Indecomposable and Noncrossed Product Division Algebras over Curves over Complete Discrete Valuation Rings

Feng Chen^{1}

Dept. of Math & CS, Emory University, Atlanta GA 30322

Abstract

Let T be a complete discrete valuation ring and X̂ a smooth projective curve over S = Spec(T) with closed fibre X. Denote by F the function field of X̂ and by F̂ the completion of F with respect to the discrete valuation defined by X, the closed fibre. In this paper, we construct indecomposable and noncrossed product division algebras over F. This is done by defining an index preserving group homomorphism s : Br(F̂)′ → Br(F)′, and using it to lift indecomposable and noncrossed product division algebras over F̂.

1. Introduction

Let X̂ be a smooth projective curve over S = Spec(T), where T is a complete discrete valuation ring with uniformizer t. Let F = K(X̂) be the function field, and let F̂ = K̂(X̂) be the completion with respect to the discrete valuation defined by the closed fibre X. We define an index-preserving homomorphism

Br(F̂)′ → Br(F)′

that splits the restriction map res : Br(F)′ → Br(F̂)′. Here Br(−) denotes the Brauer group of − and the “prime” denotes the union of the n-torsion part of Br(F), where n is prime to the characteristic of k, the residue field of T. Using the method of Brussel [5] and Brussel [4], we can construct indecomposable and noncrossed product division algebras over F̂, and lift these constructions to F using our homomorphism, generalizing the constructions in Brussel et al. [6], where indecomposable and noncrossed product division algebras over function fields of p-adic curves are constructed.

Recall that if K is a field, a K-division algebra D is a division ring that is finite-dimensional and central over K. The period or exponent of D is the order of the class [D] in Br(K), and the index of D is the square root of D’s K-dimension. A noncrossed product is a K-division algebra whose structure is not given by a Galois 2-cocycle. Noncrossed products were first constructed by Amitsur [2], settling a longstanding open problem. Since then there have been several other constructions, including Saltman [22], Jacob and Wadsworth [18], Brussel [5].

A K-division algebra is indecomposable if it cannot be expressed as the tensor product of two nontrivial K-division algebras. It is easy to see that all division algebras of period not a prime power are decomposable, so the problem of producing an indecomposable division algebra is only interesting when the period and index are unequal prime powers. Therefore we will only consider division algebras of prime power period and index in this paper. Then it is not hard to see that all division algebras of equal (prime power) period and index are trivially indecomposable. Albert constructed decomposable division algebras in the 1930’s, but indecomposable division algebras of unequal (2-power) period and index did not appear until Saltman [23]and Amitsur et al. [3]. Since then there have been several constructions, including Tignol [27], Jacob and Wadsworth [17], Jacob [16], Schofield and Van den Bergh [25], Brussel [4] and McKinnie [21].

Email address: fchen@emory.edu (Feng Chen)

Preprint submitted to Arxiv April 24, 2010

It is the author’s pleasure to thank Prof. Brussel, his thesis adviser. The author is greatly indebted to him for his patience and suggestions during the preparation of the paper. The author would also like to thank Prof. Suresh and Prof. Parimala for many instructive discussions and their most valuable comments and critiques. Finally the author thanks Prof. Harbater for reading a first draft of the paper and his valuable suggestions and comments to improve the writing.

2. Patching over Fields

Our construction is based on the method of patching over fields introduced in Harbater and Hartmann [14]. In this section, we will recall this method. Throughout this section, T will be a complete discrete valuation ring with uniformizer t, fraction field K and residue field k. Let X̂ be a smooth projective T -curve with function field F such that the reduced irreducible components of its closed fibre X is regular. (Given F, such an X̂ always exists by resolution of singularities; cf. Abhyankar [1] or Lipman [20]). Let f : X̂ → ℙ_{T}^{1} be a finite morphism such that the inverse image S of ∞ ∈ ℙ_{k}^{1} contains all the points of X at which distinct irreducible components meet. (Such a morphism exists by Harbater and Hartmann [14, Proposition 6.6]). We will call (X̂, S) a regular T -model of F.

We follow Harbater and Hartmann [14, Section 6] to introduce the notation. Given an irreducible component X_{0} of X with generic point η, consider the local ring of X̂ at η. For a (possibly empty) proper subset U of X_{0}, we let R_{U} denote the subring of this local ring consisting of rational functions that are regular at each point of U. In particular, R_{ϕ} is the local ring of X̂ at the generic point of the component X_{0}. The t-adic completion of R_{U} is denoted by R̂_{U}. If P is a closed point of X, we write R_{P} for the local ring of X̂ at P, and R̂_{P} for its completion at its maximal ideal. A height 1 prime ideal 𝔭 that contains t determines a branch of X at P, i.e., an irreducible component of the pullback of X to Spec(R̂_{P}). Similarly the contraction of 𝔭 to the local ring of X̂ at P determines an irreducible component X_{0} of X, and we say that 𝔭 lies on X_{0}. Note that a branch 𝔭 uniquely determines a closed point P and an irreducible component X_{0}. In general, there can be several branches 𝔭 on X_{0} at a point P; but if X_{0} is smooth at P then there is a unique branch 𝔭 on X_{0} at P. We write R̂_{𝔭} for the completion of the localization of R̂_{P} at 𝔭; thus R̂_{P} is contained in R̂_{𝔭}, which is a complete discrete valuation ring.

Since X̂ is normal, the local ring R_{P} is integrally closed and hence unibranched; and since T is a complete discrete valuation ring, R_{P} is excellent and hence R̂_{P} is a domain (cf. Grothendieck and Dieudonné [13, Scholie 7.8.3(ii,iii,vii)]). For nonempty U as above and Q ∈ U, R̂_{U}/t^{n}R̂_{U} → R̂_{Q}/t^{n}R̂_{Q} is injective for all n and hence R̂_{U} → R̂_{Q} is also injective. Thus R̂_{U} is also a domain. Note that the same is true if U is empty. The fraction field s of the domains R̂_{U}, R̂_{P} and R̂_{𝔭} will be denoted by F_{U}, F_{P} and F_{𝔭}.

