Chevalley groups of polynomial rings over Dedekind domains

Anastasia Stavrova

doi:10.1515/jgth-2019-0100

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Chevalley groups of polynomial rings over Dedekind domains

Anastasia Stavrova

Veröffentlicht/Copyright: 18. September 2019

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Aus der Zeitschrift Journal of Group Theory Band 23 Heft 1

Abstract

Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank ≥2. We prove that

G⁢(R⁢[x1,…,xn])=G⁢(R)⁢E⁢(R⁢[x1,…,xn]) for any⁢n≥1.

In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G=SLN,Sp2⁢N. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.

1 Introduction

A. Suslin [28, Corollary 6.5] established that, for any regular ring R of dimension ≤1, any N≥3 and any n≥1, one has

SLN⁢(R⁢[x1,…,xn])=SLN⁢(R)⁢EN⁢(R⁢[x1,…,xn]),

where EN⁢(R⁢[x1,…,xn]) is the elementary subgroup, i.e. the subgroup generated by elementary matrices I+t⁢ei⁢j, 1≤i≠j≤N, t∈R⁢[x1,…,xn]. In particular, this implies

SLN⁢(ℤ⁢[x1,…,xn])=EN⁢(ℤ⁢[x1,…,xn]).

A later theorem of A. Suslin and V. Kopeiko [29, Theorem 7.8] together with the homotopy invariance of orthogonal K-theory (see [11, Corollaire 0.8], [10, Corollary 1.12] or [22, Theorem 9.8]) implies a similar result for even orthogonal groups SO2⁢N, N≥3, under the additional assumption 2∈R×. F. Grunewald, J. Mennicke and L. Vaserstein [9] extended Suslin’s result to symplectic groups Sp2⁢N, N≥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 A𝔪 is a principal ideal ring.

We aim to extend the above results to all Chevalley–Demazure group schemes of isotropic rank ≥2. By a Chevalley–Demazure group scheme we mean a split reductive group scheme in the sense of [5]. These group schemes are defined over ℤ. We say that a Chevalley–Demazure group scheme G has isotropic rank ≥n if and only if every irreducible component of its root system has rank ≥n. For any commutative ring R with 1 and any fixed choice of a pinning, or épinglage of G in the sense of [5], 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 xα⁢(r), α∈Φ, r∈R, in the notation of [2, 14], where Φ is the root system of G. If G has isotropic rank ≥2, then E is independent of the choice of the pinning [16].

Our main result is the following theorem. Since SL2⁢(ℤ⁢[x])≠E2⁢(ℤ⁢[x]) [3], 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 ≥2. Then G⁢(R⁢[x1,…,xn])=G⁢(R)⁢E⁢(R⁢[x1,…,xn]) for any n≥1.

Theorem 1.1 for Dedekind domains was previously claimed by M. Wendt [31, Proposition 4.7]; however, his proof was incorrect [27, p. 91]. We give another proof along the lines similar to [9]. The case where R is a field was done earlier in [23] in a more general context of isotropic reductive groups.

Following [14], we say that a Dedekind domain R is of arithmetic type if R=OS 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=Z), and let G be a simply connected Chevalley–Demazure group scheme of isotropic rank ≥2. Then G⁢(R⁢[x1,…,xn])=E⁢(R⁢[x1,…,xn]) for any n≥1.

Proof.

This follows from Theorem 1.1 and [14, Théorème 12.7], which says that G⁢(R)=E⁢(R). ∎

Note that [12, p. 57] presents an example (due to J. Stallings) of a Dedekind domain D such that SL3⁢(D)≠E3⁢(D); hence Corollary 1.2 does not hold for arbitrary Dedekind domains.

A commutative R-algebra of the form A=R⁢[x1,…,xn]/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]/x⁢y. Square-free discrete Hodge algebras over a field are also called Stanley–Reisner rings.

Corollary 1.3.

Let R be a Dedekind domain and G a simply connected Chevalley–Demazure group scheme of isotropic rank ≥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 [25, Corollary 1.5] and Corollary 1.2. ∎

For non-simply connected Chevalley–Demazure group schemes, such as SOn, n≥5, we deduce the following result; see Section 3 for the proof.

Corollary 1.4.

Let R be a locally principal ideal domain and G any Chevalley–Demazure group scheme of isotropic rank ≥2. Then

G⁢(R⁢[x1,…,xn])=G⁢(R)⁢E⁢(R⁢[x1,…,xn]) for any⁢n≥1.

