# Bass's paper “Symplectic groups and modules”, used in proof of the congruence subgroup property for Sp

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?

• Two comments: 1) The IHES paper is online: numdam.org/numdam-bin/fitem?id=PMIHES_1967__33__59_0; 2) Possibly the result you want is written up in the large Benjamin lecture notes volume on algebraic K-theory by Bass published soon after? (A direct inquiry to him at U. Michigan might be worthwhile.) – Jim Humphreys Aug 19 '13 at 17:33
• @JimHumphreys : Pretty sure that it is not in Bass's book (eg on the first page of the preface it says `Time prevented me from including here a treatment of the "K-theory of symplectic modules", which I hope to publish in the near future'). – Andy Putman Aug 19 '13 at 17:40
• Yes, the subject was developing very rapidly at the time, as my answer reflects. Note that Bass and others did work out quite a bit more detail in the late 1960s, some published by his students at Columbia including M.R. Stein (later at Northwestern) and A. Bak (later at Bielefeld). – Jim Humphreys Aug 20 '13 at 13:11

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.

• Note that Mike Stein was a Ph.D. student supervised by Bass. – Jim Humphreys Aug 19 '13 at 21:27
• Thank you very much! These papers contain exactly what I need. – Andy Putman Aug 20 '13 at 21:08

Just after the limited work by Bass-Milnor-Serre was published, the work of Moore made possible a major improvement by H. Matsumoto in his Paris thesis here, treating uniformly all the split (Chevalley) group schemes including the somewhat exceptional case of symplectic groups relative to the congruence subgroup problem. I think Cor. 4.5 and the surrounding material cover the missing Bass paper, which he may have avoided completing due to Matsumoto's work. (But participants at the time could fill in that history much better than I can.)

The rank 1 group in your formulation comes from the unique long simple root for a symplectic group.

Probably the results for classical groups are also worked out in the later book by Hahn and O'Meara The Classical Groups and K-Theory (Springer, 1989), but I haven't yet tracked down an explicit reference there. Also, they don't include full details of the congruence subgroup problem, referring at times to the literature.

• Thank you also for your answer. It was hard to pick one to mark as accepted since they both contained what I wanted (but from different sources!). – Andy Putman Aug 20 '13 at 21:09
• @Andy: The literature from that period is not easy to sort out, but roughly speaking Bass and others started in the framework of linear algebra over rings for classical groups while Matsumoto (and later Prasad-Raghunathan) worked mainly with the structure theory of algebraic groups starting with Chevalley's commutation relations. – Jim Humphreys Aug 22 '13 at 0:11