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The ``Defektsatz'' for central simple algebras


Author: Joachim Gräter
Journal: Trans. Amer. Math. Soc. 330 (1992), 823-843
MSC: Primary 16H05; Secondary 16D30, 16K40, 16S70, 16W60
DOI: https://doi.org/10.1090/S0002-9947-1992-1034663-7
MathSciNet review: 1034663
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Abstract: Let $ Q$ be a central simple algebra finite-dimensional over its center $ F$ and let $ V$ be a valuation ring of $ F$. Then $ V$ has an extension to $ Q$, i.e., there exists a Dubrovin valuation ring $ B$ of $ Q$ satisfying $ V= F \cap B$. Generally, the number of extensions of $ V$ to $ Q$ is not finite and therefore the so-called intersection property of Dubrovin valuation rings $ {B_1}, \ldots,{B_n}$ is introduced. This property is defined in terms of the prime ideals and the valuation overrings of the intersection $ {B_1} \cap \cdots \; \cap {B_n}$. It is shown that there exists a uniquely determined natural number $ n$ depending only on $ V$ and having the following property: If $ {B_1}, \ldots,{B_k}$ are extensions of $ V$ having the intersection property then $ k \leq n$ and $ k= n$ holds if and only if $ {B_1} \cap \cdots \cap {B_k}$ is integral over $ V$. Let $ n$ be the extension number of $ V$ to $ Q$. There exist extensions $ {B_1}, \cdots,{B_n}$ of $ V$ having the intersection property and if $ {R_1}, \ldots,{R_n}$ are also extensions of $ V$ having the intersection property then $ {B_1} \cap \cdots \cap {B_n}$ and $ {R_1} \cap \cdots \cap {R_n}$ are conjugate. The main result regarding the extension number is the Defektsatz: $ [Q:F]= {f_B}(Q/F){e_B}(Q/F){n^2}{p^d}$, where $ {f_B}(Q/F)$ is the residue degree, $ {e_B}(Q/F)$ the ramification index, $ n$ the extension number, $ p = \operatorname{char}(V/J(V))$, and $ d$ a natural number.


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  • [A] G. Azumaya, On maximally central algebras, Nagoya Math. J. 2 (1951), 119-150. MR 0040287 (12:669g)
  • [AS] S. A. Amitsur and L. W. Small, Prime ideals in $ PI$-rings, J. Algebra 62 (1980), 358-383. MR 563234 (81c:16027)
  • [Ba] J. R. Bastida, Field extensions and Galois theory, Encyclopedia of Math, and its Appl., Vol. 22, Addison-Wesley, Reading, Mass., 1984. MR 747137 (85j:12001)
  • [Bo] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1970.
  • [BG1] H. H. Brungs and J. Gräter, Valuation rings in finite-dimensional division algebras, J. Algebra 120 (1989), 90-99. MR 977862 (90a:16005)
  • [BG2] -, Extensions of valuation rings in central simple algebras, Trans. Amer. Math. Soc. 317 (1990), 287-302. MR 946216 (90d:16023)
  • [C] P. M. Cohn, Algebra, Vol. 2, Wiley, London, 1977. MR 0530404 (58:26625)
  • [Dr] P. Draxl, Ostrowski's theorem for Henselian valued skew fields, J. Reine Angew. Math. 354 (1984), 213-218. MR 767581 (86g:12008)
  • [D1] N. I. Dubrovin, Noncommutative valuation rings, Trudy Moskov. Mat. Obsch. 45 (1982), 265-280; English transl., Trans. Moscow Math. Soc. 45 (1984), 273-287. MR 704633 (85d:16002)
  • [D2] -, Noncommutative valuation rings in simple finite-dimensional algebras over a field, Mat. Sb. 123 (1984), 496-509; English transl., Math. USSR-Sb. 51 (1985), 493-505. MR 740675 (85j:16020)
  • [E] O. Endler, Valuation theory, Springer, New York, 1972. MR 0357379 (50:9847)
  • [G1] J. Gräter, Zur Theorie nicht kommutativer Prüferringe, Arch. Math. 41 (1983), 30-36. MR 713664 (85e:16007)
  • [G2] -, Lokalinvariante Bewertungen, Math. Z. 192 (1986), 183-194. MR 840822 (87i:12021)
  • [G3] -, Valuations on finite-dimensional division algebras and their value groups, Arch. Math. 51 (1988), 128-140. MR 959388 (89h:12007)
  • [G4] -, A note on valued division algebras, J. Algebra (to appear). MR 1176896 (93f:16023)
  • [JW] B. Jacob and A. Wadsworth, Division algebras over Henselian fields, J. Algebra 128 (1990), 126-179. MR 1031915 (91d:12006)
  • [Ma] K. Mathiak, Valuations of skew fields and projective Hjelmslev spaces, Lecture Notes in Math., vol. 1175, Springer-Verlag, Berlin, Heidelberg, and New York, 1986. MR 835210 (87g:16002)
  • [M1] P. Morandi, The Henselization of a valued division algebra, J. Algebra 122 (1989), 232-243. MR 994945 (90h:12007)
  • [M2] -, An approximation theorem for Dubrovin valuation rings, Math. Z. 207 (1991), 71-82. MR 1106813 (92g:16021)
  • [MR] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, Chichester, 1987. MR 934572 (89j:16023)
  • [Re] I. Reiner, Maximal orders, Academic Press, London, 1975. MR 1972204 (2004c:16026)
  • [Ri] P. Ribenboim, Théorie des valuations, Presses Univ. Montréal, Montréal, 1968. MR 0249425 (40:2670)
  • [Rw] L. H. Rowen, Polynomial identities in ring theory, Academic Press, New York, 1980. MR 576061 (82a:16021)
  • [S1] O. F. G. Schilling, Noncommutative valuations, Bull. Amer. Math. Soc. 51 (1945), 297-304. MR 0011684 (6:201a)
  • [S2] -, The theory of valuations, Math. Surveys and Monographs, No. 4, Amer. Math. Soc., Providence, R. I., 1950. MR 0043776 (13:315b)
  • [W1] A. R. Wadsworth, Dubrovin valuation rings, Perspectives in ring theory (F. van Oystaeyen and L. LeBruyn, eds.), NATO ASI Series, Series C, Vol. 233, Kluwer, Dordrecht, 1988, pp. 359-374. MR 1048422 (91e:16053)
  • [W2] -, Dubrovin valuation rings and Henselization, Math. Ann. 283 (1989), 301-328. MR 980600 (90f:16009)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1034663-7
Keywords: Valuation rings, central simple algebras
Article copyright: © Copyright 1992 American Mathematical Society

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