LOCAL-GLOBAL PRINCIPLE FOR REDUCED NORMS OVER FUNCTION FIELDS OF p-ADIC CURVES

R. PARIMALA, R. PREETI, AND V. SURESH

Abstract. Let F be the function field of a p-adic curve. Let D be a central simple algebra over F of period n and λ ∈ F^{∗}. We show that if n is coprime to p and D · (λ) = 0 in H^{3}(F, μ_{n}^{⊗2}), then λ is a reduced norm. This leads to a Hasse principle for the SL_{1}(D), namely an element λ ∈ F^{∗} is a reduced norm from D if and only if it is a reduced norm locally at all discrete valuations of F.

1. Introduction

Let K be a p-adic field and F a function field in one-variable over K. Let Ω_{F} be the set of all discrete valuations of F. Let G be a semi-simple simply connected linear algebraic group defined over F. It was conjectured in ([5]) that the Hasse principle holds for principal homogeneous spaces under G over F with respect to Ω_{F}; i.e. if X is a principal homogeneous space under G over F with X(F_{ν}) ≠ ∅ for all ν ∈ Ω_{F}, then X(F) ≠ ∅. If G is SL_{1}(D), where D is a central simple algebra over F of square free index, it follows from the injectivity of the Rost invariant ([19]) and a Hasse principle for H^{3}(F, μ_{n}) due to Kato ([16]), that this conjecture holds. This conjecture has been subsequently settled for classical groups of type B_{n}, C_{n} and D_{n} ([14], [23]). It is also settled for groups of type ^{2}A_{n} with the assumption that n + 1 is square free ([14], [23]).

The main aim of this paper is to prove that the conjecture holds for SL_{1}(D) for any central simple algebra D over F with index coprime to p. In fact we prove the following (11.1)

Theorem 1.1. Let K be a p-adic field and F a function field in one-variable over K. Let D be a central simple algebra over F of index coprime to p and λ ∈ F^{∗}. If D · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), then λ is a reduced norm from D.

This together with Kato’s result on the Hasse principle for H^{3}(F, μ_{n}) gives the following (11.2)

Theorem 1.2. Let K be p-adic field and F the function field of a curve over K. Let Ω_{F} be the set of discrete valuations of F. Let D be a central simple algebra over F of index coprime to p and λ ∈ F^{∗}. If λ is a reduced norm from D ⊗ F_{ν} for all ν ∈ Ω_{F}, then λ is a reduced norm from D.

In fact we may restrict the set of discrete valuations to the set of divisorial discrete valuations of F; namely those discrete valuations of F centered on a regular proper model of F over the ring of integers in K.

Here are the main steps in the proof. We reduce to the case where D is a division algebra of period ℓ^{d} with ℓ a prime not equal to p. Given a central division algebra D over F of period n = ℓ^{d} with ℓ ≠ p and λ ∈ F^{∗} with D · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), we construct a degree ℓ extension L of F and μ ∈ L^{∗} such that N_{L/F} (μ) = λ, (D ⊗ L) · (μ) = 0 and the index of D ⊗ L is strictly smaller than the index of D.

Then, by induction on the index of D, μ is a reduced norm from D ⊗ L and hence N_{L/F} (μ) = λ is a reduced norm from D.

Let 𝒳 be a regular proper 2-dimensional scheme over the ring of integers in K with function field F and X_{0} the reduced special fibre of 𝒳 . By the patching techniques of Harbater-Hartman-Krashen ([9], [10]), construction of such a pair (L, μ) is reduced to the construction of compatible pairs (L_{x}, μ_{x}) over F_{x} for all x ∈ X_{0} (7.5), where for any x ∈ X_{0}, F_{x} is the field of fractions of the completion of the regular local ring at x on 𝒳 . We use local and global class field theory to construct such local pairs (L_{x}, μ_{x}). Thus this method cannot be extended to the more general situation where F is a function field in one variable over a complete discretely valued field with arbitrary residue field.

Here is the brief description of the organization of the paper. In §3, we prove a few technical results concerning central simple algebras and reduced norms over global fields. These results are key to the later patching construction of the fields L_{x} and μ_{x} ∈ L_{x} with required properties.

In §4 we prove the following local variant of (1.1)

Theorem 1.3. Let F be a complete discrete valued field with residue field κ. Suppose that κ is a local field or a global field. Let D be a central simple algebra over F of period n. Suppose that n is coprime to char(κ). Let α ∈ H^{2}(F, μ_{n}) be the class of D and λ ∈ F^{∗}. If α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), then λ is a reduced norm from D.

Let A be a complete regular local ring of dimension 2 with residue field κ finite, field of fractions F and maximal ideal m = (π, δ). Let ℓ be a prime not equal to char(κ). Let D be a central simple algebra over F of index ℓ^{n} with n ≥ 1 and α the class of D in H^{2}(F, μ_{ℓ^{n}}). Suppose that D is unramified on A except possibly at π and δ. In §5, we analyze the structure of D. We prove that index of D is equal to the period of D. A similar analysis is done by Saltman ([25]) with the additional assumption that F contains all the primitive ℓ^{n}-roots of unity, where ℓ^{n} is the index of D. Let λ ∈ F^{∗}. Suppose that λ = uπ^{r}δ^{t} for some unit u ∈ A and r, s ∈ ℤ and α·(λ) = 0 ∈ H^{3}(F, μ_{ℓ^{n}}^{⊗2}). In §6, we construct possible pairs (L, μ) with L/F of degree ℓ, μ ∈ L such that N_{L/F} (μ) = λ, ind(D ⊗ L) < ind(D) and α · (μ) = 0 ∈ H^{3}(L, μ_{ℓ^{n}}^{⊗2}).

Let K be a p-adic field and F a function field of a curve over K. Let ℓ be a prime not equal to p, D a central division algebra over F of index ℓ^{n} and α the class of D in H^{2}(F, μ_{ℓ^{n}}). Let λ ∈ F^{∗} with α · (λ) = 0 ∈ H^{3}(F, μ_{ℓ^{n}}^{⊗2}). Let 𝒳 be a normal proper model of F over the ring of integers in K and X_{0} its reduced special fibre. In §7, we reduce the construction of (L, μ) to the construction of local (L_{x}, μ_{x}) for all x ∈ X_{0} with some compatible conditions along the “branches”.

Further assume that 𝒳 is regular and ram_{𝒳} (α)∪ supp_{𝒳} (λ) ∪ X_{0} is a union of regular curves with normal crossings. We group the components of X_{0} into 8 types depending on the valuation of λ, index of D and the ramification type of D along those components. We call some nodal points of X_{0} as special points depending on the type of components passing through the point. We also say that two components of X_{0} are type 2 connected if there is a sequence of type 2 curves connecting these two components. We prove that there is a regular proper model of F with no special points and no type 2 connection between certain types of component (8.6).

Starting with a model constructed in (8.6), in §9, we construct (L_{P} , μ_{P}) for all nodal points of X_{0} (9.8) with the required properties. In §10, using the class field results of §3, we construct (L_{η}, μ_{η}) for each of the components η of X_{0} which are compatible with (L_{P} , μ_{P}) when P is in the component η.

Finally in §11, we prove the main results by piecing together all the constructions of §7, §9 and §10.

2. Preliminaries

In this section we recall a few definitions and facts about Brauer groups, Galois cohomology groups, residue homomorphisms and unramified Galois cohomology groups. We refer the reader to ([4]) and ([8]).

Let K be a field and n ≥ 1. Let _{n}Br(K) be the n-torsion subgroup of the Brauer group Br(K). Assume that n is coprime to the characteristic of K. Let μ_{n} be the group of n^{th} roots of unity. For d ≥ 1 and m ≥ 0, let H^{d}(K, μ_{n}^{⊗m}) denote the d^{th} Galois cohomology group of K with values in μ_{n}^{⊗m}. We have H^{1}(K, μ_{n}) ≃ K^{∗}/K^{∗n} and H^{2}(K, μ_{n}) ≃ _{n}Br(K). For a ∈ K^{∗}, let (a)_{n} ∈ H^{1}(K, μ_{n}) denote the image of the class of a in K^{∗}/K^{∗n}. When there is no ambiguity of n, we drop n and denote (a)_{n} by (a). If K is a product of finitely many fields K_{i}, we denote ∏ H^{d}(K_{i}, μ_{n}^{⊗m}) by H^{d}(K, μ_{n}^{⊗m}).

Every element of H^{1}(K, ℤ/nℤ) is represented by a pair (E, σ), where E/F is a cyclic extension of degree dividing n and σ a generator of Gal(E/F). Let r ≥ 1. Then (E, σ)^{r} ∈ H^{1}(K, ℤ/nℤ) is represented by the pair (E′, σ′) where E′ is the fixed field of the subgroup of Gal(E/F) generated by σ^{n/d}, where d = gcd(n, r) and σ′ = σ^{r}. In particular if r is coprime to n, then (E, σ)^{r} = (E, σ^{r}). Let (E, σ) ∈ H^{1}(K, ℤ/nℤ) and λ ∈ K^{∗}. Let (E, σ, λ) = (E/F, σ, λ) denote the cyclic algebra over K

(E, σ, λ) = E ⊕ Ly ⊕ ⋯ ⊕ Ey^{n−1}

with y^{n} = λ and ya = σ(a)y. The cyclic algebra (E, σ, λ) is a central simple algebra and its index is the order of λ in K^{∗}/N_{E/K}(E^{∗}) ([1, Theorem 18, p. 98]). The pair (E, σ) represents an element in H^{1}(K, ℤ/nℤ) and the element (E, σ)·(λ) ∈ H^{2}(K, μ_{n}) is represented by the central simple algebra (E, σ, λ). In particular (E, σ, λ) ⊗ E is a matrix algebra and hence ind(E, σ, λ) ≤ [E : F].

For λ, μ ∈ K^{∗} we have ([1, p. 97])

(E, σ, λ) + (E, σ, μ) = (E, σ, λμ) ∈ H^{2}(K, μ_{n}).

In particular (E, σ, λ^{−1}) = −(E, σ, λ).

Let (E, σ, λ) be a cyclic algebra over a field K and L/K be a field extension. Since E/K is separable, E ⊗_{K} L is a product of field extensions E_{i}, 1 ≤ i ≤ t, of L with E_{i} and E_{j} isomorphic over L and E_{i}/L is cyclic with Galois group a subgroup of the Galois group of E/K. Then (E, σ, λ) ⊗ L is Brauer equivalent to (E_{i}, σ_{i}, λ) for any i, with a suitable σ_{i}. In particular if L is a finite extension of K and EL is the composite of E and L in an algebraic closure of K, then EL/L is cyclic with Galois group isomorphic to a subgroup of the Galois group of E/K and (E, σ, λ) ⊗ L is Brauer equivalent to (EL, σ′, λ) for a suitable σ′.

Lemma 2.1. Let E/F be a cyclic extension of degree n, σ a generator of Gal(E/F) and λ ∈ F^{∗}. Let m be a factor of n and d = n/m. Let M/F be the subextension of E/F with [M : F] = m. Then (E/F, σ, λ) ⊗ F (\sqrt[d]{λ}) = (M(\sqrt[d]{λ})/F (\sqrt[d]{λ}), σ ⊗ 1, \sqrt[d]{λ}).

Proof. We have (E, σ)^{d} = (M, σ) ∈ H^{1}(F, ℤ/nℤ) and hence

(E, σ, λ) ⊗ F (\sqrt[d]{λ}) = (E(\sqrt[d]{λ})/F (\sqrt[d]{λ}), σ ⊗ 1, λ)
= (E(\sqrt[d]{λ})/F (\sqrt[d]{λ}), σ ⊗ 1, (\sqrt[d]{λ})^{d})
= (E(\sqrt[d]{λ})/F (\sqrt[d]{λ}), σ ⊗ 1)^{d} · (\sqrt[d]{λ})
= (M(\sqrt[d]{λ})/F (\sqrt[d]{λ}), σ ⊗ 1, \sqrt[d]{λ}).
□

Let K be a field with a discrete valuation ν, residue field κ and valuation ring R. Suppose that n is coprime to the characteristic of κ. For any d ≥ 1, we have the residue map ∂_{K} : H^{d}(K, μ_{n}^{⊗i}) → H^{d−1}(κ, μ_{n}^{⊗i−1}). We also denote ∂_{K} by ∂. An element α in H^{d}(K, μ_{n}^{⊗i}) is called unramified at ν or R if ∂(α) = 0. The subgroup of all unramified elements is denoted by H_{nr}^{d}(K/R, μ_{n}^{⊗i}) or simply H_{nr}^{d}(K, μ_{n}^{⊗i}). Suppose that K is complete with respect to ν. Then we have an isomorphism H^{d}(κ, μ_{n}^{⊗i}) →^{∼}

H_{nr}^{d}(K, μ_{n}^{⊗i}) and the composition H^{d}(κ, μ_{n}^{⊗i}) →^{∼} H_{nr}^{d}(K, μ_{n}^{⊗i}) ↪ H^{d}(K, μ_{n}^{⊗i}) is de-

noted by ι_{κ} or simply ι.

Let K be a complete discretely valued field with residue field κ, ν the discrete valuation on K and π ∈ K^{∗} a parameter. Suppose that n is coprime to the characteristic of κ. Let ∂ : H^{2}(K, μ_{n}) → H^{1}(κ, ℤ/nℤ) be the residue homomorphism. Let E/K be a cyclic unramified extension of degree n with residue field E_{0} and σ a generator of Gal(E/K) with σ_{0} ∈ Gal(E_{0}/κ) induced by σ. Then (E, σ, π) is a division algebra over K of degree n. For any λ ∈ K^{∗}, we have

∂(E, σ, λ) = (E_{0}, σ_{0})^{ν(λ)}.

For λ, μ ∈ K^{∗}, we have

∂((E, σ, λ) · (μ)) = (E_{0}, σ_{0}) · ((−1)^{ν(λ)ν(μ)}θ),

where θ is the image of \frac{λ^{ν(μ)}}{μ^{ν(λ)}} in the residue field.

Suppose E_{0} is a cyclic extension of κ of degree n. Then there is a unique unramified cyclic extension E of K of degree n with residue field E_{0}. Let σ_{0} be a generator of Gal(E_{0}/κ) and σ ∈ Gal(E/K) be the lift of σ_{0}. Then σ is a generator of Gal(E/K). We call the pair (E, σ) the lift of (E_{0}, σ_{0}).

Let X be an integral regular scheme with function field F. For every point x of X, let 𝒪_{X,x} be the regular local ring at x and κ(x) the residue field at x. Let 𝒪̂_{X,x} be the completion of 𝒪_{X,x} at its maximal ideal m_{x} and F_{x} the field of fractions of 𝒪̂_{X,x} . Then every codimension one point x of X gives a discrete valuation ν_{x} on F. Let n ≥ 1 be an integer which is a unit on X. For any d ≥ 1, the residue homomorphism H^{d}(F, μ_{n}^{⊗j}) → H^{d−1}(κ(x), μ_{n}^{⊗(j−1)}) at the discrete valuation ν_{x} is denoted by ∂_{x}. An element α ∈ H^{d}(F, μ_{n}^{⊗m}) is said to be ramified at x if ∂_{x}(α) ≠ 0 and unramified at x if ∂_{x}(α) = 0. If X = Spec(A) and x a point of X given by (π), π s prime element, we also denote F_{x} by F_{π} and κ(x) by κ(π).

Lemma 2.2. Let K be a complete discretely valued field and ℓ a prime not equal to the characteristic of the residue field of K. Suppose that K contains a primitive ℓ^{th} root of unity. Let L/K be a cyclic field extension or the split extension of degree ℓ. Let μ ∈ L and λ = N_{L/K}(μ) ∈ K. Then there exists θ ∈ L with N_{L/K}(θ) = 1 such that L = K(μθ) and θ is sufficiently close to 1.

Proof. Since [L : K] is a prime, if μ ∉ K, then L = K(μ). In this case θ = 1 has the required properties. Suppose that μ ∈ K. If L = ∏ K, let θ_{0} ∈ K^{∗} \ {±1} sufficiently close to 1 and θ = (θ_{0}, θ_{0}^{−1}, 1, ⋯, 1). Suppose L is a field. Let σ be a generator of Gal(L/K). Since L/K is cyclic, we have L = K(\sqrt[ℓ]{a}) for some a ∈ K^{∗}. For any sufficiently large n, θ = (1 + π^{n}\sqrt[ℓ]{a})^{−1}σ(1 + π^{n}\sqrt[ℓ]{a}) ∈ L has the required properties. □

Lemma 2.3. Let K be a field and E/K be a finite extension of degree coprime to char(K). Let L/K be a sub-extension of E/K and e = [E : L]. Suppose L/K is Galois and E = L(\sqrt[e]{π}) for some π ∈ L^{∗}. Then E/K is Galois if and only if E contains a primitive e^{th} root of unity and for every τ ∈ Gal(L/K), τ(π) ∈ E^{∗e}.

Proof. Suppose that E/K is Galois. Let f(X) = X^{e} − π ∈ L[X]. Since [E : L] = e and E = L(\sqrt[e]{π}), f(X) is irreducible in L[X]. Since f(X) has one root in E and E/L is Galois, f(X) has all the roots in E. Hence E contains a primitive e^{th} root of unity. Let τ ∈ Gal(L/K). Then τ can be extended to an automorphism τ̃ of E. We have τ(π) = τ̃(π) = (τ̃(\sqrt[e]{π}))^{e} ∈ E^{∗e}.

Conversely, suppose that E contains a primitive e^{th} root of unity and τ(π) ∈ E^{∗e} for every τ ∈ Gal(L/K). Let

g(X) = ∏_{τ ∈ Gal(L/K)}(X^{e} − τ(π)).

Then g(X) ∈ K[X] and g(X) splits completely in E. Since e is coprime to char(K), the splitting field E_{0} of g(X) over K is Galois. Since L/K is Galois and E is the composite of L and E_{0}, E/K is Galois. □

Lemma 2.4. Let F be a complete discretely valued field with residue field κ and π ∈ F a parameter. Let e be a natural number coprime to the characteristic of κ. If L/F is a totally ramified extension of degree e, then L = F (\sqrt[e]{vπ}) for some v ∈ F which is a unit in the valuation ring of F. Further if e is a power of a prime ℓ and θ ∈ F^{∗} \ F^{∗ℓ} is a norm form L, then L = F (\sqrt[e]{θ}).

Proof. Since F is a complete discretely valued field, there is a unique extension of the valuation ν on F to a valuation ν_{L} on L. Since L/F is totally ramified extension of degree e and e is coprime to char(κ), the residue field of L is κ and ν_{L}(π) = e. Let π_{L} ∈ L with ν_{L}(π_{L}) = 1. Then π = wπ_{L}^{e} for some w ∈ L with ν_{L}(w) = 0. Since the residue field of L is same as the residue field of F, there exists w_{1} ∈ F with ν(w_{1}) = 0 and the image of w_{1} is same as the image of w in the residue field κ. Since L is complete and e is coprime to char(κ), by Hensel’s Lemma, there exists u ∈ L such that w = w_{1}u^{e}. Thus π = wπ_{L}^{e} = w_{1}u^{e}π_{L}^{e} = w_{1}(uπ_{L})^{e}. In particular w_{1}^{−1}π ∈ L^{∗e} and hence L = F (\sqrt[e]{vπ}) with v = w_{1}^{−1}.

Let θ ∈ F^{∗} \ F^{∗e}. Suppose that θ is a norm from L. Let μ ∈ L with N_{L/F} (μ) = θ. Since L = F (\sqrt[e]{vπ}) with v ∈ F a unit in the valuation ring of F and π ∈ F a parameter, \sqrt[e]{vπ} ∈ L is a parameter at the valuation of L. Write μ = w_{0}(\sqrt[e]{vπ})^{s} for some w_{0} ∈ L a unit at the valuation of L and s ∈ ℤ. As above, we have w_{0} = v_{1}u_{1}^{e} for some v_{1} ∈ K and u_{1} ∈ L. Since v_{1} ∈ F, we have

θ = N_{L/F} (μ) = N_{L/F} (w_{0}(\sqrt[e]{vπ})^{s}) = N_{L/F} (v_{1}u_{1}^{e}(\sqrt[e]{vπ})^{s}) = v_{1}^{e}N_{L/F} (u_{1})^{e}(vπ)^{s}.

Since e is a power of a prime ℓ and θ ∉ F^{∗ℓ}, s is coprime to ℓ and hence L = F (\sqrt[e]{θ}). □

Lemma 2.5. Let k be a local field and ℓ a prime not equal to the characteristic of the residue field of k. Let L_{0}/k be a an extension of degree ℓ and θ_{0} ∈ k^{∗}. If θ_{0} ∉ k^{∗ℓ} and θ_{0} is a norm from L_{0}, then L_{0} = k(\sqrt[ℓ]{θ_{0}}).

Proof. Suppose that L_{0}/k is ramified. Since θ_{0} ∉ k^{∗ℓ}, by (2.4), L_{0} = k(\sqrt[ℓ]{θ_{0}}).

Suppose that L_{0}/k is unramified. Let π be a parameter in k and write θ_{0} = uπ^{r} with u a unit in the valuation ring of k. Since θ_{0} is a norm from L_{0}, ℓ divides r. Since θ_{0} not an ℓ^{th} power in k, u is not an ℓ^{th} power in k and k(\sqrt[ℓ]{θ_{0}}) = k(\sqrt[ℓ]{u}) is an unramified extension of k of degree ℓ. Since k is a local field, there is only one unramified field extension of k of degree ℓ and hence L_{0} = k(\sqrt[ℓ]{u}) = k(\sqrt[ℓ]{θ_{0}}). □

Lemma 2.6. Suppose F is a complete discretely valued field with residue feld κ a local field. Let ℓ be prime not equal to char(κ). Let L/F be a degree ℓ field extension with θ a norm from L. If θ ∉ F^{∗ℓ}, then L ≃ F (\sqrt[ℓ]{θ}).

Proof. If L/F is a ramified extension, then by (2.4), L ≃ F (\sqrt[ℓ]{θ}). Suppose that L/F is an unramified extension. Let L_{0} be the residue field of L. Then L_{0}/κ is a field extension of degree ℓ and the image \overline{θ} of θ in κ is a norm from L_{0}. Since θ ∉ F^{∗ℓ}, \overline{θ} is not an ℓ^{th} power in κ. Since κ is a local field, L_{0} ≃ κ(\sqrt[ℓ]{\overline{θ}}) (2.5) and hence L ≃ F (\sqrt[ℓ]{θ}). □

Lemma 2.7. Let F be a complete discretely valued field with residue field k a global field. Let L/F be an unramified cyclic extension of degree coprime to char(k) and L_{0} the residue field of L. Let θ ∈ F be a unit in the valuation ring of F and \overline{θ} be the image of θ in k. Suppose that θ is a norm from L. If μ_{0} ∈ L_{0} with N_{L_{0}/k}(μ_{0}) = \overline{θ}, then there exists μ ∈ L such that N_{L/F} (μ) = θ and the image of μ in L_{0} is μ_{0}.

Proof. Let σ be a generator of the Galois group of L/F and σ_{0} be the induced automorphism of L_{0}/k. Since θ ∈ F is a norm from L, there exists μ′ ∈ L with N_{L/F} (μ′) = θ. Since θ is a unit at the discrete valuation of F, μ′ ∈ L is a unit at the discrete valuation of L. Let \overline{μ}′ be the image of μ′ in L_{0}. Then N_{L_{0}/k}(\overline{μ′}) = \overline{θ} and hence \overline{μ}′μ_{0}^{−1} ∈ L_{0} is a norm one element. Thus there exist a ∈ L_{0} such that \overline{μ}′μ_{0}^{−1} = a^{−1}σ_{0}(a). Let b ∈ L be a lift of a and μ = μ′bσ(b)^{−1}. Then N_{L/F} (μ) = N_{L/F} (μ′) = θ and the image of μ in L_{0} is μ_{0}. □

For L = ∏_{1}^{ℓ} F, let σ be the automorphism of L given by σ(a_{1}, ⋯, a_{ℓ}) = (a_{2}, ⋯, a_{ℓ}, a_{1}). Then any σ^{i}, 1 ≤ i ≤ ℓ − 1 is called a generator of Gal(L/F).

Lemma 2.8. Let F be a field and ℓ a prime not equal to the characteristic of F. Let L be a cyclic extension of F or the split extension of degree ℓ and σ a generator of the Galois group of L/F. Suppose that there exists an integer t ≥ 1 such that F does not contain a primitive ℓ^{t^{th}} root of unity. Let μ ∈ L with N_{L/F} (μ) = 1 and m ≥ t. If μ ∈ L^{∗ℓ^{2m}}, then there exists b ∈ L^{∗} such that μ = b^{−ℓ^{m}}σ(b^{ℓ^{m}}).

Proof. Suppose L = ∏ F and μ ∈ L^{∗ℓ^{s}} for some s ≥ 1 with N_{L/F} (μ) = 1. Then μ = (θ_{1}^{s}, ⋯, θ_{ℓ}^{s}) ∈ L with θ_{1}^{s} ⋯ θ_{ℓ}^{s} = 1. Let b = (1, θ_{1}, ⋯, θ_{ℓ−1}) ∈ L^{∗}. Then μ = b^{−s}σ(b^{s}).

Suppose L/F is a cyclic field extension. Write μ = μ_{0}^{ℓ^{2m}} for some μ_{0} ∈ L. Let μ_{1} = μ_{0}^{ℓ^{m}}. Then μ = μ_{1}^{ℓ^{m}}. Let θ_{0} = N_{L/F} (μ_{0}) and θ_{1} = N_{L/F} (μ_{1}). Then θ_{1} = θ_{0}^{ℓ^{m}}. Since N_{L/F} (μ) = 1, we have θ_{1}^{ℓ^{m}} = N_{L/F} (μ_{1}^{ℓ^{m}}) = 1. If θ_{1} ≠ 1, then F contains a primitive ℓ^{m^{th}} root of unity. Since m ≥ t and F has no primitive ℓ^{t^{th}} root of unity, θ_{1} = 1. Hence N_{L/F} (μ_{1}) = 1 and by Hilbert 90, μ_{1} = b^{−1}σ(b) for some b ∈ L. Thus μ = μ_{1}^{ℓ^{m}} = b^{−ℓ^{m}}σ(b^{ℓ^{m}}). □

3. Global fields

In this a section we prove a few technical results concerning Brauer group of global fields and reduced norms. We begin with the following.

Lemma 3.1. Let k be a global field, ℓ a prime not equal char(k), n, d ≥ 2 and r ≥ 1 be integers. Let E_{0} be a cyclic extension of k, σ_{0} a generator of the Galois group of E_{0}/k and θ_{0} ∈ k^{∗}. Let β ∈ H^{2}(k, μ_{ℓ^{n}}) be such that rℓβ = (E_{0}, σ_{0}, θ_{0}) ∈ H^{2}(k, μ_{ℓ^{n}}). Let S be a finite set of places of k containing all the places of κ with β ⊗ k_{ν} ≠ 0. Suppose for each ν ∈ S, there is a field extension L_{ν} of k_{ν} of degree ℓ or L_{ν} is the split extension of k_{ν} of degree ℓ and μ_{ν} ∈ L_{ν}^{∗} such that

1) N_{L_{ν}/k_{ν}}(μ_{ν}) = θ_{0}

2) rβ ⊗ L_{ν} = (E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν})

3) ind(β ⊗ E_{0} ⊗ L_{ν}) < d. Then there exists a field extension L_{0}/k of degree ℓ and μ_{0} ∈ L_{0} such that

1) N_{L_{0}/k}(μ_{0}) = θ_{0}

2) rβ ⊗ L_{0} = (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0})

3) ind(β ⊗ E_{0} ⊗ L_{0}) < d

4) L_{0} ⊗ k_{ν} ≃ L_{ν} for all ν ∈ S.

5) μ_{0} is close to μ_{ν} for all ν ∈ S.

Proof. Let Ω_{k} be the set of all places of k and

S′ = S ∪ {ν ∈ Ω_{k} | θ_{0} is not a unit at ν or E_{0}/k is ramified at ν}

Let ν ∈ S′\S. Then β⊗k_{ν} = 0. Let L_{ν} be a field extension of k_{ν} of degree ℓ such that θ_{0} ∈ N(L_{ν}^{∗}). Let μ_{ν} ∈ L_{ν} with N_{L_{ν}/k_{ν}}(μ_{ν}) = θ_{0}. Since β ⊗k_{ν} = 0, ind(β ⊗E_{0} ⊗L_{ν}) = 1 < d. Since the corestriction map cor : H^{2}(L_{ν}, μ_{ℓ^{n}}) → H^{2}(k_{ν}, μ_{ℓ^{n}}) is injective (cf. [17, Theorem 10, p. 237]) and cor(E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}) = (E_{0} ⊗ k_{ν}, σ_{0} ⊗ 1, θ_{0}) = rℓβ ⊗ k_{ν} = 0, (E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}) = 0 = rβ ⊗ L_{ν}. Thus, if necessary, by enlarging S, we assume that S contains all those places ν of k with either θ_{0} is not a unit at ν or E_{0}/k is ramified at ν and that there is at least one ν ∈ S such that L_{ν} is a field extension of k_{ν} of degree ℓ.