If 𝔭 is a branch at P lying on the closure of U ⊂ X_{0}, then there are natural inclusions of R̂_{P} and R̂_{U} into R̂_{𝔭}, and hence of F_{P} and F_{U} into F_{𝔭}. The inclusion of R̂_{P} was observed above; for R̂_{U}, note that the localization of R_{U} and of R_{p} at the generic point of X_{0} are the same; and this localization is naturally contained in the t-adically complete ring R̂_{𝔭}. Thus so is R_{U} and hence its t-adic completion R̂_{U}.

In the above context, assume f : X̂ → ℙ_{T}^{1} is a finite morphism such that 𝔓 = f^{−1}(∞) contains all points at which distinct irreducible components of the closed fibre X ⊂ X̂ meet (Such an f always exists by Harbater and Hartmann [14, Proposition 6.6]). We let 𝔘 be the collection of irreducible components U of f^{−1}(𝔸_{k}^{1}), and let 𝔅 be the collection of all branches 𝔭 at all points of 𝔓.

The inclusions of R̂_{U} and of R̂_{Q} into R̂_{𝔭}, for 𝔭 = (U, Q), induce inclusions of the corresponding fraction fields F_{U} and F_{Q} into the fraction field F_{𝔭} of R̂_{𝔭}. Let I be the index set consisting of all U, Q, 𝔭 described above. Via the above inclusions, the collection of all F_{ξ}, for ξ ∈ I, then forms an inverse system with respect to the ordering given by setting U ≻ 𝔭 and Q ≻ 𝔭 if 𝔭 = (U, Q).

Under the above hypotheses, suppose that for every field extension L of F, we are given a category 𝔄(L) of algebraic structures over L(i.e. finite dimensional L-vector spaces with additional structure, e.g. associative L-algebras), along with base-change functors 𝔄(L) → 𝔄(L′) when L ⊆ L′. An 𝔄-patching problem for (X̂,S) consists of an object V_{ξ} in 𝔄(F_{ξ}) for each ξ ∈ I, together with isomorphisms ϕ_{U,𝔭} : V_{U} ⊗_{F_{U}} F_{𝔭} → V_{𝔭} and ϕ_{Q,𝔭} : V_{Q}⊗_{F_{Q}} F_{𝔭} → V_{𝔭} in 𝔄(F_{𝔭}). These patching problems form a category, denoted by PP_{𝔄}(X̂, S), and there is a base change functor 𝔄(F) → PP_{𝔄}(X̂, S).

If an object V ∈ 𝔄(F) induces a given patching problem up to isomorphism, we will say that V is a solution to that patching problem, or that it is obtained by patching the objects V_{ξ}. We similarly speak of obtaining a morphism over F by patching morphisms in PP_{𝔄}(X̂, S). The next result is given by Harbater and Hartmann [14, Theorem 7.2].

Theorem 1. Let T be a complete discrete valuation ring. Let X̂ be a smooth connected projective T -curve with closed fibre X. Let U_{1}, U_{2} ⊆ X, let U_{0} = U_{1} ∩ U_{2}, and let F_{i} := F_{U_{i}}(i = 0, 1, 2). Let U = U_{1} ∪ U_{2} and form the fibre product of groups Br(F_{1}) ×_{Br(F_{0})} Br(F_{2}) with respect to the maps Br(F_{i}) → Br(F_{0}) induced by F_{i} ↪ F_{0}. Then the base change map β : Br(F_{U}) → Br(F_{1}) ×_{Br(F_{0})} Br(F_{2}) is a group isomorphism.

The above Theorem says that giving a Brauer class over a function field F is equivalent to giving compatible division algebras over the patches. The nice thing about patching Brauer classes over a function field F is that we have good control of the index, which is stated in Harbater et al. [15, Theorem 5.1].

Theorem 2. Under the above notation, let A be a central simple F -algebra. Then ind(A) = lcm_{ξ∈𝔓∪𝔘}(ind(A_{F_{ξ}})).

To conclude this section, we record a variant of Hensel’s Lemma from Harbater et al. [15, Lemma 4.5] that will be used over and over again in the index computation.

Lemma 3. Let R be a ring and I an ideal such that R is I-adically complete. Let X be an affine R-scheme with structure morphism ϕ : X → SpecR. Let n ≥ 0. If s_{n} : Spec(R/I^{n}) → X ×_{R} (R/I^{n}) is a section of ϕ_{n} : ϕ×_{R} (R/I^{n}) and its image lies in the smooth locus of ϕ, then s_{n} may be extended to a section of ϕ.

3. Splitting Map

Let T be a complete discrete valuation ring with uniformizer t and residue field k. By a smooth curve X̂ over T, we will mean a scheme X̂ which is projective and smooth of relative dimension 1 over Spec(T). In particular, X̂ is flat and of finite presentation over Spec(T). Let F = K(X̂) be the function field of X̂. Note that since X̂ is smooth, the closed fibre X is smooth, integral, connected and of codimension 1, hence determines a discrete valuation ring on F. Let F̂ = K̂(X̂) be the completion of F with respect to this discrete valuation. Throughout the paper, n will denote an integer which is prime to the characteristic of k.

We will be using the following notation for cohomology groups in the sequel: For an integer r, we let

μ_{n}^{r} = \begin{cases}μ_{n}^{⊗r} & for r ≥ 0, \\ hom(μ_{n}^{⊗−r}, μ_{n}) & for r < 0. \end{cases}

For a fixed integer n, and for any field K, we will let H^{q}(K, r) = H^{q}(K, μ_{n}^{⊗r}) and H^{q}(K) = H^{q}(K, q − 1) = H^{q}(K, μ_{n}^{⊗q−1}). In particular, H^{2}(K) = _{n}Br(K) will be the n-torsion part of the Brauer group of K; and H^{1}(K) will be the n-torsion part of the character group of K.

