Valuation theory has classically meant the study of valuations on a commutative field. Such valuation theory has flourished for many decades, nourished by its connections with number theory and algebraic geometry. But there is also a noncommutative side of the subject in the study of valuations and valuation rings on division rings. This aspect has blossomed only in the last twenty odd years, and it is not so well known as its commutative counterpart. We give in this paper a survey of valuations and valuation theory for division rings finite dimensional over their centers. We describe the associated theory and also some of the most significant constructions that have been given as applications. The earliest use of valuations on noncommutative division rings, to my knowledge, was by Hasse in his work in [Ha] in 1931 on orders over central simple algebras over p-adic fields. In the ’40’s and ’50’s there was a little further work with valuations on division algebras over fields with complete discrete valuations. Also, there was some discussion of valuations on division algebras in Schilling’s work [Schi1], [Schi2], mostly observing that some results about valuations remain valid without assuming that the ambient field be commutative. One can speculate that there was little attention to valuations on division algebras because too often for the division algebras of interest there was no apparent valuation available. Additionally, it was not fully clear what to take as the definition of a valuation on a division ring, since the concepts of valuation and valuation ring are not equivalent in the noncommutative setting. It was not until the late 1970’s and the 1980’s that valuation theory on division algebras began to develop substantially. This was due in large measure to the realization that some of the major constructions of counterexamples in the 1970’s could best be understood using valuation theory. This applies to Amitsur’s construction of noncrossed product division algebras and also to the constructions of division algebras with nontrivial SK1 by Platonov and by Yanchevskĭi. It began to

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Valuation Theory 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 Amitsur's noncrossed product construction [Am] and in Platonov's 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 Amitsur's noncrossed product paper [Am, §2], in Saltman's 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 $}.00 + $ 25 per page

https://www.ams.org/tran/1987-302-01/S0002-9947-1987-0887507-6/S0002-9947-1987-0887507-6.pdf

Totally ramified valuations on finite-dimensional division algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

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Open problems on central simple algebras

We define and study the essential dimension of an algebraic stack We compute the essential dimension of the stacks Mgn and MgnBar of smooth, or stable, n-pointed curves of genus g We also prove a general lower bound for the essential dimension of algebraic groups with a non-trivial center Using this, we find new exponential lower bounds for the essential dimension of spin groups and new formulas for the essential dimension of some finite p-groups Finally, we apply the lower bound for spin groups to the theory of the Witt ring of quadratic forms over a field k

Essential dimension and algebraic stacks

The abelian Galois subfields of Malcev-Neumann formal series division rings are determined. The results obtained in this paper lead to a lower bound for the rank of Galois splitting fields of universal division algebras.

Kummer subfields of Malcev-Neumann division algebras

A symplectic module is a finite group with a regular antisymmetric form. The paper determines sufficient conditions for the invariants of the maximal isotropic subgroups (Lagrangians), and asymptotic values for a lower bound of a group which contains Lagrangians of all symplectic modules of a fixed orderpn. These results have application to the splitting fields of universal division algebras.

J. P. Tignol

Papers

Symplectic modules