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A stable range for quadratic forms over commutative rings. (English) Zbl 0886.11020

The quadratic stable range property is discussed. The ring \(A\) has quadratic stable range \(1\) (\(\text{qsr}(A) = 1\)) if every primitive quadratic form over \(A\) represents a unit. The property is motivated by the ring of holomorphic functions on a connected noncompact Riemann surface. The authors prove the following results:
(1) if \(\text{qsr} (A)=1\) then the stable range of \(A\) equals \(1\) and \(\text{Pic} (A)=1\).
(2) \(\text{qsr} (A)=1\) iff \(\text{Pic}(T)=1\) for every quadratic \(A\)-algebra \(T\).
They also classify quadratic forms over Bezout domains of characteristic not 2 satisfying a very strong approximation property (defined in the paper). This classification applies to the ring of holomorphic functions mentioned above.

MSC:

11E08 Quadratic forms over local rings and fields
11E12 Quadratic forms over global rings and fields
15A63 Quadratic and bilinear forms, inner products
14C22 Picard groups
46E25 Rings and algebras of continuous, differentiable or analytic functions
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References:

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