On Dubrovin valuation rings in crossed product algebras
HTML articles powered by AMS MathViewer
- by Darrell Haile and Patrick Morandi PDF
- Trans. Amer. Math. Soc. 338 (1993), 723-751 Request permission
Abstract:
Let $F$ be a field and let $V$ be a valuation ring in $F$. If $A$ is a central simple $F$-algebra then $V$ can be extended to a Dubrovin valuation ring in $A$. In this paper we consider the structure of Dubrovin valuation rings with center $V$ in crossed product algebras $(K/F,G,f)$ where $K/F$ is a finite Galois extension with Galois group $G$ unramified over $V$ and $f$ is a normalized two-cocycle. In the case where $V$ is indecomposed in $K$ we introduce a family of orders naturally associated to $f$, examine their basic properties, and determine which of these orders is Dubrovin. In the case where $V$ is decomposed we determine the structure in the case of certain special discrete, finite rank valuations.References
- S. A. Amitsur and L. W. Small, Prime ideals in PI rings, J. Algebra 62 (1980), no. 2, 358–383. MR 563234, DOI 10.1016/0021-8693(80)90188-X
- H.-H. Brungs and J. Gräter, Extensions of valuation rings in central simple algebras, Trans. Amer. Math. Soc. 317 (1990), no. 1, 287–302. MR 946216, DOI 10.1090/S0002-9947-1990-0946216-5
- N. I. Dubrovin, Noncommutative valuation rings, Trudy Moskov. Mat. Obshch. 45 (1982), 265–280 (Russian). MR 704633
- N. I. Dubrovin, Noncommutative valuation rings in simple finite-dimensional algebras over a field, Mat. Sb. (N.S.) 123(165) (1984), no. 4, 496–509 (Russian). MR 740675
- Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379
- Darrell E. Haile, Crossed-products orders over discrete valuation rings, J. Algebra 105 (1987), no. 1, 116–148. MR 871749, DOI 10.1016/0021-8693(87)90182-7
- Bill Jacob and Adrian Wadsworth, Division algebras over Henselian fields, J. Algebra 128 (1990), no. 1, 126–179. MR 1031915, DOI 10.1016/0021-8693(90)90047-R
- Patrick J. Morandi, Value functions on central simple algebras, Trans. Amer. Math. Soc. 315 (1989), no. 2, 605–622. MR 986697, DOI 10.1090/S0002-9947-1989-0986697-6
- Patrick J. Morandi and Adrian R. Wadsworth, Integral Dubrovin valuation rings, Trans. Amer. Math. Soc. 315 (1989), no. 2, 623–640. MR 986696, DOI 10.1090/S0002-9947-1989-0986696-4
- Paulo Ribenboim, Théorie des valuations, Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). Deuxième édition multigraphiée. MR 0249425
- Adrian R. Wadsworth, Dubrovin valuation rings and Henselization, Math. Ann. 283 (1989), no. 2, 301–328. MR 980600, DOI 10.1007/BF01446437 M. Westmoreland, Doctoral dissertation, University of Texas at Austin, 1990.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 338 (1993), 723-751
- MSC: Primary 16H05; Secondary 12G05, 13F30, 16G30, 16W60
- DOI: https://doi.org/10.1090/S0002-9947-1993-1104201-X
- MathSciNet review: 1104201