Using a version of Lindel’s lemma [13] and Néron–Popescu desingularization [19], 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 [23]. The following theorem is proved in Section 4.

Theorem 1.5.

Let G be a Chevalley–Demazure group scheme of isotropic rank ≥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, we have 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 [7], one concludes that G⁢(A) has Kazhdan’s property (T) for any simply connected Chevalley–Demazure group scheme G of isotropic rank ≥2 and any discrete Hodge algebra A over ℤ; in particular, G⁢(ℤ⁢[x1,…,xn]) has Kazhdan’s property (T).

Combining Corollary 1.3 with the main result of [21], 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 ≥2 and any discrete Hodge algebra A over R, where R is a Dedekind domain of arithmetic type, satisfying 2∈R× if the root system of G has components of type Cn or G2.

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 ≥2, and E denotes its elementary subgroup functor.

For any s∈R, we denote by Rs the localization of R at s, and by Fs:R→Rs the localization homomorphism and the induced homomorphism G⁢(R)→G⁢(Rs). Similarly, for any maximal ideal 𝔪 of R, we denote by F𝔪:R→R𝔪 the localization homomorphism, as well as the induced homomorphism G⁢(R)→G⁢(R𝔪).

We will need the following generalization of the Quillen–Suslin local-global principle for polynomial rings in one variable (see [28, Theorem 3.1], [29, Corollary 4.4], [16, Lemma 17], [27, Theorem 5.4]) to the case of several variables.

Lemma 2.1.

Let R be any commutative ring. Fix n≥1. If g∈G⁢(R⁢[x1,…,xn]) satisfies Fm⁢(g)∈E⁢(Rm⁢[x1,…,xn]) for any maximal ideal m of R, then we have g∈G⁢(R)⁢E⁢(R⁢[x1,…,xn]).

The proof of Lemma 2.1 uses the following three standard lemmas whose idea goes back to [20, Lemma 1].

Lemma 2.2.

Let H be any affine R-scheme of finite type. Fix 0≠s∈R, and let Fs:H⁢(R⁢[z])→H⁢(Rs⁢[z]) be the localization map. For any g⁢(z),h⁢(z)∈H⁢(R⁢[z]) such that h⁢(0)=g⁢(0) and Fs⁢(g⁢(z))=Fs⁢(h⁢(z)), there is some n≥0 such that g⁢(sn⁢z)=h⁢(sn⁢z).

Proof.

Since H is an affine R-scheme of finite type, there is a closed embedding H→𝔸Rk for some k≥0. Hence it is enough to prove the claim for H=𝔸Rk. If k=0, then we have g⁢(z)=g⁢(0)=h⁢(0)=h⁢(z). If k≥1, the claim readily reduces to the case k=1, that is, g⁢(z),h⁢(z)∈R⁢[z]. Since Fs⁢(g⁢(z))=Fs⁢(h⁢(z)), there is some n≥0 such that sn⁢g⁢(z)=sn⁢h⁢(z). Since g⁢(0)=h⁢(0), this implies g⁢(sn⁢z)=h⁢(sn⁢z). ∎

Lemma 2.3 ([27, Theorem 5.2]).

Fix s∈R, and let

Fs:G⁢(R⁢[z])→G⁢(Rs⁢[z])

be the localization homomorphism. For any g⁢(z)∈E⁢(Rs⁢[z],z⁢Rs⁢[z]), there exist h⁢(z)∈E⁢(R⁢[z],z⁢R⁢[z]) and k≥0 such that Fs⁢(h⁢(z))=g⁢(sk⁢z).

Proof.

The statement is a particular case of [27, Theorem 5.2] if the root system Φ of G is irreducible. Assume that Φ has several irreducible components Φi. By [5, Exposé XXVI, Proposition 6.1] G contains semisimple Chevalley–Demazure subgroup schemes Gi of type Φi whose elementary subgroup functors Ei are generated by elementary root unipotents corresponding to roots in Φi. Chevalley commutator relations imply that E is a direct product of all Ei. This reduces the claim to the case where Φ is irreducible. ∎

Lemma 2.4.

For any g⁢(x)∈G⁢(R⁢[x]) such that Fs⁢(g⁢(x)) lies in E⁢(Rs⁢[x]), there exists k≥0 such that

g⁢(a⁢x)⁢g⁢(b⁢x)-1∈E⁢(R⁢[x])

for all a,b∈R satisfying a≡bmodsk.

