Chevalley groups of polynomial rings over Dedekind domains
aa r X i v : . [ m a t h . K T ] J un CHEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKINDDOMAINS
A. STAVROVA
Abstract.
Let R be a Dedekind domain, and let G be a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank ≥ . We prove that G ( R [ x , . . . , x n ]) = G ( R ) E ( R [ x , . . . , x n ]) for any n ≥ . This extends the corresponding results of A. Suslin and F. Grunewald, J.Mennicke, and L. Vaserstein for G = SL N , Sp N . We also deduce some corollaries of the aboveresult for regular rings R of higher dimension and discrete Hodge algebras over R . Introduction
A. Suslin [Su, Corollary 6.5] established that for any regular ring R of dimension ≤ , any N ≥ , and any n ≥ , one has SL N ( R [ x , . . . , x n ]) = SL N ( R ) E N ( R [ x , . . . , x n ]) , where E N ( R [ x , . . . , x n ]) is the elementary subgroup, i.e. the subgroup generated by elementarymatrices I + te ij , ≤ i = j ≤ N , t ∈ R [ x , . . . , x n ] . In particular, this implies SL N ( Z [ x , . . . , x n ]) = E N ( Z [ x , . . . , x n ]) . A later theorem of A. Suslin and V. Kopeiko [SuK, Theorem 7.8] together with the homo-topy 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 N , N ≥ , un-der the additional assumption ∈ R × . F. Grunewald, J. Mennicke, and L. Vaserstein [GMV91]extended the result of Suslin to symplectic groups Sp N , N ≥ , and a slightly larger class ofrings 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 isotropicrank ≥ . By a Chevalley–Demazure group scheme we mean a split reductive group schemein the sense of [SGA3]. 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 componentof its root system has rank ≥ n . For any commutative ring R with 1 and any fixed choice ofa pinning, or ´epinglage of G in the sense of [SGA3], we denote by E the elementary subgroupfunctor of G . That is, E ( R ) is the subgroup of G ( R ) generated by elementary root unipotentelements x α ( r ) , α ∈ Φ , r ∈ R , in the notation of [Ch55, Ma], where Φ is the root system of G .If G has isotropic rank ≥ , then E is independent of the choice of the pinning [PSt].Our main result is the following theorem. Since SL ( Z [ x ]) = E ( Z [ x ]) [C], it cannot beextended to the case of isotropic rank 1. The author is a winner of the contest “Young Russian Mathematics”. The work was supported by the RFBRgrant 18-31-20044.
Theorem 1.1 (Theorem 3.4) . Let R be a locally principal ideal ring, and let G be a simplyconnected Chevalley–Demazure group scheme of isotropic rank ≥ . Then G ( R [ x , . . . , x n ]) = G ( R ) E ( R [ x , . . . , x n ]) for any n ≥ . Theorem 1.1 for Dedekind domains was previously claimed by M. Wendt [W, Proposition4.7], however, his proof was incorrect [Ste13, p. 91]. We give another proof along the linessimilar to [GMV91]. The case where R is a field was done earlier in [St14] in a more generalcontext of isotropic reductive groups.Following [Ma], we say that a Dedekind domain R is of arithmetic type, if R = O S is the ringof S -integers of a global field k with respect to a finite non-empty set S of primes containingall 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 ≥ . Then G ( R [ x , . . . , x n ]) = E ( R [ x , . . . , x n ]) for any n ≥ .Proof. This follows from Theorem 1.1 and [Ma, Th´eor`eme 12.7], which says that G ( R ) = E ( R ) . (cid:3) Note that [Lam06, p. 57] presents an example (due to J. Stallings) of a Dedekind domain D such that SL ( D ) = E ( D ) , hence Corollary 1.2 does not hold for arbitrary Dedekind domains.A commutative R -algebra of the form A = R [ x , . . . , x n ] /I , where I is an ideal generated bymonomials, 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–Reisnerrings. Corollary 1.3.
Let R be a Dedekind domain, and let G be a simply connected Chevalley–Demazure group scheme of isotropic rank ≥ . Then G ( A ) = G ( R ) E ( A ) for any discreteHodge 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. (cid:3)
For non-simply connected Chevalley–Demazure group schemes, such as SO n , n ≥ , wededuce 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–Demazuregroup scheme of isotropic rank ≥ . Then G ( R [ x , . . . , x n ]) = G ( R ) E ( R [ x , . . . , x n ]) for any n ≥ . Using a version of Lindel’s lemma [L] and N´eron-Popescu desingularization [Pop90], one mayextend the above results to higher-dimensional regular rings in place of Dedekind domains. Forequicharacteristic regular rings this was done earlier in [St14]. The following theorem is provedin § 4.
