Chevalley groups of polynomial rings over Dedekind domains

TL;DR

This paper proves a stability result for Chevalley groups over polynomial rings with coefficients in Dedekind domains, extending classical results to a broader class of algebraic groups and rings.

Contribution

It generalizes known stability theorems for special linear and symplectic groups to all simply connected Chevalley-Demazure groups over Dedekind domains.

Findings

G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for Dedekind domains R and n>=1

Extension of classical results to higher-dimensional regular rings and discrete Hodge algebras

Provides corollaries for regular rings of higher dimension

Abstract

Let R be a Dedekind domain, and let G be a simply connected Chevalley-Demazure group scheme of rank =>2. We prove that G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for any n=>1. This extends the corresponding results of A. Suslin and F. Grunewald, J. Mennicke, and L. Vaserstein for G=SL_n, Sp_2n. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

Equations

$${\mathrm{SL}}_N(R[x_1,\dots ,x_n])={\mathrm{SL}}_N(R)E_N(R[x_1,\dots ,x_n]),$$

$${\mathrm{SL}}_N(Z[x_1,\dots ,x_n])=E_N(Z[x_1,\dots ,x_n]).$$

$$g(x_{1})=\Bigl{(}\,\prod\limits_{j=0}^{N-1}g(a_{j}x_{1})g(a_{j+1}x_{1})\Bigr{)}^{-1}g(0).$$

$$G(A[x])/E(A[x])\to G(A[x{]}_f)/E(A[x{]}_f)$$

$$G(R[x_1,\dots ,x_n])=G(R)E(R[x_1,\dots ,x_n])$$

$$G\bigl{(}R_{m}(x_{1})[x_{2},\ldots,x_{n}]\bigr{)}=G\bigl{(}R_{m}(x_{1})\bigr{)}E\bigl{(}R_{m}(x_{1})[x_{2},\ldots,x_{n}]\bigr{)}.$$

$$1\to [G,G]\to G\to T\to 1,$$

$$1\to [G,G](A[x])\to G(A[x])\to T(A[x])$$

$$1\to C\stackrel{i}{\to }{G}^{\mathrm{sc}}\stackrel{\pi }{\to }G\to 1,$$

$$1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{g}\in C(A)\cdot\ker\bigl{(}\rho:G^{\mathrm{sc}}(A[x])\to G^{\mathrm{sc}}(A)\bigr{)}\subseteq C(A)E^{\mathrm{sc}}(A[x]).$$

Full text

The author is a winner of the contest ‘‘Young Russian Mathematics’’. The work was supported by the RFBR grant 18-31-20044

1. Introduction

A. Suslin [Su, Corollary 6.5] established that for any regular ring $R$ of dimension $\leq 1$, any $N\geq 3$, and any $n\geq 1$, one has

[TABLE]

where $E_{N}(R[x_{1},\ldots,x_{n}])$ is the elementary subgroup, i.e. the subgroup generated by elementary matrices $I+te_{ij}$, $1\leq i\neq j\leq N$, $t\in R[x_{1},\ldots,x_{n}]$. In particular, this implies

[TABLE]

A later theorem of A. Suslin and V. Kopeiko [SuK, Theorem 7.8] together with the homotopy invariance of orthogonal $K$-theory (see [Kar73, Corollaire 0.8], [Hor05, Corollary 1.12], or [Sch17, Theorem 9.8]) implies a similar result for even orthogonal groups $SO_{2N}$, $N\geq 3$, under the additional assumption $2\in R^{\times}$. F. Grunewald, J. Mennicke, and L. Vaserstein [GMV91] extended the result of Suslin to symplectic groups $\operatorname{Sp}_{2N}$, $N\geq 2$, and a slightly larger class of rings $R$, namely, locally principal ideal rings. One says that a (commutative associative) ring $A$ with 1 is a locally principal ideal ring, if for every maximal ideal $m$ of $A$ the localization $A_{m}$ is a principal ideal ring.