Proof.

Consider the element f⁢(z)=g⁢(x⁢(y+z))⁢g⁢(x⁢y)-1∈G⁢(R⁢[x,y,z]). Observe that Fs⁢(f⁢(z))∈E⁢(Rs⁢[x,y,z]) and f⁢(0)=1. Since Fs⁢(g⁢(x))∈E⁢(Rs⁢[x]) and f⁢(0)=1, we have Fs⁢(f⁢(z))∈E⁢(Rs⁢[x,y,z],z⁢Rs⁢[x,y,z]) (e.g. by [23, Lemma 4.1]). Now, by Lemma 2.3, there exist h⁢(z)∈E⁢(R⁢[x,y,z],z⁢R⁢[x,y,z]) and k≥1 such that Fs⁢(h⁢(z))=Fs⁢(f⁢(sk⁢z)). By Lemma 2.2, there is some l≥1 such that h⁢(sl⁢z)=f⁢(sl+k⁢z). Then g⁢(x⁢(y+sl+k⁢z))⁢g⁢(x⁢y)-1 lies in EP⁡(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

F𝔪⁢(g)∈E⁢(R𝔪⁢[x1,…,xn]),

there is some s∈R∖𝔪 such that Fs⁢(g)∈E⁢(Rs⁢[x1,…,xn]). Choose a finite set of elements s=si∈R∖𝔪i, 1≤i≤N as above so that 1=∑i=1Nci⁢si for some ci∈R. Consider g as a function g⁢(x1) of x1. By Lemma 2.4, there are ki≥1 such that g⁢(a⁢x1)⁢g⁢(b⁢x1)-1∈E⁢(R⁢[x1,…,xn]) for any a,b∈R⁢[x2,…,xn] satisfying a≡b(modsiki). Since si generate the unit ideal, their powers siki also generate the unit ideal, and we can replace si by these powers without loss of generality. Set aj=∑i=1N-jci⁢si, 0≤j≤N. Then aj+1≡aj(modsn-j), and

g⁢(x1)=(∏j=0N-1g⁢(aj⁢x1)⁢g⁢(aj+1⁢x1))-1⁢g⁢(0).

Then we have g⁢(x1)∈E⁢(R⁢[x1,…,xn])⁢g⁢(0). Since g⁢(0)∈G⁢(R⁢[x2,…,xn]), we can proceed by induction. ∎

Lemma 2.5.

Fix n≥1. One has G⁢(R⁢[x1,…,xn])=G⁢(R)⁢E⁢(R⁢[x1,…,xn]) if and only if G⁢(Rm⁢[x1,…,xn])=G⁢(Rm)⁢E⁢(Rm⁢[x1,…,xn]) for every maximal ideal m of R.

Proof.

For the direct implication, see [23, Lemma 4.2]. To prove the converse, it is enough to show that g⁢(x1,…,xn)∈G⁢(R⁢[x1,…,xn]) such that g⁢(0,…,0)=1 satisfies g⁢(x1,…,xn)∈E⁢(R⁢[x1,…,xn]). For every maximal ideal 𝔪 of R, by assumption, F𝔪(g(x1,…,xn)∈G(R𝔪)E(R𝔪[x1,…,xn]), and g⁢(0,…,0)=1 implies F𝔪⁢(g⁢(x1,…,xn))∈E⁢(R𝔪⁢[x1,…,xn]). Then Lemma 2.1 finishes the proof. ∎

3 Proof of the main theorem

The following result follows from stability results for non-stable K1-functors of Chevalley groups [26, 17].

Lemma 3.1.

Let R be a Noetherian ring of Krull dimension ≤1. If SL2⁢(R)=E2⁢(R), then G⁢(R)=E⁢(R) for any simply connected Chevalley–Demazure group scheme G over R.

Proof.

