Integral Dubrovin valuation rings

Morandi, Patrick J.; Wadsworth, Adrian R.

doi:10.1090/S0002-9947-1989-0986696-4

Integral Dubrovin valuation rings
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by Patrick J. Morandi and Adrian R. Wadsworth PDF

Trans. Amer. Math. Soc. 315 (1989), 623-640 Request permission

Abstract:

In the preceding paper, Dubrovin valuation rings integral over their centers in central simple algebras were characterized by value functions. Here, these value functions are used to give a method for extending integral Dubrovin valuation rings in generalized crossed product algebras. Several applications of this extension theorem are given, including new and more natural proofs of some theorems on valued division algebras over Henselian fields.

References

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
Peter Draxl, Ostrowski’s theorem for Henselian valued skew fields, J. Reine Angew. Math. 354 (1984), 213–218. MR 767581, DOI 10.1515/crll.1984.354.213
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
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
Wolfram Jehne, Idelklassenfaktorensysteme und verallgemeinerte Theorie der verschränkten Produkte, Abh. Math. Sem. Univ. Hamburg 18 (1952), 70–98 (German). MR 53160, DOI 10.1007/BF02941545
Wolfram Jehne, Separable adel algebras and a presentation of Weil groups, J. Reine Angew. Math. 375/376 (1987), 211–237. MR 882298, DOI 10.1515/crll.1987.375-376.211

Crossed products over simple algebras and their automorphism groups

Pat Morandi, The Henselization of a valued division algebra, J. Algebra 122 (1989), no. 1, 232–243. MR 994945, DOI 10.1016/0021-8693(89)90247-0
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
I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR 1972204

Generalized crossed products

Adrian R. Wadsworth, Dubrovin valuation rings and Henselization, Math. Ann. 283 (1989), no. 2, 301–328. MR 980600, DOI 10.1007/BF01446437
Adrian R. Wadsworth, The residue division ring of a valued division algebra, Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 307–322. Deuxième Contact Franco-Belge en Algèbre (Faulx-les-Tombes, 1987). MR 976644