Theorem 1.5.
Let G be a Chevalley–Demazure group scheme of isotropic rank ≥ . Let R be a regular ring such that every maximal localization of R is either essentially smoothover 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 Hodgealgebra A over R . HEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKIND DOMAINS 3
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 ) hasKazhdan’s property (T) for any simply connected Chevalley–Demazure group scheme G ofisotropic rank ≥ and any discrete Hodge algebra A over Z ; in particular, G ( Z [ x , . . . x n ]) hasKazhdan’s property (T).Combining Corollary 1.3 with the main result of [RR], one concludes that the congruencekernel of G ( A ) is central in G ( A ) for any simply connected Chevalley–Demazure group scheme G of isotropic rank ≥ and any discrete Hodge algebra A over R , where R is a Dedekinddomain of arithmetic type, satisfying ∈ R × if the root system of G has components of type C n or G . 2. A local–global principle
Throughout this section, R is any commutative ring with 1, G is a Chevalley–Demazuregroup scheme of isotropic rank ≥ , and E denotes its elementary subgroup functor.For any s ∈ R we denote by R s the localization of R at s , and by F s : R → R s thelocalization homomorphism, as well as the induced homomorphism G ( R ) → G ( R s ) . Similarly,for any maximal ideal m of R we denote by F m : R → R m the localization homomorphism, aswell as the induced homomorphism G ( R ) → G ( R m ) .We will need the following generalization of the Quillen–Suslin local-global principle forpolynomial rings in one variable (see [Su, Theorem 3.1], [SuK, Corollary 4.4], [PSt, Lemma17], [Ste13, Theorem 5.4]) to the case of several variables. Lemma 2.1.
Let R be any commutative ring. Fix n ≥ . If g ∈ G ( R [ x , . . . , x n ]) satisfies F m ( g ) ∈ E ( R m [ x , . . . , x n ]) for any maximal ideal m of R , then g ∈ G ( R ) E ( R [ x , . . . , x n ]) . The proof of Lemma 2.1 uses the following three standard lemmas whose idea goes backto [Q, Lemma 1].
Lemma 2.2.
Let H be any affine R -scheme of finite type. Fix = s ∈ R , and let F s : H ( R [ z ]) → H ( R s [ z ]) be the localization map. For any g ( z ) , h ( z ) ∈ H ( R [ z ]) such that h (0) = g (0) and F s ( g ( z )) = F s ( h ( z )) there is n ≥ 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 → A kR forsome k ≥ . Hence it is enough to prove the claim for H = A kR . If k = 0 , then g ( z ) = g (0) = h (0) = h ( z ) . If k ≥ , the claim readily reduces to the case k = 1 , that is, g ( z ) , h ( z ) ∈ R [ z ] .Since F s ( g ( z )) = F s ( h ( z )) , there is n ≥ such that s n g ( z ) = s n h ( z ) . Since g (0) = h (0) , thisimplies g ( s n z ) = h ( s n z ) . (cid:3) Lemma 2.3. [Ste13, Theorem 5.2]
Fix s ∈ R , and let F s : G ( R [ z ]) → G ( R s [ z ]) be the local-ization homomorphism. For any g ( z ) ∈ E ( R s [ z ] , zR s [ z ]) there exist h ( z ) ∈ E ( R [ z ] , zR [ z ]) and k ≥ 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 Φ of G is irreducible. Assume that Φ has several irreducible components Φ i . By [SGA3, Exp. XXVIProp. 6.1] G contains semisimple Chevalley–Demazure subgroup schemes G i of type Φ i whoseelementary subgroup functors E i are generated by elementary root unipotents correspondingto roots in Φ i . Chevalley commutator relations imply that E is a direct product of all E i . Thisreduces the claim to the case where Φ is irreducible. (cid:3) HEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKIND DOMAINS 4
Lemma 2.4.
