Multidimensional local skew-fields

Multidimensional local skew-fieldsDISSERTATIONzur Erlangung des akademischen Gradesdoctor rerum naturalium(Dr. rer. nat.)im Fach Mathematikeingereicht an derMathematisch-Naturwissenschaftlichen Fakultät IIHumboldt-Universität zu BerlinvonHerr Diplom-Mathematiker Alexander Zheglovgeboren am 23.08.1974 in Ivanovo, Russia.Präsident der Humboldt-Universität zu Berlin:Prof. Dr.

Jürgen MlynekDekan der Mathematisch-Naturwissenschaftlichen Fakultät II:Prof. Dr. Elmar KulkeGutachter:1. Prof. Dr. E.W.Zink2. Prof. Dr. A.N.Parshin3. Prof. Dr. V.I.Yanchevskiieingereicht am:Tag der mündlichen Prüfung:04.04.200210.07.2002ContentsIntroduction.0 The0.10.20.30.40.50.62structure of two-dimensional local skew fields.General. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .Splittable skew fields. . . . . . . . . . . . . . . . . . . . . . .Classification of two-dimensional local splittable skew fields ofteristic 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.3.1 The case α = Id. . . . .

. . . . . . . . . . . . . . . .0.3.2 The general case. . . . . . . . . . . . . . . . . . . . . .Splittable skew fields of characteristic p > 0. . . . . . . . . . .0.4.1 Wild division algebras over Laurent series fields . . . .0.4.2 Cohen’s theorem . . . . . . . . . . . . . . . . . . . . .0.4.3 Decomposition theorem . . . .

. . . . . . . . . . . . .Classes of conjugate elements . . . . . . . . . . . . . . . . . .New equations of KP-type on skew fields . . . . . . . . . . . .1 Classification of automorphisms of1.1 Basic results. . . . . . . . . . . .1.2 Proof of the theorems I and II .1.3 Proof of the theorem III . . . . .Bibliographya two-dimensional. .

. . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .6612. . . . .. . . . .charac. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .222329333435366170local field.. . . . . . . . .. . . . . . . . .. . . . . . . . .727284921071Introduction.In this work we study local skew fields, which are natural generalisation of n-dimensionallocal fields, and their applications to the theory of central division algebras overhenselian fields.Local fields appear in a natural way in algebraic geometry and algebraic numbertheory if anyone try to find a connection between local and global properties of suchobjects like algebraic number fields, arithmetic schemes and algebraic varieties.Historically the first examples of 1-dimensional local fields appeared in the 19 century in complex analisys and in algebraic number theory.

These examples are knownfields C((z)) and Qp . Now we say that 1-dimensional local field is a quotient field of acomplete discrete valuated ring.A little bit later the first examples of local skew fields were found. They were finitedimensional division algebras over classical local fields, and they were completely studied by Hasse, Brauer, Noether and Albert. At the same time there were several worksof Witt ([34]) about skew fields over discrete valuated fields, which opened researchingof skew fields over henselian fields. Basic results about a structure of such skew fieldswere recently got by Jacob and Wandsworth in ([9]).In the middle of 70-th A.N.Parshin introduced a notion of a multidimensional localfield which generalised the notion of a usual local field ([19],[24], [7]).n-dimensional local field is a complete discrete valuated field such that the residuefield is a n − 1-dimensional local field.One of the typical examples of such a field is an iterated Laurent series fieldk((z1 ))((z2 )) .

. . ((zn )). Elements z1 , . . . , zn are called local parameters of this field.Multidimensional local fields appears also as natural generalisations of local objectson 1-dimensional scheme. As an example let us consider the following construction.Consider an algebraic scheme X of dimension n.

Let Y0 ⊃ . . . ⊃ Yn be a flagof closed subschemes in X such that Y0 = X, Yn = x is a closed point on X, Yiis a codimension 1 subscheme in Yi−1 (1 ≤ i ≤ n), x is a smooth point on all Yi(0 ≤ i ≤ n). Then there exists a construction which assign in canonical way to anygiven flag a n-dimensional local field. Moreover, let X be an algebraic variety over afield k, x be a rational point over k, z1 , z2 , . .

