Total Ramification

Abstract

A symplectic module (M,a) is a finite abelian group M together with a nondegenerate alternating pairing $a\colon M\times M \to \mathbb {Q}/\mathbb {Z}$. Such pairings arise (i) on subgroups called armatures of A ^×/F ^× when the algebra A with center  F is a tensor product of symbol algebras, and (ii) on Γ _D /Γ _F when D  is a valued division algebra totally ramified over its center F. In §7.1 we consider the set Symp(Ω) of all symplectic modules on finite subgroups of an abelian torsion group Ω; we describe the canonical operation on Symp(Ω) making it into a torsion abelian group. If Γ is a torsion-free abelian group, we prove in Th. 7.22 that $\mathit{Symp}({\mathbb{T}}(\Gamma)) \cong {\mathbb{T}}(\wedge^{2} \Gamma)$, where ${\mathbb{T}}(\Gamma) = \Gamma\otimes_{\mathbb {Z}}(\mathbb {Q}/\mathbb {Z})$. In  §7.2 we consider armatures on algebras and their homogeneous counterparts on graded algebras. We show how an armature on a central simple algebra over a valued field can be used to build a gauge on the algebra. In §7.3 and §7.4 the focus is on totally ramified graded and valued division algebras. If F is an inertially closed graded field (i.e., F ₀ is separably closed), we prove a group isomorphism from Br(F) to the part of $\mathit{Symp}({\mathbb{T}}(\Gamma_{\mathsf {F}}))$ of torsion prime to $\operatorname {\mathit{char}}(\mathsf {F}_{0})$ mapping ${[\mathsf {D}] \mapsto (\Gamma_{\mathsf {D}}/\Gamma_{\mathsf {F}}, \overline{c}_{\mathsf {D}})}$, where $\overline{c}_{\mathsf {D}}$ is the canonical pairing induced by commutators. For F not inertially closed, this leads to a description of Br(F)/Br _in (F) as a subgroup of $\mathit{Symp}({\mathbb{T}}( \Gamma_{\mathsf {F}})))$ determined by the roots of unity in F ₀. The analogous result is proved for $\operatorname {\mathit{Br}}_{t}(F)/\operatorname {\mathit{Br}}_{ \mathit{in}}(F)$ for a Henselian field F.

Appendices

Exercises

Exercise 7.1

For an arbitrary integer n>1, show that

$$\mathit{Symp}\bigl((\mathbb {Z}/n\mathbb {Z})^2\bigr) \cong\, \mathbb {Z}/n\mathbb {Z}. $$

Exercise 7.2

Let Γ be a torsion-free abelian group. Given γ, δ∈Γ and an integer n≥1, prove that there exist an integer m≥1 and elements ξ, η∈Γ such that

$${\textstyle\frac{1}{n}}(\gamma\wedge\delta) \, = \, {\textstyle\frac{1}{m}} (\xi\wedge\eta) $$

and ξ, η are m-independent (i.e., for a, $b\in \mathbb {Z}$, we have aξ+bη∈mΓ if and only if a≡b≡0 (mod m)).

Exercise 7.3

Let Ω and Ω′ be torsion abelian groups, and let f:Ω→Ω′ be a group homomorphism. Let (M,a)∈Symp(Ω) with $M \cong (\mathbb {Z}/m\mathbb {Z})^{2}$. Suppose that ${\operatorname {\mathit{ker}}f\cap M \cong \mathbb {Z}/r_{1}\mathbb {Z}\times \mathbb {Z}/r_{2}\mathbb {Z}}$. Prove that the underlying abelian group of f _⋄(M,a) has invariant factors m/(r ₁ r ₂),m/(r ₁ r ₂).

Exercise 7.4

Let Γ be an arbitrary torsion-free abelian group and let  p  be a prime number. Show that if $(\varepsilon_{i})_{i\in I_{p}}$ is a family of elements in Γ such that $(\varepsilon_{i}+p\Gamma)_{i\in I_{p}}$ is an $\mathbb {F}_{p}$-vector space base of Γ/pΓ, then the p-primary component ${\mathbb{T}}(\Gamma)_{(p)}$ is a direct sum of copies of $\mathbb {Q}_{p}/\mathbb {Z}_{p}$ indexed by I _p . If the cardinalities |I _p | are the same for all  p, show that ${\mathbb{T}}(\Gamma)$ is a direct sum of |I _p |  copies of $\mathbb {T}$.