For any g ( x ) ∈ G ( R [ x ]) such that F s ( g ( x )) lies in E ( R s [ x ]) , there exists k ≥ such that g ( ax ) g ( bx ) − ∈ E ( R [ x ]) for all a, b ∈ R satisfying a ≡ b mod s k .Proof. Consider the element f ( z ) = g ( x ( y + z )) g ( xy ) − ∈ G ( R [ x, y, z ]) . Observe that F s ( f ( z )) ∈ E ( R s [ x, y, z ]) and f (0) = 1 . Since F s ( g ( x )) ∈ E ( R s [ x ]) and f (0) = 1 , we have F s ( f ( z )) ∈ 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 ) ∈ E ( R [ x, y, z ] , zR [ x, y, z ]) and k ≥ such that F s ( h ( z )) = F s ( f ( s k z )) . By Lemma 2.2 there is l ≥ such that h ( s l z ) = f ( s l + k z ) . Then g ( x ( y + s l + k z )) g ( xy ) − lies in E P ( R [ x, y, z ]) . It remainsto set y = b and to choose a suitable z depending on a . (cid:3) Proof of Lemma 2.1.
For any maximal ideal m of R , since F m ( g ) ∈ E ( R m [ x , . . . , x n ]) , there is s ∈ R \ m such that F s ( g ) ∈ E ( R s [ x , . . . , x n ]) . Choose a finite set of elements s = s i ∈ R \ m i , ≤ i ≤ N as above, so that P Ni =1 c i s i for some c i ∈ R . Consider g as a function g ( x ) of x . By Lemma 2.4 there are k i ≥ such that g ( ax ) g ( bx ) − ∈ E ( R [ x , . . . , x n ]) for any a, b ∈ R [ x , . . . , x n ] satisfying a ≡ b (mod s k i i ) . Since s i generate the unit ideal, their powers s k i i also generate the unit ideal, and we can replace s i by these powers without loss of generality.Set a j = P N − ji =1 c i s i , ≤ j ≤ N . Then a j +1 ≡ a j (mod s n − j ) , and g ( x ) = (cid:16) N − Y j =0 g ( a j x ) g ( a j +1 x ) (cid:17) − g (0) . Then g ( x ) ∈ E ( R [ x , . . . , x n ]) g (0) . Since g (0) ∈ G ( R [ x , . . . , x n ]) , we can proceed by induc-tion. (cid:3) Lemma 2.5.
Fix n ≥ . One has G ( R [ x , . . . , x n ]) = G ( R ) E ( R [ x , . . . , x n ]) if and only if G ( R m [ x , . . . , x n ]) = G ( R m ) E ( R m [ x , . . . , 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 enoughto show that g ( x , . . . , x n ) ∈ G ( R [ x , . . . , x n ]) such that g (0 , . . . ,
0) = 1 satisfies g ( x , . . . , x n ) ∈ E ( R [ x , . . . , x n ]) . For every maximal ideal m of R , by assumption, one has F m ( g ( x , . . . , x n ) ∈ G ( R m ) E ( R m [ x , . . . , x n ]) , and g (0 , . . . ,
0) = 1 implies F m ( g ( x , . . . , x n )) ∈ E ( R m [ x , . . . , x n ]) .Then Lemma 2.1 finishes the proof. (cid:3) Proof of the main theorem
The following result follows from stability results for non-stable K -funtors of Chevalleygroups [Ste78, Plo93]. Lemma 3.1.
