equation

(t_{1} − t_{4}/2)^{2} + t_{2}t_{3} + u^{p}t_{1}t_{3} = 0

which is solvable with respect to u. Nevertheless, a better choice of the twisting A(d) and of the sections s_{1}, …, s_{N} might still lead to a proof in the positive characteristic case.

3. Galois splittings

We now construct, for any λ ∈ k^{N}, a Galois covering Z_{λ} of X with group G = S_{n}, 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 in the characteristic zero case, we must construct both at the same time. We achieve this by globalizing the construction of the universal splitting algebra of a monic polynomial, which we now recall. Let R be a commutative ring and P (T) = T^{n} + b_{1}T^{n−1} + ⋯ + b_{n} a monic polynomial with coefficients in R. For 1 ≤ i ≤ n let σ_{i} be the i-th elementary symmetric function in the n variables T_{1}, …, T_{n}. The universal splitting algebra of P (T) is the quotient S of the polynomial algebra R[T_{1}, …, T_{n}] by the ideal I generated by the elements

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

We denote by τ_{1}, …, τ_{n} the classes modulo I of T_{1}, …, T_{n}. We clearly have

P (T) = (T − τ_{1}) ⋯ (T − τ_{n}) .

The symmetric group S_{n} operates on S by permuting τ_{1}, …, τ_{n}. We will use the following properties of S. (For more details and proofs see [Bou] or [EL]). P 1. The construction of S commutes with scalar extensions ([EL], 1.9). P 2. As an R-module S is free of rank n! ([EL], 1.10). P 3. For any commutative R-algebra A and any n-tuple (a_{1}, …, a_{n}) of elements of A such that p(T) = (T − a_{1}) ⋯ (T − a_{n}) in A[T] there is a unique R-homomorphism φ : S → A such that φ(τ_{i}) = a_{i} ([EL], 1.3) . P 4. The subalgebra R[τ_{n}] of S is isomorphic to R[T]/(P (T)) and S is the universal splitting algebra of P (T)/(T − τ_{n}) over R[τ_{n}] ([EL], 1.8).

P 5. If the discriminant of P (T) is a regular element of R, then S^{S_{n}} = R ([EL], 2.2). P 6. If R is a field and P (T) is separable with Galois group S_{n}, then S is a Galois extension of R with Galois group S_{n}.

We now construct Z_{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 in §2. 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 O_{U}[T_{1}, …, T_{n}] is isomorphic to T |_{U} under T_{i} ↦ f_{i}^{−1}. 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

which defines the universal splitting algebra of P_{f,U}(T). 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 3.1. Assume that λ ∈ k^{N} has been chosen such that P_{f,U}(T) = P (T) is separable and irreducible over K. 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, when S is the universal splitting algebra of P (T) over R = k[U] and show that S^{S_{n}} = R and S^{S_{n−1}} = R[T]/(P (T)). Since P (T) is separable over K the first assertion follows from property P6 and the second from properties P3 and P6.

We want to prove the following theorems.

Theorem 3.2. There exists a nonempty open set U ⊂ k^{N} such that, for any λ ∈ U, Z_{λ} is an irreducible quasi-projective surface and the natural map Z_{λ} → X is a ramified Galois cover with group S_{n}.

Theorem 3.3. 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 quasi-projective smooth surface.

The proofs require 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 3.1, 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 3.4. The dimension of Sing(Z_{n}) is at most dim(Sing(N)) + n − 1.

We now compute the dimension of Sing(N). As pointed out by George McNinch, our computation could be deduced from results already in the literature (see for instance [Ja], §7) but we prefer to be as self-contained as possible. We begin with the computation of the dimension of N.

Proposition 3.5. 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 3.6. 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 3.7. 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 3.8. 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 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 3.9. 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 3.10. 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 3.11. For n ≥ 3 the dimension of Sing(N) is n^{2} − n − 2.

Proof. By 3.9 and 3.10, 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 3.12. For n ≥ 3 the dimension of Sing(Z_{n}) is at most n^{2} − 3

Proof. This immediately follows from 3.4 and 3.11.

