# Strong approximation for principal ideal domains

A well known consequence of the strong approximation theorem for semisimple simply connected algebraic groups over a number field is that certain reduction maps are surjective, for example, the canonical projection $Sp_n({\mathbb Z}) \to Sp_n({\mathbb Z}/m{\mathbb Z})$ is surjective for any modulus $m$. This latter fact has been proven by Newman and Smart (Acta Arithmetica 9, 1964) in a rather elementary way, and their proof appears to carry over to an arbitrary principal ideal domain (with an obvious modification in the proof of Lemma 1: Replace $E^+,E^-$ by the upper triangular part of the matrix $E$ (with zeros below the diagonal) and its transpose).

I couldn't locate an explicit reference for this statement or similar results on surjectivity of reduction maps for other groups over a general principal ideal domain $R$ (or maybe other substitutes for the strong approximation theorem), except the review MR0865878 (88b:20072) of an article of Zhang and You in Dongbei Shida Xuebao (found via "Citations from reviews" in MathSciNet), which mentions in passing that the reduction map modulo an arbitrary $q$ for $Sp_n(R)$ is onto.

The question then is: Does anybody know more references for this problem?

• With a few caveats, a simply-connected Chevalley group over a Euclidean domain or DVR is generated by its elementary root subgroups. So since the canonical projection is surjective on root subgroups (use Chinese remainder theorem), you're in good shape. I'll post this as an answer, if I have time to find the precise caveats and references. – Marty Dec 4 '17 at 22:34

For $G(R) = SL_n(R)$ (see, e.g., this MO post) or $G(R) = Sp_{2n}(R)$, there is a common line of reasoning in order to prove that the reduction modulo $\mathfrak{a}$, say $\varphi_{\mathfrak{a}}: G(R) \rightarrow G(R/\mathfrak{a})$, is surjective. It consists in showing that $G(R/\mathfrak{a})$ is generated by elementary matrices any of which has a lift in $G(R)$.

For $SL_n(R)$, these elementary matrices are $E_{ij}(r) = 1_n + re_{ij}$ with $1 \le i \neq j \le n$ and where $1_n$ is the $n$-by-$n$ identity matrix and $e_{ij}$ differs from the zero matrix only by its $(i, j)$ entry, which is $r \in R$.

For $Sp_{2n}(R)$, these elementary matrices are $SE_{ij}(r) = \left\{ \begin{array}{ccc} 1_{2n} + re_{ij} & \text{ if } i = j',\\ 1_{2n} + re_{ij} - (-1)^{i + j}re_{j'i'} & \text{otherwise}. \end{array}\right.$ with $1 \le i \neq j \le n$ and where $i \mapsto i'$ is the permutation of $\mathbb{N}$ defined by $(2i)' = 2i - 1$ and $(2i - 1)' = 2i$; alternatively, this is the infinite product of transpositions $(12)(34)(56) \cdots$.

If $SL_n(R)$ is generated by the matrices $E_{ij}(r)$ with $r \in R$, the ring $R$ is said to be $GE_n$-ring in P. M. Cohn's terminology [1]. There are many examples of rings which are, or which are not $GE_n$-rings, see [1, 3] or this MO post. I don't know about the terminology for $Sp_{2n}(R)$, but here is a claim which addresses both groups.

Claim. Let $R$ be a Noetherian one-dimensional domain, e.g., $R$ is a principal ideal domain. Let $G(R) = SL_n(R)$ or $G(R) = Sp_{2n}(R)$. Then $\varphi_{\mathfrak{a}}$ is surjective for any ideal $\mathfrak{a}$ of $R$.

Proof. We can assume that $\mathfrak{a} \neq 0$. Since $R/\mathfrak{a}$ is zero-dimensional, its Bass'stable rank is $1$. Therefore $G(R/\mathfrak{a})$ is generated by elementary matrices, see [1] and [2, Theorem 7.3.b]. Because any of these matrices has a lift in $G(R)$, the map $\varphi_{\mathfrak{a}}$ is surjective.

As explained by Marty, the above should apply more generally to simply connected Chevalley groups. Let $\Phi$ be a reduced irreducible root system, $R$ be a commutative ring, and let $G = G(\Phi, R)$ be the simply connected Chevalley group of type $\Phi$ over $R$. Let $E(\Phi, R)$ be the elementary subgroup of $G$.

Claim to be checked. Let $\Phi$ be a reduced irreducible root system and let $R$ be a Noetherian one-dimensional domain. Then $\varphi_{\mathfrak{a}}$ is surjective for any ideal $\mathfrak{a}$ of $R$.

Proof. The condition on $R$ should be sufficient to ensure that $G(\Phi, R/\mathfrak{a}) = E(\Phi, R/\mathfrak{a})$ for any non-zero ideal $\mathfrak{a}$.

[1] P. Cohn, "On the structure of the $GL_2$ of a ring", 1966.
[2] L. Vaserstein, A. Suslin, "Serre's problem on projective modules over polynomial rings, and algebraic $K$-theory", 1976.
[3] F. Grunewald, J. Mennicke and L. Vaserstein, "On the groups $SL_2(\mathbb{Z}[x])$ and $SL_2(k[x,y])$", 1994.

• Both this answer and the comment above by Marty suggest proofs that are likely to work (for Sp_n, the approach of Newman and Smart also works), thank you. I am, however, still wondering whether there exist quotable references, preferably giving the most general version of this type of result - in the sixties and seventies quite a few people were working on related problems and there should exist some articles or theses written at that time. The literature on the strong approximation theorem (Eichler, Kneser, Platonov) from that time concentrates on rings of integers in global fields. – Rainer Schulze-Pillot Dec 5 '17 at 14:17
• @RainerSchulze-Pillot In this MO post mathoverflow.net/questions/265911/…, Andrei Smolensky gives recent references to closely related problems. So you may find the linked articles useful. Ideally, Andrei Smolensky would join this discussion and enlighten us! – Luc Guyot Dec 5 '17 at 16:50
• @LucGuyot I do not have a definite reference for this, but your second claim is also correct, since a zero-dimensional Noetherian ring $R/\mathfrak{a}$ is semilocal (because it is Artinian), so $G_{sc}(\Phi,R/\mathfrak{a})=E_{sc}(\Phi,R/\mathfrak{a})$, see, for example, "Chevalley groups over local rings" by Abe or "Surjective stability in dimension 0..." by Stein or Matsumoto's thesis. – Andrei Smolensky Dec 5 '17 at 23:09
• I guess I have to accept that there are no explicit references except maybe in unpublished theses somewhere out there. I had thought that it would be possible to give a proof via elementary matrices resp. transvections but am surprised that apparently nobody worked it out.So I mark the answer of Guyot as accepted now and thank for the various indirect references. – Rainer Schulze-Pillot Dec 13 '17 at 18:29
• I should have looked this up earlier: The statement for the symplectic group over an arbitrary principal ideal domain is in the book "Integral matrices" by Newman (where he notices that his earlier proof with Smart for Z carries over to the more general situation). – Rainer Schulze-Pillot Dec 20 '17 at 11:06