Adrian R. Wadsworth

Division algebras over Henselian fields

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.

Totally ramified valuations on finite-dimensional division algebras

Division algebras D with valuation v are studied, where D is lSnite-dimensional and totally ramified over its center F (i.e., the ramification index of v over wlF equals [D: F]). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When vlF is Henselian a full classification of the F-subalgebras of D is given. When F has a Henselian valuation v with separably closed residue field and A is any tame central simple F-algebra, an algorithm is given for computing the underlying division algebra of A from a suitable subgroup of A*/F*. Some examples are constructed using this valuation theory, including the first example of finitbdimensional F-central division algebras D1 and D2 with D1 XF D2 not a division ring, but D1 and D2 having no common subfield K D F. Valuation theory, long a basic tool in commutative algebra, has been relatively neglected in the study of division algebras, until quite recently. Nontheless, valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties, particularly algebras over iterated Laurent power series fields. For example, such division algebras have been key ingredients in Amitsurs noncrossed product construction [Am] and in Platonovs construction [P] of division algebras D with SK1(D) 7& 1. Valuations are not so prevalent on division algebras as on fields. But if a division algebra D does have a valuation, this structure contains a substantial amount of information about D which would scarcely be accessible otherwise. We consider here valued divisioll algebras D for which D is totQlly ramified and tame over its center F, i.e., for which the ramification index IrD rFI equals the dimension [D: F] of D over F and the characteristic char(D) does not divide [D: F]. (Here rD iS the value group of the valuation on D, and D is the residue division algebra. We assume throughout that [D: F] < oo.) Valued division algebras of this type appear, e.g., in Amitsurs noncrossed product paper [Am, §2], in Saltmans work on indecomposable division algebras [Sa], in certain of the MalcevNeumann division algebras considered by the first author and Amitsur [TA2,§4], etc. However, the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before. This may be because most past work on valued division algebras has concentrated on discrete valuations, when rD -Z; for such a valuation D is never totally ramified over its center F unless D = F (cf. (3.2) below). Received by the editors June 26, 1986. 1980 Mathematics Subject Clmsification (1985 Reon). Primary 16A39. 2Supported in part by F.N.R.S. 2Supported in part by the National Science Foundation. (:)1987 American Mathematlcal Society 0002-9947/87

Algebraic extensions of graded and valued fields

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Value Functions and Associated Graded Rings for Semisimple Algebras

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The algebra generated by two commuting matrices

If F is a field with a (Krull) valuation, then the filtration of F induced by the valuation yields an associated graded ring, which is a graded field. Conversely, if R is a graded field with totally ordered grade group, then R is an integral domain and there is a canonically associated valuation on the quotient field of R. The processes of passing from valued field to graded field and vice versa are not quite inverses of each other, but many properties in one setting are well-reflected in the other.

Value Functions on Simple Algebras, and Associated Graded Rings

We introduce a type of value function y called a gauge on a finite-dimensional semisimple algebra A over a field F with valuation v. The filtration on A induced by y yields an associated graded ring gr_y(A) which is a graded algebra over the graded field gr_v(F). Key requirements for y to be a gauge are that gr_y(A) be graded semisimple and that dim_(gr_v(F)) (gr_y(A)) = dim_F(A). It is shown that gauges behave well with respect to scalar extensions and tensor products. When v is Henselian and A is central simple over F, it is shown that gr_y(A) is simple and graded Brauer equivalent to gr_w(D), where D is the division algebra Brauer equivalent to A and w is the valuation on D extending v on F. The utility of having a good notion of value function for central simple algebras, not just division algebras, and with good functorial properties, is demonstrated by giving new and greatly simplified proofs of some difficult earlier results on valued division algebras.

Witt Rings and Brauer Groups Under Multiquadratic Extensions .2.

A simplification is given of the proof by Barria and Halmos that the algebra generated by two commuting n×n matrices over a field has dimension ≥n Using the same approach, it is shown that if M is a finitely-generated torsion module over a principal ideal domain R. and if . then the R-subalgebra of EndR(M) generated by b is isomorphic as an R-module to a submodule of M.

Graded Hermitian forms and Springer's theorem

This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsurs construction of noncrossed products and Platonovs solution of the Tannaka-Artin problem. This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased. The intended audience consists of graduate students and research mathematicians.

Pfister Ideals in Witt Rings

Let F be a field of characteristic not 2, and suppose M is a multiquadratic extension of F. That is, M/F is a finite abelian extension of exponent 2, so that M = F(@) for some finite subgroup G E F/F’, where

Dubrovin Valuation Rings

= F (0). In studying Brauer groups and products of quaternion algebras, the second author was led to consider the homology groups N,(M/F) of -a certain complex gM,F associated to the extension M/F. This complex appears in [ 11, (3.1); 301 and in (1.1) below. The purpose of the present work is to investigate the first homology group N,(M/F) and to exhibit its close connections with quadratic form theory. The higher homology groups are examined in [ 11,301. The field F is said to have property Pi(n) if N,(M/F) = 1 for every multiquadratic extension M/F with [M:F] Q 2”. Properties P,(l) and P,(2) always hold, but there are examples [29] of fields F for which P,(3) fails. These examples are generalized in Section 5.