Adopting the above notation, in this section we will define a map s : H^{2}(F̂) → H^{2}(F) and show that s has the following properties:

• s is a group homomorphism;

• s splits the restriction;

• s preserves index of Brauer classes.

Once such a map s is defined, we could use it to construct indecomposable division algebras and noncrossed product division algebras over F, as in section 5.

3.1. Construction over an Open Affine Subset

Given an element γ̂ ∈ H^{2}(F̂), we will define a lift γ_{U} to F_{U} of γ̂. Note that since F̂ is a complete discretely valued field with t a unifomizer, and with k(X) the residue field. We have an exact Witt Sequence as in Garibaldi et al. [10, II.7.10 and II.7.11],

0 → H^{2}(k(X)) → H^{2}(F̂) → H^{1}(k(X)) → 0  (1)

split (non-canonically) by the cup product with (t) ∈ H^{1}(k(X)). Hence each element γ̂ ∈ H^{2}(F̂) can be written as a sum γ_{0} + (χ_{0}, t), with γ_{0} ∈ H^{2}(k(X)) and χ_{0} ∈ H^{1}(k(X)) (Note that here we are identifying H^{r}(k(X)) as a subgroup of H^{r}(F̂), for r = 1, 2, as in Garibaldi et al. [10, II.7.10 and II.7.11]). Here we use the notation (χ_{0}, t) to denote the cup product χ_{0} ∪ (t), and we will use this notation throughout the paper without further explanation.

Let U be an open affine subset of X so that neither γ_{0} nor χ_{0} ramifies at any closed point of U. This implies that γ_{0} ∈ H^{2}(k[U]) and χ_{0} ∈ H^{1}(k[U]) by purity(cf, Colliot-Thélène [8]), where k[U] denotes the ring of regular functions of the affine scheme U.

By Cipolla [7], there exists a canonical isomorphism H^{2}(R̂_{U}) → H^{2}(k[U]) since R̂_{U} is t-adically complete and k[U] ≅ R̂_{U}/(t); therefore there is a unique lift of γ_{0} to H^{2}(R̂_{U}). At the same time, Grothendieck and Raynaud [12, Théorèm 8.3] implies that there is a unique lift of χ_{0} to H^{1}(R̂_{U}) as well. Taking γ̃_{0} and χ̃_{0} as the lifts of γ_{0} and χ_{0} to R̂_{U}, we will let

γ_{U} = γ̃_{0} + (χ̃_{0}, t)  (2)

be the lift of γ̂ to H^{2}(F_{U}).

3.2. Construction over Closed Points

Fix an open affine subset U of X and let 𝔓 = X\U. In order to apply the patching result we recalled in 2, we need to define a γ_{P} for each P ∈ 𝔓 in such a way that when 𝔭 = (U, P) is the unique branch of U at P, the restriction to F_{𝔭} of γ_{P} and γ_{U} agree with each other, i.e., res_{F_{𝔭}}(γ_{P}) = res_{F_{𝔭}}(γ_{U}) (Recall there are field embeddings F_{P} ↪ F_{𝔭} and F_{U} ↪ F_{𝔭} for 𝔭 = (U, P), as in Section 2, hence there are restrictions res : H^{2}(F_{U}) → H^{2}(F_{𝔭}) and res : H^{2}(F_{P}) → H^{2}(F_{𝔭}). For more details on these restriction maps, see Serre [26]).

Note that since X̂ is regular and the closed fibre X is smooth, the maximal ideal of the local ring R_{P} is generated by two generators, t and π. So is R̂_{P}.

We define γ_{P} in the following way: There is a field embedding F_{U} → F_{𝔭}, hence a canonical restriction res : H^{2}(F_{U}) → H^{2}(F_{𝔭}). Let γ_{𝔭} be the image of γ_{U} under this restriction. Observe that F_{𝔭} is a complete discretely valued field with residue field κ(𝔭); furthermore, κ(𝔭) is also a complete discretely valued field with residue field κ(P). Therefore, applying Garibaldi et al. [10, II.7.10 and II.7.11] twice, we get the following decomposition of H^{2}(F_{𝔭}):

H^{2}(F_{𝔭}) ≅ H^{2}(κ(P)) ⊕ H^{1}(κ(P)) ⊕ H^{1}(κ(P)) ⊕ H^{0}(κ(P)).  (3)

In other words, each element γ_{𝔭} ∈ H^{2}(F_{𝔭}) can be written as γ_{𝔭} = γ_{0,0} +(χ_{1,}π)+(χ_{2} +(π^{r}), t), where γ_{0,0} ∈ H^{2}(κ(P)), χ_{1,}χ_{2} ∈ H^{1}(κ(P)), r ∈ H^{0}(κ(P)) ≅ ℤ/nℤ and (π^{r}) denote the image in H^{1}(κ(P)) of π^{r} under the Kummer map. Note that by our notation, H^{0}(κ(P)) = H^{0}(κ(P), μ_{n}^{−1}) = ℤ/nℤ.

In order to define a lift for γ_{𝔭} to F_{P}, we first show that all characters in H^{1}(κ(𝔭)) can be lifted by proving the following lemma.

Lemma 4. Let χ ∈ H^{1}(κ(𝔭)) be a character. Then there is a unique χ̃ ∈ H^{1}(F_{P}) that lifts χ.

Proof. Since κ(𝔭) is a complete discretely valued field with residue field κ(P), we have the classical Witt’s decomposition for χ,

χ = χ_{0} + (π^{r}),

where χ_{0} ∈ H^{1}(κ(P)) and r ∈ H^{0}(κ(P)). Note that χ_{0} can be lifted without any difficulty by Grothendieck and Raynaud [12, Théorèm 8.3]; the only trouble comes from (π^{r}).

