Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K_1,2n(A, ) = G_2n(A, )/E_2n(A, ), n 3, where G_2n(A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E_2n(A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G_2n(A, ) G_2n⁰(A, ) ; G_2n¹(A, ) ... E_2n(A, ) of the general quadratic group G_2n(A, ) such that G_2n(A, )/G_2n⁰(A, ) is Abelian, G_2n⁰(A, ) G_2n¹(A, ) ... is a descending central series, and G_2n^d(A)(A, ) = E_2n(A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K_1,2n(A, ) is solvable when d(A) < .