Valuations on Division Rings

Abstract

In this chapter we introduce the central object of study in this book: valuations on a division algebra D finite-dimensional over its center  F. In §1.1 we define valuations and describe the associated structures familiar from commutative valuation theory: the valuation ring $\mathcal {O}_{D}$, its unique maximal left and maximal right ideal $\mathfrak {m}_{D}$, the residue division algebra $\overline{D}$, and the value group Γ _D . We also describe an important and distinctively noncommutative feature, namely a canonical homomorphism θ _D from  Γ _D to the automorphism group $\operatorname {\mathit{Aut}}(Z(\overline{D})\big/\,\overline{F}\,)$; θ _D is induced by conjugation by elements of D ^×. In §1.2, after proving the “Fundamental Inequality for valued division algebras, we look at valuations on D from the perspective of F. We show that a valuation on F has at most one extension to D, and prove a criterion for when such an extension exists. When this occurs, we show that the field $Z(\overline{D})$ is finite-dimensional and normal over $\overline{F}$ and that θ _D is surjective. We also describe the technical adjustments needed to apply the classical method of “composition” of valuations to division algebras. The filtration on D induced by a valuation leads to an associated graded ring $\operatorname {\mathsf {gr}}(D)$, which we describe in §1.3. Throughout the book we emphasize use of $\operatorname {\mathsf {gr}}(D)$ to help understand the valuation on D. This chapter includes many examples of division algebras with valuations.

Appendices

Exercises

Exercise 1.1

A total valuation ring not the ring of a valuation. A subring  T of a division algebra D is called a total valuation ring of D if for each d∈D ^× we have d∈T or d ⁻¹∈T. If D were commutative it is well-known that such a T is the valuation ring of a valuation in D. But, when D is not commutative, T is the ring of a valuation if and only if in addition it is closed under conjugation. This exercise gives an example of a total valuation ring that is not the ring of a valuation.

Let k be any field with $\operatorname {\mathit{char}}(k)\neq2$. Let x be transcendental over k and let F=k(x)((y)), the Laurent power series ring over k(x). Let D be the quaternion algebra $\big({1+x,y}/{F}\big)$, with its standard generators i and j satisfying i ²=1+x, j ²=y, and ij=−ji. Let $w\colon F\to \mathbb {Z}\cup\{\infty\}$ be the complete discrete rank 1 (so Henselian) y-adic valuation on F, with ${\overline{F}^{\mspace{1mu}w}=k(x)}$ and $\Gamma_{F,w}=\mathbb {Z}$. Since ${1+x \notin (\overline{F}^{\mspace{1mu}w})^{\times 2}}$ and w(y)=1∉2Γ _F,w , Ex. 1.17 Case 2 shows that D is a division algebra and w  extends to a valuation on D with $\overline{D}^{\, w} = k(x)(\sqrt{1+x})$ and $\Gamma_{D,w} = \frac{1}{2}\mathbb {Z}$. (See also Ex. 1.19.)

• (i) Let u be the x-adic valuation on k(x), which is the restriction to k(x) of the x-adic valuation on k((x)). Since u(1+x)=0 and $\overline{1+x}=\overline{1}$, which is a square in $\overline{k(x)}^{\,u}$, the valuation u has two different extensions u ₁  and  u ₂ to ${k(x)(\sqrt{1+x})=\overline{D}^{\, w}}$, say with $\overline {\sqrt {1+x}}^{\,u_{1}} = 1$ and $\overline {\sqrt {1+x}}^{\,u_{2}} = -1$. Let $\pi\colon \mathcal {O}_{D,w}\to\overline{D}^{\,w}$ be the residue map and let ${T_{\ell}=\pi^{-1}(\mathcal {O}_{\overline{D}^{\,w},u_{\ell}})}$, ℓ=1, 2. Prove that T ₁ and T ₂ are total valuation rings of D. Prove also that the inner automorphism $\operatorname {\mathit{int}}(j)$ of D given by conjugation by  j permutes T ₁ and T ₂. Hence, the T _ℓ are not invariant under inner automorphisms. Prove also that every inner automorphism of D either preserves or interchanges T ₁ and T ₂.

