Valuations of Skew Fields and Projective Hjelmslev Spaces
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1175
Karl Mathiak
Valuations of Skew Fields and Projective Hjelmslev Spaces
Springer-Verlag Berlin Heidelberg New York Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1175
Karl Mathiak
Valuations of Skew Fields and Projective Hjelmslev Spaces
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Karl Mathiak Institut fur Algebra und Zahlentheorie Technische Universitat Braunschweig Pockelsstr. 14,3300 Braunschweig, Federal Republic of Germany
Mathematics Subject Classification (1980): 10F45, 12E15, 12J10, 16A05, 16A 10, 16A39,51C05
ISBN 3·540·16099-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16099-X Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Preface
This text is intended to provide a reasonably self-contained account of the theory of valuations of skew fields and their application to projective Hjelmslev spaces. The reader is only assumed to be familiar with the basic notions of bra and the rudiments of topology and geometry. Nevertheless, the acquaintance with Krull valuations would be helpful (see for instance Endler [1] or Krull [1]). The concept of valuation used here is due to F. It is more general than that of a Schilling valuation which is characterized by the fact that the valuation ring of the valuation is invariant under the inner automorphisms of the field. In contrast to this, the valuation rings of general valuations may be invariant or not. The principal difference between general valuations and invariant valuations is the following: The value set of a general valuation is only a totally ordered set and has no algebraic structure. It contains therefore less information about the ideal structure of the valuation ring than the value set of a Krull valuation. To overcome this disadvantage, the concept of a value group is introduced which is a subgroup of the automorphism group of the value set. Using a slightly modified notion of convexity in the value group, Krull's theorems on the correspondence between prime ideals and convex subgroups may be generalized to skew fields. If a valuation is invariant, the value group can be embedded into the value set. Then the algebraic structure of the value group is carried over onto the value set. This leads back to the concept of a Krull valuation where the non-zero values form a group. There are two methods to construct fields with non-invariant valuations. These fields are formal power series fields and skew rational function fields. The e
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