Let L, L_{0}/κ(𝔭) be the field extension determined by χ, χ_{0} respectively. Then L_{0} is the maximal unramified subextension of κ(𝔭) inside L and L/L_{0} is a totally ramified extension determined by the character (π^{r}). Now Fesenko and Vostokov [9, Theorem II.3.5] implies that (π^{r}) can be lifted to H^{1}(F_{P}) in a unique fashion as well, since κ(𝔭) is a complete discretely valued field.

Now we are ready to define a lift for γ̂ in H^{2}(F_{P}). Again Cipolla [7] implies that H^{2}(κ(P)) ≅ H^{2}(R̂_{P}) and Lemma 4 implies that χ_{1}, χ_{2} + (π^{r}) can be lifted to H^{1}(R̂_{P}) uniquely. Hence each component of H^{2}(F_{𝔭}) can be lifted to R̂_{P}, and thus we will set

γ_{P} = γ̃_{0,0} + (χ̃_{1}, π) + (χ̃_{2} + (π^{r}), t).  (4)

where γ̃, χ̃_{1}, χ̃_{2} are the lifts of γ_{0,0,}χ_{1}, χ_{2} to R̂_{P} (and hence to F_{P}), respectively. Therefore this γ_{P} is a unique lift of γ_{𝔭} to F_{P}. The assignment of s_{P} (γ_{𝔭}) = s_{P} will yield a map s_{P} : H^{2}(F_{𝔭}) → H^{2}(F_{P}). It is not hard to see that s_{P} is a group homomorphism, since it is a group homomorphism on each of the components.

3.3. The Map is Well Defined

In this section we show that γ_{U} and γ_{P} that we constructed in Section 3.1 and Section3.2 are compatible in the sense of patching, that is res_{F_{𝔭}}(γ_{U}) = res_{F_{𝔭}}(γ_{P}) for each P ∈ 𝔓 = X\U when 𝔭 = (U, P) is the unique branch of U at P.

We claim that the compatibility will be proved if we can show that s_{P} splits the restriction map res_{F_{𝔭}} : H^{2}(F_{P}) → H^{2}(F_{𝔭}), or equivalently, res_{F_{𝔭}} ∘ s_{P} is the identity map. This is true because γ_{P} = s_{P} (γ_{𝔭}) = s_{P} ∘ res_{F_{𝔭}}(γ_{U}), hence we would have that res_{F_{𝔭}}(γ_{P}) = res_{F_{𝔭}}(γ_{U}) if res_{F_{𝔭}} ∘ s_{P} is the identity map. So it suffices to prove the following

Proposition 5. s_{P} as defined in 3.2 splits the restriction res : H^{2}(F_{P}) → H^{2}(F_{𝔭}), that is, res ∘ s_{P} is the identity map.

Proof. Take an arbitrary element γ_{𝔭} ∈ H^{2}(F_{𝔭}). As in section 3.2, we write γ_{𝔭} = γ_{0,0} + (χ_{1,}π) + (χ_{2} + (π^{r}), t). Therefore it is easily checked that

res ∘ s_{P} (γ_{𝔭}) = res ∘ s_{P} (γ_{0,0} + (χ_{1}, π) + (χ_{2} + (π^{r}), t))
= res(γ̃_{0,0} + (χ̃_{1}, π) + (χ̃_{2} + (π^{r}), t))
= γ_{0,0} + (χ_{1}, π) + (χ_{2} + (π^{r}), t)
= γ_{𝔭.}

Thus γ_{U}, γ_{P} will patch and yield γ ∈ H^{2}(F), by Harbater and Hartmann [14, Theorem 7.2]. But there is one more thing we have to check before we can say we have a map s : H^{2}(F̂) → H^{2}(F): we need to show that γ is independent of the choice of the open affine subset U of X. In order to do this, we prove the following

Lemma 6. Let T be a complete discrete valuation ring with residue field k; let X̂ be a smooth projective T -curve with function field F and closed fibre X. Let F̂ be the completion of F with respect to the discrete valuation induced by X, and denote by k(X) the corresponding residue field. Take an element γ̂ = γ_{0} + (χ_{0}, t) ∈ H^{2}(F̂), where γ_{0} ∈ H^{2}(k(X)) and χ_{0} ∈ H^{1}(k(X)). Assume that U_{1}, U_{2} are two open affine subsets of X so that neither γ_{0}, χ_{0} is ramified on any point of U_{1}∪U_{2}. Let 𝔓_{1}, 𝔓_{2} be the complements of U_{1}, U_{2} respectively. We construct two Brauer classes γ, γ′ ∈ H^{2}(F) by patching as we did above, while using U_{1} and U_{2} as the open affine subset in the construction, respectively. Then γ, γ′ denote the same Brauer class in H^{2}(F).

Proof. We first deal with the case where U_{1} is contained in U_{2}. In this case we have a field embedding F_{U_{2}} ↪ F_{U_{1}}. Let γ_{i} be the lift of γ_{0} to H^{2}(F_{U_{i}}), we must have γ_{1} = res_{F_{U_{1}}} (γ_{2}), since both γ_{1} and γ_{2} are the image of γ_{0}; in other words, res_{F_{U_{2}}} (γ) = res_{F_{U_{2}}} (γ′). By the construction in Section 3.2, it follows that for every P ∈ 𝔓_{2}, res_{F_{P}} (γ) = res_{F_{P}} (γ′). Therefore it follows that γ = γ′, by Harbater and Hartmann [14, Theorem 7.2]. This proves the Lemma in the case where U_{1} is contained in U_{2}.