Let ν ∈ S. By (2.2), there exists θ_{ν} ∈ L_{ν} such that N_{L_{ν}/k_{ν}}(θ_{ν}) = 1, L_{ν} = k_{ν}(θ_{ν}μ_{ν}) and θ_{ν} is sufficiently close to 1. In particular θ_{ν} ∈ L_{ν}^{ℓ^{n}} and hence rβ ⊗ L_{ν} = (E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}) = (E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}θ_{ν}). Thus, replacing μ_{ν} by μ_{ν}θ_{ν}, we assume that L_{ν} = k_{ν}(μ_{ν}). Let f_{ν}(X) = X^{ℓ} + b_{ℓ−1,ν}X^{ℓ−1} + ⋯ + b_{1,ν}X + (−1)^{ℓ}θ_{0} ∈ k_{ν}[X] be the minimal polynomial of μ_{ν} over k_{ν}.

By Chebotarev density theorem ([7, Theorem 6.3.1]), there exists ν_{0} ∈ Ω_{k} \S such that E_{0} ⊗ k_{ν_{0}} is the split extension of k_{ν_{0}}. By the strong approximation theorem ([3, p. 67]), choose b_{j} ∈ k, 0 ≤ j ≤ ℓ − 1 such that each b_{j} is sufficiently close enough to b_{j,ν} for all ν ∈ S and each b_{j} is an integer at all ν ∉ S ∪ {ν_{0}}. Let L_{0} = k[X]/(X^{ℓ} + b_{ℓ−1}X^{ℓ−1} + ⋯ + b_{1}X + (−1)^{ℓ}θ_{0}) and μ_{0} ∈ L_{0} be the image of X. We now show that L_{0} and μ_{0} have the required properties.

Since each b_{j} is sufficiently close enough to b_{j,ν} at each ν ∈ S, it follows from Krasner’s lemma that L_{0} ⊗ k_{ν} ≃ L_{ν} and the image of μ_{0} ⊗ 1 in L_{ν} is close to μ_{ν} for all ν ∈ S (cf. [26, Ch. II, §2]). Since L_{ν} is a field extension of k_{ν} of degree ℓ for at least one ν ∈ S, L_{0} is a field extension of degree ℓ over k. Since X^{ℓ}+b_{ℓ−1}X^{ℓ−1}+⋯+(−1)^{ℓ}θ_{0} is the minimal polynomial of μ_{0}, we have N(μ_{0}) = θ_{0}.

To show that ind(β ⊗ E_{0} ⊗ L_{0}) < d and rβ = (E_{0}, σ_{0}, μ_{0}) ∈ H^{2}(L_{0}, μ_{ℓ^{n}}), by Hasse-Brauer-Noether theorem (cf. [3, p. 187]), it is enough to show that for every place w of L_{0}, ind(β ⊗ E_{0} ⊗ L_{w}) < d and rβ ⊗ L_{w} = (E_{0}, σ_{0}, μ_{0}) ⊗ L_{w} ∈ H^{2}(L_{w}, μ_{ℓ^{n}}).

Let w be a place of L_{0} and ν a place of k lying below w. Suppose that ν ∈ S. Then L_{0} ⊗ k_{ν} ≃ L_{ν}. Suppose L_{ν} = ∏ k_{ν} is the split extension. Then L_{w} ≃ k_{ν}. By the assumption on L_{ν}, we have ind(β ⊗ E_{0} ⊗ k_{ν}) < d. Since μ_{ν} is close to μ_{0}, we have rβ ⊗ L_{ν} = (E_{0} ⊗ L_{ν}, σ_{0}, μ_{ν}) = (E_{0} ⊗ L ⊗ k_{ν}, σ_{0}, μ_{0}).

Suppose that L_{ν} is a field extension of k_{ν} of degree ℓ. Then L_{w} ≃ L_{0}⊗k_{ν} ≃ L_{ν} and by the assumption on L_{ν}, we have rβ⊗L_{ν} = (E_{0}, σ_{0}, μ_{ν})⊗L_{ν} and ind(β⊗E_{0}⊗L_{ν}) < d. Since μ_{0} is close to μ_{ν}, we have rβ ⊗ L_{ν} = (E_{0} ⊗ L_{ν}, σ_{0}, μ_{ν}) = (E_{0} ⊗ L ⊗ k_{ν}, σ_{0}, μ_{0}).

Suppose that ν ∉ S and ν ≠ ν_{0}. Then θ_{0} is a unit at ν, E_{0}/k is unramified at ν and β ⊗ k_{ν} = 0. Since each b_{j} is an integer at ν and μ_{0} is a root of the polynomial X^{ℓ} + b_{ℓ−1}X^{ℓ−1} + ⋯ + b_{1}X + (−1)^{ℓ}θ_{0}, μ_{0} is an integer at w. Since θ_{0} is a unit at ν, μ_{0} is a unit at w. In particular (E_{0} ⊗ L_{w}, σ_{0}, μ_{0}) = 0 = rβ ⊗ L_{w}. If ν = ν_{0}, then by the choice of ν_{0}, β ⊗ k_{ν} = 0, E_{0} ⊗ k_{ν} is the split extension of k_{ν} and hence (E_{0}, σ_{0}, μ_{0}) ⊗ L_{w} = 0 = rβ ⊗ L_{w}. □

Corollary 3.2. Let k be a global field, ℓ a prime not equal char(k), n and r ≥ 1 be integers. Let θ_{0} ∈ k^{∗}, r ≥ 1 and β ∈ H^{2}(k, μ_{ℓ^{n}}). Suppose that rℓβ = 0 ∈ H^{2}(k, μ_{ℓ^{n}}) and β ≠ 0. Then there exists a field extension L_{0}/k of degree ℓ and μ_{0} ∈ L_{0} such that N_{L_{0}/k}(μ_{0}) = θ_{0}, rβ ⊗ L_{0} = 0 and ind(β ⊗ L_{0}) < ind(β).

Proof. Let S be a finite set of places of k containing all the places of k with β ≠ 0. Let ν ∈ S. If θ_{0} ∉ k_{ν}^{∗ℓ}, then, let L_{ν} = k_{ν}(\sqrt[ℓ]{θ_{0}}) and μ_{v} = \sqrt[ℓ]{θ_{0}} ∈ L_{ν}. If θ_{0} ∈ k_{ν}^{∗ℓ}, then, let L_{ν}/k_{ν} be any field extension of degree ℓ and μ_{ν} = \sqrt[ℓ]{θ_{0}} ∈ k_{ν} ⊂ L_{ν}. In both the cases, we have N_{L_{ν}/k_{ν}}(μ_{v}) = θ_{0}. Since L_{ν}/k_{ν} is a degree ℓ field extension, ℓ divides ind(β) and k_{ν} is a local field, ind(β ⊗ L_{ν}) < ind(β) ( [3, p. 131]). Since rℓβ = 0 and L_{ν}/k_{ν} is a field extension of degree ℓ, rβ ⊗L_{ν} = 0. Let E_{0} = k. Then, by (3.1), there exist a field extension L_{0}/k of degree ℓ and μ ∈ L_{0} with required properties. □

Lemma 3.3. Let k be a global field and ℓ a prime not equal to char(k). Let E_{0}/k be a cyclic extension of degree a power of ℓ and σ_{0} a generator of Gal(E_{0}/k). Let n ≥ 1, θ_{0} ∈ k^{∗} and β ∈ H^{2}(k, μ_{ℓ^{n}}) be such that rℓβ = (E_{0}, σ_{0}, θ_{0}) for some r ≥ 1. Suppose that rβ⊗E_{0} ≠ 0. If ν is a place of k such that \sqrt[ℓ]{θ_{0}} ∉ k_{ν}, then ind(β⊗E_{0}⊗k_{ν}(\sqrt[ℓ]{θ_{0}})) < ind(β ⊗ E_{0}).

Proof. Write rℓ = mℓ^{d} with m coprime to ℓ. Then d ≥ 1. Since mℓ^{d}β = rℓβ = (E_{0}, σ_{0}, θ_{0}), we have mℓ^{d}β ⊗ E_{0} = 0. Since m is coprime to ℓ and the period of β is a power of ℓ, it follows that ℓ^{d}β ⊗ E_{0} = 0. Since rβ ⊗ E_{0} ≠ 0, ℓ^{d−1}β ⊗ E_{0} ≠ 0 and per(β ⊗ E_{0}) = ℓ^{d}.

Let ν be a place of k. Suppose that \sqrt[ℓ]{θ_{0}} ∉ E_{0} ⊗ k_{ν}. Then [E_{0} ⊗ k_{ν}(\sqrt[ℓ]{θ_{0}}) : E_{0} ⊗ k_{ν}] = ℓ and hence ind(β ⊗ E_{0} ⊗ k_{ν}(\sqrt[ℓ]{θ_{0}})) < ind(β ⊗ E_{0}) ([3, p. 131]). Suppose that \sqrt[ℓ]{θ_{0}} ∈ E_{0}⊗k_{ν}. Then E_{0}⊗k_{ν}(\sqrt[ℓ]{θ_{0}}) = E_{0}⊗k_{ν}. Write E_{0}⊗k_{ν} = ∏ E_{i} with each E_{i} a cyclic field extension of k_{ν}. Since E_{0}/k is a Galois extension, E_{i} ≃ E_{j} for all i and j and mℓ^{d}β ⊗k_{ν} = (E_{0}, σ_{0}, θ_{0})⊗k_{ν} = (E_{i}, σ_{i}, θ_{0}) for all i, for suitable generators σ_{i} of Gal(E_{i}/k_{ν}). Since \sqrt[ℓ]{θ_{0}} ∈ E_{0}⊗k_{ν}, \sqrt[ℓ]{θ_{0}} ∈ E_{i} for all i and hence θ_{0}^{[E_{i}:κ_{ν}]/ℓ} ∈ N_{E_{i}/k_{ν}}(E_{i}^{∗}). Since the period of (E_{i}, σ_{i}, θ_{0}) is equal to the order of the class of θ_{0} in the group k_{ν}^{∗}/N_{E_{i}/k_{ν}}(E_{i}^{∗}) ([1, p. 75]), per(E_{i}, σ_{i}, θ_{0}) ≤ [E_{i} : k_{ν}]/ℓ < [E_{i} : k_{ν}].

Suppose that per(β ⊗ k_{ν}) ≤ [E_{i} : k_{ν}]. Since k_{ν} is a local field, per(β ⊗ E_{i}) = 1. Thus per(β ⊗ E_{0} ⊗ k_{ν}) = per(β ⊗ E_{i}) = 1 < ℓ^{d} = per(β ⊗ E_{0}).

Suppose that per(β ⊗ k_{ν}) > [E_{i} : k_{ν}]. Since mℓ^{d}β ⊗ k_{ν} = (E_{i}, σ_{i}, θ_{0}) and m is coprime to ℓ, we have per(β ⊗ k_{ν}) ≤ ℓ^{d}per(E_{i}, σ_{i}, θ_{0}). Since k_{ν} is a local-field,

per(β ⊗E_{0}⊗k_{ν}) = per(β ⊗E_{i}) = \frac{per(β ⊗ k_{ν})}{[E_{i} : k_{ν}]} ≤ \frac{ℓ^{d}per(E_{i}, σ_{i}, θ_{0})}{[E_{i} : k_{ν}]} < ℓ^{d} = per(β ⊗E_{0}).

Since k_{ν} is a local field, period equals index and hence the lemma follows. □

Proposition 3.4. Let k be a global field and ℓ a prime not equal to char(k). Let E_{0}/k be a cyclic extension of degree a power of ℓ and σ_{0} a generator of Gal(E_{0}/k). Let θ_{0} ∈ k^{∗} and β ∈ H^{2}(k, μ_{ℓ^{n}}) be such that rℓβ = (E_{0}, σ_{0}, θ_{0}) for some r ≥ 1. Suppose that rβ ⊗ E_{0} ≠ 0. Then there exist a field extension L_{0}/κ of degree ℓ and μ_{0} ∈ L_{0} such that

1) N_{L_{0}/k}(μ_{0}) = θ_{0}

2) ind(β ⊗ E_{0} ⊗ L_{0}) < ind(β ⊗ E_{0})

2) rβ ⊗ L_{0} = (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0}).

Proof. Let S be the finite set of places of k consisting of all those places ν with β ⊗ k_{ν} ≠ 0. Let ν ∈ S. Suppose that θ_{0} ∉ k_{ν}^{ℓ}. Let L_{ν} = k_{ν}(\sqrt[ℓ]{θ_{0}}) and μ_{ν} = \sqrt[ℓ]{θ_{0}} ∈ L_{ν}. Then N_{L_{ν}/k_{ν}}(μ_{ν}) = θ_{0}. By (3.3), ind(β ⊗ E_{0} ⊗ k_{ν}) < ind(β ⊗ E_{0}). In particular ind(β ⊗E_{0} ⊗L_{ν}) ≤ ind(β ⊗E_{0} ⊗k_{ν}) < ind(β ⊗E_{0}). Since cor_{L_{ν}/k_{ν}}(rβ ⊗L_{ν}) = rℓβ = (E_{0} ⊗ k_{ν}, σ_{0}, θ_{0}) = cor_{L_{ν}/k_{ν}}(E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}) and corestriction is injective (cf. [17, Theorem 10, p. 237]), we have rβ ⊗ L_{ν} = (E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}).

Suppose that θ_{0} = μ_{ν}^{ℓ} for some μ_{ν} ∈ k_{ν}. Since k_{ν} is local field containing a primitive ℓ^{th} root of unity and E_{0} ⊗k_{ν} is a cyclic extension, there exists a cyclic field extension L_{ν}/k_{ν} of degree ℓ which is not contained in E_{0} ⊗ k_{ν}. Then N_{L_{ν}/k_{ν}}(μ_{ν}) = μ_{ν}^{ℓ} = θ_{0}. Since L_{ν} is not a subfield of E_{0} ⊗ k_{ν}, ind(β ⊗ E_{0} ⊗ L_{ν}) < ind(β ⊗ E_{0} ⊗ k_{ν}) ≤ ind(β ⊗E_{0}) ([3, p. 131]). Since cor_{L_{ν}/k_{ν}}(rβ ⊗L_{ν}) = rℓβ ⊗k_{ν} = (E_{0} ⊗k_{ν}, σ_{0} ⊗1, μ_{ν}^{ℓ}) = cor_{L_{ν}/k_{ν}}(E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}), by (cf. [17, Theorem 10, p. 237]), we have rβ ⊗ L_{ν} = (E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}).

By (3.1), we have the required L_{0} and μ_{0}. □

Proposition 3.5. Let k be a global field and ℓ a prime not equal to char(k). Let E_{0}/k be a cyclic extension of degree a positive power of ℓ and σ_{0} a generator of Gal(E_{0}/k). Let θ_{0} ∈ k^{∗} and β ∈ H^{2}(k, μ_{ℓ^{n}}) be such that rℓβ = (E_{0}, σ_{0}, θ_{0}) for some r ≥ 1. Suppose that rβ ⊗ E_{0} = 0. Let L_{0} be the unique subfield of E_{0} of degree ℓ over k. Then there exists μ_{0} ∈ L_{0} such that

1) N_{L_{0}/k}(μ_{0}) = θ_{0}

2) rβ ⊗ L_{0} = (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0}).

Proof. Since rβ ⊗ E_{0} = 0 and E_{0}/k is a cyclic extension, we have rβ = (E_{0}, σ_{0}, μ′) for some μ′ ∈ k. We have (E_{0}, σ_{0}, μ′^{ℓ}) = ℓrβ = (E_{0}, σ_{0}, θ_{0}). Thus θ_{0} = N_{E_{0}/k}(y)μ′^{ℓ}. Let μ_{0} = N_{E_{0}/L_{0}}(y)μ′ ∈ L_{0}. Since L_{0} ⊂ E_{0}, we have rβ ⊗ L_{0} = (E_{0}/L_{0}, σ_{0}^{ℓ}, μ′) = (E_{0}/L_{0}, σ_{0}^{ℓ}, N_{E_{0}/L_{0}}(y)μ′) = (E_{0}/L_{0}, σ_{0}^{ℓ}, μ_{0}) (cf. §2) and

N_{L_{0}/k}(μ_{0}) = N_{E_{0}/k}(N_{E_{0}/L_{0}}(y))μ′^{ℓ} = θ_{0}.
□

Corollary 3.6. Let k be a global field and ℓ a prime not equal to char(k). Let E_{0}/k be a cyclic extension of degree a power of ℓ and σ_{0} a generator of Gal(E_{0}/k). Let θ_{0} ∈ k^{∗} and β ∈ H^{2}(k, μ_{ℓ^{n}}) be such that rℓβ = (E_{0}, σ_{0}, θ_{0}) for some r ≥ 1. Suppose that rβ ⊗ E_{0} = 0. Let L_{0} be the unique subfield of E_{0} of degree ℓ over k. Let S be a finite set of places of k. Suppose for each ν ∈ S there exists μ_{ν} ∈ L_{0} ⊗ k_{ν} such that • N_{L_{0}⊗k_{ν}/k_{ν}}(μ_{ν}) = θ_{0}

• rβ ⊗ L_{0} ⊗ k_{ν} = (E_{0} ⊗ L_{0} ⊗ k_{ν}, σ_{0} ⊗ 1, μ_{ν}). Then there exists μ ∈ L_{0} such that

1) N_{L_{0}/k}(μ) = θ_{0}

2) rβ ⊗ L_{0} = (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ)

3) μ is close to μ_{ν} for all ν ∈ S.

Proof. By (3.5), there exists μ_{0} ∈ L_{0} such that

• N_{L_{0}/k}(μ_{0}) = θ_{0}

• rβ ⊗ L_{0} = (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0}). Let ν ∈ S. Then we have

• N_{L_{0}/k}(μ_{0}) = θ_{0} = N_{L_{0}⊗k_{ν}/k_{ν}}(μ_{ν})

• (E_{0} ⊗ L_{0} ⊗ k_{ν}, σ_{0} ⊗ 1, μ_{0}) = (E_{0} ⊗ L_{0} ⊗ k_{ν}, σ_{0} ⊗ 1, μ_{ν}). Let b_{ν} = μ_{0}μ_{ν}^{−1} ∈ L_{0} ⊗ k_{ν}. Then N_{L_{0}⊗k_{ν}/k_{ν}}(b_{ν}) = 1 and (E_{0} ⊗ L_{0} ⊗ k_{ν}, σ_{0} ⊗ 1, b_{ν}) = 1. Thus, there exists a_{ν} ∈ E_{0} ⊗ L_{0} ⊗ k_{ν} with N_{E_{0}⊗L_{0}⊗k_{ν}/L_{0}⊗k_{ν}}(a_{ν}) = b_{ν}. We have N_{E_{0}⊗L_{0}⊗k_{ν}/k_{ν}}(a_{ν}) = N_{L_{0}⊗k_{ν}/⊗k_{ν}}(b_{ν}) = 1. Since E_{0}/k is a cyclic extension with σ_{0} a generator of Gal(E_{0}/k), for each ν ∈ S, there exists c_{ν} ∈ E_{0} ⊗ L_{0} ⊗ k_{ν} such that a_{ν} = c_{ν}^{−1}(σ_{0} ⊗ 1)(c_{ν}). By the weak approximation, there exists c ∈ E_{0} ⊗ L_{0} such that c is close to c_{ν} for all ν ∈ S. Let a = c^{−1}(σ ⊗ 1)(c) ∈ E_{0} ⊗ L_{0} and μ = μ_{0}N_{E_{0}⊗L_{0}/L_{0}}(c) ∈ L_{0}. Then μ has all the required properties. □

4. complete discretely valued fields

Let F be a complete discretely valued field with residue field κ. Let D be a central simple algebra over F of period n coprime to char(κ). Let λ ∈ F^{∗} and α ∈ H^{2}(F, μ_{n}) be the class of D. In this section we analyze the condition α · (λ) = 0 and we use this analysis in the proof of our main result (§10). As a consequence, we also deduce that if κ is either a local field or a global field and α · (λ) = 0 in H^{3}(F, μ_{n}^{⊗2}), then λ is a reduced norm from D.

We use the following notation throughout this section:

• F a complete discretely valued field.

• κ the residue field of F.

• ν the discrete valuation on F.

• π ∈ F^{∗} a parameter at ν.

• n ≥ 2 an integer coprime to char(κ)

• D a central simple algebra over F of period n.

• α ∈ H^{2}(F, μ_{n}) the class representing D. Let E_{0} be the cyclic extension of κ and σ_{0} ∈ Gal(E_{0}/κ) be such that ∂(α) = (E_{0}, σ_{0}). Let (E, σ) be the lift of (E_{0}, σ_{0}) (cf. §2). The pair (E, σ) or E is called the lift of the residue of α. The following is well known.

Lemma 4.1. Let α ∈ H^{2}(F, μ_{n}), (E, σ) the lift of the residue of α. Then α = α′ + (E, σ, π) for some α′ ∈ H_{nr}^{2}(F, μ_{n}). Further α′ ⊗ E = α ⊗ E is independent of the choice of π.

Proof. Since ∂(E, σ, π) = ∂(α), α′ = α − (E, σ, π) ∈ H_{nr}^{2}(F, μ_{n}) and α = α′ + (E, σ, π). □

Lemma 4.2. Let n ≥ 2 be coprime to char(κ) and α ∈ H^{2}(F, μ_{n}). If α = α′ + (E, σ, π) as in (4.1), then ind(α) = ind(α′ ⊗ E)[E : F] = ind(α ⊗ E)[E : F].

Proof. Cf. ([6, Proposition 1(3)] and [15, 5.15]). □

Lemma 4.3. Let E be the lift of the residue of α. Suppose there exists a totally ramified extension M/F which splits α, then α ⊗ E = 0.

Proof. Write α = α′+(E, σ, π) as in (4.1). Since α′⊗E = α⊗E, we have α′⊗E⊗M = 0. Since E⊗M/E is totally ramified, the residue field of E⊗M is same as the residue field of E. Since α′ ⊗ E ⊗ M = 0 and α′ ⊗ E is unramified, it follows from ([28, 7.9 and 8.4]) that α ⊗ E = α′ ⊗ E = 0. □

Lemma 4.4. Let n ≥ 2 be coprime to char(κ). Let α ∈ H^{2}(F, μ_{n}) and (E, σ) be the lift of the residue of α. If α ⊗ E = 0, then α = (E, σ, uπ) for some u ∈ F^{∗} which is a unit at the discrete valuation and per(α) = ind(α).

Proof. We have α = α′ + (E, σ, π) as in (4.1). Since α′ ⊗ E = α ⊗ E = 0, we have α′ = (E, σ, u) for u ∈ F^{∗}. Since E/F and α′ are unramified at the discrete valuation of F, u is a unit at the discrete valuation of F. We have α = (E, σ, u) + (E, σ, π) = (E, σ, uπ). Since E/F is an unramified extension and uπ is a parameter, (E, σ, uπ) is a division algebra and its period is [E : F]. In particular ind(α) = per(α). □

Theorem 4.5. Let F be a complete discretely valued field with residue field κ. Suppose that κ is a local field. Let ℓ be a prime not equal to the characteristic of κ, n = ℓ^{d} and α ∈ H^{2}(F, μ_{n}). Then per(α) = ind(α).

Proof. Write α = α′ + (E, σ, π) as in (4.1). Then E is an unramified cyclic extension of F with ∂(α) = (E_{0}, σ_{0}) and α′ is unramified at the discrete valuation of F. Let \overline{α}′ be the image of α′ in H^{2}(κ, μ_{n})

Suppose that per(∂(α)) = per(α). Then per(∂(α)) = [E_{0} : κ]. Since F is complete discretely valued field and E/F unramified extension, we have [E_{0} : κ] = [E : F]. Thus,

0 = per(α)α
= per(α)(α′ + (E, σ, π))
= per(α)α′ + per(α)(E, σ, π)
= per(α)α′ + [E : F](E, σ, π)
= per(α)α′.

In particular, per(α′) divides per(α) = [E_{0}, κ] = [E : F]. Since κ is a local field, \overline{α}′ ⊗ E_{0} is zero ([3, p.131]) and hence α′ ⊗ E is zero. By (4.4), we have α = (E, σ, θπ) for some θ ∈ F which is a unit in the valuation ring. In particular, α is cyclic and ind(α) = per(α) = [E : F].

Suppose that per(∂(α)) ≠ per(α). Then per(∂(α)) < per(α). Since per(∂(α)) = per(E, σ, π), we have per(α) = per(α′). Since κ is a local field, per(\overline{α}′) = ind(\overline{α}′). Let E_{0} be the residue field of E. Since per(\overline{α}′) = per(α′) and per(∂(α)) = [E_{0} : κ], we have [E_{0} : κ] < per(\overline{α}′). Since κ is a local field,

ind(\overline{α}′ ⊗ E_{0}) = \frac{per(\overline{α}′)}{[E_{0} : κ]}.

Since E is a complete discrete valued field with residue field E_{0} and α′ is unramified at the discrete valuation of E, we have ind(α′ ⊗ E) = ind(\overline{α}′ ⊗ E_{0}). Thus, we have

ind(α) = ind(α′ ⊗ E)[E : F] (by(4.2))
= ind(\overline{α}′ ⊗ E_{0})[E_{0} : κ]
= \frac{per(\overline{α}′)}{[E_{0}:κ]}[E_{0} : κ]
= per(\overline{α}′) = per(α).

□

Proposition 4.6. Suppose that κ is a local field. Let n ≥ 2 be coprime to char(κ). If L/F is a finite field extension, then the corestriction homomorphism H^{3}(L, μ_{n}^{⊗2}) → H^{3}(F, μ_{n}^{⊗2}) is bijective.

Proof. Let k′ be the residue field of L. Since k and k′ are local fields, H^{3}(k, μ_{n}^{⊗2}) = H^{3}(k′, μ_{n}^{⊗2}) = 0 ([27, p. 86]). Since F and L are complete discrete valued fields, the

residue homomorphisms H^{3}(F, μ_{n}^{⊗2}) →^{∂_{F}} H^{2}(k, μ_{n}) and H^{3}(L, μ_{n}^{⊗2}) →^{∂_{L}} H^{2}(k′, μ_{n}) are

isomorphisms (cf. [28, 7.9]). The proposition follows from the commutative diagram

H^{3}(L, μ_{n}^{2}) →^{∂_{L}} H^{2}(k′, μ_{n})
↓  ↓
H^{3}(F, μ_{n}^{⊗2}) →^{∂_{F}} H^{2}(k, μ_{n}),

where the vertical arrows are the corestriction maps ([28, 8.6]). □

Lemma 4.7. Let ℓ be a prime not equal to char(κ) and n = ℓ^{d} for some d ≥ 1. Let α ∈ H^{2}(F, μ_{n}) and λ ∈ F^{∗}. Write λ = θπ^{r} for some θ, π ∈ F with ν(θ) = 0 and ν(π) = 1. Let (E, σ) be the lift of the residue of α and α = α′ + (E, σ, π) as in (4.1). Then

∂(α · (λ)) = 0 ⟺ rα′ = (E, σ, θ) ⟺ rα = (E, σ, λ).

In particular, if ∂(α · (λ)) = 0 and r = ν(λ) is coprime to ℓ, then ind(α ⊗ F (\sqrt[ℓ]{λ})) < ind(α) and α · (\sqrt[ℓ]{λ}) = 0 ∈ H^{3}(F (\sqrt[ℓ]{λ}), μ_{n}^{⊗2}).

Proof. Since rα = rα′ + (E, σ, π^{r}) and λ = θπ^{r}, rα = (E, σ, λ) if and only if rα′ = (E, σ, θ).

We have

∂(α · (λ)) = ∂((α′ + (E, σ, π)) · (θπ^{r})) = r\overline{α}′ + (E_{0}, σ_{0}, \overline{θ}^{−1}),

where ∂(α) = (E_{0}, σ_{0}).

Thus ∂(α · (λ)) = 0 if and only if r\overline{α}′ + (E_{0}, σ_{0}, \overline{θ}^{−1}) = 0 if and only if r\overline{α}′ = (E_{0}, σ_{0}, \overline{θ}) if and only if rα = (E, σ, θ) ( F being complete).

□

Lemma 4.8. Let n ≥ 2 be coprime to char(κ) and ℓ a prime which divides n. Let α ∈ H^{2}(F, μ_{n}), λ = θπ^{ℓr} ∈ F^{∗} with θ a unit in the valuation ring of F, π a parameter and α = α′ + (E, σ, π) be as in (4.1). Suppose that α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}) and there exist an extension L_{0} of κ of degree ℓ and μ_{0} ∈ L_{0} such that

• N_{L_{0}/κ}(μ_{0}) = \overline{θ},

• r\overline{α}′ ⊗ L_{0} = (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0}). Then, there exist an unramified extension L of F of degree ℓ and μ ∈ L such that • residue field of L is L_{0},

• μ a unit in the valuation ring of L, • \overline{μ} = μ_{0},

• N_{L/F} (μ) = θ,

• α · (μπ^{r}) ∈ H^{3}(L, μ_{n}^{⊗2}) is unramified.

Proof. Since ℓ is a prime and [L_{0} : κ] = ℓ, L_{0} = κ(μ′_{0}) for any μ′_{0} ∈ L_{0} \ κ. Let g(X) = X^{ℓ} + b_{ℓ−1}X^{ℓ−1} + · + b_{1}X + b_{0} ∈ κ[X] be the minimal polynomial of μ′_{0} over κ. Let a_{i} be in the valuation ring of F mapping to b_{i} and f(X) = X^{ℓ} + a_{ℓ−1}X^{ℓ−1} + ⋯ + a_{1}X + a_{0} ∈ F [X]. If μ_{0} ∉ κ, then we take μ′_{0} = μ_{0}. Since N_{L_{0}/κ}(μ_{0}) = \overline{θ}, we have b_{0} = (−1)^{ℓ}\overline{θ}. In this case we take a_{0} = (−1)^{ℓ}θ. Since g(X) is irreducible in κ[X], f(X) ∈ F [X] is irreducible. Let L = F [X]/(f). Then L/F is the unramified extension with residue field L_{0}. If μ_{0} ∈ κ, then \overline{θ} = N_{L_{0}/κ}(μ_{0}) = μ_{0}^{ℓ}. Since F is a complete discretely valued field and ℓ is coprime to char(κ), there exists μ ∈ F which is a unit in the valuation ring of F which maps to μ_{0} and μ^{ℓ} = θ. If μ_{0} ∉ κ, then let μ ∈ L be the image of X. Then the image of μ is μ_{0} and N_{L/F} (μ) = θ.

