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Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map $\text{SL}_n(\mathcal{O}_K) \rightarrow \text{SL}_n(\mathcal{O}_K/I)$ is surjective. For instance, $\mathcal{O}_K/I$ is a product of local rings, and over such rings the special linear group is generated by elementary matrices. I need this theorem in a paper I am writing, and I'd rather not spend a paragraph proving it. Does anyone know a good reference?

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    $\begingroup$ Why not giving the same argument as you gave us (while saying at the same time that the result is well-known)? After all it just takes one line: any reference will take more, and would require infinitely more work for the reader. $\endgroup$
    – Joël
    Commented May 6, 2014 at 4:03
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    $\begingroup$ btw the argument works with no change with an arbitrary noetherian domain of Krull dimension 1. $\endgroup$
    – YCor
    Commented May 6, 2014 at 13:00
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    $\begingroup$ in addition there is no need to invoke local rings and product decomposition, but just semilocal, which is the classical assumption in K-theory texts. $\endgroup$
    – YCor
    Commented May 6, 2014 at 13:10

1 Answer 1

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Platonov-Rapinchuk is the canonical reference.

EDIT A less nuclear option (and one I should have mentioned in the first place is Morris Newman's Integral Matrices. I think chapter 7 and/or 9 have a(n elementary) proof.

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    $\begingroup$ Can you tell me where in this ~650 page book the result can be found? I quickly skimmed though it and didn't find it. $\endgroup$
    – Philippe
    Commented May 6, 2014 at 2:54
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    $\begingroup$ Chapter 7. What you are looking for is "strong approximation", essentially. $\endgroup$
    – Igor Rivin
    Commented May 6, 2014 at 2:59
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    $\begingroup$ I suppose I could cite strong approximation, though that is sort of killing an ant with a nuclear weapon. I'll accept this answer if no one provides a source for a more elementary argument (e.g. like the one I sketched in my question). $\endgroup$
    – Philippe
    Commented May 6, 2014 at 3:13
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    $\begingroup$ I took a look through Newman's book, and the only relevant result is Theorem VII.6. At first glance, this looks like what I want; however, if you read a little further back you'll see that he is only considering PID's, and moreover, he just asserts without proof that the reduction map is surjective (which I guess is obvious for PID's). $\endgroup$
    – Philippe
    Commented May 6, 2014 at 4:57

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