In the general case, let U_{3} be an open affine subset of U_{1} ∩ U_{2}. Clearly γ_{0} and χ_{0} are both unramified at every point of U_{3}. Let γ′′ ∈ H^{2}(F) be the Brauer class constructed by patching as above, using U_{3} as the open affine subset in the construction. It follows that γ′′ = γ and γ′′ = γ′ since U_{3} is contained in both U_{1} and U_{2}, by what we just proved for the case where one open affine subset is contained in the other. Hence γ = γ′ = γ′′ ∈ H^{2}(F), which proves the Lemma in the general case.

3.4 s Splits the Restriction Map 7

3.4. s Splits the Restriction Map

Recall the notation: let T be a complete discrete valuation ring with residue field k and uniformizer t. Let X̂ be a smooth projective T -curve with function field F and closed fibre X. Let F̂ be the completion of F with respect to the discrete valuation induced by X. Let s : H^{2}(F̂) → H^{2}(F) be the map defined by patching as in section 3.1 and section 3.2. We will show that s splits the restriction map res : H^{2}(F) → H^{2}(F̂). Hence index of Brauer classes cannot go up under the map s, because restriction can never raise index. In particular, we prove the following Proposition.

Proposition 7. The map s is a section to the restriction map resF̂ : H^{2}(F) → H^{2}(F̂).

Proof. It suffices to show that res ∘ s is the identity map on H^{2}(F̂). Since H^{2}(F̂) ≅ H^{2}(k(X)) ⊕ H^{1}(k(X)), it suffices to show that resF̂ ∘ s is the identity map on both components; that is, given γ̂ = γ_{0} + (χ_{0}, t) where γ_{0} ∈ H^{2}(k(X)) and χ_{0} ∈ H^{1}(k(X)), the Proposition will follow if we can show that resF̂ ∘ s(γ_{0}) = γ_{0} and resF̂ ∘ s((χ_{0}, t)) = (χ_{0}, t).

Take an open affine subset U of X so that γ_{0}, χ_{0} are both unramified on every point of U; that is, we have γ_{0} ∈ H^{2}(k[U]) and χ_{0} ∈ H^{1}(k[U]). Note that we have the following commutative diagram (For a field E, H_{nr}^{2}(E) denotes the unramified part of H_{nr}^{2}(E), or equivalently, H_{nr}^{2}(E) = ∩_{v}H^{2}(E_{v}), where v runs through all discrete valuations on E, and E_{v} denotes the completion of E at v. See Colliot-Thélène [8] for more details on the unramified cohomology.):

\xymatrix{H^{2}(k(X)) \ar[r]_{f}^{∼} & H_{nr}^{2}(F̂) \ar[d]^{s} \\ H^{2}(R̂_{U}) \ar@{^{(}->}[u]^{g} \ar@{^{(}->}[d]^{h} & H^{2}(F) \ar[dl]^{_{res_{F_{U}}}} \\ H^{2}(F_{U})}

The commutativity of the above diagram follows simply from the construction of over open affine subset we outline in Section 3.1. Therefore resF̂ on s(γ_{0}) is the same as f ∘ g ∘ h^{−1} ∘ res_{F_{U}}, and thus resF̂ ∘ s(γ_{0}) = f ∘ g ∘ h^{−1} ∘ res_{F_{U}} ∘ s(γ_{0}) = γ_{0}. (Note in fact h has no inverse; however we can find an inverse image under h for res_{F_{U}} ∘ s(γ_{0}), so we write h^{−1} only merely as a shorthand notation here.)

To show that resF̂ ∘s((χ_{0}, t)) = (χ_{0}, t), it suffices to show that ram(resF̂ ∘s((χ_{0}, t))) = χ_{0}, where ram : H^{2}(F̂) → H^{1}(k(X)) denotes the ramification map on H^{2}(F̂) with respect to the valuation determined by the closed fibre X. Since χ_{0} ∈ H^{1}(k[U]), we have ram(resF̂ ∘s((χ_{0}, t))) = ram((χ̃_{0}, t)) where χ̃_{0} denotes the lift of χ_{0} to H^{1}(R̂_{U}), as we did in Section 3.1 (Since H^{1}(R̂_{U}) ≅ H^{1}(k[U]), χ̃_{0} can be viewed as as element of H^{1}(k[U]), and hence element of H^{1}(k(X)) via the injection H^{1}(k[U]) ↪ H^{1}(k(X)), and finally element of H^{1}(F̂) via the injection H^{1}(k(X)) ↪ H^{1}(F̂)). Therefore the image in H^{1}(F̂) of χ̃_{0} under the composition of these maps is in fact χ_{0}, since all these maps are injective. Then it is easy to see that ram((χ̃_{0}, t)) = χ̃_{0} = χ_{0} ∈ H^{1}(k[X]), as desired.

The following corollary is immediate:

Corollary 8. Index of Brauer classes cannot go down under the map s.

Proof. Take γ̂ ∈ H^{2}(F̂) and let γ = s(γ̂). By Proposition 7 we must have that γ̂ = resF̂(γ), therefore ind(γ̂)|ind(γ). This proves that s can never lower index of Brauer classes.

4. s Preserves Index of Brauer Classes

In this section, we will show that the splitting map s that we defined in section 3 has one more property that is crucial to the construction of indecomposable and noncrossed product division algebras over p-adic curves, that is, s preserves index of Brauer classes. In other words, ind(γ̂) = ind(γ) = ind(s(γ̂)). We make the following elementary observation, which is true for Brauer classes over an arbitrary field.

Proposition 9. Let k be an arbitrary field. Let γ ∈ H^{2}(k) be a Brauer class with the following decomposition: γ = γ_{0} + (χ, t), where γ_{0} ∈ H^{2}(k), χ ∈ H^{1}(k) and t is an arbitrary element of k. Then ind(γ)|ind(γ_{0,l}) · exp(χ), where γ_{0,l} denotes the base extension of γ_{0} to l/k, where l is the field extension determined by χ.