Exercise 7.5

Let D be a division algebra over a field F, and assume F contains a primitive s-th root of unity for some integer s. Show that every finite abelian subgroup of exponent dividing s in D ^×/F ^× is an armature.

Exercise 7.6

Let $\mathcal{A}$ be an armature in an F-algebra A. Show that if $F[ \mathit{rad}\mathcal{A}]$ is a field, then $\mathcal{A}/ \mathit{rad}\mathcal{A}$ can be canonically identified with an armature in the $F[ \mathit{rad}\mathcal{A}]$-algebra $F[\mathcal{A}]$.

Exercise 7.7

Let F be an inertially closed graded field, and let F _alg be a graded algebraic closure of F. Define a map η from the set of finite-degree graded field extensions of F to ${\mathbb{H}}(\Gamma_{\mathsf {F}})$ by K↦Γ _K . Show that η gives a one-to-one correspondence between (i) finite-degree tame graded field extensions of  F in F _alg ; and (ii) subgroups $\Delta\subseteq {\mathbb{H}}(\Gamma_{\mathsf {F}})$ containing Γ _F as a subgroup of finite index prime to $\operatorname {\mathit{char}}\mathsf {F}_{0}$. [Hint: for injectivity, use induction on the degree, Prop. 5.18, and Lemma 7.62.]

Exercise 7.8

Let (F,v) be a valued field, and let A be a central simple F-algebra with $\operatorname {\mathit{char}}\overline{F}\nmid \operatorname {\mathit{deg}}A$. Let D be the associated division algebra of A, and suppose v extends to a valuation v _D on D that is totally ramified over   v. Suppose A is spanned by an armature $\mathcal{A}$, and consider the map ${\overline{\alpha}_{\mathcal{A}}\colon{\mathcal{A}}\to \Gamma_{A}^{\times}/\Gamma_{F}}$ induced by the armature gauge $\alpha_{\mathcal{A}}$ (see §7.2.3). Let  ${\mathcal{K}=\operatorname {\mathit{ker}}\overline{\alpha}_{\mathcal{A}}}$ and let $\mathcal{K}^{\perp}$ be the orthogonal of $\mathcal{K}$ in $\mathcal{A}$ with respect to the commutator pairing $b_{\mathcal{A}}$. Prove that as symplectic modules

$$(\Gamma_D/\Gamma_F,\overline{c}_D) \,=\, \overline{\alpha}_{\mathcal{A}\, \diamond}( \mathcal{A}, b_\mathcal{A}). $$

Deduce that

$$[D{\mspace{1mu}:\mspace{1mu}}F]\,=\,\lvert \mathcal{K}^\perp{ \mspace{1mu}:\mspace{1mu}}(\mathcal{K}\cap\mathcal{K}^\perp)\rvert \qquad\text{and}\qquad \Gamma_D/\Gamma_F \,=\,\overline{\operatorname {\mathit{deg}}}(\mathcal{K}^\perp). $$

Exercise 7.9

Let $F=\mathbb{R}((x))((y))$ be the field of iterated Laurent series in two indeterminates over the field $\mathbb{R}$ of real numbers, with the (x,y)-adic valuation. Show that $\operatorname {\mathit{Br}}(F)\cong(\mathbb {Z}/2\mathbb {Z})^{4}$ and $\operatorname {\mathit{Br}}(F_{ \mathit{in}})\cong \mathbb {T}$.

Notes

§7.1.1: Symplectic modules seem to have been first investigated in relation to classification problems in topology. The earliest references where their structure is established are de Rham [201, §19] and C.T.C. Wall [259]. The structure of their Lagrangians is discussed by Tignol–Amitsur [243] for the purpose of obtaining lower bounds for the degree of Galois splitting fields of universal division algebras. Another approach to these lower bounds, also using Lagrangians in symplectic modules, is due to Reichstein–Youssin [197].