Our aim is to extend the above results to all Chevalley–Demazure group schemes of isotropic rank $\geq 2$. By a Chevalley–Demazure group scheme we mean a split reductive group scheme in the sense of [SGA3]. These group schemes are defined over $\operatorname{{\mathbb{Z}}}$. We say that a Chevalley–Demazure group scheme $G$ has isotropic rank $\geq n$ if and only if every irreducible component of its root system has rank $\geq n$. For any commutative ring $R$ with 1 and any fixed choice of a pinning, or épinglage of $G$ in the sense of [SGA3], we denote by $E$ the elementary subgroup functor of $G$. That is, $E(R)$ is the subgroup of $G(R)$ generated by elementary root unipotent elements $x_{\alpha}(r)$, $\alpha\in\Phi$, $r\in R$, in the notation of [Ch55, Ma], where $\Phi$ is the root system of $G$. If $G$ has isotropic rank $\geq 2$, then $E$ is independent of the choice of the pinning [PSt].

Our main result is the following theorem. Since $\operatorname{SL}_{2}(\operatorname{{\mathbb{Z}}}[x])\neq E_{2}(\operatorname{{\mathbb{Z}}}[x])$ [C], it cannot be extended to the case of isotropic rank 1.

Theorem 1.1** (Theorem 3.4).**

Let $R$ be a locally principal ideal ring, and let $G$ be a simply connected Chevalley–Demazure group scheme of isotropic rank $\geq 2$. Then $G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])$ for any $n\geq 1$.

Theorem 1.1 for Dedekind domains was previously claimed by M. Wendt [W, Proposition 4.7], however, his proof was incorrect [Ste13, p. 91]. We give another proof along the lines similar to [GMV91]. The case where $R$ is a field was done earlier in [St14] in a more general context of isotropic reductive groups.

Following [Ma], we say that a Dedekind domain $R$ is of arithmetic type, if $R=O_{S}$ is the ring of $S$-integers of a global field $k$ with respect to a finite non-empty set $S$ of primes containing all archimedean primes.

Corollary 1.2**.**

Let $R$ be a Dedekind domain of arithmetic type (e.g. $R=\operatorname{{\mathbb{Z}}}$), and let $G$ be a simply connected Chevalley–Demazure group scheme of isotropic rank $\geq 2$. Then $G(R[x_{1},\ldots,x_{n}])=E(R[x_{1},\ldots,x_{n}])$ for any $n\geq 1$.

Proof.

This follows from Theorem 1.1 and [Ma, Théorème 12.7], which says that $G(R)=E(R)$. ∎

Note that [Lam06, p. 57] presents an example (due to J. Stallings) of a Dedekind domain $D$ such that $\operatorname{SL}_{3}(D)\neq E_{3}(D)$, hence Corollary 1.2 does not hold for arbitrary Dedekind domains.

A commutative $R$-algebra of the form $A=R[x_{1},\ldots,x_{n}]/I$, where $I$ is an ideal generated by monomials, is called a discrete Hodge algebra over $R$. If $I$ is generated by square-free monomials, $A$ is called a square-free discrete Hodge algebra. The simplest example of such an algebra is $R[x,y]/xy$. Square-free discrete Hodge algebras over a field are also called Stanley–Reisner rings.

Corollary 1.3**.**

Let $R$ be a Dedekind domain, and let $G$ be a simply connected Chevalley–Demazure group scheme of isotropic rank $\geq 2$. Then $G(A)=G(R)E(A)$ for any discrete Hodge algebra $A$ over $R$. In particular, if $R$ is of arithmetic type, then $G(A)=E(A)$.

Proof.

This follows from [St19, Corollary 1.5] and Corollary 1.2. ∎

For non-simply connected Chevalley–Demazure group schemes, such as $\operatorname{SO}_{n}$, $n\geq 5$, we deduce the following result; see § 3 for the proof.

Corollary 1.4**.**

Let $R$ be a locally principal ideal domain, and let $G$ be any Chevalley–Demazure group scheme of isotropic rank $\geq 2$. Then $G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])$ for any $n\geq 1$.

Using a version of Lindel’s lemma [L] and Néron-Popescu desingularization [Pop90], one may extend the above results to higher-dimensional regular rings in place of Dedekind domains. For equicharacteristic regular rings this was done earlier in [St14]. The following theorem is proved in § 4.

Theorem 1.5**.**

Let $G$ be a Chevalley–Demazure group scheme of isotropic rank $\geq 2$. Let $R$ be a regular ring such that every maximal localization of $R$ is either essentially smooth over a Dedekind domain with perfect residue fields, or an unramified regular local ring. Then $G(R[x])=G(R)E(R[x])$. Moreover, $G(A)=G(R)E(A)$ for any square-free discrete Hodge algebra $A$ over $R$.

Let us mention a few other ramifications of known results yielded by Theorem 1.1.