Proof. Let E/l be a minimal extension that splits γ_{0,l}. Then [E : l] = ind(γ_{0,l}). Also there is some E′/k with [E′ : k] = exp(χ) which splits χ and hence (χ, t); therefore EE′ will split γ, furthermore it is not hard to see that [EE′ : k]|ind(γ_{0,l}) · exp(χ) and hence ind(γ)|ind(γ_{0,l}) · exp(χ).

We will apply Harbater et al. [15, Theorem 5.1], which states that ind(γ) = lcm(ind(γ_{U}), ind(γ_{P})) for each P ∈ 𝔓. Since we already showed that s can never lower index of Brauer classes as in section 3.4, we will be done if we could show that ind(γ)|ind(γ̂); therefore it suffices to show that ind(γ_{U})|ind(γ̂) and ind(γ_{P})|ind(γ̂) for each P ∈ 𝔓, respectively. We will deal with them in order.

We start by recalling the notion of Azumaya algebras and their generalized Severi-Brauer varieties. The notion of a central simple algebra over a field can be generalized to the notion of an Azumaya algebra over a domain R (cf. Saltman [24, Chapter 2], or Grothendieck [11, Part I, Section 1]). The degree of an Azumaya algebra A over R is the degree of A ⊗_{R} F as a central simple algebra over the fraction field F over R. The Brauer group of a domain R is defined as the set of equivalence classes of Azumaya algebras with the analogous operations, where one replaces the vector spaces V_{i} with projective modules in the definition of Brauer equivalences. If A is an Azumaya algebra of degree n over a domain R, and 1 ≤ i < n, there is a functorially associated smooth projective R-scheme SB_{i}(A), called the i-th generalized Severi-Brauer variety of A (cf. Van den Bergh [28, p. 334]). For each R-algebra S, the S-points of SB_{i}(A) are in bijection with the right ideals of A_{S} = A ⊗_{R} S that are direct summands of the S-module A_{S} having dimension (i.e. S-rank) ni. If R is a field F, so that A is a central simple F -algebra, and if E/F is a field extension, then SB_{i}(A)(E) ≠ ϕ if and only if ind(A_{E}) divides i (cf. Knus et al. [19, Proposition 1.17]). Here A_{E} ≅ Mat_{m}(D_{E}) for some E-division algebra D_{E} and some m ≥ 1, and the right ideals of E-dimension ni are in natural bijection with the subspaces of D_{E}^{m} of D_{E}-dimension i/ind(A_{E}) (cf. Knus et al. [19, Proposition 1.12, Definition 1.9]). Thus the F -linear algebraic group GL_{1}(A) = GL_{m}(D_{F}) acts transitively on the points of the F -scheme SB_{i}(A). We record Knus et al. [19, Proposition 1.17] here since we will be using it over and over again in the sequel.

Proposition 10. Let A be a central simple algebra over a field F. The Severi-Brauer variety SB_{r}(A) has a rational point over an extension K/F if and only if the index ind(A_{K}) divides r. In particular, SB(A) has a rational point over K if and only if K splits A.

4.1. Index Computation Over Affine Open Set

We compute ind(γ_{U}) in this section; in particular, we show that ind(γ_{U})|ind(γ̂). Thanks to Lemma 6, it suffices to show that there exists an open affine subset V ⊂ X so that ind(γ_{V})|ind(γ̂) since we could replace U by V if necessary in the construction we outlined in section 3.1 and this would not change γ ∈ H^{2}(K(X̂) by Lemma 6. Therefore we will prove the following proposition, which shows that there exists such an open affine subset V.

Proposition 11. Let T be a complete discrete valuation ring. Let X̂ be a smooth projective T -curve with closed fibre X. Let F be the function field of X̂ and F̂ the completion of F with respect to the discrete valuation determined by X. Then for every γ̂ ∈ H^{2}(F̂), there exists an affine open subset V ⊂ X such that ind(γ_{V})|ind(γ̂), where γ_{V} is the lift of γ̂ to F_{V} as defined in section 3.1.

Proof. Recall that γ̂ = γ_{0} + (χ_{0,}t) ∈ H^{2}(F̂) where γ_{0} ∈ H^{2}(k(X)) and χ_{0} ∈ H^{1}(k(X)). Therefore ind(γ̂) = ind(γ_{0,l}) · exp(χ_{0}), where l/k(X) is the field extension determined by χ_{0}, by Jacob and Wadsworth [17, Theorem 5.15], since F̂ is a complete discretely valued field.

Let U be an open affine subset of X such that neither γ_{0} nor χ_{0} ramifies on any point of U. Recall that γ_{U} = γ̃_{0} + (χ̃_{0}, t) where γ̃_{0} ∈ H^{2}(R̂_{U}) and χ̃_{0} ∈ H^{1}(R̂_{U}). Note that exp(χ̃_{0}) = exp(χ_{0}) since H^{1}(R̂_{U}) ≅ H^{1}(k(X)). By Proposition 9, we have ind(γ_{U})|ind(γ̃_{0,S}) · exp(χ̃_{0}), where S/R̂_{U} denotes the Galois cyclic extension determined by χ̃_{0}. Note when V ⊆ U, we have H^{r}(k[U]) ⊆ H^{r}(k[V]) by purity, and hence H^{r}(R̂_{U}) ⊆ H^{r}(R̂_{V}); so we have γ̃_{0} ∈ H^{2}(R̂_{V}) and χ̃_{0} ∈ H^{1}(R̂_{V}). Therefore it suffices to find some affine open subset V ⊂ U such that ind(γ̃_{0,S′})|ind(γ_{0,l}), where S′/R̂_{V} denotes the Galois cyclic extension determined by χ̃_{0}.

