Elementary subgroup of an isotropic reductive group is perfect

0.16 MB·English

by  Alexander Luzgarev

Elementary subgroup of an isotropic reductive group is
perfect
0
1
A. Luzgarev∗, A. Stavrova†
0
2
December 31, 2009
n
a
J

7

] 1 Introduction
G
A
Let R be a commutative ring with 1, and let G be an isotropic reductive algebraic group
. over R. In [7] Victor Petrov and the second author introduced a notion of an elementary
h
t subgroup E(R) of the group of points G(R). In this note we prove that, as one might
a
expectfromthe split case(e.g., [10])as wellas fromthe fieldcase (e.g.,[12]), under natural
m
assumptions the elementary subgroup of a reductive group is perfect.
[
More precisely, assume that G is isotropic in the following strong sense: it possesses a

1 parabolic subgroup that intersects properly any semisimple normal subgroup of G. Such
v a parabolic subgroup P is called strictly proper. Denote by E (R) the subgroup of G(R)
P
5
generated by the R-points of the unipotent radicals of P and of an opposite parabolic
0
subgroup P−. The main theorem of [7] states that E (R) does not depend on the choice
1 P
1 of P, as soon as for any maximal ideal M of R all irreducible components of the relative
1. root system of GRM (see [6, Exp. XXVI, §7] for the definition) are of rank ≥2. Under this
assumption,wecall E (R)the elementary subgroupofG(R)anddenote it simply byE(R).
0 P
0 In particular, E(R) is normal in G(R). This definition of E(R) generalizes the well-known
1 definition of an elementary subgroup of a Chevalley group (or, more generally, of a split
: reductive group), as well as several other definitions of an elementary subgroup of isotropic
v
i classical groups and simple groups over fields [2, 12, 13, 14, 3].
X
By the structure constants of a root system we mean the structure constants of the
r correspondingsemisimple complexLiegroup,or,inotherwords,constantsappearinginthe
a
Chevalley commutator formulas for the corresponding Chevalley group. They are among
±1, ±2, ±3.
Theorem 1. Let G be an isotropic reductive algebraic group over a commutative ring R.
Assume that for any maximal ideal M of R all irreducible components of the relative root
system of G are of rank ≥2, and, if one of the irreducible components of the (usual) root
RM
system of G is of type B =C or G , that theresidue field R /MR is not isomorphic
RM 2 2 2 M M
to F . Then E(R)=[E(R),E(R)].
2
Observethatthefirstconditionofthetheoremensuresthatthetheelementarysubgroup
E(R)ofG(R)iscorrectlydefined,whilethesecondoneessentiallyeliminatesthewell-known
cases where the elementary subgroup of a split reductive group is not perfect. Thus, the
result is the strongest possible. One should note that, if we assume only that the rank of
relative root systems of G is ≥1, the question whether individual subgroups E (R) are
RM P
perfect is of separate interest.
∗Theauthorissupported byRFBR09-01-00784, RFBR09-01-00878andRFBR09-01-90304.
†Theauthorissupported byRFBR09-01-00878andRFBR09-01-90304.
1
The proofofTheorem1 is basedonthe notionofrelativerootsubschemes(with respect
to a parabolic subgroup) of an isotropic group introduced in [7], the generalized Chevalley
commutator formula [7, Lemma 9], and localization in the Quillen—Suslin style [11]. To
shortenthe proof,wealsomakeuseoftheclassificationofTitsindicesofisotropicreductive
groups over local rings obtained in [8].
2 An abstract definition of relative roots
Inthis sectionwerecallthe notionofan(abstract)systemofrelativerootsintroducedin[7]
and prove a technical lemma.