§7.1.2: The definition of the group structure on Symp(Ω) and the map f _⋄ determined by a homomorphism f:Ω₁→Ω₂ can be explained by relating Symp(Ω) to alternating forms on the dual of Ω. Here is a sketch of the connection: Let G be an abelian topological group of one of the following types: Type I: a torsion group with the discrete topology; Type II: a profinite group. For G of either type, the $\it dual$ of G is

$$G{^{\vee}} \, =\, \operatorname {\mathit{Hom}}^c(G, \mathbb {T}), $$

the group of continuous homomorphisms from G to $\mathbb {T}= \mathbb {Q}/\mathbb {Z}$, where $\mathbb {T}$ has the discrete topology; G ^∨ is given the compact open topology. It is known that if G is of Type I or Type II, then G ^∨ is of the other type. Moreover, the canonical monomorphism G→G ^{∨ ∨} is an isomorphism of topological groups. This is a special case of the Pontrjagin duality for locally compact abelian topological groups, and it is described thoroughly in Ribes–Zaleskiĭ [205]. Further, if G ₁ and G ₂ are groups of the same type then for any continuous group homomorphism f:G ₁→G ₂, the induced map $f^{\vee}\colon G_{2}^{\vee}\to G_{1}^{\vee}$ is a continuous group homomorphism. The assignment f↦f ^∨ is a contravariant functor preserving exact sequences, and f ^{∨ ∨}=f under the canonical identification $G_{i}^{\vee}{} ^{\vee}= G_{i}$. For any closed subgroup A of G ^∨, the annihilator of A in G is

$$A^0 \, = \, \{ s\in G\mid \psi(s) = 0\ \text{ for all $\psi \in A$}\}, $$

which is a closed subgroup of G. The canonical map A ⁰→(G ^∨/A)^∨ is a topological group isomorphism.

Now, let Ω be any torsion abelian group. Let $\operatorname {\mathit{Alt}}^{c}(\Omega ^{\vee})$ denote the group of continuous alternating bilinear functions $\Omega ^{\vee}\times \Omega ^{\vee}\to \mathbb {T}$. Define a map from  $\operatorname {\mathit{Alt}}^{c}(\Omega ^{\vee})$ to Symp(Ω) as follows: Given $\alpha \in \operatorname {\mathit{Alt}}^{c}(\Omega ^{\vee})$, the continuity of  α and the compactness of Ω^∨ imply that rad α is a closed subgroup of finite index in Ω^∨. The pairing α induces a nondegenerate alternating pairing  $\overline{\alpha}$ on the finite abelian group  Ω^∨/rad α. We use $\overline{\alpha}$ to define an isomorphism ${\overline{\alpha}_{*} \colon \Omega ^{\vee}/ \mathit{rad}\, \alpha \to \big(\Omega ^{\vee}/ \mathit{rad}\, \alpha\big)^{\vee}}$ by ${\overline{\alpha}_{*}(\eta)(\rho) = \overline{\alpha}(\eta,\rho)}$ for all η,ρ∈Ω^∨/rad α. The pairing $\overline{\alpha}$ transfers via this isomorphism to a pairing on (Ω^∨/rad α)^∨, which in turn transfers to a pairing α ⁰ on rad α ⁰ using the canonical isomorphism (Ω^∨/rad α)^∨≅rad α ⁰; the alternating pairing α ⁰ is nondegenerate since $\overline{\alpha}$ is nondegenerate. We thus have a map

$$\mathcal{V}_\Omega \colon \operatorname {\mathit{Alt}}^c(\Omega ^{\vee}) \, \longrightarrow \, \mathit{Symp}(\Omega) \qquad \text{given by} \qquad \alpha \mapsto ( \mathit{rad}\, \alpha^0, \alpha^0). $$

One can show that $\mathcal{V}_{\Omega}$ is a bijection. (For the inverse map, take any (M,a)∈Symp(Ω). Then a induces an isomorphism a _∗:M→M ^∨ and thereby a corresponding nondegenerate pairing $a^{\vee}\in \operatorname {\mathit{Alt}}(M^{\vee})$ given by a ^∨(a _∗(m),a _∗(n))=a(m,n) for all m,n∈M. The inclusion M↪Ω yields an epimorphism Ω^∨→M ^∨, and a ^∨ on M ^∨ lifts to a continuous alternating pairing α _a on Ω^∨. One can check that $\mathcal{V}_{\Omega}(a_{\alpha}) = (M,a)$.) Fundamentally, this shows that there is a bijection between nondegenerate alternating pairings on finite factor groups of Ω^∨ (modulo closed subgroups) and nondegenerate alternating pairings on finite subgroups of Ω.