Let i = ind(γ_{0,l}) be the index of the restriction of γ_{0} to l. Then Proposition 10 implies that SB_{i}(γ_{0})(l) ≠ ϕ; in other words, there is an l-rational point in the i-th generalized Severi-Brauer variety of γ_{0}. Hence the Spec(k(X))-morphism π : SB_{i}(γ_{0})×_{k(X)}l → Spec(l) has a section Spec(l) → SB_{i}(γ_{0}) ×_{k(X)} l over Spec(k(X)), the generic point of the closed fibre U of Spec(R̂_{U}). Choose a Zariski dense open subset V ⊆ U such that this section over Spec(k(X)) extends to a section over V, and such that the image of this latter section lies in an open subset of SB_{i}(γ_{0}) ×_{k(X)} l that is affine over R̂_{U}. Then by Lemma 3, the section over V lifts to a section over Spec(R̂_{V}), thus we obtain an L-rational point of SB_{i}(γ̃_{0})×R̂_{V} S′, where L/F_{V} is the Galois cyclic extension determined by χ̃_{0}; or equivalently, L is the fraction field of S′. This implies that ind(γ̃_{0,S′})|i = ind(γ_{0,l}) by Proposition 10 again.

4.2. Index Computation Over Closed Points

It remains to show ind(γ_{P})|ind(γ̂). This is what we are going to do in this section. Note that γ_{P} is defined as s_{P} ∘ res_{F_{𝔭}}(γ_{U}), where res_{F_{𝔭}} can only lower index of γ_{U}. Since we have already shown that ind(γ_{U})|ind(γ̂), we have that ind(γ) will be completely determined by ind(γ_{U}) if we could show that ind(γ_{𝔭}) does not go up under the map s_{P}. Therefore we just need to show that s_{P} cannot increase index of Brauer classes, or, ind(γ_{P}) = ind(s_{P} (γ_{𝔭}))|ind(γ_{𝔭}) .

We compute ind(γ_{𝔭}) first. Since F_{𝔭} is a complete discretely valued field, we have ind(γ_{𝔭}) = ind((γ_{0,0} + (χ_{1}, π))_{M}) · exp(χ_{2} + (π^{r})), where M/κ(𝔭) is the Galois cyclic extension determined by χ_{2} + (π^{r}) ∈ H^{1}(κ(𝔭)) by Jacob and Wadsworth [17, Theorem 5.15]. It is not hard to compute ind((γ_{0,0} + (χ_{1}, π))_{M}): Since M is a finite extension of κ(𝔭), which is a complete discretely valued field, we have that M is a complete discretely valued field as well. Let e be the ramification index of M/κ(𝔭) and M̄ the residue field of M. Then by Serre [26, Exercise XII.3.2], (γ_{0,0} + (χ_{1,}π))_{M} = (γ_{0,0})M̄ +(e·χ_{1}, π′), where π′ is some uniformizer of M. Let L/κ(𝔭) be the field extension determined by e · χ_{1} and L̄ the residue field of L. Then ind((γ_{0,0} + (χ_{1,}π))_{M}) = ind((γ_{0,0})M̄ + (e · χ_{1}, π′)) = ind((γ_{0,0})M̄L̄) · exp(e · χ_{1}).

Now that we have an index formula for Brauer classes over F_{𝔭}, we are ready to show the following Proposition 12. Let T be a complete discrete valuation ring. Let X̂ be a smooth projective T -curve with closed fibre X. Suppose that U is an open affine subset of X and P ∈ X\U is a closed point. Let 𝔭 = (U, P) be the unique branch of U at P and let γ_{P} and γ_{𝔭} be defined as above. Then we have ind(γ_{P})|ind(γ_{𝔭}).

Proof. By Proposition 9 we have that ind(γ_{P})|ind((γ̃_{0,0}+(χ̃_{1}, π))M̃)·exp(χ̃_{2}+(π^{r})), where M̃/F_{P} is the Galois cyclic extension determined by χ̃_{2} +(π^{r}). We claim that exp(χ̃_{2} +(π^{r})) = exp(χ_{2})+(π^{r}): we have that exp(χ̃_{2}+(π^{r})) = lcm(exp(χ̃_{2}), exp((π^{r}))) and exp(χ_{2}+(π^{r})) = lcm(exp(χ_{2}), exp((π^{r}))). Since exp(χ̃_{2}) = exp(χ_{2}), we have proved that exp(χ̃_{2} + (π^{r})) = exp(χ_{2} + (π^{r})). Therefore this proposition will follow if we can show that ind((γ̃_{0,0} + (χ̃_{1}, π))M̃)|ind((γ_{0,0})M̄ + (e · χ_{1}, π′)).

Next we compute

(γ̃_{0,0} + (χ̃_{1}, π))M̃ = (γ̃_{0,0})M̃ + (χ̃_{1}, π)M̃
= (γ̃_{0,0})M̃ + ((χ̃_{1})M̃, π)
= (γ̃_{0,0})M̃ + ((χ̃_{1})M̃, (π′)^{e})
= (γ̃_{0,0})M̃ + (e · (χ̃_{1})M̃, π′)

By Proposition 9 again we immediately see that ind((γ̃_{0,0} + (χ̃_{1}, π))M̃)|ind((γ̃_{0,0})_{M̃L̃}) · exp(e · (χ̃_{1})M̃), where L̃/F_{P} denotes the Galois cyclic extension determined by e · χ̃_{1}. Clearly exp(e · (χ̃_{1})M̃)| exp(e · (χ_{1})), so we will be done if we can show that ind((γ̃_{0,0})_{M̃L̃})|ind((γ_{0,0})M̄L̄), which we will do in the following Lemma 13.

Lemma 13. In line with the notation in 12, we have that ind((γ̃_{0,0})_{M̃L̃})|ind((γ_{0,0})M̄L̄).