Let Φ be a reduced root system in a Euclidean space with a scalar product (−,−). Let
Π = {α ,...,α} be a fixed system of simple roots of Φ; if Φ is irreducible, we assume
1 l
that the numbering follows Bourbaki [5]. Let D be the Dynkin diagram of Φ. We identify
nodes of D with the corresponding simple roots in Π. For a subgroup Γ ⊆ Aut(D) and a
Γ-invariant subset J ⊆Π, consider the projection
π =π : ZΦ−→ZΦ/hΠ\J; α−σ(α), α∈J, σ ∈Γi.
J,Γ
The set Φ = π(Φ)\{0} is called the system of relative roots corresponding to the pair
J,Γ
(J,Γ). The rank of Φ is the rank of π(ZΦ) as a free abelian group.
J,Γ
ItisclearthatanyrelativerootA∈Φ canberepresentedasauniquelinearcombina-
J,Γ
tion ofrelative roots fromπ(Π). We say that A∈Φ is a positive (resp. negative) relative
J,Γ
root,if it is a non-negative (respectively, a non-positive) linear combinationof the elements
of π(Π). The sets of positive and negative relative roots will be denoted by Φ+ and Φ−
J,Γ J,Γ
respectively. By the level lev(A) of a relative root A we mean the sum of coefficients in its
decomposition.
Observe that Γ acts on the set of irreducible components of the root system Φ. If this
action is transitive, the system of relative roots Φ is irreducible. Clearly, any system
J,Γ
of relative roots Φ is a disjoint union of irreducible ones; we call them the irreducible
J,Γ
components of Φ .
J,Γ
We will need the following lemma.
Lemma 1. Let Φ be a root system with a fixed set of simple roots Π, Γ be a subgroup of
Aut(D), andJ beaΓ-invariant subsetofΠ. Ifarelativeroot A∈Φ lies in an irreducible
J,Γ
component of rank ≥ 2, then there exist such non-collinear B,C ∈ Φ that A = B +C
J,Γ
and all relative roots iB+jC ∈ Φ , i,j > 0, (i,j) 6= (1,1), have the same sign as A and
J,Γ
satisfy |lev(iB+jC)|>|lev(A)|.
Proof. WecanassumethattherootsystemΦisirreducible,andthatAisapositiverelative
root, that is, π−1(A)⊆Φ+.
Assume first that A = kπ(α ), where α ∈ Π is a simple root and k > 0 is a positive
r r
integer. Let α ∈ J be a simple root such that the Γ-orbits of α and α are distinct,
s s r
and α is at the least possible distance from α on the Dynkin diagram. It is easy to see
s r
that for any α ∈ π−1(A) there exists β ∈ π−1(α ) such that (α,β) < 0, and, consequently,
s
α+β ∈Φ. Indeed, we have m (α)=0 by definition, thus we can take for β the sum of all
s
simplerootsintheDynkindiagramchainbetweenα andthenearestsimplerootappearing
s
in the decomposition of α. Now set B = π(α+β) and C = π(−β). It is clear that any
root in π−1(iB +jC), i,j > 0, contains the summand iα in its decomposition, and thus
r
is a positive root. Then iB +jC is a positive relative root for any i,j > 0. Moreover,
one sees that lev(iB +jC) = lev(A) if and only if i = j = 1. Since π(α) = kπ(α ), and
r
π(−β)=−π(α ), the roots B and C are non-collinear.
s
Consider the case where A6=kπ(α ) for any α ∈J. For any α∈π−1(A) there exists a
r r
sequenceofsimplerootsβ ,...,β ∈Πsuchthatα=β +...+β andβ +...+β ∈Φfor
1 n 1 n 1 i
2
any 1 ≤ i ≤ n. Let i be the least possible index such that β ,...,β ∈ ∆. Then β ∈ J
i+1 n i
and π(β +...+β +β ) = A. Set B = π(β +...+β ) and C = π(β ). Since B and
1 i−1 i 1 i−1 i
C are positive relative roots, we have lev(iB+jC) > lev(A) for any i,j > 0 distinct from
i=j =1. The relative roots B and C are non-collinear since otherwise we would have had
A=kπ(β ) for some k >0.
i
3 Isotropic reductive groups and relative root subschemes
Inthis sectionwerecallsomebasic notionspertainingto reductivegroupsoverrings;see[6,
7, 9] for more detailed exposition.
