Let $R$ be the ring of integers in a number field. While studying the congruence subgroup property for $\text{Sp}\_{2g}(R)$ in


Bass, H.; Milnor, J.; Serre, J.-P.
Solution of the congruence subgroup problem for SLn(n≥3) and Sp2n(n≥2). 
Inst. Hautes Études Sci. Publ. Math. No. 33 1967 59–137.


they quote a theorem of Bass that says the following. First, some notation. If $R$ is a commutative ring and $q$ is an ideal of $R$, then denote by $\text{Sp}\_{2g}(R,q)$ the kernel of the map
$$\text{Sp}\_{2g}(R) \longrightarrow \text{Sp}\_{2g}(R/q).$$
Also, denote by $\text{Ep}\_{2g}(R,q)$ the normal subgroup of $\text{Sp}\_{2g}(R)$ generated by the usual elementary symplectic matrices which happen to lie in $\text{Sp}\_{2g}(R,q)$.


**Theorem** : For $R$ a Dedekind domain and $q$ an ideal of $R$ and $g \geq 2$, we have $\text{Sp}\_{2g}(R,q) = \text{Sp}\_{2}(R,q) \cdot \text{Ep}\_{2g}(R,q)$.


This is part a of Proposition 13.2 in the above paper. The reference they give is


Bass, H, Symplectic modules and groups, in preparation.


However, this paper does not seem to have ever appeared. In a paper I'm writing right now, I need a fact which is a corollary of what I assume is the proof they have in mind (alas, it doesn't just follow from the statement). Does anyone know a published account of it?



Bass's paper "Symplectic groups and modules", used in proof of the congruence subgroup property for Sp
Your question is essentially about surjective stability for *relative* symplectic $K\_1$. The latter follows from the usual (absolute) surjective stability for $K\_1$, which in symplectic case starts at $2n\geq \mathop{\mathrm{sr}}(R)$. To prove this, one can use so-called "Stein relativization", as described in M. Stein, "Relativizing Functors on Rings and Algebraic K-Theory", J.Algebra, 1971. See Corollary 1.7 therein or the remark after Theorem 4.2 in Stein's other paper, "Stability theorems for $K\_1$, $K\_2$ and related functors modelled on Chevalley groups", Japan J. Math, 1978.


It is also possible to prove it directly under somewhat weaker assumprion on a ring by explicit calculations with generators, but since you are interested in Dedekind domains, the usual stable rank condition should suffice.