Proof. Let M̃′/F_{P} be the Galois cyclic extension determined by χ_{2}. Clearly it suffices to prove that ind((γ̃_{0,0})_{M̃′L̃})|ind((γ_{0,0})M̄L̄) since ind((γ̃_{0,0})_{M̃L̃})|ind((γ̃)_{M̃′L̃}). Let i = ind((γ_{0,0})M̄L̄). By Proposition 10, we have that SB_{i}(γ_{0,0})(M̄L̄) ≠ ϕ, or equivalently, the morphism SB_{i}(γ_{0,0})×_{κ(P)} M̄L̄ has a section Spec(M̄L̄) → SB_{i}(γ_{0,0}) ×_{κ(P)} M̄L̄. By Lemma 3, this section lifts to a section over Spec(R̂_{P}); thus we obtain a M̃′L̃-rational point of SB_{i}(γ̃_{0,0})×R̂_{P} S(note that γ_{0,0} ∈ H^{2}(R̂_{P})), where S is the integral closure of R̂_{P} in M̃′L̃; or equivalently, a M̃′L̃-rational point of SB_{i}(γ̃_{0,0}) ×_{F_{P}} M̃′L̃. Therefore ind((γ̃_{0,0})_{M̃′L̃})|i again by Proposition 10, which proves this lemma.

The following Corollary is immediate:

Corollary 14. The homomorphism s : H^{2}(F̂) → H^{2}(F) preserves index of Brauer classes.

Proof. This is simply Corollary 8 plus Proposition 12.

5. Indecomposable and noncrossed product Division Algebras over Curves over complete Discrete Valuation Rings

Let T be a complete discrete valuation ring. Let X̂ be a smooth projective T -curve with closed fibre X. Let F be the function field of X̂ and F̂ the completion of F with respect to the discrete valuation determined by X. We construct indecomposable division algebras and noncrossed product division algebras over F of prime power index for all primes q where q is different from the characteristic of the residue field of T. Note that the existence of such algebras are already known when residue field of T is a finite filed, cf. Brussel et al. [6]. Our construction here is almost identical to Brussel et al. [6, Section 4], we list it here for the reader’s convenience.

5.1 Indecomposable Division Algebras over F 11

5.1. Indecomposable Division Algebras over F

First we recall the construction of indecomposable division algebras over F̂, this is done in Brussel et al. [6, Proposition 4.2].

Proposition 15. Let T be a complete discrete valuation ring and let X̂ be a smooth projective curve over Spec(T) with closed fibre X. Let F be the function field of X̂ and F̂ the completion of F with respect to the discrete valuation induced by X. Let e, i be integers satisfying l ≤ e ≤ 2e−1. For any prime q ≠ char(k), there exists a Brauer class γ̂ ∈ H^{2}(F̂) satisfying ind(γ̂) = q^{i}, exp(γ̂) = q^{e} and whose underlying division algebra is indecomposable.

Then we lift γ̂ to F by using the splitting map s we defined in section 3, and show that the lift is in fact indecomposable.

Theorem 16. In the notation of Theorem 15. Then there exists an indecomposable division algebra D over F such that ind(D) = q^{i} and exp(D) = q^{e} .

Proof. By Proposition 15, there exists γ̂ ∈ Br(F̂) with ind(γ̂) = q^{i} and exp(γ̂) = q^{e} and whose underlying division algebra is indecomposable. By Corollary 14, γ = s(γ̂) has index q^{i} too. Since s splits the restriction map, we have exp(γ) = q^{e}. We show the division algebra underlying γ is indecomposable.

We proceed by contradiction. Assume γ = β_{1} + β_{2} represents a nontrivial decomposition, then γ̂ = resF̂(β_{1}) + resF̂(β_{2}). Since the index can only go down under restriction, we have that ind(γ̂) = ind(resF̂(β_{1})) · ind(resF̂(β_{2})), which represents a nontrivial decomposition of the division algebra underlying γ̂, a contradiction.

5.2. Noncrossed Products over F

Again we will construct noncrossed product division algebras over F̂ and use the splitting map s to lift it to F and show that the lift represents a noncrossed product division algebra over F.

The construction over F̂ is in line with Brussel [5] where noncrossed products over Q(t) and Q((t)) are constructed. In order to mimic the construction in Brussel [5], we need only note that both Chebotarev density theorem and the Gruwald-Wang theorem hold for global fields which are characteristic p function fields. Then the arguments in Brussel [5] apply directly to yield noncrossed products over K̂(X̂) of index and exponent given below:

The following is Brussel et al. [6, Theorem 4.7].

Theorem 17. Let T be a complete discrete valuation ring with residue field k and let X̂ be a smooth projective curve over Spec(T). Let F be the function field of X̂ and let F̂ be the completion of F with respect to the discrete valuation induced by the closed fibre. For any positive integer a, let ϵ_{a} be a primitive a-th root of unity. Set r and s to be maximum integers such that μ_{q^{r}} ⊂ k(X)^{×} and μ_{q^{s}} ⊂ k(X)(ϵ_{q^{r+1}}). Let n, m be integers such that n ≥ 1, n ≥ m and n, m ∈ r ∪ [s, ∞). Let a, l be integers such that l ≥ n + m + 1 and 0 ≤ a ≤ 1 − n. (See Brussel [5, Page 384-385] for more information regarding these constraints.) Let q ≠ char(k) be a prime number. Then there exists noncrossed product division algebras over F̂ with index q^{l+1} and exponent q^{l}.

Corollary 18. Let R, k, X̂, X, F, F̂, q, a, l be as in Theorem 17. Then there exists noncrossed product division algebras over F of index q^{l+a} and exponent q^{l}.

5.2 Noncrossed Products over F 12

Proof. Let γ̂ be the Brauer class representing a noncrossed product over F̂ of index q^{l+a} and exponent q^{l}. Let D be the division algebra underlying the Brauer class s(γ̂). By Corollary 14, we know that ind(D) = ind(γ̂).

Assume that D is a crossed product with maximal Galois subfield M/F. Then MF̂ splits γ̂, is of degree ind(γ̂) and is Galois. This contradicts the fact that γ̂ is a noncrossed product.

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