Let R be a commutative ring with 1, and let G be a reductive group scheme, or reduc-
tive group for short, over R (see [6] for the definition). We denote by Gad and Gsc the
corresponding adjoint and simply connected semisimple groups, respectively.
Any reductive algebraic group G over R is split locally in the fpqc topology on SpecR.
If G is of constant type over R (that is, the root system of G is the same at any point
of SpecR), then G is a twisted form of a split reductive algebraic group G over R, given
0
by a cocycle ξ ∈ H1 (R,Aut(G )). Recall that the connected component of Aut(G )
fpqc 0 0
is precisely Gad. The group G is of inner type, if ξ is in the image of the natural map
0
H1 (R,Gad)→H1 (R,Aut(G )). One can always find a finite Galois extension S of R
fpqc 0 fpqc 0
such that G is of inner type. The Galois group Gal(S/R) acts on the Dynkin digram of
S
eachG , where k(s) is the algebraic closure of the residue field of a point s∈SpecR, via
k(s)
a ∗-action (see [8, 9]).
Recall that G is called isotropic, if it contains a proper parabolic subgroup P over R.
Recall that we call a parabolic subgroup P of G strictly proper, if it does not contain any
semisimple normal subgroup of G. We set
EP(R)=hUP(R),UP−(R)i,
whereP− isanyparabolicsubgroupofGoppositetoP,andUP andUP− aretheunipotent
radicals of P and P− respectively. The main theorem of [7] states that E (R) does not
P
depend on the choice ofa strictly proper parabolicsubgroup P, as soonas for any maximal
ideal M in R all irreducible components of the relative root system of G are of rank
RM
≥ 2. Under this assumption, we call E (R) the elementary subgroup of G(R) and denote
P
it simply by E(R).
Let P =P+ be a parabolic subgroup of G, and P− be an opposite parabolic subgroup.
LetL=P+∩P− betheircommonLevisubgroup. Itwasshownin[7]thatwecanrepresent
Spec(R) as a finite disjoint union
m
Spec(R)= Spec(R ),
a i
i=1
so that the following conditions hold for i=1,...,m:
• for any s∈SpecR the root system of G is the same;
i k(s)
• for any s∈SpecR the type of the parabolic subgroup P of G is the same;
i k(s) k(s)
• if S /R is a Galois extension of rings such that G is of inner type, then for any
i i Si
s ∈ SpecR the Galois group Gal(S /R ) acts on the Dynkin diagram D of G via the
i i i i k(s)
same subgroup of Aut(D ).
i
From here until the end of this section, assume that R = R for some i (or just extend
i
the base). Denote by Φ the root system of G, by Π a set of simple roots of Φ, by D the
corresponding Dynkin diagram. Then the ∗-action on D is determined by a subgroup Γ of
Aut D. Let J be the subset of Π such that Π\J is the type of P (that is, the set of
k(s)
3
simple rootsof the Levisugroup L ). ThenJ is Γ-invariant. The systemofrelativeroots
k(s)
Φ is called the system of relative roots corresponding to P and denoted also by Φ . If R
J,Γ P
is a local ring and P is a minimal parabolic subgroup of G, then Φ can be identified with
the relative root system of G in the sense of [6, Exp. XXVI §7] (see also [4] for the field
case), as was shown in [7, 9].
To any relative root A∈Φ one associates a finitely generatedprojective R-module V
P A
and a closed embedding
X :W(V )→G,
A A
where W(V ) is the affine group scheme over R defined by V , which is called a relative
A A
root subscheme of G. These subschemes possess several nice properties similar to that of
elementaryrootsubgroupsofa split group,see [7,Th. 2]. In particular,they aresubjectto
certain commutator relations which generalize the Chevalley commutator formula.
