Hermitian Forms over Rings

Stability of unitary SK1 of Azumaya algebras

“…Recall that an R-Algebra A is called an Azumaya algebra if A is a finitely generated R-module and A/mA is a central simple R/m-algebra for any maximal ideal m of R (see [7,Section III.5]). Let A be an Azumaya algebra over its center C of constant rank n 2 .…”

“…Furthermore, the element a is invertible in A if and only if Nrd A (a), the reduced norm of a, is invertible in C (see [7,III.1.2] and [12,Theorem 4.3]). Let A * be the invertible elements of A, SL(1, A) the set of elements of A with the reduced norm 1 and [A * , A * ] be the commutator subgroup of A * .…”

Norms and determinants of quaternionic line bundles

“…The determinant of a module over a ring is a well-known and important invariant in module theory. In [3] Knus implicitly showed that the determinant of a quaternion composition algebra over an arbitrary (unital associative commutative) ring R is trivial, i.e., isomorphic to R. The proof of this fact consists of several steps. Firstly, one shows that the Arf invariant of a quaternion composition algebra Q over R is trivial, i.e., isomorphic to R Â R, secondly, this implies the triviality of the discriminant module of Q, and thirdly, by method of descent, one obtains an isomorphism between the discriminant module and the determinant of Q.…”

“…Such a quadratic space can be thought of as a quaternionic line bundle. In [3] we find the notion of norm forms on right modules over quadratic resp. quaternion algebras over arbitrary (unital commutative associative) rings.…”

“…The terminology adopted in this paper is the standard one; basic facts concerning composition algebras needed in this paper can be found in [3].…”