By [1, p. 102], the maximal ideal spectrum of R is a Noetherian topological space of dimension ≤1. By [26, Theorem 1.4], this implies that R satisfies the absolute stable range condition ASR3, and hence also Bass’ stable range condition SR3 in the sense of [26, p. 86]. Then, by [26, Theorem 2.2] (see also [26, Corollary 2.3]), suitable inclusions of SL2 into G induce surjections

SL2⁢(R)/E2⁢(R)→G⁢(R)/E⁢(R)

for every simply connected Chevalley–Demazure group scheme G corresponding to an irreducible root system of classical type An, n≥1, Cn, n≥2, Dn, n≥3, or Bn, n≥2. By [26, Theorem 4.1] and [17, Corollary 3], the same also holds for G of type G2, F4, E6, E7, and E8. 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 we have 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 [12, Chapter 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

SLn⁢(R⁢(x))=En⁢(R⁢(x)) for all⁢n≥2.

If R is a discrete valuation ring, then also SLn⁢(R⁢(x))=En⁢(R⁢(x)) for all n≥2 by [12, Chapter IV, Corollary 6.3] (a corollary of [15, 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 [28, Corollary 5.7] for G=GLn.

Lemma 3.3 ([24, 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⊆A, every semisimple normal subgroup of GAm contains (Gm,Am)2. Then, for any monic polynomial f∈A⁢[x], the natural homomorphism

G⁢(A⁢[x])/E⁢(A⁢[x])→G⁢(A⁢[x]f)/E⁢(A⁢[x]f)

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

G⁢(R⁢[x1,…,xn])=G⁢(R)⁢E⁢(R⁢[x1,…,xn])

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

Proof.

For every maximal ideal 𝔪 of R, the ring R𝔪 is a local principal ideal domain, i.e. a discrete valuation ring, or a local Artinian ring. In both cases, R𝔪 is a local Noetherian ring of Krull dimension ≤1. By [12, Chapter IV, Proposition 1.2], R𝔪⁢(x1) has the same Krull dimension as R𝔪. If R𝔪 is Artinian, then R𝔪⁢(x1) is also Artinian. If R𝔪 is a discrete valuation ring, then R𝔪⁢(x1) is a principal ideal domain by [12, Chapter IV, Corollary 1.3]. Hence, by induction hypothesis,

G⁢(R𝔪⁢(x1)⁢[x2,…,xn])=G⁢(R𝔪⁢(x1))⁢E⁢(R𝔪⁢(x1)⁢[x2,…,xn]).

Then, by Lemma 3.2, G⁢(R𝔪⁢(x1)⁢[x2,…,xn])=E⁢(R𝔪⁢(x1)⁢[x2,…,xn]). By Lemma 3.3, we have G⁢(R𝔪⁢[x1,…,xn])=E⁢(R𝔪⁢[x1,…,xn]), and 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 Gsc be the simply connected cover of the adjoint semisimple group scheme Gad=G/Cent⁡(G), and let Esc 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 Gsc⁢(A⁢[x])=Gsc⁢(A)⁢Esc⁢(A⁢[x]), then G⁢(A⁢[x])=G⁢(A)⁢E⁢(A⁢[x]).

Proof.

There is a short exact sequence of ℤ-group schemes

1→[G,G]→G→T→1,

for a split ℤ-torus T. Here the group [G,G] is the algebraic derived subgroup scheme of G in the sense of [5, Exposé XXII, § 6.2]. It is a semisimple Chevalley–Demazure group scheme, and E⁢(A)≤[G,G]⁢(A). Since T⁢(A⁢[x])=T⁢(A), the exact sequence

1→[G,G]⁢(A⁢[x])→G⁢(A⁢[x])→T⁢(A⁢[x])

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

1→C⁢→𝑖⁢Gsc⁢→𝜋⁢G→1,

where C is a group of multiplicative type over ℤ, central in Gsc. Write the respective “long” exact sequences over A⁢[x] and A with respect to the fppf topology. Adding the maps induced by the homomorphism ρ:A⁢[x]→A, x↦0, we obtain a commutative diagram

Here the rightmost vertical arrow is an isomorphism by [4, Lemma 2.4]. Take any g∈ker⁡(ρ:G⁢(A⁢[x])→G⁢(A)). It is enough to show that g∈E⁢(A⁢[x]).

We have δ⁢(g)=1; hence there is some g~∈Gsc⁢(A⁢[x]) with π⁢(g~)=g. Clearly, ρ⁢(g~)∈C⁢(A), and hence

g~∈C⁢(A)⋅ker⁡(ρ:Gsc⁢(A⁢[x])→Gsc⁢(A))⊆C⁢(A)⁢Esc⁢(A⁢[x]).

Since π⁢(Esc⁢(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 R𝔪, where 𝔪 is any maximal ideal of R. Since R𝔪 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 [23].

