Dimension theory and nonstable K1 of quadratic modules

Abstract. Employing Bak’s dimension theory, we investigate the nonstable quadratic K-group K1,2n(A,) = G2n(A,)/E2n(A,), n 3, where G2n(A,) denotes the general quadratic group of rank n over a form ring (A,) and E2n(A,) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G2n(A,) ⊇ G02n(A,) ⊇ G12n(A,) ⊇ · · · ⊇ E2n(A,) of the general quadratic group G2n(A,) such that G2n(A,)/G02n(A,) is Abelian, G 0 2n(A,) ⊇ G12n(A,) ⊇ · · · is a descending central series, and Gd(A)2n (A,) = E2n(A,) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K1,2n(A,) is solvable when d(A) <∞. Mathematics Subject Classifications (2000): 20G15, 20G35. Key words: non-Abelian K1, general quadratic group, descending central series