Stabilization of unitary groups over polynomial rings (Q1893422)

Let \(*\) be an involution on a commutative ring \(R\) with 1, extended to \(A = R[X_1, \ldots, X_n]\) by \(X_i^* = X_i\). Fix the notations \(U_{2n} (A)\) for the corresponding unitary group of \((2n \times 2n)\)- matrices over \(R\), \(EU_{2n} (A)\) for the subgroup of \(U_{2n} (A)\) generated by the elementary matrices, \(U(A) = \varinjlim U_{2n} (A)\) and \(E(A) = \varinjlim EU_{2n} (A)\). Then the following stabilization theorem is derived: If \(*\) satisfies a certain condition \((\Delta)\) then the canonical mapping \(U_{2n} (A)/EU_{2n} (A) \to U(A)/EU (A) = : K_1 U(A)\) is an isomorphism for \(n \geq \max (3, \dim R + 2)\) (theorem 1.1). Furthermore as a consequence of a structure theorem for \(U_{2n} (A)\) the equality \(K_1 U(A) = K_1 U(R)\) is obtained for \(R\) being a ring of algebraic integers in a quadratic field \(\mathbb{Q} (\sqrt d)\).