Suppose Ω₁ and Ω₂ are torsion abelian groups, and f:Ω₁→Ω₂ is a group homomorphism. The map f induces a continuous homomorphism ${f^{\vee}\colon\Omega_{2}^{\vee}\to \Omega_{1}^{\vee}}$, and thereby a map $\operatorname {\mathit{Alt}}(f) \colon \operatorname {\mathit{Alt}}^{c}(\Omega_{1}^{\vee}) \to \operatorname {\mathit{Alt}}^{c}(\Omega_{2}^{\vee})$. One can then show that there is a commutative diagram

Since the assignment $f \mapsto \operatorname {\mathit{Alt}}(f)$ is clearly compatible with composition of group homomorphisms, the same must be true for f↦f _⋄. Also, the group operation on $\operatorname {\mathit{Alt}}^{c}(\Omega)$ transfers via $\mathcal{V}_{\Omega}$ to a group operation on Symp(Ω), and this coincides with the group operation defined in §7.1.2.

The group $\operatorname {\mathit{Alt}}^{c}(G)$ for G an arbitrary profinite abelian group has been studied by Brussel, who established in [44, Th. 2.4] canonical isomorphisms

$$H^2(G,\mathbb {T})\,\cong\, \operatorname {\mathit{Alt}}^c(G) \,\cong\, \wedge^2(G^{\vee}). $$

Brussel shows in [44, Th. 3.6] how for any given $\alpha\in \operatorname {\mathit{Alt}}^{c}(G)$ the order of the group G/radα can be computed from the pfaffian of a matrix representing  α with respect to a base of G/radα. See also [42, §2], where Brussel proves a similar result relating pfaffians of submatrices of an alternating matrix $A\in \operatorname {\mathit{Alt}}_{n}(\mathbb {T})$ to the order of the corresponding symplectic module ${\Upsilon_{n}(A)\in \mathit{Symp}(\mathbb {T}^{n})}$. Note however that Brussel’s result is not expressed in terms of symplectic modules, but in terms of indices of central simple algebras over a strictly Henselian field with residue field of characteristic 0 and value group $\mathbb {Z}^{n}$. These two viewpoints are equivalent by Th. 7.80.

§7.2: Armatures were first defined by Tignol in [236] for division algebras. When the base field contains enough roots of unity, the linear independence of representatives of the armature elements is automatic in this case, see [236, Lemme 1.5] and Exercise 7.5. The definition for central simple algebras was given by Tignol–Wadsworth in [245]. The notion of armature was inspired by the q-generating subsets of Amitsur et al. [11, §1] and the p-central sets of Rowen [213, §2].

If $\mathcal{A}$ is a spanning armature of a central F-algebra A, then $\mathcal{A}$ is a maximal abelian subgroup of A ^×/F ^×. To see this, consider the centralizer $C(\mathcal{A})$ of $\mathcal{A}$ in A ^×/F ^×. The commutator pairing defines a group homomorphism ${C(\mathcal{A})\to\mathcal{A}^{\vee}}$. This homomorphism is injective because the center of $F[\mathcal{A}]$ is F, hence ${\lvert C(\mathcal{A})\rvert\leq \lvert\mathcal{A}^{\vee}\rvert}$. But $\lvert\mathcal{A}^{\vee}\rvert = \lvert\mathcal{A}\rvert$ and $\mathcal{A}\subseteq C(\mathcal{A})$, so we must have $C(\mathcal{A}) = \mathcal{A}$, and therefore $\mathcal{A}$ is a maximal abelian subgroup. When F is algebraically closed of characteristic 0 and A=M _n (F), we have $A^{\times}/F^{\times}=\operatorname{PGL}_{n}(F)$. Reichstein–Youssin [197, Lemma 7.5] show that a finite abelian subgroup $\mathcal{A}\subseteq \operatorname{PGL}_{n}(F)$ is toral (i.e., embeddable in a torus) if and only if the commutator pairing is trivial on $\mathcal{A}$. They also prove [197, Lemma 7.8] that every symplectic module of order n ² embeds as an armature in $\operatorname{PGL}_{n}(F)$.