Theorem 4.1.

Let G be a Chevalley–Demazure group scheme of isotropic rank ≥2. Let R be an equicharacteristic regular domain. Then

G⁢(R⁢[x1,…,xn])=G⁢(R)⁢E⁢(R⁢[x1,…,xn]) for any⁢n≥1.

Proof.

The claim follows from [23, Theorem 1.3] since R⁢[x1,…,xn] is a regular domain containing a perfect field for any n≥1. ∎

Thus, it remains to consider the case of regular domains R of unequal characteristic. Following [31], we rely on the following generalization of Lindel’s lemma [13]. See also [18, Proposition 2.1] for a slightly weaker version.

Lemma 4.2 ([6, 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,(π)) such that K=A/m is separably generated over V/π⁢V, and let 0≠a∈m2 be such that a∉π⁢A. Then there exists a regular local subring (B,n) of (A,m), with B/n=A/m=K, and such that the following holds.

B is a localization of a polynomial ring W⁢[x1,…,xd] at a maximal ideal of the type (π,f⁢(x1),x2,…,xd), where f is a monic irreducible polynomial in W⁢[x1] and (W,(π)) is an excellent discrete valuation ring contained in A; moreover, A is an étale neighborhood of B.
There exists an element h∈B∩a⁢A such that B/h⁢B≅A/a⁢A is an isomorphism. Furthermore, h⁢A=a⁢A.

Lemma 4.3.

Let G be a Chevalley–Demazure group scheme G of isotropic rank ≥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→A factors through a localization R𝔮, for a prime ideal 𝔮 of R. If R𝔮 is equicharacteristic, we are done by Theorem 4.1. Otherwise, R𝔮=V is a discrete valuation ring of characteristic 0, and hence excellent by [8, Scholie 7.8.3]. The residue field of A is a finitely generated field extension of the perfect field R𝔮/𝔮⁢R𝔮=V/π; hence it is separably generated. Thus, all conditions of Lemma 4.2 are fulfilled.

The rest of the proof proceeds like the proof of [23, Lemma 6.3], using Lemma 4.2 instead of Lindel’s lemma, and Theorem 3.4 instead of [23, 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≥2. Then 𝔪2∖π⁢A is non-empty since A/π⁢A is an essentially smooth, hence regular, local ring over V/π, hence a domain. For any a∈𝔪2∖π⁢A, let B and h∈B∩a⁢A 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)∈G⁢(A⁢[x]) belongs to G⁢(A)⁢E⁢(A⁢[x]). Since dim⁡Ah<dim⁡A, the element g⁢(x) belongs to G⁢(Ah)⁢E⁢(Ah⁢[x]). Clearly, we can assume from the start that g⁢(0)=1; then, in fact, g⁢(x)∈E⁢(Ah⁢[x]). By Lemma 4.2, h satisfies A⁢h+B=A, A⁢h∩B=B⁢h. So, by [23, Lemma 3.4 (i)], we have g⁢(x)=g1⁢(x)⁢g2⁢(x) for some g1⁢(x)∈E⁢(A⁢[x]) and g2⁢(x)∈G⁢(Bh⁢[x]). Then g2⁢(x)∈G⁢(Bh⁢[x])∩G⁢(A⁢[x])=G⁢(B⁢[x]), so g2⁢(x)∈G⁢(B)⁢E⁢(B⁢[x]). Therefore, g⁢(x)∈G⁢(A)⁢E⁢(A⁢[x]). ∎

Lemma 4.4.

Let G be a Chevalley–Demazure group scheme of isotropic rank ≥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 0 and residual characteristic p, then, by assumption, p∉𝔪2. Then R is geometrically regular over ℤ(p) [30, p. 4], and hence a filtered inductive limit of regular local rings which are essentially smooth over ℤ(p) by [30, 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 [25, Theorem 1.3]. ∎

Communicated by Robert M. Guralnick

Funding source: Russian Foundation for Fundamental Investigations

Award Identifier / Grant number: 18-31-20044

Funding statement: The author is a winner of the contest “Young Russian Mathematics”. The work was supported by the RFBR grant 18-31-20044.

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Received: 2019-07-04

Revised: 2019-07-31

Published Online: 2019-09-18

Published in Print: 2020-01-01

Artikel in diesem Heft

https://doi.org/10.1515/jgth-2019-0100