Moreprecisely,assumethatA,B ∈Φ satisfymA6=−kB foranym,k ≥1. Thenthere
P
exists a polynomial map
N : V ×V →V ,
ABij A B iA+jB
homogeneous of degree i in the first variable and of degree j in the second variable, such
that for any R-algebra R′ and for any for any u∈V ⊗ R′, v ∈V ⊗ R′ one has
A R B R
[X (u),X (v)]= X (N (u,v)) (1)
A B Y iA+jB ABij
i,j>0
(see [7, Lemma 9]).
In a strict analogy with the split case, for any R-algebra R′ we have
E(R′)=hX (V ⊗ R′), A∈Φ i
A A R P
(see [7, Lemma 6]).
We will aslo use the following statement which is a slight extension of [7, Lemma 10].
Lemma 2. Consider A,B ∈Φ satisfying A+B ∈Φ and mA6=−kB for any m,k ≥1.
P P
Denote by Φ an irreducible component of Φ such that A,B ∈π(Φ ).
0 0
(1) In each of the following cases
(a) structure constants of Φ are invertible in R (for example, if Φ is simply laced);
0 0
(b) A6=B and A−B 6∈Φ ;
P
(c) Φ is of type B , C , or F , and π−1(A+B) consists of short roots;
0 l l 4
(d) Φ is of type B , C , or F , and there exist long roots α ∈ π−1(A), β ∈ π−1(B)
0 l l 4
such that α+β is a root;
the map N :V ×V →V is surjective.
AB11 A B A+B
(2) If A−B ∈Φ and Φ 6=G , then
P 0 2
im N +im N + im(N (−,v))=V ,
AB11 A−B,2B,1,1 X A−B,B,1,2 A+B
v∈VB
where im N =0 if 2B 6∈Φ .
A−B,2B,1,1 P
Proof. (1) By [7, Lemma 4] any γ ∈ π−1(A+B) decomposes as γ = α+β, α ∈ π−1(A),
β ∈ π−1(B). Let S be any member of an affine fpqc-covering SpecS → SpecR that
τ τ
`
splits G. Set
Ψ={iA+jB | i,j >0, (i,j)6=(1,1), iA+jB ∈Φ }.
P
Then in the notation of [7, Th. 2], over S the commutator [X (e ),X (e )], computed
τ A α B β
modulo the subgroup hX (V ), C ∈Ψi, is of the form x (±c) = X (±ce ), where c =
C C γ A+B γ
±1,±2,±3is the corresponding structure constant. If (a) holds, then c is invertible. If (b),
(c) or (d) holds, then c necessarily equals ±1. Indeed, in the only dubious case (d) one
4
should note that, due to the transitive action of the Weyl group of the Levi subgroup on
the rootsofthe sameshape (see [1]), any longrootγ ∈π−1(A+B) decomposesasa sumof
longroots. Hencec is alwaysinvertible. Thisimplies thatim(N ) =V ⊗S . Since
AB11 τ A+B τ
im N is a submodule of V defined over the base ring, we have im N =V .
AB11 A+B AB11 A+B
(2) See [7, Lemma 10].
Lemma 3. Suppose that Φ is an irreducible component of Φ such that Φ ∼= C , l >2, P
0 0 l
is a parabolic subgroup of type Π\J, where J ={α ,α }, 2i=l (α is short, α is long), so
i l i l
that Φ ∼=C . Denote π(α )=A and π(α )=A . Then
0,P 2 i 1 l 2
im(0,N )+ im f =V ⊕V ,
A1,A1+A2,1,1 X v A1+A2 2A1+A2
v∈VA
where f =(N (v,−),N (v,−)): V →V ⊕V .
v A1,A2,1,1 A1,A2,2,1 A2 A1+A2 2A1+A2
Proof. Let S be any member of an affine fpqc-covering SpecS →SpecR that splits G.
