Value Functions on Simple Algebras, and Associated Graded Rings

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.

Since our approach to structures over valued fields relies in a fundamental way on filtrations and associated graded structures, our arguments often require information that is specific to graded modules. We collect in this chapter the basic definitions and results on graded algebras and modules that will be of constant use in subsequent chapters. In §2.1 we define graded rings, modules, and the gradings on their homomorphism groups and tensor products. For finite-dimensional semisimple graded algebras A we prove in §2.2 graded analogues to the classical Wedderburn Theorems. When A is graded simple we also prove graded versions of the Double Centralizer Theorem and Skolem–Noether Theorem. For A graded simple, its degree-0 component A ₀ is semisimple, though often not simple. In §2.3 we relate the grade set Γ _A to the structure of A ₀ via the map \(\theta_{\mathsf {A}}\colon \Gamma^{\times}_{\mathsf {A}}\to \operatorname {\mathcal {G}}(Z(A_{0})/(Z(\mathsf {A}))_{0})\) induced by inner automorphisms of homogeneous units. We also describe inertial graded algebras A, which are completely determined by A ₀.

In this chapter we define value functions on vector spaces over valued division algebras and on algebras over valued fields. These value functions yield associated graded structures that are graded vector spaces and graded algebras. We emphasize value functions whose associated graded structures have good properties: those called norms on vector spaces and gauges on algebras. The use of gauges is a key aspect of our approach to valuation theory on division algebras and simple algebras; this aspect will be emphasized throughout the rest of the book. This chapter provides foundational material on norms and gauges and their behavior with respect to subspaces and subalgebras, tensor products, spaces of homomorphisms, and coarsenings of valuations.

This chapter is crucial for the study of gauges. Let F be a field with valuation v, and let (F _h ,v _h ) be the Henselization of (F,v). In §4.1 we prove Morandi’s criterion, Th. 4.1, that v extends to a division algebra D with center F if and only if D⊗ _F F _h is a division algebra. Among the applications are a primary decomposition theorem and an Ostrowski-type defect theorem, Th. 4.3, for valued division algebras. The valuation v is said to be defectless in D if equality holds in the Fundamental Inequality for the extension of v to D. In  §4.2 we extend the notion of defectlessness of v to any semisimple F-algebra  A by considering the simple components of A⊗ _F F _h . In §4.3 we prove key classification results for gauges. We show in Th. 4.26 that if S is a central simple F _h -algebra, then any v _h -gauge on S is an \(\operatorname {\mathit{End}}\)-gauge as in  Prop. 3.​34. We also show how v-gauges on a semisimple F-algebra A are built from gauges on the simple components A _i of A for the extensions of v to  Z(A _i ). §4.4 is devoted to proving Th. 4.50, which says that a semisimple F-algebra has a v-gauge if and only if v is defectless in A.

We pursue in this chapter the investigation of valuations through the graded structures associated to the valuation filtration. The graded field of a valued field is an enhanced version of the residue field, inasmuch as it encapsulates information about the value group in addition to the residue field. It thus captures much of the structure of the field, particularly in the Henselian case. This point is made clear in §5.2, where we show that—when the ramification is tame—Galois groups and their inertia subgroups of Galois extensions of valued fields can be determined from the corresponding extension of graded fields. Henselian fields are shown to satisfy a tame lifting property from graded field extensions, generalizing the inertial lifting property. In §5.1, we lay the groundwork for the subsequent developments by an independent study of graded fields, their algebraic extensions and their Galois theory.

This chapter has two parts. In the first section, we organize the central simple graded algebras over a given graded field into a Brauer group and obtain analogues of the general results on the Brauer group of fields. Graded fields have a special structure however, since the degree 0 component is a canonically-defined subfield. This structure is reflected in a canonical filtration of the Brauer group of graded fields. The second section aims to reproduce this canonical filtration for the Brauer group of valued fields. For this, special types of gauges are defined on central simple algebras over valued fields. They lead to the definition of the tame and the inertial part of the Brauer group. These developments culminate with Th. 6.64, which establishes a canonical index-preserving isomorphism between the tame part of the Brauer group of a Henselian-valued field and the Brauer group of its graded field. This result is fundamental for the use of graded algebras to study division algebras over Henselian fields. An application is given to obtain a split exact sequence for the inertially split part of the Brauer group of a Henselian field. Another application of the theorem yields Witt’s description of the Brauer group of a field with a complete discrete rank 1 valuation.

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.