Let R be a Noetherian ring of Krull dimension ≤ . If SL ( R ) = E ( 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 spaceof dimension ≤ . By [Ste78, Theorem 1.4] this implies that R satisfies the absolute stablerange condition ASR , and hence also Bass’ stable range condition SR in the sense of [Ste78,p. 86]. Then by [Ste78, Theorem 2.2] (see also [Ste78, Corollary 2.3]) suitable inclusions of SL into G induce surjections SL ( R ) /E ( R ) → 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 ≥ , C n , n ≥ , D n , n ≥ , or B n , n ≥ . By [Ste78, Theorem 4.1] and [Plo93, Corollary 3]the same also holds for G of type G , F , E , E , and E . Consequently, G ( R ) = E ( R ) for anysimply connected Chevalley–Demazure group scheme G over R . (cid:3) HEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKIND DOMAINS 5
For any commutative ring R with 1, denote by R ( x ) the localization of R [ x ] at the set of allmonic 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 hencea finite product of local rings. Then SL n ( R ( x )) = E n ( R ( x )) for all n ≥ . If R is a discretevaluation ring, then also SL n ( R ( x )) = E n ( R ( x )) for all n ≥ by [Lam06, Ch. IV, Corollary6.3] (a corollary of [Mur66, Proposition 1’]). Hence by Lemma 3.1 one has G ( R ( x )) = E ( R ( x )) in both cases. (cid:3) We will also use the following lemma, that was established in [Su, Corollary 5.7] for G = GL n . Lemma 3.3. [St15, Lemma 2.7]
Let A be a commutative ring, and let G be a reductive groupscheme over A , such that every semisimple normal subgroup of G is isotropic. Assume more-over that for any maximal ideal m ⊆ A , every semisimple normal subgroup of G A m contains ( G m ,A m ) . 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 [ x , . . . , x n ]) = G ( R ) E ( R [ x , . . . , x n ]) for any simply connected Chevalley–Demazure group scheme G over R of isotropic rank ≥ and any n ≥ .Proof. For every maximal ideal m of R , the ring R m is a local principal ideal domain, i.e. adiscrete valuation ring, or a local Artinian ring, In both cases R m is a local Noetherian ringof Krull dimension ≤ . By [Lam06, Ch. IV, Proposition 1.2] R m ( x ) has the same Krulldimension as R m . If R m is Artinian, then R m ( x ) is also Artinian. If R m is a discrete valuationring, then R m ( x ) is a principal ideal domain by [Lam06, Ch. IV, Corollary 1.3]. Hence byinduction hypothesis G (cid:0) R m ( x )[ x , . . . , x n ] (cid:1) = G (cid:0) R m ( x ) (cid:1) E (cid:0) R m ( x )[ x , . . . , x n ] (cid:1) . Then by Lemma 3.2 G (cid:0) R m ( x )[ x , . . . , x n ] (cid:1) = E (cid:0) R m ( x )[ x , . . . , x n ] (cid:1) . Then by Lemma 3.3 wehave G (cid:0) R m [ x , . . . , x n ] (cid:1) = E (cid:0) R m [ x , . . . , x n ] (cid:1) . Then Lemma 2.5 finishes the proof. (cid:3) To pass from simply connected Chevalley–Demazure group schemes to general ones, we usethe following reduction lemma.
Lemma 3.5.
Let G be a Chevalley–Demazure group scheme, and let E be an elementarysubgroup functor of G . Let G sc be the simply connected cover of the adjoint semisimple groupscheme G ad = G/ Cent( G ) , and let E sc be its elementary subgroup functor corresponding to thepinning compatible with that of G . Let A be a normal Noetherian integral domain. If one has G sc ( A [ x ]) = G sc ( A ) E sc ( A [ x ]) , then G ( A [ x ]) = G ( A ) E ( A [ x ]) . HEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKIND DOMAINS 6
Proof.
There is a short exact sequence of Z -group schemes → [ G, G ] → G → T → , for a split Z -torus T . Here the group [ G, G ] is the algebraic derived subgroup scheme of G inthe sense of [SGA3, Exp. 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 → [ 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 → C i −→ G sc π −→ G → , where C is a group of multiplicative type over Z , central in G sc . Write the respective “long”exact sequences over A [ x ] and A with respect to fppf topology. Adding the maps induced bythe homomorphism ρ : A [ x ] → A , x , we obtain a commutative diagram / / C ( A [ x ]) ρ (cid:15) (cid:15) i / / G sc ( A [ x ]) ρ (cid:15) (cid:15) π / / G ( A [ x ]) ρ (cid:15) (cid:15) δ / / H ( A [ x ] , C ) ∼ = (cid:15) (cid:15) / / C ( A ) i / / G sc ( A ) π / / G ( A ) δ / / H ( A, C ) Here the rightmost vertical arrow is an isomorphism by [CTS, Lemma 2.4]. Take any g ∈ ker (cid:0) ρ : G ( A [ x ]) → G ( A ) (cid:1) . It is enough to show that g ∈ E ( A [ x ]) .We have δ ( g ) = 1 , hence there is ˜ g ∈ G sc ( A [ x ]) with π (˜ g ) = g . Clearly, ρ (˜ g ) ∈ C ( A ) , andhence ˜ g ∈ C ( A ) · ker (cid:0) ρ : G sc ( A [ x ]) → G sc ( A ) (cid:1) ⊆ C ( A ) E sc ( A [ x ]) . Since π ( E sc ( A [ x ])) = E ( A [ x ]) , this proves the claim. (cid:3) Proof of Corollary 1.4.