. , zn ∈ k(X) be fixed local parameters suchthat zn−i+1 = 0 is an equation of Yi on Yn−1 in a neiborhood of the point x (1 ≤ i ≤ n).Then our n-dimensional local field can be identified with k((z1 ))((z2 )) . . . ((zn )) ([24],[7]).Using this assignment a number of results known earlier only for the case of a curvewas generalised to a higher dimensional case.

These are such well-known results asmulti-dimensional reciprocity lows of Parshin-Lomadze ([19], [15], [20], [7]).During the last 25 years there was opened another direction in the theory of localfields. This is an application to the theory of integrable systems connected with theKrichever-Sato-Wilson correspondence on a curve (for more details on the Krichever2correspondence see [6], [29], [16], [27]).Recently there were issued several papers [17], [18], [23], where the ideas of theKrichever-Sato-Wilson correspondence on a curve were developed to the case of varieties of higher dimension.

In particular, A.N. Parshin pointed out one class of noncommutative local fields arising in differential equations and showed that these skewfields possesses many features of commutative fields. He defined a skew field of formal pseudo-differential operators in n variables and studied some of their properties.He raised a problem of classifying non-commutative local skew fields. It was the firstargument to begin to study such skew fields.A generalisation of a notion ”local field” looks very natural:n-dimensional local skew field is a complete discrete valuated skew field such that theresidue skew field is a n − 1-dimensional local skew field.In this work we try to study n-dimensional local skew fields bearing in mind only thedefinition.

Unfortunately, there appear very hard obstructions already on the first stepswhich leads us to some restrictions. So, we study only skew fields with commutativeresidue skew field. By the way, a number of results valid in general case (see, forexample, proposition 0.7 and corollary 1) and a number of results can be generalisedto the case of skew fields with residue skew field finite dimensional over its centre (see,for example, section 1.4).Some applications of developed theory to the Krichever correspondence we get insection 1.6. Namely, we get some generalisations of the classical KP-equations (hierarchy).Surprisingly the studying of local skew fields leads to some new unexpected resultsin the valuation theory on finite dimensional division algebras.

Using general formulasfor splittable local skew fields (i.e. for skew fields such that the residue skew field canbe embedded into the valuation ring) we get a decomposition theorem for a class ofsplittable wild division algebras over a Laurent series field with arbitrary residue field ofcharacteristic greater than two. This theorem is a generalisation of the decompositiontheorem for tame division algebras given by Jacob and Wadsworth in [9]. An extensiveanalysis of the wild division algebras of degree p over a field F with complete discreterank 1 valuation with char(F̄ ) = p was given by Saltman in [28] ( Tignol in [32]analysed more general case of the defectless division algebras of degree p over a field Fwith Henselian valuation).

In his recent revue [33] Wadsworth pointed out that for mostof the specific examples and applications it is suffice to consider Henselian valued fieldslike iterated Laurent series fields, that is n-dimensional local fields. So, we get in somesense the complete picture of a local structure of the Brauer group over such fields. Asa corollary we get the positive answer on the following conjecture: the exponent of Ais equal to its index for any division algebra A over a C2 -field F = F1 ((t2 )), where F1is a C1 -field (see [26], 3.4.5.).From the other hand, the problem of classification of local skew fields leads tothe problem of classification of conjugacy classes in the automorphism group of an3n-dimensional local (commutative) field.

We solve this problem for the group of continuous automorphisms.We note that the automorphism group of a local field of positive characteristic isintensively studied now in the algebraic number theory (we mean recent applications tothe problem of description the Galois group of an arithmetically profinite extension).Moreover, the automorphism group of the field Fq ((t)) (so called Nottingham group)is now carefully studied in the group theory (for more details see papers [5], [3], [12],[8], [13], [14], [10], [11], [36].

We hope that our results on the automorphism group willbe applied in the future to obtain some useful results about the Galois group of anarithmetically profinite extension.Here is a brief overview of this thesis. It consists of two chapters.The first chapter consists of five paragraphs. In §1 we give general definitions of alocal skew field, of a splitness and of an isomorphism of local skew fields. Also we studysome general properties of splittable skew fields.Thereafter except §4 we study mostly two-dimensional local skew fields with commutative residue skew field.