Since L/F, E/F and α′ are unramified at the discrete valuation of F, we have ∂_{L}(α′ · (μπ^{r})) = r\overline{α}′ ⊗ L_{0} and ∂_{L}((E, σ, π) · (μπ^{r})) = ∂_{L}((E ⊗ L, σ ⊗ 1, μ^{−1}) · (π)) = (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0}^{−1}). Since α = α′ + (E, σ, π), we have

∂_{L}(α · (μπ^{r})) = ∂_{L}((α′ ⊗ L) · (μπ^{r})) + ∂_{L}((E, σ, π) · (μπ^{r}))
= r\overline{α}′ ⊗ L_{0} + (E_{0} ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0}^{−1})
= 0
□

Lemma 4.9. Suppose that κ is a local field. Let ℓ be prime not equal to char(k) and n = ℓ^{d}. Let α ∈ H^{2}(F, μ_{n}) and λ ∈ F^{∗}. Suppose λ ∉ F^{∗ℓ}, α ≠ 0 and α · (λ) = 0. Then ind(α ⊗ F (\sqrt[ℓ]{λ})) < ind(α) and α · (\sqrt[ℓ]{λ}) = 0 ∈ H^{3}(F (\sqrt[ℓ]{λ}), μ_{n}^{⊗2}).

Proof. Suppose ν(λ) is coprime to ℓ. Then, by (4.7), we have ind(α ⊗ F (\sqrt[ℓ]{λ})) < ind(α) and α · (\sqrt[ℓ]{λ}) = 0 ∈ H^{3}(F (\sqrt[ℓ]{λ}), μ_{n}^{⊗2}).

Suppose that ν(λ) is divisible by ℓ. Write λ = θπ^{ℓd} with θ ∈ F a unit in the valuation ring of F.

Write α = α′ + (E, σ, π) as in (4.1). Then ind(α) = ind(α′ ⊗ E)[E : F] and ind(α ⊗ F (\sqrt[ℓ]{θ})) ≤ ind(α′ ⊗ E(\sqrt[ℓ]{θ}))[E(\sqrt[ℓ]{θ}) : F (\sqrt[ℓ]{θ})] (cf. 4.2). If \sqrt[ℓ]{θ} ∈ E, then F (\sqrt[ℓ]{θ}) ⊂ E = E(\sqrt[ℓ]{θ}). In particular [E(\sqrt[ℓ]{θ}) : F (\sqrt[ℓ]{θ})] = [E : F (\sqrt[ℓ]{θ})] < [E : F]. Since ind(α′ ⊗ E(\sqrt[ℓ]{θ})) ≤ ind(α′ ⊗ E), it follows that ind(α ⊗ F (\sqrt[ℓ]{θ})) < ind(α).

Suppose \sqrt[ℓ]{θ} ∉ E. Since E is unramified extension of F and θ is a unit in the valuation ring of E, E(\sqrt[ℓ]{θ}) is an unramified extension of F with residue field E_{0}(\sqrt[ℓ]{\overline{θ}}), where E_{0} is the residue field of E and \overline{θ} is the image of θ in the residue field. Since F is a complete discretely valued field and θ is not an ℓ^{th} power in E, \overline{θ} is not an ℓ^{th} power in E_{0} and [E_{0}(\sqrt[ℓ]{\overline{θ}}) : E_{0}] = ℓ.

Suppose α′ ⊗ E ≠ 0. Then \overline{α}′ ⊗ E_{0} ≠ 0. Since E_{0} is a local field and ind(\overline{α}′) is a power of ℓ, ind(\overline{α}′ ⊗ E_{0}(\sqrt[ℓ]{\overline{θ}})) < ind(\overline{α}′ ⊗ E_{0}) ([3, p. 131]). Hence ind(α′ ⊗ E(\sqrt[ℓ]{θ})) < ind(α′ ⊗ E) and ind(α ⊗ F (\sqrt[ℓ]{θ})) < ind(α).

Suppose that α′ ⊗ E = 0. Then, by (4.4), α = (E, σ, uπ) for some unit u in the valuation ring of F. Since α · (λ) = 0, (E, σ, uπ) · (λ) = 0. Since E/F is unramified with residue field E_{0}, u, θ are units in the valuation ring of F and π a parameter, by taking the residue of α · (λ) = 0, we see that (E_{0}, σ_{0}, \overline{θ}^{−1}\overline{u}^{ℓd}) = 0 ∈ H^{2}(κ, μ_{n}) (cf. 4.7). In particular, \overline{θ}\overline{u}^{−ℓd} is a norm from E_{0}. Since [E_{0} : k] is a power of ℓ and E_{0}/κ is cyclic, there exists a sub extension L of E_{0} such that [L : κ] = ℓ. Then \overline{θ}\overline{u}^{−ℓd} is a norm from L and hence \overline{θ} is a norm from L. Since \overline{θ} is not in κ^{∗ℓ}, by (2.5), L = κ(\sqrt[ℓ]{\overline{θ}}). In particular \sqrt[ℓ]{\overline{θ}} ∈ E_{0} and hence \sqrt[ℓ]{θ} ∈ E. Thus ind(α ⊗ F (\sqrt[ℓ]{θ})) = ind((E, σ, uπ) ⊗ F (\sqrt[ℓ]{θ})) < ind(E, σ, uπ) = ind(α). □

Lemma 4.10. Suppose κ is a local field. Let ℓ be a prime not equal to char(κ) and n = ℓ^{d}. Let α ∈ H^{2}(F, μ_{n}) and λ ∈ F^{∗}. Suppose that κ contains a primitive ℓ^{th} root of unity. If α ≠ 0 and α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), then there exist a cyclic field extension L/F of degree ℓ and μ ∈ L^{∗} such that N_{L/F} (μ) = λ, ind(α ⊗ L) < ind(α) and α · (μ) = 0 ∈ H^{3}(L, μ_{n}^{⊗2}). Further, if λ ∈ F^{∗ℓ}, then L/F is unramified and μ ∈ F.

Proof. Suppose λ is not an ℓ^{th} power in F. Let L = F (\sqrt[ℓ]{λ}) and μ = \sqrt[ℓ]{λ}. Then, by (4.9), ind(α ⊗ L) < ind(α) and α · (μ) = 0 ∈ H^{3}(L, μ_{n}^{⊗2}).

Suppose λ = μ^{ℓ} for some μ ∈ F^{∗}. Write α = α′ + (E, σ, π) as in (4.1).

Suppose that α′ ⊗ E = 0. Then, by (4.4), α = (E, σ, uπ) for some u ∈ F^{∗} which is a unit in the valuation ring of F. Let L be the unique subfield of E with L/F of degree ℓ. Then ind(α ⊗ L) < ind(α). Since cor_{L/F} (α · (μ)) = α · (μ^{ℓ}) = α · (λ) = 0, by (4.6), α · (μ) = 0 in H^{3}(L, μ_{n}). We also have λ = μ^{ℓ} = N_{L/F} (μ).

Suppose that α′ ⊗ E ≠ 0. Let E_{0} be the residue field of E. Then E_{0}/κ is a cyclic field extension of κ of degree equal to the degree of E/F. Since κ is a local field and contains a primitive ℓ^{th} root of unity, there are at least three distinct cyclic field extensions of κ of degree ℓ. Since E_{0}/κ is a cyclic extension, there is at most one sub extension of E_{0} of degree ℓ over κ. Thus there exists a cyclic field extension L_{0}/κ of degree ℓ such that E_{0} ⊗ L_{0} is a field. Let L/F be the unramified extension with residue field L_{0}. Then E ⊗ L is a field. Let \overline{α}′ be the image of α′ in H^{2}(κ, μ_{n}). Since E is a complete discretely valued field, \overline{α}′ ⊗ E_{0} ≠ 0. Since E_{0} ⊗ L_{0}/E_{0} is a field extension of degree ℓ and κ is a local field, ind(\overline{α}′ ⊗ E_{0} ⊗ L_{0}) < ind(\overline{α}′ ⊗ E_{0}) ([3, p. 131]). Since E is a complete discretely valued field, ind(α′ ⊗ E ⊗ L) < ind(α′ ⊗ E). Since L/F is unramified, ∂(α ⊗ L) = ∂(α) ⊗ L_{0} (cf. [4, Proposition 3.3.1]) and hence the decomposition α ⊗ L = α′ ⊗ L + (E ⊗ L, σ ⊗ 1, π) is as in (4.1). Thus, by (4.2), ind(α ⊗ L) < ind(α). As above, we also have λ = N_{L/F} (μ) and α · (μ) = 0 ∈ H^{3}(L, μ_{n}^{⊗2}). □

Lemma 4.11. Suppose κ is a global field. Let ℓ be a prime not equal to char(κ) and n = ℓ^{d}. Let α ∈ H^{2}(F, μ_{n}) and λ ∈ F^{∗}. If α ≠ 0 and α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), then there exist a field extension L/F of degree ℓ and μ ∈ L^{∗} such that N_{L/F} (μ) = λ, ind(α ⊗ L) < ind(α) and α · (μ) = 0 ∈ H^{3}(L, μ_{n}^{⊗2}).

Proof. Suppose that ν(λ) is coprime to ℓ. Then, by (4.7), L = F (\sqrt[ℓ]{λ}) and μ = \sqrt[ℓ]{λ} has the required properties.

Suppose that ν(λ) is divisible by ℓ. Let π be a parameter in F. Then λ = θπ^{rℓ} with ν(θ) = 0. Write α = α′ + (E, σ, π) as in (4.1). Let \overline{α}′ be the image of α′ in H^{2}(κ, μ_{n}) and θ_{0} the image of θ in κ. Since α · (λ) = 0, by (4.7), we have rℓ\overline{α}′ = (E_{0}, σ_{0}, θ_{0}), where E_{0} is the residue field of E and σ_{0} induced by σ.

Suppose that r\overline{α}′ ⊗E_{0} ≠ 0. Then, by (3.4) , there exist a extension L_{0}/κ of degree ℓ and μ_{0} ∈ L_{0} such that N_{L_{0}/κ}(μ_{0}) = θ_{0}, ind(\overline{α}′ ⊗ E_{0} ⊗ L_{0}) < ind(\overline{α}′ ⊗ E_{0}) and r\overline{α}′ ⊗ L_{0} = (E_{0} ⊗ L_{0}, σ_{0}, μ_{0}).

Suppose that r\overline{α}′ ⊗ E_{0} = 0. Suppose that E_{0} ≠ κ. Let L_{0} be the unique subfield field of E_{0} of degree ℓ over κ. Then, by (3.5), there exists μ_{0} ∈ L_{0} such that N_{L_{0}/κ}(μ_{0}) = θ_{0} and r\overline{α}′ ⊗ L_{0} = (E_{0}, σ_{0}, μ_{0}).

Suppose that E_{0} = κ. Then, by (3.2), there exist a field extension L_{0}/κ of degree ℓ and μ_{0} ∈ L_{0} such that N_{L_{0}/k}(μ_{0}) = θ_{0} and ind(\overline{α}′ ⊗ L_{0}) < ind(\overline{α}′).

By (4.8), we have the required L and μ. □

Theorem 4.12. Let F be a complete discrete valued field with residue field κ. Suppose that κ is a local field or a global field. Let D be a central simple algebra over F of period n. Suppose that n is coprime to char(κ). Let α ∈ H^{2}(F, μ_{n}) be the class of D and λ ∈ F^{∗}. If α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), then λ is a reduced norm from D.

Proof. Write n = ℓ_{1}^{d_{1}} ⋯ ℓ_{r}^{d_{r}}, ℓ_{i} distinct primes, d_{i} > 0, D = D_{1} ⊗ ⋯ ⊗ D_{r} with each D_{i} a central simple algebra over F of period power of ℓ_{i} ([1, Ch. V, Theorem 18]). Let α_{i} be the corresponding cohomology class of D_{i}. Since ℓ_{i}’s are distinct primes, α · (λ) = 0 if and only if α_{i} · (λ) = 0 and λ is a reduced norm from D if and only if λ is a reduced norm from each D_{i}. Thus without loss of generality we assume that per(D) = ℓ^{d} for some prime ℓ.

We prove the theorem by the induction on the index of D. Suppose that deg(D) = 1. Then every element of F^{∗} is a reduced norm from D. We assume that deg(D) = n = ℓ^{d} ≥ 2.

Let λ ∈ F^{∗} with α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}). Let ρ be a primitive ℓ^{th} root of unity. Since [F (ρ) : F] is coprime to n, λ is a reduced norm from F is and only if λ is a reduced from D ⊗ F (ρ). Thus, replacing F by F (ρ), we assume that ρ ∈ F.

Since κ is either a local field or a global field, by (4.10, 4.11), there exist an extension L/F of degree ℓ and μ ∈ L^{∗} such that N_{L/F} (μ) = λ, α · (μ) = 0 and ind(α ⊗ L) < ind(α). Thus, by induction, μ is a reduced norm from D ⊗ L. Since N_{L/F} (μ) = λ, λ is a reduced norm from D. □

The following technical lemma is used in §6.

Lemma 4.13. Let κ be a finite field and K a function field of a curve over κ. Let u, v, w ∈ κ^{∗} and θ ∈ K^{∗}. Let ℓ a prime not equal to char(κ) and θ = wuλ. If κ contains a primitive ℓ^{th} root of unity and w ∉ κ^{∗ℓ}, then for r ≥ 1, the element (v, \sqrt[ℓ^{r}]{θ})_{ℓ} is H^{2}(K( \sqrt[ℓ^{e}]{θ}) is trivial over K( \sqrt[ℓ^{r}]{θ}, \sqrt[ℓ]{v + uλ}).

Proof. Let L = K( \sqrt[ℓ^{r}]{θ}, \sqrt[ℓ]{v + uλ}) and β = (v, \sqrt[ℓ^{r}]{θ})_{ℓ}. Since L is a global field, to show that β ⊗ L is trivial, it is enough to show that β ⊗ L_{ν} is trivial for every discrete valuation ν of L. Let ν be a discrete valuation of L. Since v ∈ κ^{∗}, v is a unit at ν. If θ is a unit at ν, then β ⊗ L is unramified at ν and hence β ⊗ L_{ν} is trivial. Suppose that θ is not a unit at ν. Since u and v are units at ν, λ is not a unit. Suppose that ν(λ) > 0. Then v ∈ L_{ν}^{∗ℓ} and hence β ⊗ L_{ν} is trivial. Suppose that ν(λ) < 0. Then \sqrt[ℓ]{uλ} ∈ L_{ν}. Since r ≥ 1, θ = uwλ and \sqrt[ℓ^{r}]{θ} ∈ L_{ν}, we have \sqrt[ℓ]{θ} = \sqrt[ℓ]{wuλ} ∈ L_{ν}. Hence \sqrt[ℓ]{w} ∈ L_{ν}. Since w ∈ κ^{∗} \ κ^{∗ℓ}, v ∈ κ^{∗} and κ is a finite field, \sqrt[ℓ]{v} ∈ κ(\sqrt[ℓ]{w}). Since κ(\sqrt[ℓ]{w}) ⊂ L_{ν}, β ⊗ L_{ν} is trivial. □

We end this section with the following well known facts.

Lemma 4.14. Let F be a complete discrete valued field with the residue field κ. Let α ∈ Br(F) and L/F an unramified extension with residue field L_{0}. Suppose that per(α) is coprime to char(κ). Let ∂(α) = (E_{0}, σ_{0}). If ∂(α ⊗ L) is trivial, then E_{0} is isomorphic to a subfield of L_{0}.

Proof. Let L_{0} be the residue field of L. Since L/F is unramified, (E_{0}, σ_{0}) ⊗ L_{0} = ∂(α) ⊗ L_{0} = ∂(α ⊗ L) (cf. [4, Proposition 3.3.1]). Since α ⊗ L = 0, ∂(α ⊗ L) = 0 and hence E_{0} is isomorphic to a subfield of L_{0}. □

Corollary 4.15. Let L/F be a cyclic extension of degree n, τ a generator of Gal(L/F) and θ ∈ F^{∗}. If ν(θ) is coprime to n and ind(L/F, τ, θ) = [L : F], then [L : F] = per(∂(L/F, τ, θ)).

Proof. Let β = (L/F, τ, θ) and m = per(∂(β)). Since n = [L : F] = ind(β), m divides n. Since ν(θ) is coprime to n, F ( \sqrt[m]{θ})/F is a totally ramified extension of degree m with residue field equal to the residue field κ of F. Since ∂(β ⊗ F ( \sqrt[m]{θ})) = m∂(β), β ⊗ F ( \sqrt[m]{θ}) is unramified. Since F (\sqrt[n]{θ})/F ( \sqrt[m]{θ}) is totally ramified and β ⊗ F (\sqrt[n]{θ}) is trivial, β ⊗ F ( \sqrt[m]{θ}) is trivial (cf. 4.3). Hence n = m. □

5. Brauer group - complete two dimensional regular local rings

Through out this section A denotes a complete regular local ring of dimension 2 with residue field κ and F its field of fractions. Let ℓ be a prime not equal to the characteristic of κ and n = ℓ^{d} for some d ≥ 1. Let m = (π, δ) be the maximal ideal of A. For any prime p ∈ A, let F_{p} be the completion of the field of fractions of the completion of the local ring A_{(p)} at p and κ(p) the residue field at p.

Lemma 5.1. Let E_{π} be a Galois extension of F_{π} of degree coprime to char(κ). Then there exists a Galois extension E of F of degree [E_{π} : F_{π}] which is unramified on A except possibly at δ and Gal(E/F) ≃Gal(E_{π}/F_{π}).

Proof. Since A is complete and m = (π, δ), κ(π) is a complete discretely valued field with residue field κ and the image \overline{δ} of δ as a parameter. Let E_{0} be the residue field of E_{π}. Then E_{0}/κ(π) is a Galois extension with Gal(E_{0}/κ(π)) ≃ Gal(E_{π}/F_{π}). Let L_{0} be the maximal unramified extension of κ(π) contained in E_{0}. Then L_{0} is also a complete discretely valued field with \overline{δ} as a parameter. Since E_{0}/L_{0} is a totally ramified extension of degree coprime to char(κ), we have E_{0} = L_{0}(\sqrt[e]{v\overline{δ}}) for some v ∈ L_{0} which is a unit at the discrete valuation of L_{0} (cf. 2.4).

Since E_{0}/κ(π) is a Galois extension, E_{0}/L_{0} and L_{0}/κ(π) are Galois extensions. Let κ_{0} be the residue field of E_{0}. Then the residue field of L_{0} is also κ_{0}. Since κ_{0} is a Galois extension of κ and A is complete, there exists a Galois extension L of F which is unramified on A with residue field κ_{0}. Let B be the integral closure of A in L. Then B is a regular local ring with residue field κ_{0} (cf. [21, Lemma 3.1]). Let u ∈ B be a lift of v.

Let E = L(\sqrt[e]{uδ}). Since L/F is unramified on A, E/F is unramified on A except possibly at δ. In particular E/F is unramified at π with residue field E_{0}. By the construction [E : F] = [E_{0} : κ(π)]. Hence E ⊗ F_{π} ≃ E_{π}.

Since L/F is a Galois extension which is unramified at π, we have Gal(L/F) ≃ Gal(L_{0}/κ(π)). Let τ ∈ Gal(L/F) and \overline{τ} ∈Gal(L_{0}/κ(π)) be the image of τ. Since E_{0}/κ(π) is Galois and E_{0} = L_{0}(\sqrt[e]{v\overline{δ}}), by (2.3), E_{0} contains a primitive e^{th} root of unity ρ and \overline{τ}(v\overline{δ})) ∈ E_{0}^{e}. In particular ρ ∈ κ_{0}. Since B is complete with residue field κ_{0}, ρ ∈ B and hence ρ ∈ L ⊆ E. Since \overline{τ}(v\overline{δ}) = \overline{τ}(v)\overline{δ} and v\overline{δ}, \overline{τ}(v\overline{δ}) ∈ E_{0}^{e}, \overline{τ}(v)/v ∈ E_{0}^{e}. Since \overline{τ}(v) and v are units at the discrete valuation of L_{0} and E_{0}/L_{0} is totally ramified, \overline{τ}(v)/v ∈ L_{0}^{e}. Since B is complete and the image of τ(u)/u in L_{0} is \overline{τ}(v)/v, τ(u)/u ∈ L^{e}. Since E = L(\sqrt{uδ}), τ(uδ) ∈ E^{e}. Thus, by (2.3), E/F is Galois. Since E ⊗ F_{π} ≃ E_{π}, Gal(E/F) ≃Gal(E_{π}/F_{π}). □

Since A is complete and (π, δ) is the maximal ideal of A, A/(π) is a complete discrete valuation ring with \overline{δ} is a parameter and A/(δ) is a complete discrete valuation ring with \overline{π}. The following follows from ([16, Proposition 1.7]).

Lemma 5.2. ([16, Proposition 1.7]) Let m ≥ 1 and α ∈ H^{m}(F, μ_{n}^{⊗(m−1)}). Suppose that α is unramified on A except possibly at π and δ. Then

∂\overline{δ}(∂_{π}(α)) = −∂\overline{π}(∂_{δ}(α)).

Let H_{nr}^{m}(F, μ_{n}^{⊗(m−1)}) be the intersections of the kernels of the residue homomorphisms ∂_{θ} : H^{m}(F, μ_{n}^{⊗(m−1)}) → H^{m−1}(κ(θ), μ_{n}^{⊗(m−2)}) for all primes θ ∈ A. The following lemma follows from the purity theorem of Gabber.

Lemma 5.3. For m ≥ 1, H_{nr}^{m}(F, μ_{n}^{⊗(m−1)}) ≃ H^{m}(κ, μ_{n}^{⊗(m−1)}).

Proof. By the purity theorem of Gabber (cf. [24, CH. XVI]), we have H_{nr}^{m}(F, μ_{n}^{⊗(m−1)}) ≃ H_{ét}^{m}(A, μ_{n}^{⊗(m−1)}). Since A is complete, we have H_{ét}^{m}(A, μ_{n}^{⊗(m−1)}) ≃ H^{m}(κ, μ_{n}^{⊗(m−1)}) (cf. [20, Corollary 2.7, p.224]). □

Lemma 5.4. Let m ≥ 1 and α ∈ H^{m}(F, μ_{n}^{⊗(m−1)}). Suppose that α is unramified except possibly at π. Then there exist α_{0} ∈ H^{m}(F, μ_{n}^{⊗(m−1)}) and β ∈ H^{m−1}(F, μ_{n}^{⊗(m−2)}) which are unramified on A such that

α = α_{0} + β · (π).

Proof. Let β_{0} = ∂_{π}(α). By (5.2), β_{0} ∈ H^{m−1}(κ(π), μ_{n}^{⊗(m−2)}) is unramified on A/(π). Since A/(π) is a complete discrete valuation ring with residue field κ, we have H_{nr}^{m−1}(κ(π), μ_{n}^{⊗(m−2)}) ≃ H^{m−1}(κ, μ_{n}^{⊗(m−2)}). Since A is a complete regular local ring of dimension 2, H_{nr}^{m−1}(F, μ_{n}^{⊗(m−2)}) ≃ H^{m−1}(κ, μ_{n}^{⊗(m−2)}) (5.3). Thus, there exists β ∈ H_{nr}^{m−1}(F, μ^{⊗(m−1)}) which is a lift of β_{0}. Then α_{0} = α − β · (π) is unramified on A. Hence α = α_{0} + β · (π). □

Corollary 5.5. Let m ≥ 1 and α ∈ H^{m}(F, μ_{n}^{⊗(m−1)}) is unramified on A except possibly at π and δ. If α ⊗ F_{δ} = 0, then α = 0. In particular if α_{1}, α_{2} ∈ H^{m}(F, μ_{n}^{⊗(m−1)}) unramified on A except possibly at π and δ and α_{1} ⊗ F_{δ} = α_{2} ⊗ F_{δ}, then α_{1} = α_{2}.

Proof. Since α ⊗ F_{δ} = 0, α is unramified at δ. Thus α is unramified on A except possibly at π. By (5.4), we have α = α_{0} + β · (π) for some α_{0} ∈ H^{m}(F, μ_{n}^{⊗(m−1)}) and β ∈ H^{m−1}(F, μ_{n}^{⊗(m−2)}) which are unramified on A. Since α ⊗ F_{δ} = 0, we have (β · (π)) ⊗ F_{δ} = −α_{0} ⊗ F_{δ}. Since β · (π) and α_{0} are unramified at δ, we have \overline{β} · (\overline{π}) = −\overline{α}_{0}, where denotes the image over κ(δ). Since κ(δ) is a complete discrete valued field with \overline{π} as a parameter, by taking the residues, we see that the image of β is 0 in H^{m−1}(κ, μ_{n}^{⊗(m−2)}). Since A is a complete regular local ring, β = 0 (5.3). Hence α = α_{0} is unramified on A. Since α ⊗ F_{δ} = 0, \overline{α} = 0 ∈ H^{m}(κ(δ), μ_{n}^{⊗(m−1)}). In particular the image of α in H^{m}(κ, μ_{n}^{⊗(m−1)}) is zero. Since A is a complete regular local ring, α = 0 (5.3). □

Corollary 5.6. Let m ≥ 1 and α ∈ H^{m}(F, μ_{n}^{m−1}). If α is unramified on A except possibly at π and δ, then per(α) = per(α ⊗ F_{π}) = per(α ⊗ F_{δ}).

Proof. Suppose t = per(α ⊗ F_{δ}). Then tα ⊗ F_{δ} = 0 and hence, by (5.5), tα = 0. Since per(α⊗F_{δ}) ≤ per(α), it follows that per(α) = per(α⊗F_{δ}). Similarly, per(α) = per(α ⊗ F_{π}). □

Corollary 5.7. Suppose that κ is a finite field. Let α ∈ H^{2}(F, μ_{n}). If α is unramified except at π and δ, then there exist a cyclic extension E/F and σ ∈ Gal(E/F) a generator, u ∈ A a unit, and 0 ≤ i, j < n such that α = (E, σ, uπ^{i}δ^{j}) with E/F is unramified on A except at δ and i = 1 or E/F is unramified on A except at π and j = 1.

Proof. Since n is a power of the prime ℓ and nα = 0, per(∂_{π}(α)) and per(∂_{δ}(α)) are powers of ℓ. Let d′ be the maximum of per(∂_{π}(α)) and per(∂_{δ}(α)). Then ∂_{π}(d′α) = d′∂_{π}(α) = 0 and ∂_{δ}(d′α) = d′∂_{δ}(α) = 0. In particular d′α is unramified on A. Since κ is a finite field, d′α = 0. Hence per(α) divides d′ and d′ = per(α). Thus per(α) = per(∂_{π}(α)) or per(∂_{δ}(α)).

Suppose that per(α) = per(∂_{π}(α)). Since ∂_{π}(α⊗F_{π}) = ∂_{π}(α), we have per(∂_{π}(α)) ≤ per(α ⊗ F_{π}) ≤ per(α). Thus per(α ⊗ F_{π}) = per(∂_{π}(α ⊗ F_{π})). Thus, by (4.4), we have α ⊗ F_{π} = (E_{π}/F_{π}, σ, θπ) for some cyclic unramified extension E_{π}/F_{π} and θ ∈ F_{π} a unit in the valuation ring of F_{π}.

By (5.1), there exists a Galois extension E/F which is unramified on A except possibly at (δ) such that E ⊗ F_{π} ≃ E_{π}. Since E_{π}/F_{π} is cyclic, E/F is cyclic. Since θ ∈ F_{π} is a unit in the valuation ring of F_{π} and the residue field of F_{π} is a complete discrete valued field with \overline{δ} as parameter, we can write θ = uδ^{j}θ_{1}^{n} for some unit u ∈ A, θ_{1} ∈ F_{π} and 0 ≤ j ≤ n − 1. Then α ⊗ F_{π} ≃ (E, σ, uδ^{j}π) ⊗ F_{π}. Thus, by (5.5), we have α = (E, σ, uδ^{i}π).

If per(α) = per(∂_{δ}(α)), then, as above, we get α = (E, σ, uπ^{j}δ) for some cyclic extension E/F which is unramified on A except possibly at π. □

The following is proved in ([29, 2.4]) under the assumption that F contains a primitive n^{th} root of unity.

Proposition 5.8. Suppose that κ is a finite field. Let α ∈ H^{2}(F, μ_{n}). If α is unramified on A except possibly at (π) and (δ). Then ind(α) = ind(α ⊗ F_{π}) = ind(α ⊗ F_{δ}).

Proof. Suppose that α is unramified on A except possibly at (π) and (δ). Then, by (5.7), we assume without loss of generality that α = (E/F, σ, πδ^{j}) with E/F unramified on A except possibly at δ. Then ind(α) ≤ [E : F]. Since E/F is unramified on A expect possibly at δ, we have [E : F] = [E_{π} : F_{π}] and ind(α ⊗ F_{π}) = [E_{π} : F_{π}]. Thus [E : F] = [E_{π} : F_{π}] = ind(α ⊗ F_{π}) ≤ ind(α) ≤ [E : F] and hence [E : F] = ind(α ⊗ F_{π}) = ind(α).