Proof of Theorem 3.2. It suffices to show that for a general λ the fibre Z_{λ} is irreducible. 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 and q are flat. As in the previous section we set Z̃_{K} = Z̃ ×_{X} Spec(K) and q_{K} : Z̃_{K} → 𝔸_{K}^{N} the restriction of q to Z̃_{K}. We first note that, by the choice of s_{N} made above, the fibre q_{K}^{−1}(0, …, 0, 1) is integral. In fact, by construction, its coordinate algebra is the universal splitting algebra of the characteristic polynomial P_{s_{N} /f}(T) of s_{N}/f. Since the Galois group of P_{s_{N} /f}(T) is S_{n}, its universal splitting algebra, by property P6, is a field . We can now complete the proof exactly as we did in the proof of Theorem 2.4. By Theorem 9.7.7 of [Gr], it suffices to show that the geometric generic fibre of q is integral. Let Ω, S, Λ and X̃_{Λ} be as in section 2 and define Z̃_{Ω}, Z̃_{Λ}, π_{Ω} and π_{Λ} as we did there for Ỹ_{Ω} and so on. The proof given in section 2 goes through once we remark that the universal splitting algebra Z̃_{Λ} is reduced. This is a special case of the following lemma.

Lemma 3.13. Let R be a domain, K its field of fractions and P (T) ∈ R[T] a monic polynomial. Assume that P (T) is separable over K. Then the universal splitting algebra of P (T) over R is reduced.

Proof. Let S be the universal splitting algebra of P (T) over R. It is a free R-algebra of degree n!. The construction of the universal splitting algebra commutes with scalar extensions (property P1), hence S ⊗_{R} K is the splitting algebra of P (T) over K. Since P (T) is separable over K, it follows immediately from property P4 that S ⊗_{R} K is étale over K, in particular reduced. By Lemma 2.5 S is reduced too.

Proof of Theorem 3.3. 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 1.11. By 1.12 the singularities of Z̃ are contained in the union of the singularities of the fibers of p. Since, by Theorem 3.12, the singularities of the closed fibres of p are at worst in codimension 3, we can argue exactly as in the proof of Theorem 1.11 and conclude that q is generically smooth, from which the assertion of Theorem 3.3 immediately follows.

4. Deformations

We now construct a flat family of surfaces over 𝔸_{k}^{1} = Spec(k[t]) that deforms the surface Y constructed in §2 into a union of copies of X. As we mentioned in the introduction, this is a crucial step in the proof of the main result of [dJ] (see also [CT]).

Let \overline{k}(t) be an algebraic closure of k(t)

Proposition 4.1. Let X be a smooth projective surface over an algebraically closed field k of characteristic zero and A an Azumaya algebra over X, of rank n^{2}. There exists a diagram

[figure: W; g; f; X; 𝔸^{1}]

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 an isomorphism for every i,

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

(5) Y = W_{0} = f^{−1}(0) is an irreducible smooth projective surface, g|_{Y} : Y → X is finite and flat, and g^{∗}(A)|_{Y} is trivial in Br(Y).

Proof. We fix a projective embedding of X and choose global sections s_{1}, …, s_{N} of a suitable twist A(d), as we did in §2. Let s = λ_{1}s_{1} + ⋯ + λ_{N}s_{N} with λ = (λ_{1}, …, λ_{N}) ∈ k^{N} and, denoting O_{X}(d) by L, let J_{s} ∈ Sym(L^{−1}) be 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} is irreducible, smooth and splits A. 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 2.3, 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}. 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}. The properties in (5) are clear from the construction of W. To prove property (1) we observe that f^{−1}(0) = Y is integral and that the polynomial defining k(W) over k(X ×_{k} 𝔸_{k}^{1}) is separable, hence the integrality of the geometric generic fibre of f can be proved as we did in §2 for Ỹ → 𝔸_{k}^{N}.

5. A splitting criterion

We now show that the flat family of Proposition 4.1 can be used to show the triviality of an Azumaya algebra.

Proposition 5.1. Let X be an integral projective d-dimensional variety 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 (d + 1)-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 tale 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.

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Manuel Ojanguren, IGAT, EPFL, CH-1015 Lausanne, Switzerland E-mail address: manuel.ojanguren@epfl.ch

Raman Parimala, Emory University, Atlanta, GA, USA E-mail address: parimala@mathcs.emory.edu