• (ii) Let the valuation v on F be the restriction of the (x,y)-adic valuation on  k((x))((y)), so v is the composition u∗w. Since $\overline {1+x}^{\,v} = 1 \in (\overline{F}^{\mspace{1mu}v})^{\times 2}$, Ex. 1.17 Case 2 shows that v does not extend to a valuation on D. Hence, there can be no total valuation ring V of D invariant under inner automorphisms with $V\cap F = \mathcal {O}_{F,v}$. Prove that in fact T ₁  and  T ₂ are the only total valuation rings of D whose intersection with F is  $\mathcal {O}_{F,v}$.

Exercise 1.2

Composite valuations on division rings of Laurent series. Suppose that D is a division ring with valuation u and that τ is an automorphism of D preserving  u, in the sense that $u\bigl(\tau(d)\bigr)=u(d)$ for all d∈D. Let E=D((y;τ)) with its y-adic valuation v _y . Build a composite valuation w=u∗v _y as follows: give $\Gamma_{D,u}\times \mathbb {Z}$ the right-to-left lexicographic ordering in which

$$(\gamma,i)\,\leq\,(\delta,j)\quad\text{if and only if}\quad i\,< \,j\,\text{ or ($i\,=\,j$ and $\gamma\,\leq\,\delta$)}. $$

Then define $w\colon E\to\bigl(\Gamma_{D,u}\times \mathbb {Z}\bigr)\cup\{\infty\}$ by

$$ w\bigl(\operatorname*{ \mathchoice{\textstyle\sum}{\sum}{\sum}{\sum}}_{i=k}^\infty d_iy^i\bigr)\,= \,(u(d_j),j) \quad\text{where $j$ is minimal with $d_{j}\neq0$, if some $d_{i}\neq0$}, $$

and w(0)=∞. Prove that w is a valuation on E with $\overline{E}^{\, w}=\overline{D}^{\, u}$ and that ${\Gamma_{E,w}=\Gamma_{D,u}\times \mathbb {Z}}$. Prove also that the canonical homomorphism ${\theta_{E,w}\colon\Gamma_{D,u}\times \mathbb {Z}\to \operatorname {\mathit{Aut}}\bigl(Z(\overline{D}^{\, u})\bigr)}$ is given by

$$(u(d),j)\, \mapsto \, \theta_{D,u}\bigl(u(d)\bigr)\circ (\overline{\tau}\rvert_{Z(\overline{D}^u)})^j, $$

where $\overline{\tau}$ is the automorphism of $\overline{D}$ induced by τ.

Exercise 1.3

The Gaussian valuation on D(x). Let D be any division ring, let D[x] be the polynomial ring over D (with x commuting with the elements of D). Since D[x] is a right Ore domain, it has a right ring of quotients

$$D(x) \,=\,{\{fg^{-1}\mid f,g \in D[x],\ g\ne 0\}}. $$

Suppose v is a valuation on D.

• (i) Extend v first to a map on D[x] by defining

  $$v\big(\operatorname*{ \mathchoice{\textstyle\sum}{\sum}{\sum}{\sum}}_{i=0}^n d_i x^i\big) \,=\, \operatorname* {\mathit{min}}\limits_{0\le i\le n}v(d_i). $$

  Prove that this v on D[x] satisfies the axioms for a valuation, including the multiplicative property v(fg)=v(f)+v(g) for all f, g∈D[x].

• (ii) Extend v to D(x) by defining v(fg ⁻¹)=v(f)−v(g) for all f,g∈D[x], g≠0. Prove that v is well-defined and is a valuation on D(x). This valuation is called the Gaussian valuation on D(x) obtained from v on  D.

• (iii) Prove that Γ _D(x)=Γ _D and $\overline {D(x)} = \overline{D}(\overline{x})$, which is the right ring of quotients of the polynomial ring $\overline{D}[\overline{x}]$.