□

Corollary 5.9. Suppose that κ is a finite field. Let α ∈ H^{2}(F, μ_{n}). If α is unramified on A except possibly at (π) and (δ). Then ind(α) = per(α).

Proof. By (5.6), per(α) = per(α ⊗ F_{π}) and by (4.5), ind(α ⊗ F_{π}) = per(α ⊗ F_{π}). Thus per(α) = ind(α ⊗ F_{π}). By (5.8), we have ind(α) = per(α). □

The following follows from ([11] and [13]).

Proposition 5.10. Let α ∈ H^{2}(F, μ_{n}). Let ϕ : 𝒳 → Spec(A) be a sequence of blow-ups and V = ϕ^{−1}(m). Then ind(α) = l.c.m{ind(α ⊗ F_{x}) | x ∈ V }.

Proof. Follows from similar arguments as in the proof of ([11, Theorem 9.11]) and using ([13, Theorem 4.2.1]). □

We end this section with the following well known result Lemma 5.11. Let E/F be a cyclic extension of degree ℓ^{d} for some d ≥ 1. If E/F is unramified on A except possibly at δ, then there exist a subextension E_{nr} of E/F and w ∈ E_{nr} which is a unit in the integral closure of A in E_{nr} such that E_{nr}/F is unramified on A and E = E_{nr}( \sqrt[ℓ^{e}]{wδ}). Further if κ is a finite field, κ contains a primitive ℓ^{th} root of unity and 0 < e < d, then N_{E_{nr}/F} (w) ∈ A is not an ℓ^{th} power in A.

Proof. Let E(π) be the residue field of E at π. Since E/F is unramified at A except possibly at δ, by (5.6), [E(π) : κ(π)] = [E : F]. Since E/F is cyclic, E(π)/κ(π) is cyclic. As in the proof of (5.1), there exist a cyclic extension E_{0}/F unramified on A and a unit w in the integral closure of A in E_{0} such that the residue field of E_{0}( \sqrt[ℓ^{e}]{wδ}) at π is E(π). By (5.5), we have E ≃ E_{0}( \sqrt[ℓ^{e}]{wδ}). Let E_{nr} = E_{0}. Then E_{nr} has the required properties.

Suppose that κ is a finite field and contains a primitive ℓ^{th} root of unity. Let B be the integral closure of A in E_{nr}. Then B is a complete regular local ring with residue field κ′ a finite extension of κ.

Let w_{0} = N_{E_{nr}/F} (w) ∈ A^{∗} and \overline{w}_{0} ∈ κ^{∗}. Suppose that w_{0} ∈ A^{∗ℓ}. Then \overline{w}_{0} ∈ κ^{∗ℓ}. Since κ contains a primitive ℓ^{th} root of unity, we have | κ^{′∗}/κ^{′∗ℓ} |=| κ^{∗}/κ^{∗ℓ} |= ℓ. Since norm map is surjective from κ′ to κ, the norm map induces an isomorphism from κ^{′∗}/κ^{′∗ℓ} → κ^{∗}/κ^{∗ℓ}. Thus the image of w in κ′ is an ℓ^{th} power. Since B is a complete regular local ring, w ∈ B^{∗ℓ}. Suppose 0 < e < f. Then \sqrt[ℓ]{δ} ∈ E. Since E_{nr}/F is nontrivial unramified extension and F (\sqrt[ℓ]{δ})/F is a nontrivial totally ramified extension of F, we have two distinct degree ℓ subextensions of E/F, which is a contradiction to the fact that E/F is cyclic. Hence w_{0} ∉ A^{∗ℓ}. □

6. Reduced norms - complete two dimensional regular local rings

Throughout this section we fix the following notation:

• A a complete two dimensional regular local ring

• F the field of fractions of A

• m = (π, δ) the maximal ideal of A

• κ = A/m a finite field

• ℓ a prime not equal to char(κ)

• n = ℓ^{d}

• α ∈ H^{2}(F, μ_{n}) is unramified on A except possibly at (π) and (δ)

• λ = wπ^{s}δ^{t}, w ∈ A a unit and s, t ∈ ℤ with 1 ≤ s, t < n.

The aim of this section is to prove that if α ≠ 0 and α · (λ) = 0, then there exist an extension L/F of degree ℓ and μ ∈ L such that ind(α ⊗ L) < ind(α) and N_{L/F} (μ) = λ. We assume that

• F contains a primitive ℓ^{th} root of unity.

We begin with the following

Lemma 6.1. If α · (λ) = 0, then sα = (E, σ, λ) for some cyclic extension E of F which is unramified on A except possibly at δ. In particular, if s is coprime to ℓ, then α = (E′, σ′, λ) for some cyclic extension E′ of F which is unramified on A except possibly at δ.

Proof. By (4.7), there exists an unramified cyclic extension E_{π} of F_{π} such that sα ⊗ F_{π} = (E_{π}, σ, λ). Let E(π) be the residue field of E_{π}. Then E(π) is a cyclic extension of κ(π). By (5.1), there exists a cyclic extension E of F which is unramified on A except possibly at δ with E ⊗ F_{π} ≃ E_{π}. Since E/F is unramified on A except possibly at δ and λ = wπ^{s}δ^{t} with w a unit in A, (E, σ, λ) is unramified on A except possibly at (π) and (δ). Since α is unramified on A except possibly at (π) and (δ), sα − (E, σ, λ) is unramified on A except possibly at (π) and (δ). Since sα ⊗ F_{π} = (E_{π}, σ, λ) = (E, σ, λ) ⊗ F_{π}, by (5.5), sα = (E, σ, λ). □

Lemma 6.2. Suppose that α·(λ) = 0 and λ ∉ F^{∗ℓ}. If α ≠ 0, then ind(α⊗F (\sqrt[ℓ]{λ})) < ind(α) and α · (\sqrt[ℓ]{λ}) = 0 ∈ H^{3}(F (\sqrt[ℓ]{λ}), μ_{n}^{⊗2}).

Proof. Suppose that s is coprime to ℓ. Then, by (6.1), α = (E′, σ′, λ) for some cyclic extension E′ of F which is unramified on A except possibly at δ. Since ν_{π}(λ) = s is coprime to ℓ and E′/F is unramified at π, it follows that ind(α) = [E′ : F]. In particular, ind(α ⊗ F (\sqrt[ℓ]{λ})) ≤ [E′ : F]/ℓ < ind(α). Similarly, if t is coprime to ℓ, then ind(α ⊗ F (\sqrt[ℓ]{λ})) < ind(α). Further α · (\sqrt[ℓ]{λ}) = (E′, σ′, λ) · (\sqrt[ℓ]{λ}) = 0.

Suppose that s and t are divisible by ℓ. Since λ = wπ^{s}δ^{t}, we have F (\sqrt[ℓ]{λ}) = F (\sqrt[ℓ]{w}). Let L = F (\sqrt[ℓ]{λ}) = F (\sqrt[ℓ]{w}) and B be the integral closure of A in L. Since w is a unit in A, by ([21, Lemma 3.1]), B is a complete regular local ring with maximal ideal generated by π and δ. Since w is not an ℓ^{th} power in F and A is a complete regular local ring, the image of w in A/m is not an ℓ^{th} power. Since A/(π) is also a complete regular local ring with residue field A/m, the image of w in A/(π) is not an ℓ^{th} power. Since F_{π} is a complete discrete valued field with residue field the field of fractions of A/(π), w is not an ℓ^{th} power in F_{π}. Since α·(λ) = 0 and the residue field of F_{π} is a local field, by (4.9), ind(α ⊗ L_{π}) < ind(α). Hence, by (5.8), ind(α ⊗ L) < ind(α).

Since L_{π} = L ⊗ F_{π} and L_{δ} = L ⊗ F_{δ} are field extension of degree ℓ over F_{π} and F_{δ} respectively and cores(α · (\sqrt[ℓ]{λ})) = α · (λ) = 0, by (4.6), (α · (\sqrt[ℓ]{λ})) ⊗ L_{π} = 0 and (α · (\sqrt[ℓ]{λ})) ⊗ L_{δ} = 0. Hence, by (5.5), α · (\sqrt[ℓ]{λ}) = 0. □

Lemma 6.3. Suppose α = (E/F, σ, uπδ^{ℓm}) for some m ≥ 0, u a unit in A, E/F a cyclic extension of degree ℓ^{d} which is unramified on A except possibly at δ and σ a generator of Gal(E/F). Let ℓ^{e} be the ramification index of E/F at δ and f = d − e. Let i ≥ 0 be such that ℓ^{f} + ℓ^{di} > ℓm. Let v ∈ A be a unit which is not in F^{∗ℓ} and L = F (\sqrt[ℓ]{vδ^{ℓ^{f} +ℓ^{di}−ℓm} + uπ}). If f > 0, then ind(α ⊗ L) < ind(α)

Proof. Let B be the integral closure of A in L and r = ℓ^{f} + ℓ^{di} − ℓm. Since ℓ^{f} + ℓ^{di} > ℓm, L = F (\sqrt[ℓ]{vδ^{r} + uπ}) and vδ^{r} + uπ is a regular prime in A. Thus B is a complete regular local ring (cf. [21, Lemma 3.2]) and π, δ remain primes in B. Note that π and δ may not generate the maximal ideal of B. Let L_{π} and L_{δ} be the completions of L at the discrete valuations given by π and δ respectively. Since v ∉ F^{∗ℓ}, F (\sqrt[ℓ]{v}) is the unique degree ℓ extension of F_{π} which is unramified on A. Since f > 0, there is a subextension E of degree ℓ over F which is unramified on A and hence F (\sqrt[ℓ]{v}) ⊂ E.

Since E/F is unramified on A except possibly at δ, by (5.8), [E : F] = [E_{π} : F_{π}] and hence ind(α) = per(α) = [E : F].

Since r is divisible by ℓ, L_{π} ≃ F_{π}(\sqrt[ℓ]{v}) and hence L_{π} ⊂ E_{π}. Thus ind(α ⊗ L_{π}) < ind(α). Since r > 0, L_{δ} ≃ F_{δ}(\sqrt[ℓ]{uπ}). Since α = (E/F, σ, uπδ^{ℓm}), ind(α ⊗ L_{δ}) < [E⊗L_{δ} : L_{δ}] ≤ [E : F]. In particular by (4.5), per(α⊗F_{π}) < ind(α) and per(α⊗F_{δ}) < ind(α). Since α⊗L is unramified on B except possibly at π and δ and H^{2}(B, μ_{ℓ}) = 0, per(α⊗L) < ind(α). If d = 1, then per(α⊗L) < ind(α) = ℓ and hence per(α⊗L) = ind(α ⊗ L) = 1 < ind(α). Suppose that d ≥ 2.

Let ϕ : 𝒳 → Spec(B) be a sequence of blow-ups such that the ramification locus of α ⊗ L is a union of regular curves with normal crossings. Let V = ϕ^{−1}(P). To show that ind(α ⊗ L) < ind(α), by (5.10), it is enough to show that for every point x of V, ind(α ⊗ L_{x}) < ind(α).

Let x ∈ V be a closed point. Then, by (5.9), ind(α ⊗ L_{x}) = per(α ⊗ L_{x}). Since per(α ⊗ L_{x}) < ind(α), ind(α ⊗ L_{x}) < ind(α).

Let x ∈ V be a codimension zero point. Then ϕ(x) is the closed point of Spec(B). Let ν̃ be the discrete valuation of L given by x. Then κ(ν̃) ≃ κ′(t) for some finite extension κ′ over κ and a variable t over κ. Let ν be the restriction of ν̃ to F.

Suppose that ν(δ^{r}) < ν(π). Then L ⊗ F_{ν} = F_{ν}(\sqrt[ℓ]{vδ^{r}}). Since ℓ divides r, L ⊗ F_{ν} = F_{ν}(\sqrt[ℓ]{v}). Since F (\sqrt[ℓ]{v}) ⊂ E, ind(α ⊗ L ⊗ F_{ν}) < ind(α). Suppose that ν(δ^{r}) > ν(π). Then L ⊗ F_{ν} = F_{ν}(\sqrt[ℓ]{uπ}) and as above ind(α ⊗ L ⊗ F_{ν}) < ind(α). Suppose that ν(δ^{r}) = ν(π). Let λ = π/δ^{r}. Then λ is a unit at ν and Lν̃ = F_{ν}(\sqrt[ℓ]{v + uλ}). We have uπδ^{ℓm} = uλδ^{r+ℓm} = uλδ^{ℓ^{f} +ℓ^{di}} and

α ⊗ F_{ν} = (E ⊗ F_{ν}/F_{ν}, σ ⊗ 1, uπδ^{ℓm}) = (E ⊗ F_{ν}/F_{ν}, σ ⊗ 1, uλδ^{ℓ^{f} +ℓ^{di}}).

Since [E : F] = ℓ^{d}, α ⊗ F_{ν} = (E ⊗ F_{ν}/F_{ν}, σ ⊗ 1, uλδ^{ℓ^{f}}). Suppose that f = d. Then E/F is unramified and hence every element of A^{∗} is a norm from E. Thus (E⊗F_{ν}/F_{ν}, σ⊗1, w_{0}uλ) with w_{0} ∈ A^{∗}\A^{∗ℓ}. Suppose that f < d. Then e = d−f > 0 and hence by (5.11), we have E = E_{nr}( \sqrt[ℓ^{e}]{wδ}), for some unit w in the integral closure of A in E_{nr}, with N( \sqrt[ℓ^{d}]{wδ}) = w_{1}δ^{ℓ^{f}} with w_{1} ∈ A^{∗} \ A^{∗ℓ}. Thus

α ⊗ F_{ν} = (E ⊗ F_{ν}/F_{ν}, σ ⊗ 1, uλδ^{ℓ^{f}}) = (E ⊗ F_{ν}/F_{ν}, σ ⊗ 1, w_{0}uλ).

with w_{0} = w_{1}^{−1}.

If E ⊗ F_{ν} is not a field, then ind(α ⊗ F_{ν}) < [E : F]. Suppose E ⊗ F_{ν} is a field. Let θ = w_{0}uλ. Since α ⊗ F_{ν} = (E ⊗ F_{ν}/F_{ν}, σ ⊗ 1, θ), ind(α ⊗ L ⊗ F_{ν}) ≤ ind(α ⊗ L ⊗ F_{ν}( \sqrt[ℓ^{d−1}]{θ})) · [L ⊗ F_{ν}( \sqrt[ℓ^{d−1}]{θ}) : L ⊗ F_{ν}]. Since [L ⊗ F_{ν}( \sqrt[ℓ^{d−1}]{θ}) : L ⊗ F_{ν}] ≤ ℓ^{d−1} < [E : F], it is enough to show that α ⊗ L ⊗ F_{ν}( \sqrt[ℓ^{d−1}]{θ}) is trivial.

Since F (\sqrt[ℓ]{v})/F is the unique subextension of E/F degree ℓ and [E : F] = ℓ^{d}, we have α ⊗ F_{ν}( \sqrt[ℓ^{d−1}]{θ}) = (F_{ν}( \sqrt[ℓ^{d−1}]{θ}, \sqrt[ℓ]{v})/F ( \sqrt[ℓ^{d−1}]{θ}), σ, \sqrt[ℓ^{d−1}]{θ}) (cf. 2.1). Let M = F_{ν}( \sqrt[ℓ^{d−1}]{θ}). Since κ contains a primitive ℓ^{th} root of unity, we have α⊗M = (v, \sqrt[ℓ^{d−1}]{θ})_{ℓ}. Then M is a complete discrete valuation field. Since λ is a unit at ν, θ is a unit at ν. Hence the residue field of M is κ(ν)( \sqrt[ℓ^{d−1}]{\overline{θ}}). Since θ and v are units at ν, α ⊗ M = (v, \sqrt[ℓ^{d−1}]{θ}) is unramified at the discrete valuation of M. Hence it is enough to show that the specialization β of α ⊗ M is trivial over κ(ν)( \sqrt[ℓ^{d−1}]{\overline{θ}}) ⊗ L_{0}, where L_{0} is the residue field of L ⊗ F_{ν} at ν.

Suppose that Lν̃/F_{ν} is ramified. Since Lν̃ = F_{ν}(\sqrt[ℓ]{u + vλ}), v + uλ is not a unit at ν. Thus v = −uλ modulo F_{ν}^{∗ℓ^{d}} and θ = w_{0}uλ = −w_{0}v modulo F_{ν}^{∗ℓ^{d}}. In particular \sqrt[ℓ^{d−1}]{θ} = \sqrt[ℓ^{d−1}]{−w_{0}v} modulo M^{∗ℓ}. Since \overline{v}, \overline{w_{0}} ∈ κ and κ a finite field, β = (\sqrt[ℓ]{\overline{v}}, \sqrt[ℓ^{d−1}]{θ}) = (\sqrt[ℓ]{\overline{v}}, \sqrt[ℓ^{d−1}]{−\overline{w}_{0}\overline{v}}) is trivial.

Suppose that Lν̃/F_{ν} is unramified. Then L_{0} = κ(π)(\sqrt[ℓ]{\overline{v} + \overline{uλ}}). Since κ(π) is a global field and d − 1 ≥ 1, by (4.13), β ⊗ L_{0}( \sqrt[ℓ^{d−1}]{\overline{θ}}) = 0. □

Lemma 6.4. Suppose L_{π}/F_{π} and L_{δ}/F_{δ} are unramified cyclic field extensions of degree ℓ and μ_{π} ∈ L_{π}, μ_{δ} ∈ L_{δ} such that

• ind(α ⊗ L_{π}) < d_{0} for some d_{0}, • λ = N_{L_{π}/F_{π}}(μ_{π}) and λ = N_{L_{δ}/F_{δ}}(μ_{δ}),

• α · (μ_{π}) = 0 ∈ H^{3}(L_{π}, μ_{n}^{⊗2}), α · (μ_{δ}) = 0 ∈ H^{3}(L_{δ}, μ_{n}^{⊗2}),

• if λ ∈ F_{P}^{∗ℓ} and α = (E/F, σ, vπ) for some cyclic extension E/F which is unramified on A except possibly at δ, then L_{δ}/F_{δ} = F_{δ}(\sqrt[ℓ]{vπ}). Then there exists a cyclic extension L/F of degree ℓ and μ ∈ L such that

• ind(α ⊗ L) < d_{0},

• λ = N_{L/F} (μ),

• α · (μ) = 0 ∈ H^{3}(L, μ_{n}^{⊗2}),

• L ⊗ F_{π} ≃ L_{π} and L ⊗ F_{δ} ≃ L_{δ}.

Proof. Since α · (μ_{π}) = 0 ∈ H^{3}(L_{π}, μ_{n}^{⊗2}) and λ = N_{L_{π}/F_{π}}(μ_{π}), by taking the corestriction, we see that α · (λ) = 0 ∈ H^{3}(F_{π}, μ_{n}^{⊗2}). Since α · (λ) is unramified on A except possibly at π and δ, by (5.5), α · (λ) = 0.

Suppose that λ ∉ F^{∗ℓ}. Then, by (2.6) and (6.2), L = F (\sqrt[ℓ]{λ}) and μ = \sqrt[ℓ]{λ} have the required properties.

Suppose that λ ∈ F^{∗ℓ}. Let L(π) and L(δ) be the residue fields of L_{π} and L_{δ} respectively. Since L_{π}/F_{π} and L_{δ}/F_{δ} are unramified cyclic extensions of degree ℓ, L(π)/κ(π) and L(δ)/κ(δ) are cyclic extensions of degree ℓ. Since F contains a primitive ℓ^{th} root of unity, we have L(π) = κ(π)[X]/(X^{ℓ} −a) and L(δ) = κ(δ)[X]/(X^{ℓ} −b) for some a ∈ κ(π) and b ∈ κ(δ). Since κ(π) is a complete discretely valued field with \overline{δ} a parameter, without loss of generality we assume that a = \overline{u_{1}}\overline{δ}^{ϵ} for some unit u_{1} ∈ A and ϵ = 0 or 1. Similarly we have b = \overline{u_{2}}\overline{π}^{ϵ′} for some unit u_{2} ∈ A and ϵ′ = 0 or 1.

By (5.7), we assume that α = (E/F, σ, uπδ^{j}) for some cyclic extension E/F which is unramified on A except possibly at δ, u a unit in A and j ≥ 0. Then ind(α) = [E : F]. Let E_{0} be the residue field of E at π. Then [E : F] = [E_{0} : κ(π)]. Since ∂_{π}(α) = (E_{0}/κ(π), \overline{σ}), per(∂_{π}(α)) = [E : F] = ind(α). Since L_{π}/F_{π} is an unramified cyclic extension of degree ℓ and ind(α ⊗ L_{π}) < ind(α), the residue field L(π) of L_{π} is the unique degree ℓ subextension of E_{0}/κ(π).

Suppose that ϵ = ϵ′ = 0. Since L_{π} and L_{δ} are fields, u_{1} and u_{2} are not ℓ^{th} powers. Let L/F be the unique cyclic field extension of degree ℓ which is unramified on A. Then L ⊗ F_{π} ≃ L_{π} and L ⊗ F_{δ} ≃ L_{δ}. Let B be the integral closure of A in L. Then B is a regular local ring with maximal ideal (π, δ) and hence by (5.8) ind(α ⊗ L) < ind(α).

Suppose ϵ = 1. Then L_{π} = F_{π}(\sqrt[ℓ]{u_{1}δ}) and L(π) = κ(π)(\sqrt[ℓ]{\overline{u_{1}}\overline{δ}}). Since E_{0}/κ(π) is a cyclic extension containing a totally ramified extension, E_{0}/κ(π) is a totally ramified cyclic extension. Thus κ(π) contains a primitive ℓ^{d^{th}} root of unity and E_{0} = κ(π)( \sqrt[ℓ^{d}]{\overline{u_{1}}\overline{δ}}). In particular F contains a primitive ℓ^{d^{th}} root of unity and α = (u_{1}δ, uπδ^{j}) = (u_{1}δ, u′π). Since L_{δ}/F_{δ} is an unramified extension of degree ℓ with ind(α ⊗ L_{δ}) < ind(α), as above, we have L_{δ} = F_{δ}(\sqrt{u′π}) and hence α = (u_{1}δ, u_{2}π). Let L = F (\sqrt[ℓ]{u_{1}δ + u_{2}π}). Then L ⊗ F_{π} ≃ L_{π} and L ⊗ F_{δ} ≃ L_{δ}. Since for any a, b ∈ F^{∗}, (a, b) = (a+b, −a^{−1}b), we have α = (u_{1}π+u_{2}δ, −u_{1}^{−1}π^{−1}u_{2}δ). In particular ind(α ⊗ L) < ind(α).

Suppose that ϵ = 0 and ϵ′ = 1. Suppose j is coprime to ℓ. Then, by (4.15), ind(α) = per(∂_{δ}(α)) and as in the proof of (5.7), we have α = (E′/F, σ′, vδπ^{j′}) for some cyclic extension E′/F which is unramified on A except possibly at π. Thus, we have the required extension as in the case ϵ = 1.

Suppose j is divisible by ℓ. Since ϵ = 0, L_{π} = F_{π}(\sqrt[ℓ]{u_{1}}). Since the residue field L_{π}(π) of L_{π} is contained in the residue field E_{0} of E at π, F (\sqrt[ℓ]{u_{1}}) ⊂ E and hence E/F is not totally ramified at δ. Since E/F is unramified on A except possibly at δ, by (5.11), E = E_{nr}( \sqrt[ℓ^{e}]{wδ}) for some unit w in the integral closure of A in E_{nr}. Suppose e = 0. Then E = E_{nr}/F is unramified on A. Since κ is a finite field and A is complete, every unit in A is a norm from E/F. Thus multiplying uπδ^{j} by a norm from E/F we assume that α = (E/F, σ, u_{2}πδ^{j}). Suppose that e > 0. Then, by (5.11), N_{E/F} (wδ) = w_{1}δ^{ℓ^{f}} with w_{1} ∈ A^{∗} \ A^{∗ℓ}. Since A^{∗}/A^{∗ℓ} is a cyclic group of order ℓ, we have α = (E/F, σ, u_{2}πδ^{j+j′ℓ^{f}}) for some j′. Since j is divisible by ℓ and f ≥ 1, j + j′ℓ^{f} is divisible by ℓ. Hence, we assume that α = (E/F, σ, u_{2}πδ^{ℓm}) for some m. Thus, by (6.3), there exists i ≥ 0 such that ind(α ⊗ L) < ind(α) for L = F (\sqrt[ℓ]{u_{1}δ^{ℓ^{f+di}} + u_{2}πδ^{ℓm}}) .

By the choice, we have L/F is the unique unramified extension or L = F (\sqrt[ℓ]{u_{1}δ + u_{2}π}) or L = F (\sqrt[ℓ]{u_{1}δ^{ℓ^{f+di}} + u_{2}πδ^{ℓm}}) with ℓ^{f+di} > ℓm. Let B be the integral closure of A in L. Then B is a complete regular local ring with π and δ remain prime in B. Since λ = wπ^{s}δ^{t} and λ ∈ F_{P}^{∗ℓ}, we have λ = w_{0}^{ℓ}π^{ℓs_{1}}δ^{ℓt_{1}} for some unit w_{0} ∈ A. Let μ = w_{0}π^{s_{1}}δ^{t_{1}} ∈ F. Then N_{L/F} (μ) = μ^{ℓ} = λ. Since α · (λ) = 0, by (4.6), α · (μ) = 0 in H^{3}(L_{π}, μ_{n}^{⊗2}) and H^{3}(L_{δ}, μ_{n}^{⊗2}). Hence α · (μ) is unramified at all height one prime ideals of B. Since B is a complete regular local ring with residue field κ finite, α · (μ) = 0 (5.3). □

Lemma 6.5. Suppose that ν_{π}(λ) is divisible by ℓ, α is unramified on A except possibly at π and δ, and α · (λ) = 0. Let L_{π} be a cyclic unramified or split extension of F_{π} of degree ℓ, μ_{π} ∈ L_{π} and d_{0} ≥ 2 such that

• N_{L_{π}/F_{π}}(μ_{π}) = λ,

• ind(α ⊗ L_{π}) < d_{0},

• α · (μ_{π}) = 0 in H^{3}(L_{π}, μ_{n}). Then there exists an extension L over F of degree ℓ and μ ∈ L such that

• N_{L/F} (μ) = λ,

• ind(α ⊗ L) < d_{0},

• α · (μ) = 0 ∈ H^{3}(L, μ_{n}^{⊗2}) and

• there is an isomorphism ϕ : L_{π} → L ⊗ F_{π} with

ϕ(μ_{π})(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{π})^{ℓ^{m}},

for all m ≥ 1.

Proof. Since ν_{π}(λ) is divisible by ℓ, λ = wπ^{rℓ}δ^{s} for some w ∈ A a unit.

Suppose that L_{π} = ∏ F_{π} is a split extension. Let L = ∏ F be the split extension of degree ℓ. Since μ_{π} ∈ L_{π}, we have μ_{π} = (μ_{1}, ⋯, μ_{ℓ}) with μ_{i} ∈ F_{π}. Write μ_{i} = θ_{i}π^{r_{i}} with θ_{i} ∈ F_{π} a unit at its discrete valuation. Since N_{L_{π}/F_{π}}(μ_{π}) = λ = wπ^{rℓ}, θ_{1} ⋯ θ_{ℓ} = w and π^{r_{1}+···+r_{ℓ}} = π^{rℓ}. For 2 ≤ i ≤ ℓ, let θ̄_{i} be the image of θ_{i} in the residue field κ(π) of F_{π}. Since κ(π) is the field of fractions of A/(δ) and A/(δ) is a complete discrete valuation ring with δ̄ as a parameter, we have θ̄_{i} = ū_{i}δ̄^{s_{i}} for some unit u_{i} ∈ A. For 2 ≤ i ≤ ℓ, let θ̃_{i} = u_{i}δ^{s_{i}} ∈ F, θ̃_{1} = wθ̃_{2}^{−1} ⋯ θ̃_{ℓ}^{−1} and μ = (θ̃_{1}π^{r_{1}}, ⋯, θ̃_{ℓ}π^{r_{ℓ}}) ∈ L = ∏ F. Then N_{L/F} (μ) = λ. Since θ̃_{i}π^{r_{i}}μ_{i}^{−1} is a unit at π with image 1 in κ(π), θ̃_{i}π^{r_{i}}μ_{i}^{−1} ∈ F_{π}^{ℓ^{m}} for any m ≥ 1. In particular α·(θ̃_{i}π^{r_{i}}) = α·(μ_{i}) = 0 ∈ H^{3}(F_{π}, μ_{n}^{⊗2}). Since α is unramified on A except possibly at π and δ and θ̃_{i} = u_{i}δ^{t_{i}} with u_{i} ∈ A a unit, α·(θ̃_{i}π^{r_{i}}) is unramified on A except possibly at π and δ. Thus, by (5.5), α · (θ̃_{i}π^{r_{i}}) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}). Since ind(α ⊗ L_{π}) < d_{0} and L_{π} is the split extension, ind(α ⊗ F_{π}) < d_{0}. Since α is unramified on A except possibly at π and δ, by (5.8), ind(α) < d_{0}. Thus L and μ = (θ̃_{1}π^{r_{1}}, ⋯, θ̃_{ℓ}π^{r_{ℓ}}) ∈ L have the required properties.

