Dimension theory and nonstable K₁ of quadratic modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K^1,2n(A, Λ) = G_2n(A, Λ)/E _2n(A, Λ), n ≥ 3, where G_2n(A, Λ) denotes the general quadratic group of rank n over a form ring (A, Λ) and E _2n(A, Λ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G_2n(A, Lamda;) ⊇ G_2n¹(A, Lamda;) ⊇ ⋯ ⊇ E_2n (A, Lamda;) of the general quadratic group G _2n(A, Lamda;) such that G_2n(A, Lamda;)/G_2n ⁰ (A, Lamda;) is Abelian, G_2n⁰(A, Lamda;) ⊇ G_2n¹(A, Lamda;) ⊇ ⋯ is a descending central series, and G_2n^d(A) (A, Lamda;) = E _2n(A, Lamda;) whenever d(A) = (Bass-Serre dimension of A) is finite. In particular K_1,2n(A, Lamda;) is solvable when d(A) < ∞.