Combining Corollary 1.3 with the main result of [EJZK17], one concludes that $G(A)$ has Kazhdan’s property (T) for any simply connected Chevalley–Demazure group scheme $G$ of isotropic rank $\geq 2$ and any discrete Hodge algebra $A$ over $\operatorname{{\mathbb{Z}}}$; in particular, $G(\operatorname{{\mathbb{Z}}}[x_{1},\ldots x_{n}])$ has Kazhdan’s property (T).

Combining Corollary 1.3 with the main result of [RR], one concludes that the congruence kernel of $G(A)$ is central in $G(A)$ for any simply connected Chevalley–Demazure group scheme $G$ of isotropic rank $\geq 2$ and any discrete Hodge algebra $A$ over $R$, where $R$ is a Dedekind domain of arithmetic type, satisfying $2\in R^{\times}$ if the root system of $G$ has components of type $C_{n}$ or $G_{2}$.

2. A local–global principle

Throughout this section, $R$ is any commutative ring with 1, $G$ is a Chevalley–Demazure group scheme of isotropic rank $\geq 2$, and $E$ denotes its elementary subgroup functor.

For any $s\in R$ we denote by $R_{s}$ the localization of $R$ at $s$, and by $F_{s}:R\to R_{s}$ the localization homomorphism, as well as the induced homomorphism $G(R)\to G(R_{s})$. Similarly, for any maximal ideal $m$ of $R$ we denote by $F_{m}:R\to R_{m}$ the localization homomorphism, as well as the induced homomorphism $G(R)\to G(R_{m})$.

We will need the following generalization of the Quillen–Suslin local-global principle for polynomial rings in one variable (see [Su, Theorem 3.1], [SuK, Corollary 4.4], [PSt, Lemma 17], [Ste13, Theorem 5.4]) to the case of several variables.

Lemma 2.1**.**

Let $R$ be any commutative ring. Fix $n\geq 1$. If $g\in G(R[x_{1},\ldots,x_{n}])$ satisfies $F_{m}(g)\in E(R_{m}[x_{1},\ldots,x_{n}])$ for any maximal ideal $m$ of $R$, then $g\in G(R)E(R[x_{1},\ldots,x_{n}])$.

The proof of Lemma 2.1 uses the following three standard lemmas whose idea goes back to [Q, Lemma 1].

Lemma 2.2**.**

Let $H$ be any affine $R$-scheme of finite type. Fix $0\neq s\in R$, and let $F_{s}:H(R[z])\to H(R_{s}[z])$ be the localization map. For any $g(z),h(z)\in H(R[z])$ such that $h(0)=g(0)$ and $F_{s}(g(z))=F_{s}(h(z))$ there is $n\geq 0$ such that $g(s^{n}z)=h(s^{n}z)$.

Proof.

Since $H$ is an affine $R$-scheme of finite type, there is a closed embedding $H\to\mathbb{A}^{k}_{R}$ for some $k\geq 0$. Hence it is enough to prove the claim for $H=\mathbb{A}^{k}_{R}$. If $k=0$, then $g(z)=g(0)=h(0)=h(z)$. If $k\geq 1$, the claim readily reduces to the case $k=1$, that is, $g(z),h(z)\in R[z]$. Since $F_{s}(g(z))=F_{s}(h(z))$, there is $n\geq 0$ such that $s^{n}g(z)=s^{n}h(z)$. Since $g(0)=h(0)$, this implies $g(s^{n}z)=h(s^{n}z)$. ∎

Lemma 2.3**.**

[Ste13, Theorem 5.2]** Fix $s\in R$, and let $F_{s}:G(R[z])\to G(R_{s}[z])$ be the localization homomorphism. For any $g(z)\in E(R_{s}[z],zR_{s}[z])$ there exist $h(z)\in E(R[z],zR[z])$ and $k\geq 0$ such that $F_{s}(h(z))=g(s^{k}z)$.

Proof.

The statement is a particular case of [Ste13, Theorem 5.2] if the root system $\Phi$ of $G$ is irreducible. Assume that $\Phi$ has several irreducible components $\Phi_{i}$. By [SGA3, Exp. XXVI Prop. 6.1] $G$ contains semisimple Chevalley–Demazure subgroup schemes $G_{i}$ of type $\Phi_{i}$ whose elementary subgroup functors $E_{i}$ are generated by elementary root unipotents corresponding to roots in $\Phi_{i}$. Chevalley commutator relations imply that $E$ is a direct product of all $E_{i}$. This reduces the claim to the case where $\Phi$ is irreducible. ∎

Lemma 2.4**.**

For any $g(x)\in G(R[x])$ such that $F_{s}(g(x))$ lies in $E(R_{s}[x])$, there exists $k\geq 0$ such that $g(ax)g(bx)^{-1}\in E(R[x])$ for all $a,b\in R$ satisfying $a\equiv b\mod s^{k}$.