Suppose that L_{π} is a field extension of F_{π}. By (5.1), there exists a cyclic extension L of F of degree ℓ which is unramified on A except possibly at δ with L ⊗ F_{π} ≃ L_{π}. Let B be the integral closure of A in L. By the construction of L, either L/F is unramified A or L = F (\sqrt[ℓ]{uδ}) for some unit u ∈ A. Replacing δ by uδ, we assume that L/F is unramified on A or L = F (\sqrt[ℓ]{δ}). In particular, B is a regular local ring with maximal ideal generated by (π, δ′) for δ′ = δ or δ′ = \sqrt[ℓ]{δ} (cf. [21, Lemma 3.2]). Since α is unramified on A except possibly at π and δ, α ⊗ L is unramified on B except possibly at π and δ′. Since ind(α ⊗ L_{π}) < d_{0}, by (5.8), ind(α ⊗ L) = ind(α ⊗ L_{π}) < d_{0}.

Since L_{π}/F_{π} is unramified and N_{L_{π}/F_{π}}(μ_{π}) = λ = wπ^{rℓ}δ^{s}, we have μ_{π} = θ_{π}π^{r} for some θ_{π} ∈ L_{π} which is a unit at its discrete valuation. Let \overline{θ}_{π} be the image of θ_{π} in L(π). Since L(π) is the field of fractions of the complete discrete valuation ring B/(π) and \overline{δ′} is a parameter in B/(π), we have \overline{θ}_{π} = \overline{v}\overline{δ′}^{t} for some unit v ∈ B. Since N_{L(π)/κ(π)}(\overline{θ}_{π}) = \overline{w}\overline{δ}^{s}, it follows that N_{L(π)/κ(π)}(\overline{v}) = \overline{w}. Since w is a unit in A, there exists a unit ṽ ∈ B with N_{L/F} (ṽ) = w and \overline{ṽ} = \overline{v}. Let μ = ṽπ^{r}δ′^{t} ∈ L. Then μμ_{π}^{−1} ∈ L_{π} is a unit in the valuation ring at π with the image 1 in the residue field L(π) and hence μμ_{π}^{−1} ∈ L_{π}^{ℓ^{m}} for all m ≥ 1. In particular α · (μ) = α · (μ_{π}) = 0 ∈ H^{3}(L_{π}, μ_{n}^{⊗2}). Since ind(α ⊗ L_{π}) < d_{0}, by (5.8), ind(α ⊗ L) < d_{0}. Since α · (μ) = 0 in H^{3}(L_{π}, μ_{n}^{⊗2}), α is unramified on A except possibly at π and the support of μ on A is at most π, by (5.5), α · (μ) = 0 in H^{3}(L, μ_{n}^{⊗2}). □

Lemma 6.6. Suppose that α·(λ) = 0 and ν_{δ}(λ) = sℓ. Suppose that α = (E/F, σ, πδ^{m}) for some cyclic extension E/F which is unramified on A except possibly at δ. Let E_{δ} be the lift of the residue of α at δ. If sα ⊗ E_{δ} = 0, then there exists an integer r_{1} ≥ 0 such that w_{1}δ^{mr_{1}−s} is a norm from the extension E/F for some unit w_{1} ∈ A.

Proof. Write α ⊗ F_{δ} = α′ + (E_{δ}/F_{δ}, σ_{δ}, δ) as in (4.1). Since α ⊗ E_{δ} = α′ ⊗ E_{δ}, sα′ ⊗ E_{δ} = 0. Hence sα′ = (E_{δ}, σ, θ) for some θ ∈ F_{δ}. Since α′ and E_{δ}/F_{δ} are unramified at δ, we assume that θ ∈ F_{δ} is a unit at δ. Since the residue field κ(δ) of F_{δ} is a complete discrete valued field with the image of π as a parameter, without loss of generality we assume that θ = w_{0}π^{r_{1}} for unit w_{0} ∈ A and r_{1} ≥ 0. Let λ_{1} = w_{0}π^{r_{1}}δ^{s}. Since sα′ = (E_{δ}, σ_{δ}, θ), by (4.7), α · (λ_{1}) = 0 ∈ H^{3}(F_{δ}, μ_{n}^{⊗2}). Since α is unramified on A except possibly at π,δ and λ_{1} = w_{0}π^{r_{1}}δ^{s} with w_{0} ∈ A a unit, α · (λ_{1}) is unramified in A except possibly at π and δ. Hence, by (5.5), α ·(λ_{1}) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}). We have

0 = ∂_{π}(α·(λ_{1})) = ∂_{π}((E/F, σ, uπδ^{m})·(w_{0}π^{r_{1}}δ^{s})) = (E(π)/κ(π), \overline{σ}, (−1)^{r_{1}} \overline{u}^{r_{1}} \overline{w}_{1}^{−1}\overline{δ}^{mr_{1}−s}).

Since (E/F, σ, (−1)^{r_{1}}u^{r_{1}}w_{0}^{−1}δ^{mr_{1}−s}) is unramified on A except possibly at π and δ, by (5.5), (E/F, σ, (−1)^{r_{1}}u^{r_{1}}w_{0}^{−1}δ^{mr_{1}−s}) = 0. In particular (−1)^{r_{1}}u^{r_{1}}w_{0}^{−1}δ^{mr_{1}−s} is a norm from the extension E/F. □

Lemma 6.7. Suppose that α · (λ) = 0 and λ = wπ^{r}δ^{sℓ} for some unit w ∈ A and r coprime to ℓ. Let E_{δ} be the lift of the residue of α at δ. If sα ⊗ E_{δ} = 0, then there exists θ ∈ A such that

• α · (θ) = 0,

• ν_{π}(θ) = 0,

• ν_{δ}(θ) = s.

Proof. Since r is coprime to ℓ, by (6.1), α = (E/F, σ, λ) for some cyclic extension E/F which is unramified on A except possible at δ. Let t = [E : F]. Since t is a power of ℓ and r is coprime to ℓ, there exists an integer r′ ≥ 1 such that rr′ ≡ 1 modulo t. We have

α = α^{rr′} = (E/F, σ, wπ^{r}δ^{sℓ})^{rr′} = (E/F, σ)^{r} · (wπ^{r}δ^{sℓ})^{r′} = (E/F, σ)^{r} · (w^{r′}πδ^{r′sℓ}).

Since r is coprime to ℓ, we also have (E/F, σ)^{r} = (E/F, σ^{r′}) (cf. §2) and hence α = (E/F, σ^{r}, πδ^{r′sℓ}). Thus, by (6.6), there exist a unit w_{1} ∈ A and r_{1} ≥ 0 such that w_{1}δ^{r′sℓr_{1}−s} is a norm from E/F. Since r′ℓr_{1} − 1 is coprime to ℓ, r′ℓr_{1} − 1 is coprime to t and hence there exists an integer r_{1} ≥ 0 such that (r′ℓr_{1} − 1)r_{2} ≡ 1 modulo t. In particular w_{1}^{r_{2}}δ^{s} ≡ (w_{1}δ^{r′sℓr_{1}−s})^{r_{2}} modulo F^{∗t} and hence w_{1}^{r_{2}}δ^{s} is a norm from E/F. Thus θ = w_{1}^{r_{2}}δ^{s} has the required properites. □

Lemma 6.8. Let E_{π} and E_{δ} be the lift of the residues of α at π and δ respectively. Suppose that α · (λ) = 0 and λ = wπ^{rℓ}δ^{sℓ} for some unit w ∈ A. If α · (λ) = 0, rα ⊗ E_{π} = 0 and sα ⊗ E_{δ} = 0, then there exists θ ∈ A such that

• α · (θ) = 0,

• ν_{π}(θ) = r,

• ν_{δ}(θ) = s.

Proof. By (5.7), we assume that α = (E/F, σ, uπδ^{m}) for some extension E/F which is unramified on A except possible at δ and m ≥ 0. Without loss of generality, we assume that 0 ≤ m < [E : F]. By (6.6), there exists an integer r_{1} ≥ 0 such that w_{1}δ^{mr_{1}−s} is a norm from E/F. Let t = [E − F] and θ = (−uπ + δ^{t−m})^{r_{1}−r}w_{1}^{−1}(−u)^{r}π^{r}δ^{s}. Since t − m > o, we have ν_{π}(θ) = r and ν_{δ}(θ) = s.

Now we show that α · (θ) = 0. Since t − m > 0, we have (−uπ + δ^{t−m})^{r_{1}−r} = (−uπ)^{r_{1}−r} modulo δ and hence θ ≡ (−u)^{r_{1}−r}π^{r_{1}−r}w_{1}^{−1}(−u)^{r}π^{r}δ^{s} = w_{1}^{−1}(−u)^{r_{1}}π^{r_{1}}δ^{s} modulo F_{δ}^{∗t}. Since w_{1}δ^{mr_{1}−s} is a norm from E/F, we have

(α · (θ)) ⊗ F_{δ} = (E/F, σ, uπδ^{m}) · (w_{1}^{−1}(−u)^{r_{1}}π^{r_{1}}δ^{s}) ⊗ F_{δ}
= (E/F, σ, uπδ^{m}) · (w_{1}^{−1}(−u)^{r_{1}}π^{r_{1}}δ^{s}w_{1}δ^{mr_{1}−s}) ⊗ F_{δ}
= (E/F, σ, uπδ^{m}) · ((−u)^{r_{1}}π^{r_{1}}δ^{mr_{1}}) ⊗ F_{δ}
= (E/F, σ, uπδ^{m}) · ((−uπδ^{m})^{r_{1}}) ⊗ F_{δ} = 0.

Thus α · (θ) is unramified at δ.

We have (−uπ + δ^{t−m})^{r_{1}−r} ≡ δ^{t(r_{1}−r)+m(r−r_{1})} modulo π and hence

θ ≡ δ^{t(r_{1}−r)+m(r−r_{1})}w_{1}^{−1}(−u)^{r}π^{r}δ^{s} ≡ (−uπδ^{m})^{r}(w_{1}δ^{mr_{1}−s})^{−1} modulo F_{π}^{∗t}.

Since w_{1}δ^{mr_{1}−s} is a norm from E/F and t = [E : F], we have

\begin{aligned}(α · (θ)) ⊗ F_{π} & = & (E/F, σ, uπδ^{m}) · ((−uπδ^{m})^{r}(w_{1}δ^{mr_{1}−s})^{−1}) ⊗ F_{π} \\ & = & (E/F, σ, uπδ^{m}) · ((−uπδ^{m})^{r}) ⊗ F_{π} = 0. \end{aligned}

In particular α · (θ) is unramified at δ.

Let γ be a prime in A with (γ) ≠ (π) and (γ) ≠ (δ). Since α is unramified on A except possibly at π and δ, if γ does not divide θ, then α · (θ) is unramified at γ. Suppose γ divides θ. Then γ = −uπ + δ^{t−m}. Thus uπδ^{m} ≡ δ^{t} modulo γ. Since ∂_{γ}(α · (θ)) = (E(θ), \overline{σ}, \overline{uπ}\overline{δ}^{m}), where E(θ) is the residue field of E at θ and¯ denotes the image modulo γ, we have ∂_{γ}(α · (θ)) = (E(θ), \overline{σ}, \overline{uπ}\overline{δ}^{m}) = (E(θ), \overline{σ}, \overline{δ}^{t}) = 0.

Hence α · (θ) is unramified on A. Since α · (θ) ⊗ F_{π} = 0, by (5.5), we have α · (θ) = 0. □

7. Patching

We fix the following data:

• R a complete discrete valuation ring,

• K the field of fractions of R,

• κ the residue field of R,

• ℓ a prime not equal to char(κ) and n = ℓ^{d} for some d ≥ 1.

• X a smooth projective geometrically integral variety over K,

• F the function field of X,

• α ∈ H^{2}(F, μ_{n}), α ≠ 0,

• λ ∈ F^{∗} with α · (λ) = 0,

• 𝒳 a normal proper model of X over R and X_{0} the reduced special fibre of 𝒳 .

• 𝒫_{0} a finite set of closed points of X_{0} containing all the points of intersection of irreducible components of X_{0}.

For x ∈ 𝒳 , let Â_{x} be the completion of the regular local ring at x on 𝒳 , F_{x} the field of fractions of Â_{x} and κ(x) the residue field at x. Let η ∈ X_{0} be a codimension zero point and P ∈ X_{0} be a closed point such that P is in the closure of η. For abuse of the notation we denote the closure of η by η and say that P is a point of η. A pair (P, η) of a closed point P and a codimension zero point of X_{0} is called a branch if P is in η. Let (P, η) be a branch. Let F_{P,η} be the completion of F_{P} at the discrete valuation on F_{P} associated to η. Then F_{x} and F_{P} are subfields of F_{P,x}. Since κ(η) is the function field of the curve η, any closed point of η gives a discrete valuation on κ(η). The residue field κ(η)_{P} of F_{P,η} is the completion of κ(η) at the discrete valuation on κ(η) given by P. Let η be a codimension zero point of X_{0} and U ⊂ η be a non-empty open subset. Let A_{U} be the ring of all those functions in F which are regular at every closed point of U. Let t be parameter in R. Then t ∈ R_{U_{η}}. Let Â_{U} be the (t)-adic completion of A_{U} and F_{U} be the field of fractions of Â_{U}. Then F ⊆ F_{U} ⊆ F_{η}.

We begin with the following result, which follows from ([11, Theorem 9.11]) ( cf. proof of [22, Theorem 2.4]).

Proposition 7.1. For each irreducible component X_{η} of X_{0}, let U_{η} be a non-empty proper open subset of X_{η} and 𝒫 = X_{0} \ ∪_{η}U_{η}, where η runs over the codimension zero points of X_{0}. Suppose that 𝒫_{0} ⊆ 𝒫. Let L be a finite extension of F. Suppose that there exists N ≥ 1 such that for each codimension zero point η of X_{0}, ind(α ⊗ L ⊗ F_{U_{η}}) ≤ N and for every closed point P ∈ 𝒫, ind(α ⊗ L ⊗ F_{P}) ≤ N. Then ind(α ⊗ L) ≤ N.

Proof. Let 𝒴 be the integral closure of 𝒳 in L and ϕ : 𝒴 → 𝒳 be the induced map. Let 𝒫′ be a finite set if closed points of 𝒴 containing points of the intersection of distinct irreducible curves on the special fibre Y_{0} of 𝒴 and inverse image of 𝒫 under ϕ. Let U be an irreducible component of Y_{0} \ 𝒫′_{0}. Then ϕ(U) ⊂ U_{η} for some U_{η} and there is a homomorphism of algebras from L⊗F_{U_{η}} to L_{U}. (Note that L⊗F_{U_{η}} may be a product of fields). Since ind(α ⊗ L ⊗ F_{U_{η}}) ≤ d, we have ind(α ⊗ L_{U}) ≤ N. Let Q ∈ 𝒫′. Suppose ϕ(Q) = P ∈ 𝒫. Then there is a homomorphism of algebras from L ⊗ F_{P} to L_{Q}. (Once again note that L ⊗ F_{P} may be a product of fields). Since ind(α⊗L⊗F_{P}) ≤ N, ind(α⊗L_{Q}) ≤ N. Suppose that ϕ(Q) ∈ U_{η} for some U_{η}. Then there is a homomorphism of algebras from L ⊗ F_{U_{η}} to L_{Q}. Thus ind(α ⊗ L_{Q}) ≤ N. Therefore, by ([11, Theorem 9.11]), ind(α ⊗ L) ≤ N. □

Lemma 7.2. Let η be a codimension zero point of X_{0}. Suppose there exists a field extension or split extension L_{η}/F_{η} of degree ℓ and μ_{η} ∈ L_{η} such that

1) N_{L_{η}/F_{η}}(μ_{η}) = λ

2) ind(α ⊗ L_{η}) < ind(α)

3) α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2}). Then there exists a non-empty open subset U_{η} of η, a split or field extension L_{U_{η}}/F_{U_{η}} of degree ℓ and μ_{U_{η}} ∈ L_{U_{η}} such that

1) N_{L_{Uη} /F_{Uη}} (μ_{U_{η}}) = λ

2) ind(α ⊗ L_{η}) < ind(α)

3) α · (μ_{U_{η}}) = 0 ∈ H^{3}(L_{U_{η}}, μ_{n}^{⊗2})

4) there is an isomorphism ϕ_{U_{η}} : L_{U_{η}} ⊗ F_{η} → L_{η} with ϕ_{U_{η}}(μ_{U_{η}} ⊗ 1)μ_{η}^{−1} ≡ 1 modulo the radical of the integral closure of R̂_{η} in L_{η}. Further if L_{η}/F_{η} is cyclic, then L_{U_{η}}/F_{U_{η}} is cyclic.

Proof. Suppose L_{η} = ∏ F_{η} is the split extension of degree ℓ. Write μ_{η} = (μ_{1}, ⋯, μ_{ℓ}) with μ_{i} ∈ F_{η}. Then λ = N_{L_{η}/F_{η}}(μ_{η}) = μ_{1} ⋯ μ_{ℓ}. Since ind(α ⊗ L_{η}) = ind(α ⊗ F_{η}) < ind(α), by ([11, Proposition 5.8], [18, Proposition 1.17]), there exists a non-empty open subset U_{η} of η such that ind(α) ⊗ F_{U_{η}} < ind(α). Since F_{η} is the completion of F at the discrete valuation given by η, there exist θ_{i} ∈ F^{∗}, 1 ≤ i ≤ ℓ, such that θ_{i}μ_{i}^{−1} ≡ 1 modulo the maximal ideal of R̂_{η}. Let L_{U_{η}} = ∏ F_{U_{η}} and μ_{U_{η}} = (λ(θ_{2} ⋯ θ_{ℓ})^{−1}, θ_{2}, ⋯, θ_{ℓ}) ∈ L_{U_{η}}. Then N_{L_{Uη/F_{Uη}}} (μ_{U_{η}}) = λ. Since α · (θ_{i}) ∈ H^{3}(F_{U_{η}}, μ_{n}^{⊗2}) and α·(θ_{i}) = 0 ∈ H^{3}(F_{η}, μ_{n}^{⊗2}), by ([12, Proposition 3.2.2]), there exists a non-empty open subset V_{η} ⊆ U_{η} such that α·(θ_{i}) = 0 ∈ H^{3}(F_{V_{η}}, μ_{n}^{⊗2}). By replacing U_{η} by V_{η}, we have the required L_{U_{η}} and μ_{U_{η}} ∈ L_{U_{η}}.

Suppose that L_{η}/F_{η} is a field extension of degree ℓ. Let F_{η}^{h} be the henselization of F at the discrete valuation η. Then there exists a field extension L_{η}^{h}/F_{η}^{h} of degree ℓ with an isomorphism ϕ_{η}^{h} : L_{η}^{h} ⊗_{F_{η}^{h}} F_{η} → L_{η}. We identify L^{h} with a subfield of L_{η} through ϕ^{h}. Further if L_{η}/F_{η} is cyclic extension, then L^{h}/F^{h} is also a cyclic extension. Let π̃_{η} ∈ L^{h} be a parameter. Then π̃_{η} is also a parameter in L_{η}.Write μ_{η} = u_{η}π̃_{η}^{r} for some u_{η} ∈ L_{η} a unit at η. Since N_{L_{η}/F_{η}}(μ_{η}) = λ, we have λ = N_{L_{η}/F_{η}}(u_{η})N_{L_{η}/F_{η}}(π̃_{η}). Since u_{η} ∈ L_{η} is a unit at η, N_{L_{η}/F_{η}}(u_{η}) ∈ F_{η} is a unit at η. By ([2, Theorem 1.10]), there exists u^{h} ∈ L_{η}^{h} such that N_{L_{η}^{h}/F_{η}^{h}}(u_{η}^{h}) = N_{L_{η}/F_{η}}(u_{η}). Let μ_{η}^{h} = u_{η}^{h}π̃_{η} ∈ L_{η}^{h}. Since F_{η}^{h} is the filtered direct limit of the fields F_{V}, where V ranges over the non-empty open subset of η ([12, Lemma 2.2.1]), there exists a non-empty open subset U_{η} of η, a field extension L_{U_{η}}/F_{U_{η}} of degree ℓ and μ_{U_{η}} ∈ L_{U_{η}} such that N_{L_{Uη} /F_{Uη}} (μ_{U_{η}}) = λ and there is an isomorphism ϕ_{U_{η}}^{h} : L_{U_{η}} ⊗ F_{η} ≃ L_{η}^{h} with ϕ_{U_{η}}^{h}(μ_{U_{η}}) = μ_{η}^{h}. By shrinking U_{η}, we assume that α · (μ_{U_{η}}) = 0 ∈ H^{3}(L_{U_{η}}, μ_{n}^{⊗2}) ([12, Proposition 3.2.2]). □

Lemma 7.3. Suppose that for each codimension zero point η of X_{0} there exist a field (not necessarily cyclic) or split extension L_{η}/F_{η} of degree ℓ, μ_{η} ∈ F_{η} and for every closed point P of X_{0} there exist a cyclic or split extension L_{P} /F_{P} of degree ℓ and μ_{P} ∈ L_{P} such that for every point x of X_{0}

1) N_{L_{x}/F_{x}}(μ_{x}) = λ

2) α · (μ_{x}) = 0 ∈ H^{3}(L_{x}, μ_{x}^{⊗2})

3) ind(α ⊗ L_{x}) < ind(α)

4) for any branch (P, η) there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} such that for a generator σ of Gal(L_{P} ⊗ F_{P,η}/F_{P,η}) there exists θ_{P,η} ∈ L_{P} ⊗ F_{P,η} such that ϕ_{P,η}(μ_{η})μ_{P}^{−1} = θ_{P,η}^{−ℓ^{d}}σ(θ_{P,η})^{ℓ^{d}}. Then there exist

• a field extension L/F of degree ℓ

• a non-empty open subset U_{η} of η for every codimension zero point η of X_{0} with θ_{U_{η}} ∈ L ⊗ F_{U_{η}}

• for every P ∈ 𝒫 = X_{0} \ ∪U_{η}, θ_{P} ∈ L ⊗ F_{P} such that

1) ind(α ⊗ L) < ind(α)

2) N_{L⊗F_{Uη} /F_{Uη}} (θ_{U_{η}}) = λ and α · (θ_{U_{η}}) = 0 ∈ H^{3}(L ⊗ F_{U_{η}}, μ_{n}^{⊗2}) for all codimension zero points η of X_{0}

3) N_{L⊗F_{P} /F_{P}} (θ_{P}) = λ and α · (θ_{P}) = 0 ∈ H^{3}(L ⊗ F_{P} , μ_{n}^{⊗2}) for all P ∈ 𝒫

4) for any branch (P, η), L ⊗ F_{P,η}/F_{P,η} is cyclic or split and for a generator σ of Gal(L ⊗ F_{P,η}/F_{P,η}) there exists γ_{P,η} ∈ L ⊗ F_{P} such that θ_{U_{η}}θ_{P}^{−1} = γ_{P,η}^{−ℓ^{d}}σ(γ_{P,η})^{ℓ^{d}}. Further if for each x ∈ X_{0}, L_{x}/F_{x} is cyclic or split, then L/F is cyclic.

Proof. Let η be a codimension zero point of X_{0}. By the assumption, there exist a cyclic or split extension L_{η}/F_{η} and μ_{η} ∈ L_{η} such that N_{L_{x}/F_{x}}(μ_{x}) = λ, α · (μ_{x}) = 0 ∈ H^{3}(L_{x}, μ_{x}^{⊗2}) and ind(α ⊗ L_{η}) < ind(α). By (7.2), there exist a non-empty open set U_{η} of η, a cyclic or split extension L_{U_{η}}/F_{U_{η}} of degree ℓ and μ_{U_{η}} ∈ L_{U_{η}} such that N_{L_{Uη} /F_{Uη}} (μ_{η}) = λ, α · (μ_{x}) = 0 ∈ H^{3}(L_{x}, μ_{x}^{⊗2}), ind(α ⊗ L_{U_{η}}) < ind(α), ϕ_{η} : L_{U_{η}} ⊗F_{η} → L_{η} an isomorphism and ϕ_{η}(μ_{U_{η}}) = μ_{η}. By shrinking U_{η}, if necessary, we assume that 𝒫_{0} ∩ U_{η} = ∅.

Let 𝒫 = X_{0} \ ∪_{η}U_{η} and P ∈ 𝒫. Then, by the assumption we have a cyclic or split extension L_{P} /F_{P} of degree ℓ and for every branch (P, η) there is an isomorphism ϕ_{P,η} : L_{η} ⊗F_{P,η} → L_{P} ⊗F_{P,η}. Thus ϕ_{P,U_{η}} = ϕ_{P,η}(ϕ_{η} ⊗1) : L_{U_{η}} ⊗F_{η} ⊗F_{P,η} → L_{P} ⊗F_{P,η} is an isomorphism. Thus, by ([9, Theorem 7.1]), there exists an extension L/F of degree ℓ with isomorphisms ϕ_{U_{η}} : L ⊗ F_{U_{η}} → L_{U_{η}} for all codimension zero points η of X_{0} and ϕ_{P} : L ⊗ F_{P} → L_{P} for all P ∈ 𝒫 such that the following commutative diagram

L ⊗ F_{U_{η}} ⊗ F_{P,η} →^{ϕ_{Uη} ⊗1} L_{U_{η}} ⊗ F_{η} ⊗ F_{P,η}
↓  ↓ ϕ_{P,U_{η}}
L ⊗ F_{P} ⊗ F_{P,η} →^{ϕ_{P} ⊗1}  L_{P} ⊗ F_{P,η}

where the vertical arrow on the left side is the natural map. Further if each L_{x}/F_{x} is cyclic for all x ∈ X_{0}, then L/F is cyclic ([9, Theorem 7.1]).

Since ind(α ⊗L⊗F_{U_{η}}) < ind(α) for all codimension zero points of X_{0} and ind(α ⊗ L ⊗ F_{P}) < ind(α) for all P ∈ 𝒫, by (7.1), ind(α ⊗ L) < ind(α). In particular L is a field.

For every codimension zero point η of X_{0}, let θ_{U_{η}} = (ϕ_{U_{η}})^{−1}(μ_{U_{η}}) ∈ L⊗F_{U_{η}} and for every P ∈ 𝒫, let θ_{P} = (ϕ_{P})^{−1}(μ_{P}) ∈ L ⊗ F_{P}. Since ϕ_{U_{η}} and ϕ_{P} are isomorphisms, we have the required properties. □

Proposition 7.4. Suppose that for each point x of X_{0} there exist a cyclic or split extension L_{x}/F_{x} of degree ℓ and μ_{x} ∈ L_{x} such that

1) N_{L_{x}/F_{x}}(μ_{x}) = λ

2) α · (μ_{x}) = 0 ∈ H^{3}(L_{x}, μ_{x}^{⊗2})

3) ind(α ⊗ L_{x}) < ind(α)

4) for any branch (P, η) there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} such that for generator σ of Gal(L_{P} ⊗ F_{P,η}/F_{P,η}) there exists θ_{P,η} ∈ L_{P} ⊗ F_{P,η} such that ϕ_{P,η}(μ_{η})μ_{P}^{−1} = θ_{P,η}^{−ℓ^{d}}σ(θ_{P,η})^{ℓ^{d}}. Then there exist a cyclic extension L of degree ℓ and μ ∈ L^{∗} such that

• N_{L/F} (μ) = λ and

• α · (μ) = 0 ∈ H^{3}(L, μ_{n}^{⊗2})

• ind(α ⊗ L) < ind(α).

Proof. Let L/F, U_{η}, 𝒫, θ_{U_{η}} and θ_{P} be as in (7.3). Since each L_{x}/F_{x} is cyclic or split, L/F is cyclic. Let σ be a generator of Gal(L/F). Let (P, η) be a branch. By (7.3), there exists γ_{(P,η)} ∈ L ⊗ F_{P,η} such that μ_{U_{η}}μ_{P}^{−1} = γ_{P,η}^{−ℓ^{d}}σ(γ_{P,η}^{ℓ^{d}}). Applying ([10, Theorem 3.6]) for the rational group GL_{1}, there exist γ_{U_{η}} ∈ L⊗F_{U_{η}} and γ_{P} ∈ L⊗F_{P} for every codimension zero point η of X_{0} and P ∈ 𝒫 such that for every branch (P, η), γ_{P,η} = γ_{U_{η}}γ_{P}.

Let μ′_{U_{η}} = μ_{U_{η}}γ_{U_{η}}^{ℓ^{d}}σ(γ_{U_{η}}^{−ℓ^{d}}) ∈ L ⊗ F_{U_{η}} and μ′_{P} = μ_{P} γ_{P}^{−ℓ^{d}}σ(γ_{P}^{ℓ^{d}}) ∈ L ⊗ F_{P}. If (P, η) is a branch, then we have

μ′_{U_{η}} = μ_{U_{η}}γ_{U_{η}}^{ℓ^{d}}σ(γ_{U_{η}}^{−ℓ^{d}})
= μ_{P} θ_{P,η}^{−ℓ^{d}}σ(θ_{P,η}^{ℓ^{d}})γ_{U_{η}}^{ℓ^{d}}σ(γ_{U_{η}}^{−ℓ^{d}})
= μ_{P} γ_{P}^{−ℓ^{d}}σ(γ_{P}^{ℓ^{d}})
= μ′_{P} ∈ L ⊗ F_{P,η}.

