\(K_ 2\) d'un anneau euclidien (Q801027)

Let A be a Dedekind domain with field of fractions F. The map \(K_ 2(A)\to K_ 2(F)\) is known to be injective when the residue fields of A are algebraic over a finite field, when A is a discrete valuation ring, or when A is the polynomial ring in one variable over a field. In this note the author shows that \(K_ 2(A)\to K_ 2(F)\) is injective whenever A is Euclidean. His argument is based on a lifting trick inspired by the Tate-Gauss computation of \(K_ 2{\mathbb{Q}}\).