§7.4: The first result on the structure of tame totally ramified central division algebras over Henselian fields is due to Draxl [64, Th. 1], who showed that these algebras are tensor products of symbol algebras. Tame totally ramified division algebras (over fields that are not necessarily Henselian) were further studied by Tignol–Wadsworth in [245], where the canonical pairing on the value group is defined. By contrast, it had already been observed by Scharlau that for a Henselian field F, the part of $\operatorname {\mathit{Br}}(F_{ \mathit{in}})$ of torsion prime  to  $\operatorname {\mathit{char}}\overline{F}$ is generated by symbol algebras [225, Kor. 3.4]. In the same paper, Scharlau purports to describe the quotients of the filtration of $\operatorname {\mathit{Br}}(F)$ as in Th. 6.66 and Th. 7.84. However, his Satz 4.1 is flawed: the image of $\operatorname {\mathit{Br}}_{ \mathit{tr}}(F)$ in $\operatorname {\mathit{Br}}_{ \mathit{tr}}(F_{ \mathit{in}})$ is only the $\mu(\overline{F})$-torsion part of $\operatorname {\mathit{Br}}_{ \mathit{tr}}(F_{ \mathit{in}})$, not the full $\operatorname {\mathit{Br}}_{ \mathit{tr}}(F_{ \mathit{in}})$; see Th. 7.84 and Exercise 7.9.

That symbol algebras generate the Brauer group of a strictly Henselian field of residue characteristic 0 can also be derived by an argument of Saltman [220, Th. 2.1] from the fact that the absolute Galois group of these fields is abelian. For F a strictly Henselian field with $\operatorname {\mathit{char}}\overline{F}=0$ and $\Gamma_{F}\cong \mathbb {Z}^{n}$, Brussel sets up in [42, §1] an isomorphism $\operatorname {\mathit{Br}}(F)\cong \operatorname {\mathit{Alt}}_{n}(\mathbb {T})$ depending on a base of Γ _F ; his map is the isomorphism Σ _F of Prop. 7.79 given the identification ${\mathbb{T}}(\wedge^{2}\Gamma_{F})\cong \operatorname {\mathit{Alt}}_{n}(\mathbb {T})$ of (7.21). In [44], Brussel uses the same ideas more generally to describe the relative Brauer group of a Galois extension with abelian Galois group. The isomorphism $\operatorname {\mathit{Br}}(F)\cong \operatorname {\mathit{Alt}}_{n}(\mathbb {T})$ for F strictly Henselian with residue field k of characteristic zero and $\Gamma_{F}\cong \mathbb {Z}^{n}$ can also be obtained as follows: Since every central division algebra over F is tame, Th. 6.64 yields an isomorphism $\operatorname {\mathsf {gr}}\colon \operatorname {\mathit{Br}}(F)\stackrel{\sim}{\to} \mathsf{Br}(\operatorname {\mathsf {gr}}(F))$. As observed in the notes to Ch. 6, we have $\mathsf{Br}(\operatorname {\mathsf {gr}}(F)) = \operatorname {\mathit{Br}}(\operatorname {\mathsf {gr}}(F)^{\natural})$ because $\operatorname {\mathit{char}}\operatorname {\mathsf {gr}}(F)=0$. But $\operatorname {\mathsf {gr}}(F)^{\natural}\cong k[t_{1},t_{1}^{-1},\ldots,t_{n},t_{n}^{-1}]$ is the coordinate ring of an n-dimensional torus over k, so $\operatorname {\mathit{Br}}(\operatorname {\mathsf {gr}}(F)^{\natural})\cong(\mathbb {Q}/\mathbb {Z})^{n(n-1)/2}$ by Magid’s computation of the Brauer group of a torus [129, Cor. 7]. (See also Gille–Pianzola [83, Prop. 3.1(2), (4.2)] and Gille–Semenov [85, Th. 2.8] for other approaches to Magid’s computation.)

Example 7.77 was given in a more general form in Tignol–Wadsworth [245]. For any integers r, s>1, and t∈{1,…,rs}, Prop. 5.8 of that paper yields a division algebra of degree and exponent rs that is tame and totally ramified over its center, which contains a primitive rs-th root of unity, with relative value group isomorphic to $(\mathbb {Z}/\ell \mathbb {Z})^{2}\times(\mathbb {Z}/d\mathbb {Z})^{2}$ where $\ell=\operatorname {\mathit{lcm}}(rs, rt, st)/t$ and d=rs/ℓ. In particular, if r and s are not relatively prime, then the division algebra does not contain any armature. It is therefore not a symbol algebra, nor a tensor product of symbol algebras.