Notes

The early history of commutative valuation theory—up to the work of Krull—is ably recounted by Roquette in [207]. To our knowledge the earliest use of valuations on noncommutative division algebras was by Hasse in [98] in his analysis of maximal orders in finite-dimensional division algebras over local fields. Schilling in [226] and [227] was the first to consider valuations on arbitrary division algebras. He showed in [227, Th. 9, Th. 10, pp. 53–54] that a Henselian valuation (which he called a ‘relatively complete” valuation, see [227, Def. 17, p. 52]) on the center of a finite-dimensional division algebra extends uniquely to a valuation on the division algebra.

Invariant valuation rings. Let w be a valuation on an division ring D. For the associated valuation ring $\mathcal {O}_{D}$ observe that for every d∈D ^× we have $d\in \mathcal {O}_{D}$ or $d^{-1}\in \mathcal {O}_{D}$. Furthermore, for every d∈D ^× we have ${d\mathcal {O}_{D}d^{-1}=\mathcal {O}_{D}}$. Because of this invariance under conjugation, $\mathcal {O}_{D}$ is called an invariant valuation ring. The invariance is equivalent to $d\mathcal {O}_{D}=\mathcal {O}_{D}d$ for all d∈D ^×; hence, $\mathcal {O}_{D}$ is a duo ring, i.e., every left ideal is a right ideal, and vice versa. Furthermore, the left=right=two-sided ideals of $\mathcal {O}_{D}$ are linearly ordered by inclusion. We have

$$\mathcal {O}_D^\times\,= \,\{d\in D\mid w(d)=0\}\, = \, \operatorname {\mathit{ker}}(w\rvert_{D^\times}). $$

Thus, w induces an isomorphism

$$D^\times\big/\mathcal {O}_D^\times\,\xrightarrow{\,\sim\,}\,\Gamma_D. $$

Since in addition w(d)≥w(e) if and only if $d\mathcal {O}_{D}\subseteq e\mathcal {O}_{D}$, it follows that the ring $\mathcal {O}_{D}$ fully determines the valuation w (up to an ordered group isomorphism of Γ _D ). Indeed, if R is any subring of D such that for each d∈D, (i)  d  or  d ⁻¹∈R, and (ii)  dRd ⁻¹=R, and D ^×/R ^× is abelian, then the relation ≤ on D ^×/R ^× given by

$$dR^\times\leq eR^\times \quad\text{if and only if}\quad dR\supseteq eR $$

makes D ^×/R ^× into a totally ordered abelian group; moreover the map y:D→D ^×/R ^×∪{∞} given by y(d)=dR ^× if d≠0 and y(0)=∞ is a valuation on D with $\mathcal {O}_{D}=R$.

Other valuation rings. For fields, there are three basic structures in valuation theory: (i) valuations; (ii) valuation rings; and (iii) places. These are equivalent in that a valuation v on F determines its valuation ring $\mathcal {O}_{F}$, and conversely $\mathcal {O}_{F}$ determines v by the inclusion ordering on its principal ideals. The associated place of v is the map $\pi_{v} \colon F \to \overline{F} \cup \{\infty \}$ given by $c \mapsto \overline{c}$ if  $c\in \mathcal {O}_{F}$ and c↦∞ otherwise. Thus, $\mathcal {O}_{F}$ determines the place π _v , but also π _v determines $\mathcal {O}_{F}$, since $\mathcal {O}_{F} = \pi_{v}^{-1}(\overline{F})$. (The terminology “composition” of valuations comes from the composition as functions of the associated places.) People working with division algebras and central simple algebras have found generalizations of each of these structures, but they are no longer equivalent. Generalizing (i) we have the notion of a valuation on a division algebra as defined in §1.1, for which the associated rings are the invariant valuation rings described above. Another natural generalization of valuation rings on fields, in the spirit of (ii), is given by total valuation rings: a subring T of a division algebra D is called a total valuation ring if d∈T or d ⁻¹∈T for every d∈D ^×. Total valuation rings have an associated “value set” of right ideals, which is totally ordered by inclusion, but this set is typically not a group. Every invariant valuation ring of D is a total valuation ring, but the converse is not true, as Exercise 1.1 illustrates. For more on total valuation rings on division algebras see Brungs–Gräter  [34] and [35], Mathiak [134], [135], [136], Wadsworth [255]. It is known, for example, see [255, Th. G] that if D is a division algebra finite-dimensional over its center F and V is a valuation ring of F (i.e., with quotient field F), then there are at most finitely many total valuation rings T ₁, …, T _m of D with T _i ∩F=V. (But there may well be no such  T _i at all.) When the T _i exist, they are all conjugate in  D, and the number m of T _i equals the matrix size of D⊗ _F F _h , where F _h  is the Henselization of F with respect to v. Also, if m>1, then V has Krull dimension at least 2.