Hence, by ([9, Proposition 6.3]), there exists μ ∈ L such that μ = μ′_{U_{η}} and μ = μ′_{P} for every codimension zero point η of X_{0} and P ∈ 𝒫. Clearly N_{L/F} (μ) = λ over F. Let P ∈ 𝒫. Since α · (μ_{P}) = 0 and α · (γ_{P}^{ℓ^{d}}) = 0, α · (μ) = 0 ∈ H^{3}(L ⊗ F_{P} , μ_{n}^{⊗2}). Similarly α · (μ) = 0 ∈ H^{3}(L ⊗ F_{U_{η}}, μ_{n}^{⊗2}) for every codimension zero point η of X_{0}. Hence, by ([12, Theorem 3.1.5]), α · (μ) = 0 in H^{3}(L, μ_{n}^{⊗2}). □

Proposition 7.5. Suppose that for each codimension zero point η of X_{0} there exist a field (not necessarily cyclic) or split extension L_{η}/F_{η} of degree ℓ, μ_{η} ∈ F_{η} and for every closed point P of X_{0} there exist a cyclic or split extension L_{P} /F_{P} of degree ℓ and μ_{P} ∈ L_{P} such that for every point x of X_{0}

1) N_{L_{x}/F_{x}}(μ_{x}) = λ

2) α · (μ_{x}) = 0 ∈ H^{3}(L_{x}, μ_{x}^{⊗2})

3) ind(α ⊗ L_{x}) < ind(α)

4) for any branch (P, η) there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} such that for a generator σ of Gal(L_{P} ⊗ F_{P,η}/F_{P,η}) there exists θ_{P,η} ∈ L_{P} ⊗ F_{P,η} such that ϕ_{P,η}(μ_{η})μ_{P}^{−1} = θ_{P,η}^{−ℓ^{d}}σ(θ_{P,η})^{ℓ^{d}}. Then there exist a field extension N/F of degree coprime to ℓ, a field extension L/F of degree ℓ and μ ∈ (L ⊗ N)^{∗} such that

• N_{L⊗N/N}(μ) = λ and

• α · (μ) = 0 ∈ H^{3}(L ⊗ N, μ_{n}^{⊗2})

• ind(α ⊗ L) < ind(α).

Proof. Let L/F, U_{η}, 𝒫, θ_{U_{η}}, θ_{P} and γ_{P,η} be as in (7.3). Since L/F is a degree ℓ extension, there exists a field extension N/F of degree coprime to ℓ such that L ⊗ N/N is a cyclic extension.

Let 𝒴 be the integral closure of 𝒳 in N and Y_{0} the reduced special fibre of 𝒴 . Let ϕ : Y_{0} → X_{0} be the induced morphism. Let y ∈ Y_{0} and x = ϕ(y) ∈ X_{0}. Then the inclusion F ⊂ N, induces an inclusion F_{x} ⊂ N_{y}. Let L′_{y} = L⊗_{F} F_{x} ⊗_{F_{x}} N_{y}. Since L ⊗ N/N is a cyclic extension of degree ℓ, L′_{y}/N_{y} is either cyclic or split extension of degree ℓ.

Let η′ ∈ Y_{0} be a codimension zero point. Then η = ϕ(η′) ∈ X_{0} is a codimension zero point. Then F_{η} ⊂ F L_{η′}, L⊗F_{U_{η}} ⊂ L⊗F_{η} and θ_{U_{η}} ∈ L⊗F_{η}. Let μ_{η′} = θ_{U_{η}} ⊗1 ∈ L ⊗ F_{η} ⊗_{F_{η}} N_{η′} = L′_{η′}.

Let Q ∈ Y_{0} be a closed point and P = ϕ(Q) ∈ X_{0}. Then P is a closed point of X_{0} and F_{P} ⊂ N_{Q}. Suppose that P ∈ U_{η} for some codimension zero point η of X_{0}. Then F_{U_{η}} ⊂ F_{P}, L ⊗ F_{U_{η}} ⊂ L ⊗ F_{P} and θ_{U_{η}} ∈ L ⊗ F_{P}. Let μ_{Q} = θ_{U_{η}} ⊗ 1 ∈ L ⊗ F_{P} ⊗_{F_{P}} N_{Q} = L′_{Q}. Suppose that P is not in U_{η} for any codimension zero point η of X_{0}. Let μ_{Q} = θ_{P} ⊗ 1 ∈ L ⊗ F_{P} ⊗_{F_{P}} N_{Q}.

Let y ∈ Y_{0} and x = ϕ(x) ∈ X_{0}. Since N_{L_{x}/F_{x}}(μ_{x}) = λ, α · (μ_{x}) = 0 ∈ H^{3}(L_{x}, μ_{n}^{⊗2}) and ind(α ⊗ F_{x}) < ind(α), it follows that N_{L′_{y}/N_{y}}(μ_{y}) = λ, α · (μ_{y}) = 0 ∈ H^{3}(L′_{y}, μ_{n}^{⊗2}) and ind(α ⊗ L′_{y}) < ind(α).

Let (Q, η′) be a branch in Y_{0} and P = ϕ(Q), η = ϕ(η′). Then (P, η) is a branch in X_{0}. The isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} induces an isomorphism ϕ′_{Q,η′} : L′_{η′} ⊗ N_{Q,η′} → L′_{Q} ⊗ N_{Q,η′}. By the choice of μ_{η′} and μ_{Q} it follows that for any generator σ of Gal(L′_{Q} ⊗N_{Q,η′}/N_{Q,η′}) there exists θ_{Q,η′} such that ϕ′_{Q,η′}(μ_{η′})μ_{Q}^{−1} = θ_{Q,η′}^{−ℓ^{d}}σ(θ_{Q,η′})^{ℓ^{d}}. Thus, by (7.4), there exists a cyclic extension L′/N and μ′ ∈ L′ such that N_{L′/N}(μ′) = λ, ind(α ⊗ L′) < ind(α ⊗ N) and α · (μ′) = 0 ∈ H^{3}(L′, μ_{n}^{⊗2}). By the construction we have L′ = L ⊗ N. □

8. Types of points, special points and type 2 connections

Let F, α ∈ H^{2}(F, μ_{n}), λ ∈ F^{∗} with α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), 𝒳 and X_{0} be as in (§7). Further assume that

• 𝒳 is regular such that ram_{𝒳} (α)∪ supp_{𝒳} (λ)∪X_{0} is a union of regular curves with normal crossings.

• the intersection of any two distinct irreducible curves in X_{0} is at most one closed point. We fix the following notation.

• 𝒫 is the set of points of intersection of distinct irreducible curves in X_{0}.

• 𝒪_{𝒳 ,𝒫} is the semi-local ring at the points of 𝒫 on 𝒳 .

• if a codimension zero point η of X_{0} contains a closed point P ∈ 𝒫, then π_{η} ∈ 𝒪_{𝒳 ,𝒫} is a prime defining η on 𝒪_{𝒳 ,𝒫} .

Let η be a codimension zero point of X_{0}. For the rest of this paper, let (E_{η}, σ_{η}) denote the lift of the residue of α at η. Since α ∈ H^{2}(F, μ_{n}) with n a power of ℓ, [E_{η} : F_{η}] is a power of ℓ. If α is unramified at η, then E_{η} = F_{η} and let M_{η} = F_{η}. If α is ramified at η, then E_{η} ≠ F_{η} and there is a unique subextension of E_{η} of degree ℓ and we denote it by M_{η}.

Remark 8.1. Let η be a codimension zero point of X_{0}. Suppose α is ramified at η. Since ind(α ⊗ F_{η}) = ind(α ⊗ E_{η})[E_{η} : F_{η}] (cf. 4.2) and M_{η} ⊂ E_{η}, it follows that ind(α ⊗ M_{η}) < ind(α).

We divide the codimension zero points η of X_{0} as follows:

Type 1: ν_{η}(λ) is coprime to ℓ and ind(α ⊗ F_{η}) = ind(α) Type 2: ν_{η}(λ) is coprime to ℓ and ind(α ⊗ F_{η}) < ind(α) Type 3: ν_{η}(λ) = rℓ, rα ⊗ E_{η} ≠ 0 and ind(α ⊗ F_{η}) = ind(α) Type 4: ν_{η}(λ) = rℓ, rα ⊗ E_{η} ≠ 0 and ind(α ⊗ F_{η}) < ind(α) Type 5: ν_{η}(λ) = rℓ, rα ⊗ E_{η} = 0 and ind(α ⊗ F_{η}) = ind(α)

Type 6: ν_{η}(λ) = rℓ, rα ⊗ E_{η} = 0 and ind(α ⊗ F_{η}) < ind(α).

Let P be a closed point of 𝒳 . Suppose P is the point of intersection of two distinct codimension zero points η_{1} and η_{2} of X_{0}. We say that the point P is a 1) special point of type I if η_{1} is of type 1 and η_{2} is of type 2,

2) special point of type II if η_{1} is of type 1 and η_{2} is of type 4,

3) special point of type III if η_{1} is of type 3 or 5 and η_{2} is of type 4,

4) special point of type IV if η_{1} is of type 1, 3 or 5 and η_{2} is of type 5 with M_{η_{2}} ⊗ F_{P,η_{2}} not a field.

Lemma 8.2. Suppose that η_{1} and η_{2} are two distinct codimension zero points of X_{0} and P a point of intersection of η_{1} and η_{2}. Suppose that α is ramified at η_{1}. Let (E_{η_{1}}, σ_{1}) be the lift of residue of α at η_{1}. If E_{η_{1}} ⊗ F_{P,η_{1}} is not a field, then ind(α ⊗ F_{P}) < ind(α).

Proof. Suppose that E_{η_{1}} ⊗ F_{P,η_{1}} is not a field. Since E_{η_{1}}/F_{η_{1}} is a cyclic extension, E_{η_{1}} ⊗ F_{P,η_{1}} ≃ ∏ E_{η_{1},P} with [E_{η_{1},P} : F_{P,η_{1}}] < [E_{η_{1}} : F_{η_{1}}]. We have (E_{η_{1}}, σ_{1}, π_{η_{1}}) ⊗ F_{P,η_{1}} = (E_{η_{1},P} , σ_{1}, π_{η_{1}}) (cf. §2).

Write α ⊗ F_{η_{1}} = α_{1} + (E_{η_{1}}, σ_{1}, π_{η_{1}}) as in (4.1). Then α ⊗ F_{P,η_{1}} = α_{1} ⊗ F_{P,η_{1}} + (E_{η_{1},P} , σ_{1}, π_{η_{1}}). By (4.2), we have ind(α ⊗ F_{η_{1}}) = ind(α_{1} ⊗ E_{η_{1}})[E_{η_{1}} : F_{η_{1}}]. We have

ind(α ⊗ F_{P,η_{1}}) ≤ ind(α_{1} ⊗ E_{η_{1},P})[E_{η_{1},P} : F_{P,η_{1}}]
≤ ind(α_{1} ⊗ E_{η_{1}})[E_{η_{1},P} : F_{P,η_{1}}]
< ind(α_{1} ⊗ E_{η_{1}})[E_{η_{1}} : F_{η_{1}}]
= ind(α ⊗ F_{η_{1}}).

Thus, by (5.8), ind(α ⊗ F_{P}) < ind(α). □

Lemma 8.3. Let η ∈ X_{0} be a point of codimension zero and P a closed point on η. Let 𝒳_{P} → 𝒳 be the blow-up at P and γ the exceptional curve in 𝒳_{P}. If E_{η} ⊗ F_{P,η} is not a field or η is of type 2, 4 or 6, then γ is of type 2, 4 or 6.

Proof. If E_{η} ⊗ F_{P,η} is not a field, then by (8.2), ind(α ⊗ F_{P}) < ind(α). If η is of type 2, 4 or 6, then ind(α ⊗F_{η}) < ind(α) and hence by (5.8), ind(α ⊗F_{P}) < ind(α). Since F_{P} ⊂ F_{γ}, we have ind(α ⊗ F_{γ}) ≤ ind(α ⊗ F_{P}) < ind(α). Hence γ is of type 2, 4 or 6. □

Lemma 8.4. Let η_{1} and η_{2} be two distinct codimension zero points of X_{0} intersecting at a closed point P. Suppose that η_{1} is of type 1 or 2 and η_{2} is of type 2. Then there exists a sequence of blow-ups ψ : 𝒳 ′ → 𝒳 such that if η̃_{i} are the strict transforms of η_{i}, then

1) ψ : 𝒳 ′ \ ψ^{−1}(P) → 𝒳 \ {P } is an isomorphism

2) ψ^{−1}(P) is the union of irreducible regular curves γ_{1}, ⋯, γ_{m}

3) η̃_{1} ∩ γ_{1} = {P_{0}}, γ_{i} ∩ γ_{i+1} = {P_{i}}, γ_{m} ∩ η̃_{2} = {P_{m}}, η̃ ∩ γ_{i} = ∅ for all i > 1, η̃_{2} ∩ γ_{i} = ∅ for all i < m, η̃_{1} ∩ η̃_{2} = ∅, γ_{i} ∩ γ_{j} = ∅ for all j ≠ i + 1,

4) γ_{1} and γ_{m} are of type 6 and γ_{i}, 1 < i < m are of type 2, 4 or 6,

5) ψ^{−1}(P) has no special points.

Proof. Let 𝒳_{P} → 𝒳 be the blow-up of 𝒳 at P and γ the exceptional curve in 𝒳_{P}. Let η̃_{i} be the strict transform of η_{i}. Then η̃_{1} intersects γ only at one point P_{0} and η̃_{2} intersects γ at only one point P_{1}. Since η_{2} is of type 2, by (8.3), γ is of type 2, 4 or 6 and hence P_{1} is not a special point.

Let s_{1} = ν_{η_{1}}(λ), s_{2} = ν_{η_{2}}(λ). Then ν_{γ}(λ) = s_{1} + s_{2}. Suppose s_{1} + s_{2} = ℓ^{d+1}r_{0} for some integer r_{0}, where ℓ^{d} = ind(α). Since ℓ^{d}α = 0, ℓ^{d}r_{0}α = 0. Thus, γ is of type 6. Hence P_{0} is not a special point and 𝒳_{P} has all the required properties.

Suppose s_{1} + s_{2} = ℓ^{t}r_{0} with t ≤ d and r_{0} coprime to ℓ. Then, blow-up the points P_{0} and P_{1} and let γ_{1} and γ_{2} be the exceptional curves in this blow-up. Then we have η_{γ_{1}}(λ) = 2s_{1} + s_{2} and η_{γ_{2}}(λ) = s_{1} + 2s_{2}. If 2s_{1} + s_{2} is not of the form ℓ^{d+1}r_{1} for some r_{1} ≥ 1, then blow-up, the point of intersection of the strict transform of η_{1} and γ_{1}. If s_{1} + 2s_{2} is not of the form ℓ^{d+1}r_{2} for some r_{2} ≥ 1, then blow-up, the point of intersection of the strict transform of η_{2} and γ_{2}. Since s_{1} and s_{2} are coprime to ℓ, there exist i and j such that is_{1} + s_{2} = ℓ^{d+1}r and s_{1} + js_{2} = ℓ^{d+1}r′ for some r, r′ ≥ 1. Thus, we get the required finite sequence of blow-ups. □

Proposition 8.5. There exists a regular proper model of F with no special points.

Proof. Let P ∈ 𝒫. Then there exist two codimension zero points η_{1} and η_{2} of X_{0} intersecting at P.

Suppose that P is a special point of type I. Let ψ : 𝒳 ′ → 𝒳 be a sequence of blow-ups as in (8.4). Then there are no special points in ψ^{−1}(P). Since there are only finitely many special points in 𝒳 , replacing 𝒳 by a finite sequence of blow ups at all special points of type I, we assume that 𝒳 has no special points of type I.

Suppose P is a special point of type II. Without loss of generality we assume that, η_{1} is of type 1 and η_{2} is of type 4. Let 𝒳_{P} → 𝒳 be the blow-up of 𝒳 at P and γ the exceptional curve in 𝒳_{P}. Since η_{2} is of type 4, by (8.3), γ is of type 2, 4 or 6. Since η_{1} is of type 1 and η_{2} is of type 4, ν_{η_{1}}(λ) is coprime to ℓ and ν_{η_{2}}(λ) is divisible by ℓ. Since ν_{γ}(λ) = ν_{η_{1}}(λ) + ν_{η_{2}}(λ), ν_{γ}(λ) is coprime to ℓ and hence γ is of type 2. Let η̃_{i} be the strict transform of η_{i} in 𝒳_{P}. Then η̃_{i} and γ intersect at only one point Q_{i}. Since γ is of type 2, Q_{1} is a special point of type I and Q_{2} is not a special point. Thus, as above, by replacing 𝒳 by a sequence of blow-ups of 𝒳 , we assume that 𝒳 has no special points of type I or II.

Suppose P is a special point of type III. Without loss of generality assume that η_{1} is of type 3 or 5 and η_{2} of type 4. Let 𝒳_{P} → 𝒳 be the blow-up of 𝒳 at P, γ, η̃_{i}, and Q_{i} be as above. Since η_{2} is of type 4, by (8.3), γ is of type 2, 4 or 6. Since ν_{η_{1}}(λ) and ν_{η_{2}}(λ) are divisible by ℓ, ν_{γ}(λ) = ν_{η_{1}}(λ) + ν_{η_{2}}(λ) is divisible by ℓ. Thus γ is of type 4 or 6. Hence Q_{2} is not a special point. By (5.7), α ⊗ F_{P} = (E_{P} , σ, uπ_{η_{1}}^{d_{1}}π_{η_{2}}^{d_{2}}) for some cyclic extension E_{P} /F_{P} and u ∈ Â_{P} a unit and at least one of d_{i} is coprime to

ℓ (in fact equal to 1). In particular, α ⊗F_{P} is split by the extension F_{P} ( \sqrt[m]{uπ_{η_{1}}^{d_{1}}π_{η_{2}}^{d_{2}}} ), where m is the degree of E_{P} /F_{P} which is a power of ℓ. Suppose d_{1} + d_{2} is coprime

to ℓ. Since ν_{γ}(π_{η_{1}}^{d_{1}}π_{η_{2}}^{d_{2}}) = d_{1} + d_{2}, F_{P} ( \sqrt[m]{uπ_{η_{1}}^{d_{1}}π_{η_{2}}^{d_{2}}} ) is totally ramified at γ. Thus, by (4.3), γ is of type 6. Hence Q_{1} is not a special point. Suppose that d_{1} +d_{2} is divisible by ℓ. Let π_{γ} be a prime defining γ at Q_{1}. Then, we have uπ_{η_{1}}^{d_{1}}π_{η_{2}}^{d_{2}} = w_{1}π_{η_{1}}^{d_{1}}π_{γ}^{d_{1}+d_{2}} for some unit w_{1} at Q_{1}. Since one of d_{i} is coprime to ℓ and d_{1} + d_{2} is divisible by ℓ, d_{i} are not divisible by ℓ. In particular 2d_{1} + d_{2} is coprime to ℓ. Let 𝒳_{Q_{1}} be the blow-up of 𝒳_{P} at Q_{1} and γ′ be the generic point of the exceptional curve in 𝒳_{Q_{1}}. Then ν_{γ′}(uπ_{η_{1}}^{d_{1}}π_{η_{2}}^{d_{2}}) = ν_{γ′}(w_{1}π_{η_{1}}^{d_{1}}π_{γ}^{d_{1}+d_{2}}) = 2d_{1} + d_{2}. Since 2d_{1} + d_{2} is coprime to ℓ, once again by (4.3), γ′ is of type 6. In particular no point on the exceptional curve in 𝒳_{Q_{1}} is a special point. Thus, replacing 𝒳 by a sequence of blow-ups, we assume that 𝒳 has no special points of type I, II or III.

Suppose P is a special point of type IV. Without loss of generality assume that, η_{1} is of type 1, 3 or 5 and η_{2} is of type 5, with M_{η_{2}} ⊗ F_{P,η_{2}} not a field. Let 𝒳_{P} → 𝒳 be the blow-up of 𝒳 at P and γ, η̃_{i}, Q_{i} be as above. Since M_{η_{2}} ⊗ F_{P,η_{2}} is not a field, by (8.3), γ is of type 2, 4 or 6. If γ is of type 6, then Q_{1} and Q_{2} are not special points. Suppose γ is of type 2 or 4. Then Q_{1} and Q_{2} are special points of type I, II or III. Thus, as above, by replacing 𝒳 by a sequence of blow-ups of 𝒳 , we assume that 𝒳 has no special points. □

Let η and η′ be two codimension zero points of X_{0} (may not be distinct). We say that there is a type 2 connection from η to η′ if one of the following holds

• one of η or η′ is of type 2

• there exist distinct codimension zero points η_{1}, ⋯, η_{n} of X_{0} of type 2 such that η intersects η_{1}, η′ intersects η_{n}, η_{i} intersects η_{i+1} for all 1 ≤ i ≤ n − 1, η does not intersect η_{i} for i > 1, η′ does not intersect η_{i} for i < n and η_{i} does not intersect η_{j} for j ≠ i + 1.

Proposition 8.6. There exists a regular proper model 𝒳 of F such that

1) 𝒳 has no special points

2) if η_{1} and η_{2} are two (not necessarily distinct) codimension zero points of X_{0} with η_{1} of type 3 or 5 and η_{2} of type 3, 4 or 5, then there is no type 2 connection between η_{1} and η_{2}.

Proof. Let 𝒳 be a regular proper model with no special points (8.5). Let m(𝒳 ) be the number of type 2 connections between a point of type 3 or 5 and a point of type 3, 4 or 5. We prove the proposition by induction on m(𝒳 ). Suppose m(𝒳 ) ≥ 1. We show that there is a sequence of blow-ups 𝒳 ′ of 𝒳 with no special points and m(𝒳 ′) < m(𝒳 ).

Let η be a codimension zero point of X_{0} of type 3 or 5 and η′ a codimension zero point of X_{0} of types 3, 4 or 5. Suppose η and η′ have a type 2 connection. Then there exist η_{1}, ⋯, η_{n} codimension zero points of X_{0} of type 2 with η intersecting η_{1}, η′ intersecting η_{n} and η_{i} intersecting η_{i+1} for i = 1, ⋯, n − 1.

Suppose n = 1. Let Q be the point of the intersection of η and η_{1}. Let 𝒳_{Q} → 𝒳 be the blow-up of 𝒳 at Q and γ the exceptional curve in 𝒳_{Q}. Since η_{1} is of type 2, by (8.3), γ is of type 2, 4 or 6. Since η is of type 3 or 5 and η_{1} is of type 2, ℓ divides ν_{η}(λ) and ℓ does not divide ν_{η_{1}}(λ). Since ν_{γ}(λ) = ν_{η}(λ) + ν_{η_{1}}(λ), ν_{γ}(λ) is not divisible by ℓ and hence γ is of type 2. Let η̃ and η̃_{1} be the strict transform of η and η_{1} in 𝒳_{Q}. Since γ is a point of type 2, the points of intersection of η̃ and η̃_{1} with γ are not special points. Hence 𝒳_{Q} has no special points. By replacing 𝒳 by 𝒳_{Q} we assume that n ≥ 2 and 𝒳 has no special points.

Let P be the point of intersection of η_{1} and η_{2}. Let 𝒳 ′ be as in (8.4). Then 𝒳 ′ has no special points and all the exceptional curves in 𝒳 ′ are of type 2, 4 or 6 and the exceptional curves which intersect the strict transforms of η_{1} and η_{2} are of type 6. In particular the number of type 2 connections between the strict transforms of η and η′ is one less than the number of type 2 connections between η and η′. Since all the exceptional curves in 𝒳 ′ are of type 2, 4 or 6, m(𝒳 ′) = m(𝒳 ) − 1. Thus, by induction, we have a regular proper model with required properties. □

Lemma 8.7. Let 𝒳 be as in (8.6) and X_{0} the special fibre of 𝒳 . Let η be a codimension zero point of X_{0} of type 2 and η′ a codimension zero point of X_{0} of type 3 or 5. Suppose there is a type 2 connection from η to η′. If there is a type 2 connection from η to a type 3 or 5 point η′′, then η′ = η′′. Further, if η_{1}, ⋯, η_{n} are codimension zero points of X_{0} of type 2 giving a type 2 connection from η to η′ and γ_{1}, ⋯, γ_{m} codimension zero points of X_{0} of type 2 giving another type 2 connection from η to η′, then n = m and η_{i} = γ_{i} for all i.

Proof. Suppose η′′ is a codimension zero point of X_{0} of type 3 or 5 with type 2 connection to η. Since η is of type 2 and there is a type 2 connection from η′ to η′′. Since no two points of type 3 or 5 have a type 2 connection (cf. 8.6), η′ = η′′. Suppose γ_{1}, ⋯, γ_{m} is of type 2 connection from η to η′. If η_{i} is not equal to γ_{i}, then we will have type 2 connection from η′ to η′ and hence a contradiction to the choice of 𝒳 (cf. 8.6). Thus n = m and η_{i} = γ_{i} for all i. □

Let η be a codimension zero point of X_{0} of type 2 and η′ be a codimension zero point of X_{0} of type 3 or 5. Suppose there is a type 2 connection η_{1}, ⋯, η_{n} from η to η′. Then, by (8.7), η′ and η_{n} are uniquely defined by η. We call this point of intersection of η_{n} with η′ as the point of type 2 intersection of η and η′. Once again note that such a closed point is uniquely defined by η.

9. Choice of L_{P} and μ_{P} at closed points

Let F, α ∈ H^{2}(F, μ_{n}), λ ∈ F^{∗} with α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), 𝒳 and X_{0} be as in (§7 and §8). Throughout this section we assume that 𝒳 has no special points and if η_{1} and η_{2} are two (not necessarily distinct) codimension zero points of X_{0} with η_{1} is of type 3 or 5 and η_{2} is of type 3, 4 or 5, then there is no type 2 connection between η_{1} and η_{2}.

Let η be a codimension zero point of X_{0} of type 5. Then we call η of type 5a if α is unramified at η and of type 5b if α is ramified at η. Suppose η is of type 5b. Then α is ramified and hence M_{η} is the unique subextension of E_{η} of degree ℓ, where (E_{η}, σ_{η}) is the lift of the residue of α.

For the rest of the paper we assume that κ is a finite field.

Lemma 9.1. Let η be a codimension zero point of X_{0} of type 5b. Then ind(α⊗M_{η}) < ind(α) and there exists μ_{η} ∈ M_{η} such that N_{M_{η}/F_{η}}(μ_{η}) = λ and α · (μ_{η}) = 0 ∈ H^{3}(M_{η}, μ_{n}^{⊗2}).

Proof. Since η is of type 5b, α is ramified at η, ν_{η}(λ) = rℓ and rα⊗E_{η} = 0. By (8.1), ind(α ⊗ M_{η}) < ind(α). Write α ⊗ F_{η} = α′ + (E_{η}, σ, π_{η}) as in (4.1) and λ = θ_{η}π_{η}^{rℓ} where θ_{η} is a unit at η and π_{η} is a parameter at η. Let β_{0} be the image of α′ in H^{2}(κ(η), μ_{n}). Since α′ ⊗ E_{η} = α ⊗ E_{η} and rα ⊗ E_{η} = 0, rβ_{0} ⊗ E(η) = 0. Let θ_{0} be the image of θ_{η} in κ(η). Then, by (3.5), there exists μ_{0} ∈ M_{η}(η), such that N_{M_{η}(η)/κ(η)}(μ_{0}) = θ_{0} and rβ_{0} ⊗ M_{η}(η) = (E_{η}(η), σ, μ_{0}). Thus, by (4.8), there exists μ_{η} ∈ M_{η} with the required properties. □

Lemma 9.2. Let P ∈ 𝒫, η_{1} and η_{2} be codimension zero points of X_{0} containing P. Suppose that η_{1} and η_{2} are of type 5. Then there exist a cyclic field extension L_{P} /F_{P} of degree ℓ and μ_{P} ∈ L_{P} such that

1) N_{L_{P} /F_{P}} (μ_{P}) = λ,

2) ind(α ⊗ L_{P}) < ind(α),

3) α · (μ_{P}) = 0 ∈ H^{3}(L_{P} , μ_{n}^{⊗2}),

4) if η_{i} is of type 5a, then L_{P} ⊗ F_{P,η_{i}}/F_{P,η_{i}} is an unramified field extension,

5) if η_{i} is of type 5b, then L_{P} ⊗ F_{P,η_{i}} ≃ M_{η_{i}} ⊗ F_{P,η_{i}}.

Proof. Since 𝒳 has no special points, P is not a special point of type IV. Since η_{1} and η_{2} are of type 5 intersecting at P, M_{η_{1}} ⊗F_{P,η_{1}} and M_{η_{2}} ⊗F_{P,η_{2}} are fields. Suppose η_{i} is of type 5a. If α ⊗ F_{P,η_{i}} = 0, then let L_{P,η_{i}}/F_{P,η_{i}} be any cyclic unramified field extension with λ a norm and μ_{η_{i}} ∈ L_{P,η_{i}} with N_{L_{P,η_{i}}/F_{P,η_{i}}}(μ_{η_{i}}) = λ. If α ⊗ F_{P,η_{i}} ≠ 0, then let L_{P,η_{i}}/F_{P,η_{i}} be a cyclic unramified field extension of degree ℓ and μ_{η_{i}} be as in (4.10). Suppose η_{i} is of type 5b. Let L_{P,η_{i}} = M_{η_{i}} ⊗ F_{P,η_{i}} and μ_{η_{i}} ∈ M_{η_{i}} be as in (9.1). Then, by choice L_{P,η_{i}}/F_{P,η_{i}} are unramified field extensions. By applying (6.4) to L_{P,η_{i}} and μ_{η_{i}}, there exist a cyclic field extension L_{P} /F_{P} and μ_{P} ∈ L_{P} with required properties. □

Lemma 9.3. Let η be a codimension zero point of X_{0} of type 3 and P a closed point on the closure of η. Then, there exists a cyclic field extension L_{P,η}/F_{P,η} of degree ℓ such that if α ⊗ E_{η} ⊗ F_{P,η} ≠ 0, then ind(α ⊗ E_{η} ⊗ L_{P,η}) < ind(α ⊗ E_{η} ⊗ F_{P,η}).