Proof.

Consider the element $f(z)=g(x(y+z))g(xy)^{-1}\in G(R[x,y,z])$. Observe that $F_{s}(f(z))\in E(R_{s}[x,y,z])$ and $f(0)=1$. Since $F_{s}(g(x))\in E(R_{s}[x])$ and $f(0)=1$, we have $F_{s}(f(z))\in E(R_{s}[x,y,z],zR_{s}[x,y,z])$ (e.g. by [St14, Lemma 4.1]). Now by Lemma 2.3 there exist $h(z)\in E(R[x,y,z],zR[x,y,z])$ and $k\geq 1$ such that $F_{s}(h(z))=F_{s}(f(s^{k}z))$. By Lemma 2.2 there is $l\geq 1$ such that $h(s^{l}z)=f(s^{l+k}z)$. Then $g(x(y+s^{l+k}z))g(xy)^{-1}$ lies in $\operatorname{E}_{P}(R[x,y,z])$. It remains to set $y=b$ and to choose a suitable $z$ depending on $a$. ∎

Proof of Lemma 2.1.

For any maximal ideal $m$ of $R$, since $F_{m}(g)\in E(R_{m}[x_{1},\ldots,x_{n}])$, there is $s\in R\setminus m$ such that $F_{s}(g)\in E(R_{s}[x_{1},\ldots,x_{n}])$. Choose a finite set of elements $s=s_{i}\in R\setminus m_{i}$, $1\leq i\leq N$ as above, so that $1=\sum_{i=1}^{N}c_{i}s_{i}$ for some $c_{i}\in R$. Consider $g$ as a function $g(x_{1})$ of $x_{1}$. By Lemma 2.4 there are $k_{i}\geq 1$ such that $g(ax_{1})g(bx_{1})^{-1}\in E(R[x_{1},\ldots,x_{n}])$ for any $a,b\in R[x_{2},\ldots,x_{n}]$ satisfying $a\equiv b\pmod{s_{i}^{k_{i}}}$. Since $s_{i}$ generate the unit ideal, their powers $s_{i}^{k_{i}}$ also generate the unit ideal, and we can replace $s_{i}$ by these powers without loss of generality. Set $a_{j}=\sum_{i=1}^{N-j}c_{i}s_{i}$, $0\leq j\leq N$. Then $a_{j+1}\equiv a_{j}\pmod{s_{n-j}}$, and

[TABLE]

Then $g(x_{1})\in E(R[x_{1},\ldots,x_{n}])g(0)$. Since $g(0)\in G(R[x_{2},\ldots,x_{n}])$, we can proceed by induction. ∎

Lemma 2.5**.**

Fix $n\geq 1$. One has $G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])$ if and only if $G(R_{m}[x_{1},\ldots,x_{n}])=G(R_{m})E(R_{m}[x_{1},\ldots,x_{n}])$ for every maximal ideal $m$ of $R$.

Proof.

For the direct implication, see [St14, Lemma 4.2]. To prove the converse, it is enough to show that $g(x_{1},\ldots,x_{n})\in G(R[x_{1},\ldots,x_{n}])$ such that $g(0,\ldots,0)=1$ satisfies $g(x_{1},\ldots,x_{n})\in E(R[x_{1},\ldots,x_{n}])$. For every maximal ideal $m$ of $R$, by assumption, one has $F_{m}(g(x_{1},\ldots,x_{n})\in G(R_{m})E(R_{m}[x_{1},\ldots,x_{n}])$, and $g(0,\ldots,0)=1$ implies $F_{m}(g(x_{1},\ldots,x_{n}))\in E(R_{m}[x_{1},\ldots,x_{n}])$. Then Lemma 2.1 finishes the proof. ∎

3. Proof of the main theorem

The following result follows from stability results for non-stable $K_{1}$-funtors of Chevalley groups [Ste78, Plo93].