τ τ
Suppose that γ ∈ π−1(2A +A ) is a short root. We c`an find short roots α ∈ π−1(A ),
1 2 1
β ∈π−1(A +A ) such that γ =α+β. Therefore,
1 2
[X (e ),X (e )]=x (±1)=X (±e ).
A1 α A1+A2 β γ 2A1+A2 γ
Hence, e ∈ im(N ) . Now let γ ∈ π−1(2A +A ) be a long root. Take α ∈
γ A1,A1+A2,1,1 τ 1 2
π−1(A ), β ∈ π−1(A ) such that β 6= α and γ = 2α+β (note that α is short, β is long).
1 2 l
Therefore
[X (e ),X (e )]=x (±1)x (±1)=X (±e )X (±e ).
A1 α A2 β α+β 2α+β A1+A2 α+β 2A1+A2 γ
Finally, any γ ∈ π−1(A + A ) is a short root, so there exist short roots α ∈ π−1(A ),
1 2 1
β ∈ π−1(A ) such that α+β = γ, hence [X (e ),X (e )] = X (±e ). Combining
2 A1 α A2 β A1+A2 γ
these results and noting that the modules in question are defined overthe base ring, we are
done.
4 Proof of Theorem 1
Let R be a commutative ring with 1, G be a reductive groupoverR, P be a strictly proper
parabolic subgroup of G. For any ideal I ⊆R we write
EP(I)=hUP(I),UP−(I)i≤G(R).
We denote by R[Y,Z] a ring of polynomials in two variables Z and Y over R.
Lemma 4. Let G be a reductive group scheme over a commutative ring R, and let P and
P′ be two strictly proper parabolic subgroups of G such that P ≤ P′ or P′ ≤ P. Then for
any integer m>0 there exists an integer k >0 such that
EP(ZkR[Z])≤EP′(ZmR[Z]).
Proof. Without loss of generality, we can assume that over R we have two sets of relative
root subschemes XA(VA), A ∈ ΦP, and XB(VB), B ∈ ΦP′, corresponding to P and P′
respectively.
Then,ifP ≤P′,by[7,Lemma12]thereexistsanintegerk >0suchthatforanyA∈Φ
P
andanyv ∈VA one canfind relativerootsBi ∈ΦP′, elements vi ∈VBi, andintegersni >0
(1≤i≤m), such that
m
X (Ykv)= X (Yniv ),
A Y Bi i
i=1
5
and hence XA(Ykv)∈EP′(YR[Y]). Substituting R by R[Z] and Y by Zm, we obtain
EP(ZkR[Z])≤EP′(ZmR[Z]).
If, conversely, P′ ≤ P, we have UP ≤ UP′. Let Ψ± ⊆ Φ+ be two closed sets of roots
corresponding to UP±. Then we have π(Ψ±) ⊆ Φ±P′, where π : Φ → ΦP′ is the canonical
projection. By [7, Lemma 6] the map
XΨ: W(cid:16) M VA(cid:17)→UP±, (vA)A 7→YXA(vA),
A∈π(Ψ±) A
where the product is takeninany fixedorderrespecting the levelofrelativeroots inΦP′, is
an isomorphism of schemes over R. Therefore, UP±(ZmR[Z])≤UP′±(ZmR[Z]).
Lemma 5. In the setting of Theorem 1, assume moreover that R is a local ring. Then for
any integer m>0 there exists an integer k >0 such that for any R-algebra R′ one has
E (ZkR′[Z])⊆[E (ZmR′[Z]),E (ZmR′[Z])].
