Suslin's Stability Theorem for Chevalley Groups

Question

I am looking for a version of Suslin's Stability Theorem for Chevalley groups. The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary matrices generate $G$. I think by embedding many copies of $SL_n$, one can prove a similar statement, but probably with a bad bound, for all Chevalley groups over ${\mathbb Z}[x_1, \dots , x_m]$. I was wondering if a sharp bound has already been worked out in the literature.

Answer 1

I am somewhat puzzled by your version of "Suslin stability theorem". What you are referring to is a combination of usual stabilization for $SK_1$ and an estimate for the stable rank of integer polynomials. Suslin's theorem states something much stronger, in particular Corollary 6.6 in his paper On the structure of the special linear group over polynomial rings is as follows (note that it doesn't depend on the number of variables):

Let $A$ be a regular ring such that $SK_1(A)=0$ (for example, the ring of integers in an algebraic number field). Then $SL_r(A[x_1,\ldots,x_n])$ is generated by elementary matrices for $r\geqslant\max(3,\dim A+2)$.

Corollary 7.10 extends this result to rings of the form $A[x_1^\pm,\ldots,x_k^\pm,x_{k+1},\ldots,x_n]$ for $A$ regular.

For other Chevalley groups the situation is complicated. One has the stability theorems for $K_1(\Phi)$ in terms of stable rank (or its ramifications such as absolute stable rank or $\Lambda$-stable rank), but they give pretty bad bounds for polynomial rings.

There is, however, the following version of Suslin's theorem for symplectic group in a paper On symplectic groups over polynomial rings by F. Grunewald, J. Mennicke and L. Vaserstein:

Let $A$ be a locally principal ring. For an integer $m\geqslant0$ put $R=A[x_1,\ldots,x_m]$. Then $Sp_{2n}(R)=Sp_{2n}(A)\cdot Ep_{en}(R)$ for any $n\geqslant2$.

By locally principal ring they mean a commutative ring such that its localization at any maximal ideal is a principal ideal ring.

For euclidean ring $A$ this gives $K_1(\mathsf{C}_\ell,R)=0$.

As a byproduct they also prove a stronger version of Suslin's theorem for $SL$ and a locally principal ring.

They also have a version for Laurent polynomial rings and claim that by using stability theorems for $K_1$ as in M. Stein's papers one can prove the same results for classical simple algebraic groups of relative rank $\geqslant2$, but the latter has never been written in full details.

UPDATE 14.02.2019 The proof for all simply-connected Chevalley groups is given by A. Stavrova in Chevalley groups of polynomial rings over Dedekind domains. Namely, she proves the following theorem:

Let $R$ be a locally principal ring, and let $G$ be a Chevalley—Demazure group scheme of isotropic rank $\geq2$. Then $G(R[x_1,\ldots,x_n])=G(R)E(R[x_1,\ldots,x_n])$ for any $n\geq1$.

If $R$ is a Dedekind ring of arithmetic type (for example, $R=\mathbb{Z}$), it follows that $G(R[x_1,\ldots,x_n])=E(R[x_1,\ldots,x_n])$.

Answer 2

I don't know what happens over $\mathbb{Z}$, but for reductive groups over a field, it turns out that there is no analogue to Suslin's theorem in general. Indeed consider the following result (due to A.Stavrova, Homotopy invariance of non-stable K_1-functors, http://arxiv.org/abs/1111.4664):

For a split reductive group $G$ over a ring $R$, let me denote by $K^G_1(R)$ the quotient of $G(R)$ by the subgroup $E(R)$ of elementary matrices (defined via a Chevalley presentation, and which turns out to be a normal subgroup). Then for any field $k$, any $G$ such that any semi-simple normal subgroup of $G$ is of rank at least $2$ (e.g. $G$ simple of rank at least $2$) and any $n\geq 0$. Then $$ K_1^G(k[X_1,\ldots,X_n])\simeq K^G_1(k) $$ So the obstruction for elementary matrices to generate is stable.

This does not contradict Suslin's result because, as is well-known, $K^{SL_n}_1(k)$ is trivial (as is the case for all semi-simple simply-connected split groups).