Hermitian Morita Theory and Unitary Groups of Hyperbolic Quadratic Modules over Maximal Orders in Central Simple Algebras

Abstract

Let R = Z with quotient field say K. Let Λ and Λ1 be maximal R-orders with antistructure in central simple K-algebras A = M n (D) and A 1 = M n (D) where D and D 1 are separable over K. Let (M, h) and (N, k) be hyperbolic quadratic, hermitian Λ and Λ1 modules, respectively. Let be an arbitrary isomorphism of the corresponding projective unitary groups. Under some minimal conditions there is a category equivalence from the category of hermitian Λ-modules to that of hermitian Λ1-modules which arises fronm a hermitian Morita context and produces Φ by its action on morphisms.