Lemma 3.1**.**

Let $R$ be a Noetherian ring of Krull dimension $\leq 1$. If $\operatorname{SL}_{2}(R)=E_{2}(R)$, then $G(R)=E(R)$ for any simply connected Chevalley–Demazure group scheme $G$ over $R$.

Proof.

By [Bas68, p. 102] the maximal ideal spectrum of $R$ is a Noetherian topological space of dimension $\leq 1$. By [Ste78, Theorem 1.4] this implies that $R$ satisfies the absolute stable range condition $ASR_{3}$, and hence also Bass’ stable range condition $SR_{3}$ in the sense of [Ste78, p. 86]. Then by [Ste78, Theorem 2.2] (see also [Ste78, Corollary 2.3]) suitable inclusions of $\operatorname{SL}_{2}$ into $G$ induce surjections $\operatorname{SL}_{2}(R)/E_{2}(R)\to G(R)/E(R)$ for every simply connected Chevalley—Demazure group scheme $G$ corresponding to an irreducible root system of classical type $A_{n}$, $n\geq 1$, $C_{n}$, $n\geq 2$, $D_{n}$, $n\geq 3$, or $B_{n}$, $n\geq 2$. By [Ste78, Theorem 4.1] and [Plo93, Corollary 3] the same also holds for $G$ of type $G_{2}$, $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. Consequently, $G(R)=E(R)$ for any simply connected Chevalley–Demazure group scheme $G$ over $R$. ∎

For any commutative ring $R$ with 1, denote by $R(x)$ the localization of $R[x]$ at the set of all monic polynomials.

Lemma 3.2**.**

Let $R$ be a discrete valuation ring or a local Artinian ring. Then $G(R(x))=E(R(x))$ for any simply connected Chevalley–Demazure group scheme $G$ over $R$.

Proof.

Since $R$ is a commutative Noetherian ring, by [Lam06, Ch. IV, Proposition 1.2] $R(x)$ has the same Krull dimension as $R$. If $R$ is Artinian, then $R(x)$ is also Artinian, and hence a finite product of local rings. Then $SL_{n}(R(x))=E_{n}(R(x))$ for all $n\geq 2$. If $R$ is a discrete valuation ring, then also $SL_{n}(R(x))=E_{n}(R(x))$ for all $n\geq 2$ by [Lam06, Ch. IV, Corollary 6.3] (a corollary of [Mur66, Proposition 1’]). Hence by Lemma 3.1 one has $G(R(x))=E(R(x))$ in both cases. ∎

We will also use the following lemma, that was established in [Su, Corollary 5.7] for $G=\operatorname{GL}_{n}$.

Lemma 3.3**.**

[St15, Lemma 2.7]** Let $A$ be a commutative ring, and let $G$ be a reductive group scheme over $A$, such that every semisimple normal subgroup of $G$ is isotropic. Assume moreover that for any maximal ideal $m\subseteq A$, every semisimple normal subgroup of $G_{A_{m}}$ contains $(\operatorname{{\mathbf{G}}_{m}}_{,A_{m}})^{2}$. Then for any monic polynomial $f\in A[x]$ the natural homomorphism

[TABLE]

is injective.

Now we are ready to establish the main theorem for simply connected semisimple Chevalley–Demazure group schemes.

Theorem 3.4**.**

Let $R$ be a locally principal ideal ring or Artinian ring. Then

[TABLE]

for any simply connected Chevalley–Demazure group scheme $G$ over $R$ of isotropic rank $\geq 2$ and any $n\geq 1$.

Proof.

For every maximal ideal $m$ of $R$, the ring $R_{m}$ is a local principal ideal domain, i.e. a discrete valuation ring, or a local Artinian ring, In both cases $R_{m}$ is a local Noetherian ring of Krull dimension $\leq 1$. By [Lam06, Ch. IV, Proposition 1.2] $R_{m}(x_{1})$ has the same Krull dimension as $R_{m}$. If $R_{m}$ is Artinian, then $R_{m}(x_{1})$ is also Artinian. If $R_{m}$ is a discrete valuation ring, then $R_{m}(x_{1})$ is a principal ideal domain by [Lam06, Ch. IV, Corollary 1.3]. Hence by induction hypothesis

[TABLE]

Then by Lemma 3.2 G\bigl{(}R_{m}(x_{1})[x_{2},\ldots,x_{n}]\bigr{)}=E\bigl{(}R_{m}(x_{1})[x_{2},\ldots,x_{n}]\bigr{)}. Then by Lemma 3.3 we have G\bigl{(}R_{m}[x_{1},\ldots,x_{n}]\bigr{)}=E\bigl{(}R_{m}[x_{1},\ldots,x_{n}]\bigr{)}. Then Lemma 2.5 finishes the proof. ∎

To pass from simply connected Chevalley–Demazure group schemes to general ones, we use the following reduction lemma.