In this chapter we focus on the tame division algebras D with center a field F with Henselian valuation v. As usual, we approach this by first obtaining results for graded division algebras, then lifting back from \(\operatorname {\mathsf {gr}}(D)\) to D. This is facilitated by results in §8.1 on existence and uniqueness of lifts of tame subalgebras from \(\operatorname {\mathsf {gr}}(D)\) to D. In §8.2, we describe four fundamental canonical (up to conjugacy) subalgebras of D that reflect its valuative structure. The rest of the chapter is devoted to Brauer group factorizations of D corresponding to the noncanonical direct product decomposition of \(\operatorname {\mathit{Br}}_{t}(F)\) given in Cor. 7.​85. The factor \(\operatorname {\mathit{Hom}}^{c}(\operatorname {\mathcal {G}}(\overline{F}), {\mathbb{T}}(\Gamma_{F}))\) is represented by a type of division algebra N called decomposably semiramified, defined in §8.3, and characterized by the property that N contains a maximal subfield inertial over F and another totally ramified over F. We show in §8.4 that every tame division algebra D is Brauer equivalent to some S⊗ _F T where S is inertially split and T is tame and totally ramified over F. We show further that every inertially split division algebra S is Brauer equivalent to some I⊗ _F N, where I is inertial over F and N is decomposably semiramified. The classes \([\,\overline{I}\,]\) for the I appearing in the I⊗ _F N decompositions of S are shown to range over a single coset of \(\mathit{Dec}(Z(\overline{S})/\overline{F})\) in \(\operatorname {\mathit{Br}}(\overline{F})\), called the specialization coset of S. In the final subsection,  §8.4.6, we summarize what happens in the special case that v is discrete of rank 1, where substantial simplifications occur.

In this chapter we apply the machinery developed in previous chapters to analyze the subfields and splitting fields of division algebras over a Henselian field F. In §9.1 we give properties of the splitting fields of tame division algebra D with center F, with particularly strong criteria proved if D is inertial or totally ramified over F. This leads to explicit constructions of several interesting examples of division algebras, including noncyclic division algebras of degree p ² with no maximal subfield of the form \(F(\!\sqrt[p^{2}]{a})\) in Examples 9.15, 9.17, and 9.18; noncyclic p-algebras in Ex. 9.26; noncrossed product algebras including universal division algebras in Th.  9.30 and division algebras over Laurent series over \(\mathbb {Q}\), noncrossed products whose degree exceeds the exponent in Cor. 9.46; and crossed product division algebras with only one Galois group for all maximal subfields Galois over the center in Prop. 9.28[9.28].

A central division algebra D over a field F is said to be decomposable if D=D ₁⊗ _F D ₂ for some proper subalgebras D ₁, D ₂ of D; otherwise, it is indecomposable. In this chapter we give examples of indecomposable algebras, emphasizing constructions that use valuation theory. In light of the primary decomposition, we restrict attention to division algebras of prime power degree. Indecomposable algebras of exponent p ² or higher are relatively easy to construct as “p-th roots” of other division algebras. Such constructions are discussed in §10.1, where we also give an example of an indecomposable division algebra D that becomes decomposable after a scalar extension of degree prime to \(\operatorname {\mathit{deg}}D\). §10.2 focuses on the more difficult case of indecomposables of prime exponent. We give in  §10.2.1 a criterion of Jacob to test the decomposability of a tame semiramified division algebra of prime exponent over a Henselian field. This criterion yields examples of exponent 2 and degree 8 in  §10.2.2, and of exponent p≠2 and degree p ^r with r≥2 in §10.2.4. The last section, §10.3, deals with complete decompositions into tensor products of symbol algebras. The main result is Th. 10.26, which relates armatures in an inertially split division algebra over a Henselian field to armatures in special representatives of its specialization coset.

For a central simple algebra A, the group SK ₁(A) is defined to be the quotient \(\{a\in A^{\times}\mid \operatorname {\mathit{Nrd}}_{A}(a) = 1\}\big/ [A^{\times},A^{\times}]\), where \(\operatorname {\mathit{Nrd}}_{A}\) is the reduced norm and [A ^×,A ^×] is the commutator group of A ^×. This group  SK ₁(A) is a subtle invariant of A ^× that has been of longstanding interest to those working with central simple algebras, algebraic groups, and K-theory. Since SK ₁ is a Brauer class invariant, it suffices to compute SK ₁ for division algebras. We prove here nearly all the known formulas for SK ₁(D) for D a tame central division algebra over a Henselian field. We do this by first obtaining in §11.1 formulas for SK ₁ in the graded setting, where calculations are substantially easier and more transparent. We then recover in §11.2 the corresponding formulas in the valued setting by proving in Th. 11.21 that \(\mathit{SK}_{1}(D) \cong \mathit{SK}_{1}(\operatorname {\mathsf {gr}}(D))\) for F=Z(D) Henselian and D tame over F.

Roughly speaking, the essential dimension of an algebraic structure measures the complexity of the structure by giving the number of independent parameters needed to define it. For a prime number p, the essential p-dimension is a variation of essential dimension that takes into account simplifications in structure that can occur after field extensions of degree prime to p. In §12.1 we define the basic notions of essential dimension and essential p-dimension. We also give examples, which emphasize the relation with the decomposability of central simple algebras into tensor products of symbols. Merkurjev in [150] and Baek–Merkurjev in [21] recognized that the presence of a valuation on a division algebra D implies some level of complexity to D, and they used this to give a lower bound on the essential p-dimension of D. In §12.2 we give a version of the valuation-theoretic part of their arguments. Then in §12.3 we sketch their method to obtain lower bounds on the essential p-dimension of central simple algebras. In the final §12.4, we discuss consequences of these lower bounds for the decomposability of division algebras into tensor products of symbol algebras.