Proof. Since E_{η}/F_{η} is a cyclic unramified field extension of degree a power of ℓ, E_{η} ⊗ F_{P,η} ≃ ∏ E_{P,η} for some cyclic field extension E_{P,η}/F_{P,η} of degree a power of ℓ. Let E(η)_{P} be the residue field of E_{P,η}. Then E(η)_{P} /κ(η)_{P} is a cyclic extension of degree a power of ℓ. Note that either E(η)_{P} = κ(η)_{P} or E(η)_{P} /κ(η)_{P} is a cyclic extension of degree a positive power of ℓ. If E(η)_{P} = κ(η)_{P}, then let L(η)_{P} /κ(η)_{P} be any cyclic extension of degree ℓ. Suppose E(η)_{P} ≠ κ(η)_{P}. Since E(η)_{P} /κ(η)_{P} is a cyclic extension of degree a positive power of ℓ, there is only one subextension of E(η)_{P} which is cyclic over κ(η)_{P} of degree ℓ. Since κ(η)_{P} is a local field containing primitive ℓ^{th} root of unity, there are at least 2 non-isomorphic cyclic field extensions of κ(η)_{P} of degree ℓ. Thus there exists a cyclic field extension L(η)_{P} /κ(η)_{P} of degree ℓ which is not isomorphic to a subfield of E(η)_{P}. Let L_{P,η}/F_{P,η} be the unramified extension of degree ℓ with residue field L(η)_{P}.

Suppose α ⊗ E_{η} ⊗ F_{P,η} ≠ 0. Then α ⊗ E_{P,η} ≠ 0. Since E_{P,η} and L_{P,η} are cyclic field extensions of F_{P,η} and L_{P,η} is not isomorphic to a subfield of E_{P,η}, E_{P,η} ⊗ L_{P,η} is a field and [E_{P,η} ⊗ L_{P,η} : E_{P,η}] = [L_{P,η} : F_{P,η}] = ℓ. In particular E(η)_{P} ⊗ L(η)_{P} is a field and [E(η)_{P} ⊗ L(η)_{P} : E(η)_{P}] = ℓ. Write α ⊗ F_{η} = α′ + (E_{η}, σ_{η}, π_{η}) as in (4.1). Since α ⊗ E_{η} = α′ ⊗ E_{η}, α ⊗ E_{η} is unramified at η. Let β_{P} be the image of α ⊗ E_{η} ⊗ F_{P,η} in H^{2}(E(η)_{P} , μ_{n}). Since α ⊗ E_{P,η} ≠ 0 and E_{P,η} is a completely discretely valued field with residue field E(η)_{P}, β_{P} ≠ 0. Since E(η)_{P} is a local field, ind(β_{P} ⊗ E(η)_{P} ⊗ L(η)_{P}) < ind(β_{P}) and hence ind(α ⊗ E_{η} ⊗ L_{P,η}) = ind(α ⊗ E_{P,η} ⊗ L_{P,η}) < ind(α ⊗ E_{P,η}) = ind(α ⊗ E_{η} ⊗ F_{P,η}). □

Lemma 9.4. Let P ∈ 𝒫, η_{1} and η_{2} be codimension zero points of X_{0} containing P. Suppose that η_{1} is of type 2 and η_{2} is of type 5 or 6. Then there exist μ_{i} ∈ F_{P}, 1 ≤ i ≤ ℓ, such that

1) μ_{1} ⋯ μ_{ℓ} = λ,

2) ν_{η_{1}}(μ_{1}) = ν_{η_{1}}(λ), ν_{η_{1}}(μ_{i}) = 0 for i ≥ 2,

3) ν_{η_{2}}(μ_{i}) = ν_{η_{2}}(λ)/ℓ for all i ≥ 1,

4) α · (μ_{i}) = 0 ∈ H^{3}(F_{P} , μ_{n}^{⊗2}).

Proof. Since η_{1} is of type 2 and η_{2} is of type 5 or 6, we have λ = wπ_{η_{1}}^{r_{1}}π_{η_{2}}^{r_{2}ℓ} with r_{1} coprime to ℓ and r_{2}α ⊗ E_{η_{2}} = 0. Hence, by (6.7), there exists θ ∈ F_{P} such that α · (θ) = 0, ν_{η_{1}}(θ) = 0 and ν_{η_{2}}(θ) = r_{2}. For i ≥ 2, let μ_{i} = θ and μ_{1} = λθ^{1−ℓ}. Then μ_{i} have the required properties. □

Lemma 9.5. Let P ∈ 𝒫, η_{1} and η_{2} be codimension zero points of X_{0} containing P. Suppose that η_{1} and η_{2} are of type 5 or 6. Then there exist μ_{i} ∈ F_{P}, 1 ≤ i ≤ ℓ, such that 1) μ_{1} ⋯ μ_{ℓ} = λ,

2) ν_{η_{j}}(μ_{i}) = ν_{η_{j}}(λ)/ℓ for all i ≥ 0 and j = 1, 2,

4) α · (μ_{i}) = 0 ∈ H^{3}(F_{P} , μ_{n}^{⊗2}).

Proof. Since η_{1} and η_{2} are of type 5 or 6, by (6.8), there exists θ ∈ F_{P} such that α · (θ) = 0 and ν_{η_{i}}(θ) = ν_{η_{i}}(λ)/ℓ for i = 1, 2. For i ≥ 2, let μ_{i} = θ ∈ F_{P} and μ_{1} = λθ^{1−ℓ} ∈ F_{P}. Then μ_{i} have the required properties. □

Lemma 9.6. Let P ∈ 𝒫, η_{1} be a codimension zero point of X_{0} of type 3 and η_{2} a codimension zero point of X_{0} of type 5. Suppose η_{1} and η_{2} intersect at P. Then there exist a cyclic field extension L_{P} /F_{P} of degree ℓ and μ_{P} ∈ L_{P} such that

1) N_{L_{P} /F_{P}} (μ_{P}) = λ

2) ind(α ⊗ L_{P}) < ind(α)

3) α · (μ_{P}) = 0 ∈ H^{3}(L_{P} , μ_{n}^{⊗2})

4) L_{P} ⊗ F_{P,η_{i}}/F_{P,η_{i}} is an unramified field extension

5) if λ ∈ F_{P}^{∗ℓ} and α ⊗ E_{η_{1}} ⊗ F_{P,η_{1}} ≠ 0, then ind(α ⊗ (E_{η_{1}} ⊗ F_{P,η_{1}}) ⊗ (L_{P} ⊗ F_{P,η_{1}})) < ind(α ⊗ E_{η_{1}} ⊗ F_{P,η_{1}})

6) if η_{2} is of type 5b, then L_{P} ⊗ F_{P,η_{2}} ≃ M_{η_{2}} ⊗ F_{P,η_{2}}.

Proof. Suppose λ ∉ F_{P}^{∗ℓ}. Let L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}. Then N_{L_{P} /F_{P}} (μ_{P}) = λ and by (6.2) 2) and 3) are satisfied. Since η_{i} is of type 3 or 5, ν_{η_{i}}(λ) is divisible by ℓ and hence 4) is satisfied. Since λ ∉ F_{P}^{∗ℓ}, the case 5) does not arise. Suppose that η_{2} is of type 5b. Since 𝒳 has no special points, M_{η_{2}} ⊗ F_{P,η_{2}} is a field. Since λ is a norm from M_{η_{2}} (9.1), by (2.6), we have L_{P} ⊗ F_{P,η_{2}} ≃ M_{η_{2}} ⊗ F_{P,η_{2}}.

Suppose that λ ∈ F_{P}^{∗ℓ}. Let L_{P,η_{1}} be as in (9.3) and μ_{P,η_{1}} = \sqrt[ℓ]{λ}. Write α ⊗ F_{η_{1}} = α_{1} ⊗ (E_{η_{1}}, σ_{1}, π_{η_{1}}) as in (4.1). Then by (4.2), we have ind(α ⊗ F_{η_{1}}) = ind(α ⊗ E_{η_{1}})[E_{η_{1}} : F_{η_{1}}]. Since η_{1} is of type 3, ind(α) = ind(α ⊗ F_{η_{1}}) and r_{1}α ⊗ E_{η_{1}} ≠ 0, where ν_{η_{1}}(λ) = r_{1}ℓ. In particular α ⊗ E_{η_{1}} ≠ 0. By the choice of L_{P,η_{1}} as in (9.3), we have either α ⊗ E_{η_{1}} ⊗ F_{P,η_{1}} = 0 or ind(α ⊗ E_{η_{1}} ⊗ L_{P,η_{1}}) < ind(α ⊗ E_{η_{1}} ⊗ F_{P,η_{1}}). Thus ind(α ⊗ E_{η_{1}} ⊗ F_{P,η_{1}}) < ind(α ⊗ E_{η_{1}}). We have ind(α ⊗ L_{P,η_{1}}) ≤ ind(α ⊗ E_{η_{1}} ⊗ L_{P,η_{1}})[E_{η_{1}} ⊗ L_{P,η_{1}} : L_{P,η_{1}}] < ind(α ⊗ E_{η_{1}})[E_{η_{1}} : F_{η_{1}}] = ind(α). Since L_{P,η_{1}} is a field and cores_{L_{P,η_{1}}/F_{P,η_{1}}}(α·(μ_{P,η_{1}})) = α·(λ) = 0, by (4.6), α·(μ_{P,η_{1}}) = 0 ∈ H^{3}(L_{P,η_{1}}, μ_{n}^{⊗2}). Since 𝒳 has no special points, M_{η_{2}} ⊗ F_{P,η_{2}} is a field. Let L_{P,η_{2}} = M_{η_{2}} ⊗ F_{P,η_{2}} and μ_{P,η_{2}} = \sqrt[ℓ]{λ}. Then N_{L_{P,η_{2}}/F_{P,η_{2}}}(μ_{P,η_{2}}) = λ and by (9.1), ind(α ⊗ L_{P,η_{2}}) < ind(α) and α · (μ_{P,η_{2}}) = 0. Then, by (6.4), there exist L_{P} and μ_{P} with required properties. □

Lemma 9.7. Let P ∈ 𝒫, η_{1} and η_{2} be codimension zero points of X_{0} of type 3, 4 or 6. Suppose η_{1} and η_{2} intersect at P. Then there exist a cyclic field extension L_{P} /F_{P} of degree ℓ and μ_{P} ∈ L_{P} such that

1) N_{L_{P} /F_{P}} (μ_{P}) = λ

2) ind(α ⊗ L_{P}) < ind(α)

3) α · (μ_{P}) = 0 ∈ H^{3}(L_{P} , μ_{n}^{⊗2})

4) L_{P} ⊗ F_{P,η_{i}}/F_{P,η_{i}} is an unramified field extension,

5) if η_{i} is of type 3, λ ∈ F_{P}^{∗ℓ} and α ⊗ E_{η_{i}} ⊗ F_{P,η_{i}} ≠ 0, then ind(α ⊗ (E_{η_{i}} ⊗ F_{P,η_{i}}) ⊗ (L_{P} ⊗ F_{P,η_{i}})) < ind(α ⊗ E_{η_{i}} ⊗ F_{P,η_{i}}).

Proof. Suppose λ ∉ F_{P}^{∗ℓ}. Let L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}. Then N_{L_{P} /F_{P}} (μ_{P}) = λ and by (6.2), 2) and 3) are satisfied. Since η_{i} are of type 3, 4 or 6, ν_{η_{i}}(λ) is divisible by ℓ and hence 4) is satisfied. Since λ ∉ F_{P}^{∗ℓ}, 5) does not arise.

Suppose that λ ∈ F_{P}^{∗ℓ}. If η_{i} is of type 3, then let L_{P,η_{1}} be as in (9.3) and μ_{P,η_{1}} = \sqrt[ℓ]{λ}. Then, as in (9.6), N_{L_{P} /F_{P}} (μ_{P}) = λ, ind(α ⊗ L_{P,η_{i}}) < ind(α) and α · (μ_{P}) = 0 ∈ H^{3}(L_{P} , μ_{n}^{⊗2}). Suppose that η_{i} is of type 4 or 6. Let L_{P,η_{i}}/F_{P,η_{i}} be a cyclic unramified field extension of degree ℓ and μ_{P,η_{i}} be as in (4.10).

Then, by (6.4), there exist L_{P} and μ_{P} with required properties. □

Proposition 9.8. Let P ∈ 𝒫. Then there exist a cyclic field extension or split extension L_{P} /F_{P} of degree ℓ and μ_{P} ∈ L_{P} such that

1) N_{L_{P} /F_{P}} (μ_{P}) = λ

2) ind(α ⊗ L_{P}) < ind(α)

3) α · (μ_{P}) = 0 ∈ H^{3}(L_{P} , μ_{n}^{⊗2}) Further, suppose η is a codimension zero point of X_{0} containing P.

4) If η is of type 1, then L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}.

5) Suppose η is of type 2 with a type 2 connection to a type 5 point η′. Let Q be the type 2 intersection of η and η′. If M_{η′} ⊗ F_{Q,η′} is not a field, then L_{P} = ∏ F_{P} and μ_{P} = (θ_{1}, ⋯, θ_{ℓ}) with θ_{i} ∈ F_{P}, ν_{η}(θ_{1}) = ν_{η}(λ) and ν_{η}(θ_{i}) = 0 for i ≥ 2.

6) Suppose η is of type 2 with a type 2 connection to a type 5 point η′. Let Q be the type 2 intersection of η and η′. If M_{η′} ⊗ F_{Q,η′} is a field, then L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}.

7) Suppose η is of type 2 and there is no type 2 connection from η to any type 5 point. Then L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}.

8) If η is of type 3, then L_{P} ⊗ F_{P,η}/F_{P,η} is an unramified field extension and further if λ ∈ F_{P}^{∗ℓ} and α ⊗ E_{η} ⊗ F_{P,η} ≠ 0, then ind(α ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P} ⊗ F_{P,η})) < ind(α ⊗ E_{η} ⊗ F_{P,η}).

9) If η is of type 4, then L_{P} ⊗ F_{P,η}/F_{P,η} is an unramified field extension.

10) If η is of type 5a, then L_{P} ⊗ F_{P,η}/F_{P,η} is an unramified field extension.

11) If η is of type 5b, then L_{P} ⊗ F_{P,η} ≃ M_{η} ⊗ F_{P,η} and if L_{P} = ∏ F_{P}, then μ_{P} = (θ_{1}, ⋯, θ_{ℓ}) with ν_{η}(θ_{i}) = ν_{η}(λ)/ℓ.

12) If η is of type 6, then either L_{P} ⊗ F_{P,η}/F_{P,η} is an unramified field extension or L_{P} = ∏ F_{P}, with μ_{P} = (θ_{1}, ⋯, θ_{ℓ}) and ν_{η}(θ_{i}) = ν_{η}(λ)/ℓ.

Proof. Let η_{1} and η_{2} be two codimension zero points intersecting at P. By the choice of 𝒳 , X_{0} is a union of regular curves with normal crossings and hence there are no other codimension zero points of X_{0} passing through P.

Case I. Suppose that either η_{1} or η_{2}, say η_{1}, is of type 1. Then ν_{η_{1}}(λ) is coprime to ℓ and hence λ ∉ F_{P}^{∗ℓ}. Let L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}. Then, by (6.2), L_{P} and μ_{P} satisfy 1), 2) and 3). By choice 4) is satisfied. Since 𝒳 has no special points, η_{2} is not of type 2 or 4. Thus 5), 6), 7) and 9) do not arise. Suppose η_{2} is of type 3, 5 or 6. Then ν_{η_{2}}(λ) is divisible by ℓ and hence L_{P} ⊗ F_{P,η_{2}}/F_{P,η_{2}} is an unramified field extension. Thus 8), 10) and 12) are satisfied. Suppose η_{2} is of type 5b. Since 𝒳 has no special points and η_{1} is of type 1, M_{η_{2}} ⊗ F_{P,η_{2}} is a field. Since λ is a norm from the extension M_{η_{2}}/F_{η_{2}} (9.1) and λ ∉ F_{P,η_{2}}^{∗ℓ}, by (2.6 ), M_{η_{2}} ⊗ F_{P,η_{2}} ≃ F_{P,η_{2}}(\sqrt[ℓ]{λ}) and hence 11) is satisfied.

Case II. Suppose neither η_{1} nor η_{2} is of type 1. Suppose either η_{1} or η_{2} is of type 2, say η_{1} is of type 2. Then ν_{η_{1}}(λ) is coprime to ℓ and hence λ ∉ F_{P}^{∗ℓ}.

Suppose that η_{1} has type 2 connection to a codimension zero point η′ of X_{0} of type 5. Let Q be the closed point on η′ which is the type 2 intersection point of η_{1} and η′. By the choice of 𝒳 , η_{2} is of type 2, 5 or 6. Note that if η_{2} is also of type 2, then Q is also the point of type 2 intersection of η_{2} and η′. Thus if both η_{1} and η_{2} are of type 2, η′ and Q do not depend on whether we start with η_{1} or η_{2}.

Suppose that M_{η′} ⊗ F_{Q,η′} is not a field. Let L_{P} = ∏ F_{P}. Suppose η_{2} is of type 2. Then, let μ_{P} = (λ, 1, ⋯, 1) ∈ L_{P} = ∏ F_{P}. Suppose η_{2} is of type 5. Then by the assumption on 𝒳 , η_{2} = η′, Q = P. Thus M_{η_{2}} ⊗F_{P,η_{2}} = M_{η′} ⊗F_{Q,η′} is not a field and hence η_{2} is of type 5b. Let μ_{i} ∈ F_{P} be as in (9.4) and μ_{P} = (μ_{1}, ⋯, μ_{ℓ}). Suppose η_{2} is of type 6. Let μ_{i} ∈ F_{P} be as in (9.4) and μ_{P} = (μ_{1}, ⋯, μ_{ℓ}) ∈ L_{P}. Then, L_{P} and μ_{P} satisfy 1) and 3). Since η_{1} is of type 2, ind(α ⊗ F_{η_{1}}) < ind(α) and hence, by (5.8), ind(α ⊗ F_{P}) < ind(α) and 2) is satisfied. Since neither η_{1} nor η_{2} is of type 1, the case 4) does not arise. By choice L_{P} satisfies 5). Since there is only one type 5 point with a type 2 connection to η_{1} or η_{2}, the case 6) does not arise. Clearly the case 7) does not arise. Since η_{2} is not of type 3, 4 or 5a, the cases 8), 9) and 10) do not arise. By choice of L_{P} and μ_{P}, 11) and 12) are satisfied.

Suppose M_{η′} ⊗ F_{Q,η′} is a field. Let L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}. Since λ ∉ F_{P}^{∗ℓ}, by (6.2), L_{P} and μ_{P} satisfy 1), 2) and 3). As above the cases 4), 5), 7) and 8) do not arise. By choice 6) is satisfied. Suppose η_{2} is of type 5. Then η_{2} = η′, Q = P and ν_{η_{2}}(λ) is divisible by ℓ and hence 9) is satisfied. Suppose η_{2} is of type 5b. Since M_{η_{2}} ⊗F_{P,η_{2}} is a field, as in case I, M_{η_{2}} ⊗F_{P,η_{2}} ≃ L_{P} ⊗F_{P,η_{2}} and hence 10) is satisfied. Since λ ∉ F_{P}^{∗ℓ} and if η_{2} is of type 6, ν_{η_{2}}(λ) is divisible by ℓ, L_{P} ⊗ F_{P,η_{2}}/F_{P,η_{2}} is an unramified field extension and hence 11) is satisfied.

Suppose that neither η_{1} nor η_{2} have a type 2 connection to a point of type 5. In particular η_{2} is not of type 5. Then, let L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}. Then, by (6.2), L_{P} and μ_{P} satisfy 1), 2) and 3). Since neither η_{1} nor η_{2} is of type 1, the case 4) does not arise. Since neither η_{1} nor η_{2} has type 2 connection to a point of type 5,

5) and 6) do not arise. By the choice of L_{P} and μ_{P}, 7) is satisfied. If η_{2} is of type 4 or 6, ν_{η_{2}}(λ) is divisible by ℓ, 8), 9) and 12) are satisfied. Since neither η_{1} nor η_{2} is of type 5, 10) and 11) do not arise.

Case III. Suppose neither of η_{i} is of type 1 or 2. Suppose that one of the η_{i}, say η_{1}, is of type 3. Since 𝒳 has no special points, η_{2} is not of type 4 and hence η_{2} is of type 3, 5 or 6. If η_{2} is of type 5, let L_{P} and μ_{P} be as in (9.6). If η_{2} is of type 3 or 6, let L_{P} and μ_{P} be as in (9.7). Then, 1), 2), 3), 8), 9), 10), 11) and 12) are satisfied and other cases do not arise.

Case IV. Suppose neither of η_{i} is of type 1, 2 or 3. Suppose that one of the η_{i}, say η_{1}, is of type 4. Since 𝒳 has no special points, η_{2} is not of type 5. Hence η_{2} is of type 4 or 6. Let L_{P} and μ_{P} be as in (9.7). Then L_{P} and μ_{P} have the required properties.

Case V. Suppose neither of η_{i} is of type 1, 2, 3 or 4. Suppose that one of the η_{i} is of type 5, say η_{1} is of type 5. Then η_{2} is of type 5 or 6. Suppose that η_{2} is of type 5. Since 𝒳 has no special points, M_{η_{i}} ⊗ F_{P,η_{i}} are fields for i = 1, 2. Let L_{P} and μ_{P} be as in (9.2). Then L_{P} and μ_{P} have the required properties. Suppose that η_{2} is of type 6. Suppose M_{η_{1}} ⊗ F_{P,η_{1}} is a field. Let L_{P,η_{1}} = M_{η_{1}} ⊗ F_{P,η_{1}} and μ_{η_{1}} ∈ M_{η_{1}} with N_{M_{η_{1}}/F_{η_{1}}}(μ_{η_{1}}) = λ (cf., 9.1). Let L_{P} and μ_{P} be as in (6.5) with L_{P} ⊗ F_{P,η_{1}} ≃ L_{P,η_{1}}.

Then L_{P} and μ_{P} have the required properties. Suppose that M_{η_{1}} ⊗ F_{P,η_{1}} is not a field. Let L_{P} = ∏ F_{P} and μ_{i} ∈ F_{P} be as in (9.5) and μ_{P} = (μ_{1}, ⋯, μ_{ℓ}) ∈ L_{P}. Then L_{P} and μ_{P} have the required properties.

Case VI. Suppose neither of η_{i} is of type 1, 2, 3, 4 or 5. Then, η_{1} and t_{P} η_{2} are of type 6. Let L_{P} and μ_{P} be as in (9.7). Then L_{P} and μ_{P} have the required properties. □

10. Choice of L_{η} and μ_{η} at codimension zero points.

Let F, n = ℓ^{d}, α ∈ H^{2}(F, μ_{n}), λ ∈ F^{∗} with α ≠ 0, α · (λ) = 0 ∈ H^{3}(F, μ_{n}^{⊗2}), 𝒳 , X_{0} and 𝒫 be as in (§7, §8 and §9). Assume that 𝒳 has no special points and there is no type 2 connection between a codimension zero point of X_{0} of type 3 or 5 and a codimension zero point of X_{0} of type 3, 4 or 5. Further we assume that for every closed point P of X_{0}, the residue field κ(P) at P is a finite field. Let P be a closed point P of X_{0}. Then there exists t_{P} ≥ d such that there is no primitive ℓ^{t_{P}^{th}} root of unity in κ(P).

For a codimension zero point η of X_{0}, let 𝒫_{η} = η ∩ 𝒫.

Proposition 10.1. Let η be a codimension zero point of X_{0} of type 1. For each P ∈ 𝒫_{η}, let (L_{P} , μ_{P}) be chosen as in (9.8) and L_{η} = F_{η}(\sqrt[ℓ]{λ}) and μ_{η} = \sqrt[ℓ]{λ} ∈ L_{η}. Then

1) N_{L_{η}/F_{η}}(μ_{η}) = λ

2) α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2})

3) ind(α ⊗ L_{η}) < ind(α)

4) for P ∈ 𝒫_{η}, there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} = 1.

Proof. By choice, we have N_{L_{η}/F_{η}}(μ_{η}) = λ. Since η is of type 1, ν_{η}(λ) is coprime to ℓ and hence by (4.7), L_{η} and μ_{η} satisfies 2) and 3). Let P ∈ 𝒫_{η}. Since η is of type 1, by the choice of L_{P} and μ_{P} (cf. 9.8(4)), we have L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}. Hence L_{η} and μ_{η} satisfy 4). □

Lemma 10.2. Let η be a codimension zero point of X_{0}. For each P ∈ 𝒫_{η}, let θ_{P} ∈ F_{P} with α · (θ_{P}) = 0 ∈ H^{3}(F_{P,η}, μ_{n}^{⊗2}). Suppose ν_{η}(θ_{P}) = 0 for all P ∈ 𝒫_{η}. Then there exists θ_{η} ∈ F_{η} such that

1) α · (θ_{η}) = 0 ∈ H^{3}(F_{η}, μ_{n}^{⊗2})

2) for P ∈ 𝒫_{η}, θ_{P}^{−1}θ_{η} ∈ F_{P,η}^{ℓ^{2t_{P}}}.

Proof. Let π_{η} ∈ F_{η} be a parameter. Write α ⊗ F_{η} = α′ + (E_{η}, σ_{η}, π_{η}) as in (4.1). Let E(η) be the residue field of E_{η}. Since α · (θ_{P}) = 0 ∈ H^{3}(F_{P,η}, μ_{n}^{⊗2}) and ν_{η}(θ_{P}) = 0, by (4.7), we have (E(η) ⊗ κ(η)_{P} , σ_{0}, \overline{θ}_{P}) = 0 ∈ H^{2}(κ(η)_{P} , μ_{n}), where \overline{θ}_{P} is the image of θ_{P} ∈ κ(η)_{P}. Hence \overline{θ}_{P} is a norm from E(η) ⊗ κ(η)_{P} for all P ∈ 𝒫_{η}. For P ∈ 𝒫_{η}, let θ̃_{P} ∈ E(η) ⊗ κ(η)_{P} with N_{E(η)⊗κ(η)_{P} /κ(η)_{P}} (θ̃_{P}) = \overline{θ}_{P}. By weak approximation, there exists θ̃ ∈ κ(η) which is sufficiently close to θ̃_{P} for all P ∈ 𝒫_{η}. Let θ_{0} = N_{E(η)/κ(η)}(θ̃) ∈ κ(η). Then θ_{0} is sufficiently close to \overline{θ}_{P} for all P ∈ 𝒫_{η}. In particular, θ_{0}^{−1}\overline{θ}_{P} ∈ κ(η)_{P}^{ℓ^{2t_{P}}}. Let θ_{η} ∈ F_{η} have image θ_{0} in κ(η). Then (E_{η}, σ_{η}, θ_{η}) = 0 and hence, by (4.7), α · (θ_{η}) = 0. Since θ_{0}^{−1}\overline{θ}_{P} ∈ κ(η)_{P}^{ℓ^{2t_{P}}} and F_{P,η} is a complete discretely valuated field with residue field κ(η)_{P}, it follows that θ_{η}^{−1}θ_{P} ∈ F_{P,η}^{ℓ^{2t_{P}}}. □

Proposition 10.3. Let η be a codimension zero point of X_{0} of type 2. Suppose there is a type 2 connection between η and a codimension zero point η′ of X_{0} of type 5. Let Q be the point of type 2 intersection of η and η′. Suppose that M_{η′} ⊗ F_{Q,η′} is not a field. For each P ∈ 𝒫_{η}, let μ_{P} = (θ_{1}^{P} , ⋯, θ_{ℓ}^{P}) ∈ L_{P} = ∏ F_{P} be as in (9.8). Let L_{η} = ∏ F_{η}. Then there exists μ_{η} = (θ_{1}^{η}, ⋯, θ_{ℓ}^{η}) ∈ L_{η} such that

1) N_{L_{η}/F_{η}}(μ_{η}) = λ

2) α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2})

3) ind(α ⊗ L_{η}) < ind(α),

4) μ_{P}^{−1}μ_{η} ∈ (L_{η} ⊗ F_{P,η})^{ℓ^{2t_{P}}} for all P ∈ 𝒫_{η}.

Proof. Let i ≥ 2. By choice (cf. 9.8(5)), we have ν_{η}(θ_{i}^{P}) = 0 and α · (θ_{i}^{P}) = 0 ∈ H^{3}(F_{P} , μ_{n}^{⊗2}) for all P ∈ 𝒫_{η}. By (10.2), there exists θ_{i}^{η} ∈ F_{η} such that α · (θ_{i}^{η}) = 0 ∈ H^{3}(F_{η}, μ_{n}^{⊗2}) and (θ_{i}^{P})^{−1}θ_{i}^{η} ∈ F_{P,η}^{ℓ^{2t_{P}}} for all P ∈ 𝒫_{η}. Let θ_{1}^{η} = λ(θ_{2}^{η} ⋯ θ_{ℓ}^{η})^{−1}. Then θ_{1}^{η} ⋯ θ_{ℓ}^{η} = λ and (θ_{1}^{P})^{−1}θ_{1}^{η} ∈ F_{P,η}^{ℓ^{2t_{P}}}. Since α · (λ) = 0 and α · (θ_{i}^{η}) = 0 ∈ H^{3}(F_{η}, μ_{n}^{⊗2}) for i ≥ 2, we have α · (θ_{1}) = 0 ∈ H^{3}(F_{η}, μ_{n}^{⊗2}). Hence L_{η} = ∏ F_{η} and μ_{η} = (θ_{1}^{η}, ⋯, θ_{ℓ}^{η}) ∈ L_{η}. Since η is of type 2, ind(α ⊗ F_{η}) < ind(α) and hence L_{η}, μ_{η} have the required properties. □

Proposition 10.4. Let η be a codimension zero point of X_{0} of type 2. For each P ∈ 𝒫_{η}, let (L_{P} , μ_{P}) be chosen as in (9.8). Suppose one of the following holds.