Invariant and total valuation rings exist only in division algebras. For matrix rings over division algebras, Dubrovin has defined and studied in depth in  [68] and [69] a family of rings based on the idea of places, (iii)  above, for the category of simple Artinian rings; such rings are now called Dubrovin valuation rings. Invariant valuation rings and total valuation rings are Dubrovin valuation rings, as are matrix rings over such rings. Moreover, Dubrovin’s rings have good extension behavior from the center: If A is a central simple algebra over a field F and V is a valuation ring of F, then Dubrovin showed in  [69] that there is a Dubrovin valuation ring B of A with B∩F=V; moreover, Brungs and Gräter in [35] (for V of finite rank) and Wadsworth in [255] (in general) showed that such a B is unique up to conjugacy in A. See the Notes to Ch. 3 and to Ch. 4 for the definition and more properties of Dubrovin valuation rings and connections between these rings and the rings associated with gauges on central simple algebras. A good reference for the substantial theory of Dubrovin valuation rings is the book [132] by Marubayashi et al.

This book is about the most restrictive of these three kinds of noncommutative valuation ring. Invariant valuation rings and their associated valuations have proved the most fruitful in applications. Such a valuation ring may not exist in a given division algebra D, as Th. 1.4 demonstrates; but the presence of such a valuation ring in D is often a source of considerable insight into properties of D.

§1.1: Value groups. In our definition of a valuation on a division ring D, the target group Γ is assumed to be abelian. This might seem an unnatural restriction since D itself is typically noncommutative. If we drop from the definition of a valuation the requirement that the totally ordered group Γ be abelian, there is still an associated valuation ring with unique maximal left and maximal right ideal and residue division ring and value group Γ _D , and associated graded ring. The Mal’cev–Neumann construction of §1.1.4 still works with nonabelian Γ, and shows that any totally ordered group can be the value group of a valued division algebra (see Cohn [55, §2.4]). However, if  D  is finite-dimensional over its center, it turns out (cf. Wadsworth [253, p. 21]) that Γ _D must be abelian, even if that were not initially assumed. Since the division algebras considered in this book are essentially always finite-dimensional over their centers, there is no loss in assuming at the outset that Γ is abelian.

Mal’cev–Neumann series algebras were originally defined independently by Mal’cev in [131] and Neumann in [172].

§1.2: Theorem 1.4 was proved by Ershov in [76] and rediscovered independently by Wadsworth in [253]. It had earlier been proved by Cohn in [54] for rank  1 valuations. The theorem generalizes readily to the case where the division algebra D is infinite-dimensional over its center F but locally finite, i.e., every finite subset of D lies in some finite-dimensional F-subalgebra of D. A possible further generalization was proved by Mahdavi-Hezavehi in [130]: Let D ^×′ be the multiplicative commutator subgroup of D ^×. He proved that if D is algebraic over F, then a valuation v on F extends to D if and only if (i) v has a unique extension to each subfield L of D containing F, and (ii)   ${D^{\times \prime}\cap F^{\times}\subseteq \mathcal {O}_{F}^{\times}}$. Whether this result is actually more general is uncertain because it is still unknown whether there exist algebraic division algebras that are not locally finite.

Proposition 1.16 is due to Cohn [54, Th. 1].

§1.3: Early contributions to the study of graded rings associated to filtrations defined by valuations are due to Krasner [117] and Dedecker [58].