∧-1-ford稳定环上二次型群的K1

Prestability for Quadratic K

1

of Λ-1-Fold

Stable Rings

Hong You

1

1) Department of Mathematics, Suzhou University, Suzhou 215006, P. R. China

2) Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China

Shengkui Ye

n

1

1

stable range condition on the ring of entries. In this article we give a description for KU

under the Λ-1-fold stable range condition on the ring of entries.

is defined on R such that

ε

= {x − xε| x ∈ R}, R

ε

= {x ∈ R | x = −xε} . We fix an additive subgroup Λ of

R with the following properties:

U

2n

(R, Λ) =

α β

γ δ

2n

R | αδ + γεβ = I

n

, αγ, βδ ∈ Λ

n

if and only if the following conditions hold:

n

σ = ϕ

n

where ϕ

n

=

0 I

εI 0

α β

γ δ

. (1.1)

There is an obvious embedding

α 0 β 0

0 1 0 0

γ 0 δ 0

0 0 0 1

2n

U

a ∈ R and 1 ≤ i 6= j ≤ 2n we define the elementary quadratic matrices ρ

(a) and ρ

with j 6= σi as follows: ρ

with a ∈ Λ when n + 1 ≤ i and

a ∈ Λ when

i ≤ n. ρ

(a) = ρ

(−a

) = I

+ aE

2n

(R, Λ) generated by these matrices is denoted by EU

2n

(R, Λ). The

stabilization map U

2n

2(n+1)

(R, Λ) above induces a stabilization map EU

2n

(R, Λ) →

EU

2(n+1)

(R, Λ) which will be used frequently to identify EU

EU

(R, Λ).