Abstract
Let R be a regular local ring, K its field of fractions and A1, A2 two Azumaya algebras with involutions over R. We show that if A1 ⊗ R K and A1 ⊗ R K are isomorphic over K, then A1 and A2 are isomorphic over R. In particular, if two quadratic spaces over the ring R become similar over K then these two spaces are similar already over R. The results are consequences of three facts:(a) rationally isomorphic hermitian spaces are locally isomorphic, (b) the results hold for discrete valuation ring, (c) a purity theorem hold for multipliers.
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References
J.-L. Colliot-Thélène, M. Ojanguren: Espaces Principaux Homogènes Localement Triviaux, Publ. Math. IHES 75 (1992), no. 2, 97–122. MR
J.-L. Colliot-Thélène, J.-J. Sansuc: Principal Homogeneous Spaces under Flasque Tori: Applications, J. Algebra 106 (1987), 148–205. MR
D. Eisenbud: Commutative Algebra, Graduate Texts in Math., no. 150, Springer-Verlag, 1994. MR
G. Harder: Halbeinfache Gruppenschemata ueber Dedekindringen, Inventiones Math. 4 (1967), 165–191. MR
M.A. Knus, A.S. Merkurjev, M. Rost, J.-R Tignol: Book of Involutions, Colloquium Publication, no. 44, AMS, 1998. MR
Y. Nisnevich: Rationally Trivial Principal Homogeneous Spaces and Arithmetic of Reductive Group Schemes Over Dedekind Rings, C. R. Acad. Sc. Paris, Serie I, 299, (1984), no. 1, 5–8. MR
M. Ojanguren: Quadratic forms over regular rings, J. Indian Math. Soc., 44, (1980), 109–116. MR
M. Ojanguren, I. Panin: A Purity Theorem for the Witt Group, Ann. Sci. Ecole Norm. Sup., Serie 4, 32, (1999), 71–86. MR
M. Ojanguren, I. Panin: Rationally trivial hermitian spaces are locally trivial, Math. Z., 237, (2001), 181–198. MR
D. Popescu: General Néron desingularization and aproximation, Nagoya Math. Journal, 104, (1986), 85–115. MR
I. Panin, A. Suslin: On a conjecture of Grothendieck concerning Azumaya algebras, St.Petersburg Math. J., 9, (1998), no. 4, 851–858. MR
R. G. Swan: Néron-Popescu desingularization, Algebra and Geometry (Taipei, 1995), Lect. Algebra Geom. Vol. 2, Internat. Press, Cambridge, MA, 1998, pp. 135–192. MR
M. S. Raghunathan: Principal bundles admitting a rational section, Inventiones Math., 116, (1994), 409–423. MR
K. Zainoulline: On Grothendieck Conjecture about Homogeneous Spaces for some Classical Algebraic Groups, St.Petersburg Math. J., 12, (2001), no. 1, 117–143. MR
K. Zainoulline: The Purity Problem for Functors with Transfers, K-theory, 22, (2001), no. 4, 303–333. MR
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© 2005 Hindustan Book Agency
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Panin, I. (2005). Purity for multipliers. In: Tandon, R. (eds) Algebra and Number Theory. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-23-1_5
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DOI: https://doi.org/10.1007/978-93-86279-23-1_5
Publisher Name: Hindustan Book Agency, Gurgaon
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