Lemma 3.5**.**

Let $G$ be a Chevalley–Demazure group scheme, and let $E$ be an elementary subgroup functor of $G$. Let $G^{\mathrm{sc}}$ be the simply connected cover of the adjoint semisimple group scheme $G^{\mathrm{ad}}=G/\operatorname{Cent}(G)$, and let $E^{\mathrm{sc}}$ be its elementary subgroup functor corresponding to the pinning compatible with that of $G$. Let $A$ be a normal Noetherian integral domain. If one has $G^{\mathrm{sc}}(A[x])=G^{\mathrm{sc}}(A)E^{\mathrm{sc}}(A[x]),$ then $G(A[x])=G(A)E(A[x]).$

Proof.

There is a short exact sequence of $\operatorname{{\mathbb{Z}}}$-group schemes

[TABLE]

for a split $\operatorname{{\mathbb{Z}}}$-torus $T$. Here the group $[G,G]$ is the algebraic derived subgroup scheme of $G$ in the sense of [SGA3, Exp. XXII, §6.2]. It is a semisimple Chevalley–Demazure group scheme, and $E(A)\leq[G,G](A)$. Since $T(A[x])=T(A)$, the exact sequence

[TABLE]

implies that it is enough to prove the claim for $[G,G]$. In other words, we may assume that $G$ is semisimple. Then there is a short exact sequence of algebraic groups

[TABLE]

where $C$ is a group of multiplicative type over $\operatorname{{\mathbb{Z}}}$, central in $G^{\mathrm{sc}}$. Write the respective ‘‘long’’ exact sequences over $A[x]$ and $A$ with respect to fppf topology. Adding the maps induced by the homomorphism $\rho:A[x]\to A$, $x\mapsto 0$, we obtain a commutative diagram

[TABLE]

Here the rightmost vertical arrow is an isomorphism by [CTS, Lemma 2.4]. Take any g\in\ker\bigl{(}\rho:G(A[x])\to G(A)\bigr{)}. It is enough to show that $g\in E(A[x])$.

We have $\delta(g)=1$, hence there is $\tilde{g}\in G^{\mathrm{sc}}(A[x])$ with $\pi(\tilde{g})=g$. Clearly, $\rho(\tilde{g})\in C(A)$, and hence

[TABLE]

Since $\pi(E^{\mathrm{sc}}(A[x]))=E(A[x])$, this proves the claim. ∎

Proof of Corollary 1.4.

By Lemma 2.5 it is enough to prove the claim for $R_{m}$, where $m$ is any maximal ideal of $R$. Since $R_{m}$ is a discrete valuation ring, the claim follows from Lemma 3.5 and Theorem 1.1. ∎

4. Extension to higher dimensional regular rings

In this section we discuss extensions of Theorem 1.1 to rings of polynomials over higher dimensional regular rings $R$. Note that the following result is contained in [St14].

Theorem 4.1**.**

Let $G$ be a Chevalley–Demazure group scheme of isotropic rank $\geq 2$. Let $R$ be an equicharacteristic regular domain. Then $G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])$ for any $n\geq 1$.

Proof.

The claim follows from [St14, Theorem 1.3], since $R[x_{1},\ldots,x_{n}]$ is a regular domain containig a perfect field, for any $n\geq 1$. ∎

Thus, it remains to consider the case of regular domains $R$ of unequal characteristic. Following [W], we rely on the following generalization of Lindel’s lemma [L]. See also [Pop89, Proposition 2.1] for a slightly weaker version.

Lemma 4.2**.**

[Dut00, Theorem 1.3]** Let $(A,m)$ be a regular local ring of dimension $d+1$, essentially of finite type and smooth over an excellent discrete valuation ring $(V,(\pi))$ such that $K=A/m$ is separably generated over $V/\pi V$. Let $0\neq a\in m^{2}$ be such that $a\not\in\pi A$. Then there exists a regular local subring $(B,n)$ of $(A,m)$, with $B/n=A/m=K$, and such that

1. (1)

$B$* is a localization of a polynomial ring $W[x_{1},...,x_{d}]$ at a maximal ideal of the type $(\pi,f(x_{1}),x_{2},\ldots,x_{d})$ where $f$ is a monic irreducible polynomial in $W[x_{1}]$ and $(W,(\pi))$ is an excellent discrete valuation ring contained in $A$; moreover $A$ is an étale neighborhood of $B$.* 2. (2)

There exists an element $h\in B\cap aA$ such that $B/hB\cong A/aA$ is an isomorphism. Furthermore $hA=aA$.