• There is a type 2 connection between η and codimension zero point η′ of X_{0} of type 5 with Q the point of type 2 intersection of η and η′ and M_{η′} ⊗ F_{Q,η′} is a field.

• There is no type 2 connection between η and any codimension zero point of X_{0} of type 5. Let L_{η} = F_{η}(\sqrt[ℓ]{λ}) and μ_{η} = \sqrt[ℓ]{λ}. Then

1) N_{L_{η}/F_{η}}(μ_{η}) = λ

2) α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2})

3) ind(α ⊗ L_{η}) < ind(α)

4) for P ∈ 𝒫_{η}, there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} = 1.

Proof. Since ν_{η}(λ) is coprime to ℓ, by (4.7), α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2}) and ind(α ⊗ L_{η}) < ind(α). Clearly N_{L_{η}/F_{η}}(μ_{η}) = λ. By the choice of (L_{P} , μ_{P}) (cf. 9.8), for P ∈ 𝒫_{η}, we have L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ}. Thus L_{η} and μ_{η} have the required properties. □

Lemma 10.5. Let η be a codimension zero point of X_{0} of type 3, 4 or 5a. Let P ∈ η. Suppose there exists L_{P,η}/F_{P,η} a degree ℓ unramified field extension and μ_{P,η} ∈ L_{P,η} such that

1) N_{L_{P,η}/F_{P,η}}(μ_{P,η}) = λ,

2) ind(α ⊗ L_{P,η}) < ind(α),

3) α · (μ_{P,η}) = 0 ∈ H^{3}(L_{P,η}, μ_{n}^{⊗2}),

4) If η is of type 3, λ ∈ F_{P}^{∗ℓ} and α⊗E_{η}⊗F_{P,η} ≠ 0, then ind(α⊗(E_{η}⊗F_{P,η})⊗(L_{P,η})) < ind(α ⊗ E_{η} ⊗ F_{P,η}). Then ind(α ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P,η})) < ind(α)/[E_{η} : F_{η}].

Proof. Write α ⊗ F_{η} = α′ + (E_{η}, σ_{η}, π_{η}) as in (4.1). Then, by (4.2), ind(α ⊗ F_{η}) = ind(α′ ⊗ E_{η})[E_{η} : F_{η}] = ind(α ⊗ E_{η})[E_{η} : F_{η}]. Let t = [E_{η} : F_{η}] and β be the image of α′ in H^{2}(κ(η), μ_{n}).

Suppose η is of type 4. Then ind(α ⊗ F_{η}) < ind(α) and hence ind(α ⊗ E_{η}) = ind(α ⊗ F_{η})/t < ind(α)/t. We have ind(α ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P,η})) ≤ ind(α ⊗ E_{η}) < ind(α)/t.

Suppose that η is of type 5a. Then α is unramified at η and hence E_{η} = F_{η} and t = 1. The lemma is clear if α ⊗ F_{P,η} = 0. Suppose α ⊗ F_{P,η} ≠ 0. Then β ≠ 0. Since L_{P,η} is a an unramified field extension, the residue field L_{P} (η) of L_{P,η} is a field extension of κ(η)_{P} of degree ℓ. Since κ(η)_{P} is a local field and ind(β) is divisible by ℓ, ind(β ⊗ L_{P} (η)) < ind(β) ([3, p. 131]). In particular ind(α ⊗ L_{P,η}) < ind(α).

Suppose that η is of type 3. Then rα ⊗ E_{η} ≠ 0 and hence α′ ⊗ E_{η} = α ⊗ E_{η} ≠ 0. In particular ind(α ⊗ F_{η}) > t and β ⊗ E(η) ≠ 0. If α ⊗ E_{η} ⊗ F_{P,η} = 0, then ind(α ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P,η})) = 1 < ind(α)/t. Suppose that α ⊗ E_{η} ⊗ F_{P,η} ≠ 0. Suppose λ ∈ F_{P}^{∗ℓ}. Then, by the choice of L_{P,η}, ind(α ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P,η})) < ind(α ⊗ E_{η} ⊗ F_{P,η}) ≤ ind(α ⊗ E_{η}) = ind(α)/t. Suppose λ ∉ F_{P}^{∗ℓ}. Then λ ∉ F_{P,η}^{∗ℓ}. Since L_{P,η} is a field extension of degree ℓ and λ is a norm from L_{P,η}, by (2.6), L_{P,η} ≃ F_{P,η}(\sqrt[ℓ]{λ}). Since η is of type 3, ν_{η}(λ) = rℓ and λ = θ_{η}π_{η}^{rℓ} with θ_{η} ∈ F_{η} a unit

at η. Let \overline{θ}_{η} be the image of θ_{η} in κ(η). Then \overline{θ}_{η} ∉ κ(η)_{P}^{ℓ} and L_{P} (η) = κ(η)_{P} (\sqrt[ℓ]{\overline{θ}_{η}}). Since α · (λ) = 0, by (4.7), rℓα′ = (E_{η}, σ_{η}, θ_{η}) and hence rℓβ = (E(η), σ_{0}, \overline{θ}_{η}). Thus, by (3.3), ind(β ⊗ E(η)_{P} ⊗ L_{P} (η)) < ind(β ⊗ E(η)). Thus

ind(α ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P,η})) = ind(α′ ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P,η}))
= ind(β ⊗ E(η)_{P} ⊗ L_{P} (η))
< ind(β ⊗ E(η)) = ind(α′ ⊗ E_{η})
= ind(α ⊗ E_{η}) = ind(α)/t.
□

Proposition 10.6. Let η be a codimension zero point of X_{0} of type 3, 4 or 5a. For each P ∈ 𝒫_{η}, let (L_{P} , μ_{P}) be chosen as in (9.8). Then there exist a field extension L_{η}/F_{η} of degree ℓ and μ_{η} ∈ L_{η} such that

1) N_{L_{η}/F_{η}}(μ_{η}) = λ

2) α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2})

3) ind(α ⊗ L_{η}) < ind(α)

4) for P ∈ 𝒫_{η}, there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{P,η})^{ℓ^{2t_{P}}} .

Proof. Write α ⊗ F_{η} = α′ + (E_{η}, σ_{η}, π_{η}) as in (4.1). By (4.7), rℓα′ = (E_{η}, σ_{η}, θ_{η}). Let β be the image of α′ in H^{2}(κ(η), μ_{n}) and E(η) the residue field of E_{η}. Then rℓβ = (E(η), σ_{0}, θ_{0}) ∈ H^{2}(κ(η), μ_{n}), where σ_{0} is the automorphism of E(η) induced by σ_{η} and θ_{0} is the image of θ_{η} in κ(η).

Let S be a finite set of places of κ(η) containing the places given by closed points of 𝒫_{η} and places ν of κ(η) with β ⊗ κ(η)_{ν} ≠ 0. For each ν ∈ S, we now give a cyclic field extension L_{ν}/κ(η)_{ν} of degree ℓ and μ_{ν} ∈ L_{ν} satisfying the conditions of (3.1) with E_{0} = E(η) and d = ind(α)/t.

Let ν ∈ S. Then ν is given by a closed point P of η. If P ∈ 𝒫, let L_{P,η} = L_{P} ⊗F_{P,η} and μ_{P,η} = μ_{P} ⊗ 1 ∈ L_{P,η}. Suppose P ∉ 𝒫. Suppose that λ ∉ F_{P}^{∗ℓ}. Then λ ∉ F_{P,η}^{∗ℓ}. Let L_{P,η} = F_{P,η}(\sqrt[ℓ]{λ}) and μ_{P,η} = \sqrt[ℓ]{λ}. Suppose that λ ∈ F_{P}^{∗ℓ}. If η is of type 3, then let L_{P,η}/F_{P,η} be a cyclic unramified field extension of degree ℓ as in (9.3) and μ_{P,η} = \sqrt[ℓ]{λ}. If η is of type 4 or 5a, then let L_{P,η}/F_{P,η} be a cyclic unramified field extension of degree ℓ as in (4.10) and μ_{P,η} = \sqrt[ℓ]{λ}.

Since L_{P,η}/F_{P,η} is an unramified field extension of degree ℓ, the residue field L_{P} (η) is a degree ℓ field extension of κ(η)_{P}. Let L_{ν} = L_{P} (η). We have ν_{η}(λ) = rℓ for some integer r and λ = θ_{η}π_{η}^{rℓ} for some parameter π_{η} at η and θ_{η} ∈ F_{η} a unit at η. Further π_{η} is a parameter in L_{P,η}. Since N_{L_{P,η}/F_{P,η}}(μ_{P,η}) = λ, μ_{P,η} = θ_{P,η}π_{η}^{r} for some θ_{P,η} ∈ L_{P} ⊗ F_{P,η} which is a unit at η. Let μ_{ν} be the image of θ_{P,η} in L_{ν} = L_{P} (η). Then N_{L_{ν}/κ(η)_{ν}}(μ_{ν}) = θ_{0}. Since the corestriction map H^{2}(L_{ν}, μ_{n}) → H^{2}(κ(η)_{ν}, μ_{n}) is injective, rβ ⊗ L_{ν} = (E_{0} ⊗ L_{ν}, σ_{0} ⊗ 1, μ_{ν}). Let t = [E_{η} : F_{η}]. By (10.5), we have ind(α ⊗ (E_{η} ⊗ F_{P,η}) ⊗ L_{P,η}) < ind(α)/t. Since α ⊗ E_{η} = α′ ⊗ E_{η}, we have ind(α′ ⊗ (E_{η} ⊗ F_{P,η}) ⊗ L_{P,η}) < ind(α)/ℓ^{d}. Since ind(β ⊗ E_{0} ⊗ L_{μ}) = ind(α′ ⊗ (E_{η} ⊗ F_{P,η}) ⊗ (L_{P} ⊗ F_{P,η})), ind(β ⊗ E_{0} ⊗ L_{ν}) < ind(α)/t.

Since κ(η) is a global-field, by (3.1), there exist a field extension L_{0}/κ(η) of degree ℓ and μ_{0} ∈ L_{0} such that

1) N_{L_{0}/k}(μ_{0}) = θ_{0}

2) rβ ⊗ L_{0} = (E(η) ⊗ L_{0}, σ_{0} ⊗ 1, μ_{0})

3) ind(β ⊗ E(η) ⊗ L_{0}) < ind(α)/t

4) L_{0} ⊗ κ(η)_{P} ≃ L_{P} (η) for all P ∈ 𝒫_{η}

5) μ_{0} is close to \overline{θ}_{P} for all P ∈ 𝒫_{η}. Then, by (4.8), there exist a field extension L_{η}/F_{η} of degree ℓ and μ ∈ L_{η} such that • residue field of L_{η} is L_{0},

• μ a unit in the valuation ring of L_{η},

• \overline{μ} = μ_{0},

• N_{L_{η}/F_{η}}(μ) = θ_{η},

• α · (μπ_{η}^{r}) ∈ H^{3}(L_{η}, μ_{n}^{⊗2}) is unramified. Since L_{η} is a complete discretely valued field with residue field L_{0} a global field, H_{nr}^{3}(L_{η}, μ_{n}^{⊗2}) = 0 ([27, p. 85]) and hence α·(μπ_{η}^{r}) = 0. Since L_{η}/F_{η} is unramified and α ⊗L_{η} = α′ ⊗L_{η} +(E_{η} ⊗L_{η}, σ_{η}, π_{η}), ind(α ⊗L_{η}) ≤ ind(α′ ⊗E_{η} ⊗L_{η})[E_{η} ⊗L_{η} : L_{η}] = ind(β ⊗ E(η) ⊗ L_{0})t < ind(α) . Thus L_{η} and μ_{η} = μπ_{η}^{r} ∈ L_{η} have the required properties. □

Proposition 10.7. Let η be a codimension zero point of X_{0} of type 5b. Let (E_{η}, σ_{η}) be the residue of α at η and M_{η} be the unique subfield of E_{η} with M_{η}/F_{η} a cyclic extension of degree ℓ. For each P ∈ 𝒫_{η}, let L_{P} and μ_{P} be as in (9.8). Then there exists μ_{η} ∈ M_{η} such that

1) N_{M_{η}/F_{η}}(μ_{η}) = λ

2) α · (μ_{η}) = 0 ∈ H^{3}(M_{η}, μ_{n}^{⊗2})

3) ind(α ⊗ M_{η}) < ind(α)

4) for P ∈ 𝒫_{η}, there is an isomorphism ϕ_{P,η} : M_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{P,η})^{ℓ^{2t_{P}}} .

Proof. Let E(η) and M(η) be the residue fields of E_{η} and M_{η} at η. Since η is of type 5b, M(η) is the unique subfield of E(η) with M(η)/κ(η) a cyclic field extension of degree ℓ. Let π_{η} be a parameter at η. Since η is of type 5, ν_{η}(λ) = rℓ and λ = θ_{η}π_{η}^{rℓ} for some θ_{η} ∈ F a unit at η. Let \overline{θ}_{η} be the image of θ_{η} in κ(η). Let P ∈ 𝒫_{η}. Suppose M_{η} ⊗ F_{P,η} is a field. Since N_{M_{η}⊗F_{P,η}/F_{P,η}}(μ_{P}) = λ = θ_{η}π_{η}^{rℓ}, we have μ_{P} = μ′_{P} π_{η}^{r} with μ′_{P} ∈ M_{η} ⊗ F_{P,η} a unit at η and N_{M_{η}⊗F_{P,η}/F_{P,η}}(μ′_{P}) = θ_{η}. Suppose M_{η} ⊗ F_{P,η} is not a field. Then, by the choice of μ_{P} (cf. 9.8(10)), we have μ_{P} = μ′_{P} π_{η}^{r}, where μ′_{P} = (θ′_{1}, ⋯, θ′_{ℓ}) ∈ M_{η} ⊗ F_{P,η′} = ∏ F_{P,η} with each θ′_{i} ∈ F_{P,η} is a unit at η. Let \overline{μ′}_{P} be the image of μ′_{P} in the residue field M(η) ⊗ κ(η)_{P} of M_{η} ⊗ F_{P,η} at η. Write α⊗F_{η} = α′ +(E_{η}, σ_{η}, π_{η}) as in (4.1). Let β be the image of α′ in H^{2}(κ(η), μ_{n}). Since α · (λ) = 0, by (4.7), rℓβ = (E(η), σ_{η}, \overline{θ}_{η}). Since α · (μ_{P}) = 0 in H^{3}(M_{η} ⊗ F_{P,η}, μ_{n}^{⊗2}), once again by (4.7), rβ ⊗ κ(η)_{P} = (E(η) ⊗ M(η) ⊗ κ(η)_{P} , σ_{η}, \overline{μ′}_{P}). Since κ(η) is a global field, by (3.6), there exists μ′_{η} ∈ M(η) such that

1) N_{M(η)/κ(η)}(μ′_{η}) = \overline{θ}_{η}

2) rβ ⊗ M(η) = (E(η) ⊗ M(η), σ_{η}, μ′_{η})

3) \overline{μ′}_{P} is close to μ′_{η} for all P ∈ 𝒫_{η}. Since M_{η} is complete, there exists μ̃_{η} ∈ M_{η} such that N_{M_{η}/F_{η}}(μ̃′_{η}) = θ_{η} and the image of μ̃′_{η} in M(η) is μ′_{η}. Let μ_{η} = μ̃′_{η}π_{η}^{r}. Since M_{η}/F_{η} is of degree ℓ, ind(α ⊗ M_{η}) < ind(α ⊗ F_{η}) (cf. 8.1 ). Thus μ_{η} has the required properties. □

Proposition 10.8. Let η be a codimension zero point of X_{0} of type 6. For each P ∈ 𝒫_{η}, let L_{P} and μ_{P} be as in (9.8). Then there exist a field extension L_{η}/F_{η} of degree ℓ and μ_{η} ∈ L_{η} such that

1) N_{L_{η}/F_{η}}(μ_{η}) = λ

2) α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2})

3) ind(α ⊗ L_{η}) < ind(α)

4) for P ∈ 𝒫_{η}, there is an isomorphism ϕ_{P,η} : M_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{P,η})^{ℓ^{2t_{P}}} .

Proof. Let P ∈ 𝒫_{η}. Suppose L_{P} ⊗ F_{P,η} is a field. Let L_{P} (η), \overline{θ}_{P} ∈ L_{P} (η), θ_{0} ∈ κ(η) and β be as in the proof of (10.6). Then, as in the proof of (10.6), we have N_{L_{P} (η)/κ(η)_{P}} (\overline{θ}_{P}) = θ_{0} and ind(β ⊗ E_{0} ⊗ L_{P} (η)) < ind(α)/[E_{η} : F_{η}]. As in the proof of (10.7), we have rβ ⊗ L_{P} (η) = (E_{0} ⊗ L_{P} (η), σ_{0} ⊗ 1, \overline{θ}_{P}).

If L_{P} /F_{P} is not a field, by choice (cf. 9.8(11)), we have μ_{P} = (θ_{1}π_{η}^{r}, ⋯, θ_{ℓ}π_{η}^{r}). Since α · (μ_{P}) = 0 in H^{3}(L_{P} , μ_{n}^{⊗}) = ∏ H^{3}(F_{P} , μ_{n}^{⊗2}), we have α · (θ_{i}π_{η}^{r}) = 0 ∈ H^{3}(F_{P} , μ_{n}^{⊗2}). Thus, by (4.7), we have rβ ⊗ κ(η)_{P} = (E_{0}, σ_{0} ⊗ 1, \overline{θ}_{i}) for all i. Since L_{P} (η) = ∏ κ(η)_{P} and \overline{θ}_{P} = (\overline{θ}_{1}, ⋯, \overline{θ}_{ℓ}), we have rβ ⊗ L_{P} (η) = (E_{0} ⊗ L_{P} (η), σ_{0} ⊗ 1, \overline{θ}_{P}).

As in the proof of (10.6), we construct L_{η} and μ_{η} with the required properties. □

Lemma 10.9. Let η be a codimension one point of X_{0} and P a closed point on η. Suppose there exist θ_{η} ∈ F_{η} such that α · (θ_{η}) = 0 ∈ H^{3}(F_{η}, μ_{n}^{⊗2}). Then there exists θ_{P} ∈ F_{P} such that α · (θ_{P}) = 0 ∈ H^{3}(F_{P} , μ_{n}^{⊗2}), ν_{η}(θ_{P}) = ν_{η}(θ_{η}) and θ_{P}^{−1}θ_{η} ∈ F_{P,η}^{∗ℓ^{2t_{P}}}.

Proof. Let π be a prime representing η at P. Since η is regular on 𝒳 , there exists a prime δ at P such that the maximal ideal at P is generated by π and δ. Since F_{P,η} is a complete discrete valued field with π as a parameter, θ_{η} = wπ^{s} for some w ∈ F_{η} unit at η. Since the residue field κ(η)_{P} of F_{P,η} is a complete discrete valued field with \overline{δ} as a parameter, we have \overline{w} = \overline{u}\overline{δ}^{r} for some u ∈ F_{P} unit at P. Let θ_{P} = uδ^{r}π^{s}. Then clearly ν_{η}(θ_{η}) = ν_{η}(θ_{P}) and θ_{P}^{−1}θ_{η} ∈ F_{P,η}^{ℓ^{2t_{P}}}. Since α · (θ_{P}) is unramified at P except possibly at π and δ and α · (θ_{P}) = α · (μ_{P}) = 0 ∈ H^{3}(F_{P,η}, μ_{n}^{⊗2}), by (5.5), α · (θ_{P}) = 0 ∈ H^{3}(F_{P} , μ_{n}^{⊗2}). □

11. The main theorem

Theorem 11.1. Let K be a local field with residue field κ and F the function field of a curve over K. Let D be a central simple algebra over F of exponent n, α its class in H^{2}(F, μ_{n}), and λ ∈ F^{∗}. If α · (λ) = 0 and n is coprime to char(κ), then λ is a reduced norm from D^{∗}.

Proof. As in the proof of (4.12), we assume that n = ℓ^{d} for prime ℓ with ℓ ≠char(κ) and F contains a primitive ℓ^{th} root of unity. We prove the theorem by induction on ind(D).

Suppose that ind(D) = 1.Then D is a matrix algebra and hence every element of F is a reduced norm. Assume that ind(D) > 1.

Without loss of generality we assume that K is algebraically closed in F. Let X be a smooth projective geometrically integral curve over K with K(X) = F. Let R be the ring of integers in K and κ its residue field. Let 𝒳 be a regular proper model of F over R such that the union of ram_{𝒳} (α), supp_{𝒳} (λ) and the special fibre X_{0} of 𝒳 is a union of regular curves with normal crossings. By (8.6), we assume that 𝒳 has no special points, and there is no type 2 connection between a codimension zero point of X_{0} of type 3, or 5 and codimension zero point of X_{0} of type 3, 4 or 5.

Let 𝒫 be the set of nodal points of X_{0}. For each P ∈ 𝒫, let L_{P} and μ_{P} be as in (9.8). Let η be a codimension zero point of X_{0} and 𝒫_{η} = 𝒫 ∩ η. Let L_{η} and μ_{η} be as in 10.1, 10.3, 10.4, 10.6, 10.7 or 10.8 depending on the type of η. Then L_{η}/F_{η} is an extension of degree ℓ and μ_{η} ∈ L_{η} such that

1) N_{L_{η}/F_{η}}(μ_{η}) = λ

2) α · (μ_{η}) = 0 ∈ H^{3}(L_{η}, μ_{n}^{⊗2})

3) ind(α ⊗ L_{η}) < ind(α)

4) for P ∈ 𝒫_{η}, there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{P,η})^{ℓ^{2t_{P}}} .

Let P ∈ 𝒳 be a closed point with P ∉ 𝒫. Then there is a unique codimension zero point η of X_{0} with P ∈ η. We give a choice of a cyclic or split extension L_{P} /F_{P} of degree ℓ and μ_{P} ∈ L_{P}^{∗} such that

1) N_{L_{P} /F_{P}} (μ_{P}) = λ,

2) ind(α ⊗ L_{P}) < ind(α),

3) α · (μ_{P}) = 0 ∈ H^{3}(L_{P} , μ_{n}^{⊗2}),

4) there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{P,η})^{ℓ^{2t_{P}}} .

Suppose that η is of type 1. Then, by the choice of L_{η} and μ_{η} ( 10.1), L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ} have the required properties.

Suppose that η is of type 2. Suppose that there is a type 2 connection to a codimension zero point η′ of X_{0} of type 5. Let Q be the point of type 2 intersection η and η′. Suppose that M_{η′} ⊗ F_{Q,η′} not a field. Then, by choice (cf. 10.3), we have L_{η} = ∏ F_{η} and μ_{η} = (θ_{1}, ⋯, θ_{ℓ}). Since α · (μ_{η}) = 0, we have α · (θ_{i}) = 0. For each i, 2 ≤ i ≤ ℓ, by (10.9), there exists θ_{i}^{P} ∈ F_{P} such that α · (θ_{i}^{P}) = 0 ∈ H^{3}(F_{P} , μ_{n}^{⊗2}) and θ_{i}^{−1}θ_{i}^{P} ∈ F_{P,η}^{∗ℓ^{2t_{P}}}. Let θ_{1}^{P} = λ(θ_{2}^{P} ⋯ θ_{ℓ}^{P})^{−1}. Then L_{P} = ∏ F_{P} and μ_{P} = (θ_{1}^{P} , ⋯, θ_{ℓ}^{P}) have the required properties. Suppose that M_{η′} ⊗ F_{Q,η′} is a field or there is no type 2 connection from η to any point of type 5. Then, by the choice (10.4), we have L_{η} = F_{η}(\sqrt[ℓ]{λ}) and μ_{η} = \sqrt[ℓ]{λ}. Hence L_{P} = F_{P} (\sqrt[ℓ]{λ}) and μ_{P} = \sqrt[ℓ]{λ} ∈ L_{P} have the required properties.

Suppose that η is not of type 1 or 2. Then, by choice L_{η}/F_{η} is an unramified field extension of degree ℓ or the split extension of degree ℓ. Let Â_{P} be the completion of the local ring at P and π a prime in Â_{P} defining η at P. Since P ∉ 𝒫 and ram_{𝒳} (α) is union of regular curves with normal crossings, there exists a prime δ ∈ Â_{P} such that α is unramified on Â_{P} except possibly at π and δ. Further, λ = wπ^{r}δ^{s} for some unit u ∈ Â_{P}. Since η is not of type 1 or 2, ν_{η}(λ) = r is divisible by ℓ. Thus, by (6.5), there exist a cyclic extension L_{P} /F_{P} and μ_{P} ∈ L_{P} such that

1) L_{P} ⊗ F_{P,η} ≃ L_{η} ⊗ F_{P,η},

2) ind(α ⊗ L_{P}) < ind(α),

3) α · (μ_{P}) = 0 ∈ H^{3}(L_{P} , μ_{n}^{⊗2}),

4) there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and

ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{P,η})^{ℓ^{2t_{P}}} .

Thus for every x ∈ X_{0}, we have chosen an extension L_{x}/F_{x} of degree ℓ and μ_{x} ∈ L_{x} such that

1) N_{L_{x}/F_{x}}(μ_{x}) = λ

2) α · (μ_{x}) = 0 ∈ H^{3}(L_{x}, μ_{x}^{⊗2})

3) ind(α ⊗ L_{x}) < ind(α)

4) for any branch (P, η), there is an isomorphism ϕ_{P,η} : L_{η} ⊗ F_{P,η} → L_{P} ⊗ F_{P,η} and ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} ∈ (L_{P} ⊗ F_{P,η})^{ℓ^{2t_{P}}}. Further if P is a closed point of X_{0}, then L_{P} /F_{P} is cyclic or the split extension.

Let (P, η) be a branch. Since κ(P) has no ℓ^{2t_{P}^{th}} primitive root of unity and κ(η)_{P} is a complete discretely valued field with residue field κ(P), κ(η)_{P} has no ℓ^{2t_{P}^{th}} primitive root of unity. Since F_{P,η} is a complete discretely valued field with residue field κ(η)_{P}, F_{P,η} has no ℓ^{2t_{P}^{th}} primitive root of unity. Since ϕ_{P,η}(μ_{η} ⊗1)(μ_{P} ⊗1)^{−1} ∈ (L_{P} ⊗F_{P,η})^{ℓ^{2t_{P}}} and t_{P} ≥ d, by (2.8), for a generator σ of Gal(L_{P} ⊗ F_{P,η}/F_{P,η}), there exists θ_{P,η} ∈ L_{P} ⊗ F_{P,η} such that ϕ_{P,η}(μ_{η} ⊗ 1)(μ_{P} ⊗ 1)^{−1} = θ_{P,η}^{−ℓ^{d}}σ(θ_{P,η})^{ℓ^{d}}.

By (7.5), there exist extensions L/F of degree ℓ, N/F of degree coprime to ℓ, and μ ∈ L ⊗ N such that

• N_{L⊗N/F} (μ) = λ and

• α · (μ) = 0 ∈ H^{3}(L ⊗ N, μ_{n}^{⊗2})

• ind(α ⊗ L) < ind(α).

Since L ⊗ N is also a function field of a curve over a p-adic field, by induction hypotheses, μ is a reduced norm from D ⊗ L ⊗ N and hence λ = N_{L⊗N/N}(μ) is a reduced norm from D. Since N_{N/F} (λ) = λ^{[N:F]}, λ^{[N:F]} is a norm from D. Since [N : F] is coprime to ℓ, λ is a reduced norm from D. □

Corollary 11.2. Let K be a local field with residue field κ and F the function field of a curve over K. Let Ω be the set of divisorial discrete valuations of F. Let D be a central simple algebra over F of index coprime to char(κ) and λ ∈ F. If λ is a reduced norm from D ⊗ F_{ν} for all ν ∈ Ω, then λ is a reduced norm from D.

Proof. Since λ is a reduced norm from F_{ν} for all ν ∈ Ω_{F}, α · (λ) = 0 in H^{3}(F_{ν}, μ_{n}^{⊗2}) for all ν ∈ Ω. Thus, by ([16, Proposition 5.2]), α · (λ) = 0 in H^{3}(F, μ_{n}^{⊗2}) and by (11.1), λ is a reduced norm from D. □

Acknowledgments: We thank Nivedita for the wonderful conversations during the preparation of the paper. The first author is partially supported by National Science Foundation grants DMS-1401319 and DMS-1463882 and the third author is partially supported by National Science Foundation grants DMS-1301785 and DMS-1463882.

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Department of Mathematics & Computer Science, Emory University, 400 Dowman Drive NE, Atlanta, GA 30322, USA

E-mail address: parimala@mathcs.emory.edu

Department of Mathematics, Indian Institute of Technology (Bombay), Powai, Mumbai-400076, India

E-mail address: preeti@math.iitb.ac.in

Department of Mathematics & Computer Science, Emory University, 400 Dowman Drive NE, Atlanta, GA 30322, USA

E-mail address: suresh@mathcs.emory.edu