Sofie Beke: Specialisation and good reduction for algebras with involution

Submission: 2013, Jan 8, revised: 2013, Dec 16

Given a place from one field to another, the isotropy behaviour of Azumaya algebras with involution over the valuation ring corresponding to the place is studied. In particular, it is shown that isotropic right ideals specialise in an appropriate way. This provides a natural analogue to the existing specialisation theory for symmetric bilinear spaces. We devote particular attention to the case of a Henselian valuation ring in which 2 is invertible, where the specialisation results can be strenghened. In turn, this allows us to show that isomorphism of Azumaya algebras with involution over the Henselian valuation ring can be detected rationally. We use this to define a notion of good reduction with respect to places for algebras with involution.

2010 Mathematics Subject Classification: 16W10, 16W60, 16H05, 11E39

Azumaya algebras with involution, central simple algebras with involution, Brauer group, (skew-)hermitian spaces, bilinear spaces, (Henselian) valuation rings, value functions

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