Lemma 4.3**.**

Let $G$ be a Chevalley–Demazure group scheme $G$ of isotropic rank $\geq 2$. Let $R$ be a Dedekind domain with perfect residue fields. Let $A$ be a regular $R$-algebra that is essentially smooth over $R$. Then $G(A[x])=G(A)E(A[x])$.

Proof.

By Lemma 2.5 we can assume that $A$ is local. Then, in particular, $R$ is a regular domain, and hence we can assume that $G$ is simply connected by Lemma 3.5. The map $R\to A$ factors through a localization $R_{q}$, for a prime ideal $q$ of $R$. If $R_{q}$ is equicharacteristic, we are done by Theorem 4.1. Otherwise $R_{q}=V$ is a discrete valuation ring of characteristic [math], and hence excellent by [Gro65, Scholie 7.8.3]. The residue field of $A$ is a finitely generated field extension of the perfect field $R_{q}/qR_{q}=V/\pi$, hence it is separably generated. Thus, all conditions of Lemma 4.2 are fulfilled.

The rest of the proof proceeds as the proof of [St14, Lemma 6.3], using Lemma 4.2 instead of Lindel’s lemma, and Theorem 3.4 instead of [St14, Theorem 1.2]. Namely, one proceeds by induction on $\dim A=d+1$. If $\dim A=1$, we are in the setting of Theorem 3.4. Assume $\dim A\geq 2$. Then $m^{2}\setminus\pi A$ is non-empty, since $A/\pi A$ is an essentially smooth, hence regular, local ring over $V/\pi$, hence a domain. For any $a\in m^{2}\setminus\pi A$, let $B$ and $h\in B\cap aA$ be as in Lemma 4.2. Since $B$ is a localization of a polynomial ring over a discrete valuation ring, which is subject to Theorem 3.4, by Lemma 2.5 one has $G(B[x])=G(B)E(B[x])$. We need to show that any element $g(x)\in G(A[x])$ belongs to $G(A)E(A[x])$. Since $\dim A_{h}<\dim A$, the element $g(x)$ belongs to $G(A_{h})E(A_{h}[x])$. Clearly, we can assume from the start that $g(0)=1$, then in fact $g(x)\in E(A_{h}[x])$. By Lemma 4.2 $h$ satisfies $Ah+B=A$, $Ah\cap B=Bh$. Hence by [St14, Lemma 3.4 (i)] we have $g(x)=g_{1}(x)g_{2}(x)$ for some $g_{1}(x)\in E(A[x])$ and $g_{2}(x)\in G(B_{h}[x])$. Then $g_{2}(x)\in G(B_{h}[x])\cap G(A[x])=G(B[x])$. Then we have $g_{2}(x)\in G(B)E(B[x])$. Therefore, $g(x)\in G(A)E(A[x])$.

∎

Lemma 4.4**.**

Let $G$ be a Chevalley–Demazure group scheme of isotropic rank $\geq 2$. Let $R$ be a regular ring such that every maximal localization of $R$ is an unramified regular local ring. Then $G(R[x])=G(R)E(R[x])$.

Proof.

By Lemma 2.5 we can assume that $R$ is an unramified regular local ring with maximal ideal $m$. If $R$ is equicharacteristic, we are done by Theorem 4.1. If $R$ has characteristic [math] and residual characteristic $p$, then by assumption $p\not\in m^{2}$. Then $R$ is geometrically regular over $\operatorname{{\mathbb{Z}}}_{(p)}$ [Sw, p. 4], and hence a filtered inductive limit of regular local rings which are essentially smooth over $\operatorname{{\mathbb{Z}}}_{(p)}$ by [Sw, Corollary 1.3]. Then Lemma 4.3 finishes the proof. ∎

Proof of Theorem 1.5.

For the first claim, combine Lemma 2.5, Lemma 4.3, and Lemma 4.4. For the second claim, add [St19, Theorem 1.3]. ∎