By Lemma 2.5 it is enough to prove the claim for R m , where m is anymaximal ideal of R . Since R m is a discrete valuation ring, the claim follows from Lemma 3.5and Theorem 1.1. (cid:3) Extension to higher dimensional regular rings
In this section we discuss extensions of Theorem 1.1 to rings of polynomials over higherdimensional 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 ≥ . Let R bean equicharacteristic regular domain. Then G ( R [ x , . . . , x n ]) = G ( R ) E ( R [ x , . . . , x n ]) for any n ≥ .Proof. The claim follows from [St14, Theorem 1.3], since R [ x , . . . , x n ] is a regular domaincontainig a perfect field, for any n ≥ . (cid:3) Thus, it remains to consider the case of regular domains R of unequal characteristic. Fol-lowing [W], we rely on the following generalization of Lindel’s lemma [L]. See also [Pop89,Proposition 2.1] for a slightly weaker version. HEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKIND DOMAINS 7
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, ( π )) such that K = A/m is separably generated over
V /πV . Let = a ∈ m be such that a πA . Then thereexists a regular local subring ( B, n ) of ( A, m ) , with B/n = A/m = K , and such that (1) B is a localization of a polynomial ring W [ x , ..., x d ] at a maximal ideal of the type ( π, f ( x ) , x , . . . , x d ) where f is a monic irreducible polynomial in W [ x ] and ( W, ( π )) isan excellent discrete valuation ring contained in A ; moreover A is an ´etale neighborhoodof B . (2) There exists an element h ∈ B ∩ aA such that B/hB ∼ = A/aA is an isomorphism.Furthermore hA = aA . Lemma 4.3.
Let G be a Chevalley–Demazure group scheme G of isotropic rank ≥ . Let R bea Dedekind domain with perfect residue fields. Let A be a regular R -algebra that is essentiallysmooth 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 factorsthrough a localization R q , for a prime ideal q of R . If R q is equicharacteristic, we are doneby Theorem 4.1. Otherwise R q = V is a discrete valuation ring of characteristic , and henceexcellent by [Gro65, Scholie 7.8.3]. The residue field of A is a finitely generated field extensionof the perfect field R q /qR q = V /π , hence it is separably generated. Thus, all conditions ofLemma 4.2 are fulfilled.The rest of the proof proceeds as the proof of [St14, Lemma 6.3], using Lemma 4.2 insteadof Lindel’s lemma, and Theorem 3.4 instead of [St14, Theorem 1.2]. Namely, one proceeds byinduction on dim A = d + 1 . If dim A = 1 , we are in the setting of Theorem 3.4. Assume dim A ≥ . Then m \ πA is non-empty, since A/πA is an essentially smooth, hence regular,local ring over
V /π , hence a domain. For any a ∈ m \ πA , let B and h ∈ B ∩ aA be as inLemma 4.2. Since B is a localization of a polynomial ring over a discrete valuation ring, whichis subject to Theorem 3.4, by Lemma 2.5 one has G ( B [ x ]) = G ( B ) E ( B [ x ]) . We need to showthat any element g ( x ) ∈ 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 infact g ( x ) ∈ E ( A h [ x ]) . By Lemma 4.2 h satisfies Ah + B = A , Ah ∩ B = Bh . Hence by [St14,Lemma 3.4 (i)] we have g ( x ) = g ( x ) g ( x ) for some g ( x ) ∈ E ( A [ x ]) and g ( x ) ∈ G ( B h [ x ]) .Then g ( x ) ∈ G ( B h [ x ]) ∩ G ( A [ x ]) = G ( B [ x ]) . Then we have g ( x ) ∈ G ( B ) E ( B [ x ]) . Therefore, g ( x ) ∈ G ( A ) E ( A [ x ]) . (cid:3) Lemma 4.4.
Let G be a Chevalley–Demazure group scheme of isotropic rank ≥ . Let R be aregular 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 maximalideal m . If R is equicharacteristic, we are done by Theorem 4.1. If R has characteristic andresidual characteristic p , then by assumption p m . Then R is geometrically regular over Z ( p ) [Sw, p. 4], and hence a filtered inductive limit of regular local rings which are essentiallysmooth over Z ( p ) by [Sw, Corollary 1.3]. Then Lemma 4.3 finishes the proof. (cid:3) 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]. (cid:3)
HEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKIND DOMAINS 8
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