P P P
Proof. Letder(G)bethealgebraicderivedsubgroupofthereductivegroupschemeG(see[6,
Exp. XXII, 6.2]). Then, clearly, E (R) ⊆ der(G)(R). Since der(G) is a semisimple group,
P
we can assume that G is semisimple. Moreover, since the canonical projection Gsc → G,
where Gsc is the simply connected semisimple group corresponding to G, is surjective on
U±(R),wecanassumethatGissimplyconnected. Anysimplyconnectedsemisimplegroup
is a direct product of simply connected semisimple groups that cannot be decomposed into
a productofsmaller semisimple groups. These groupsG , i=1,...,n, areWeil restrictions
i
ofcertainsimplereductivegroupsG′ overafinite´etaleextensionS ofR: G =R (G′)[6,
i i S/R i
Exp. XIV Prop. 5.10]. Note that the group of R-points G (R) is canonically isomorphic to
i
thegroupofS-pointsG′(S). ThisisomorphismalsorespectstheembeddingP (R)→G (R),
i i i
for any parabolic subgroup P of G . Then, clearly, we can assume from the very beginning
i i
that G is a simple reductive group, and the root system Φ of G is irreducible.
Note that by Lemma 4 we can substitute P by any other strictly proper parabolic
subgroup P′ of G such that P ≤ P′ or P′ ≤ P. Further, over R we have a set of relative
root subschemes X (V ), A∈Φ , corresponding to P.
A A P
Wearegoingtoshowbyinductionon|levA|thatforanyA∈Φ thereexistsaninteger
P
k =k(A)>0 such that for any R-algebra R′ and any v ∈V ⊗ R′ one has
A R
X (Zkv)∈[E (ZR′[Z]),E (ZR′[Z])]. (2)
A P P
The claim of the lemma then follows by substituting Z by Zm and R′ by R[Z], and taking
the final k to be the maximum of all k(A), A∈Φ .
P
Recall that by Lemma 1 there exist non-collinear relative roots B,C ∈ Φ such that
P
A = B +C and and all roots iB+jC ∈ Φ , i,j > 0, (i,j) 6= (1,1), have the same sign
J,Γ
as A and satisfy |lev(iB+jC)| > |lev(A)|. Assume that the map N : V ×V →V
BC11 B C A
is surjective. Then by the generalized Chevalley commutator formula (1) we have that for
any R-algebra R′, and any v ∈V ⊗ R′[Z],
A R
X (Zkv)=[X (Zu ),X (Zk−1u )]· X (Zi+j(k−1)u )
A B 10 C 01 Y iB+jC ij
i,j>0;
(i,j)6=(1,1)
for some u ∈V ⊗ R′[Z], i,j >0, (i,j)6=(1,1). Then, by the inductive hypothesis,
ij iB+jC R
(2) holds for k large enough.
6
Now for anyrelativerootA∈Φ (for a suitable choice ofthe parabolicsubgroupP)we
P
either show that for any decomposition A = B+C, where B and C are non-collinear, the
map N is surjective; or provide an explicit decomposition of X (Zkv), v ∈ V ⊗ R′,
BC11 A A R
into a product of commutators in E (ZR′[Z]), so that (2) is satisfied for k large enough.
P
Assume first that all structure constants of the root system Φ of G are invertible in
R; this includes the case where Φ is simply laced. Then by Lemma 2 the map N is
BC11
surjective for any decomposition A=B+C, where B and C are non-collinear.
Consider the case where Φ = Φ = C , so G is split. Let Π = {A ,A }, Φ+ =
P 2 1 2
{A ,A ,A +A ,2A +A }. Let M be the maximal ideal of R. By the hypothesis of
1 2 1 2 1 2
Theorem1, R/M 6∼=F , hence we cantakeε∈R\M suchthat ε2−ε∈R\M =R∗. If the
2
root A∈Φ is long, we can assume that A=2A +A . Let
P 1 2
g (s,t)=[X (s),X (t)]=X (st)X (s2t)
1 A1 A2 A1+A2 2A1+A2
and
g (s,t,u)=[X (u),[X (s),X (t)]]=X (−stu)X (−s2t2u).
2 A2 A1+A2 −A2 A1+A2 2A1+A2
Therefore,
g (Z2,−Zk−4ε(ε2−ε)−1v)·g (Z,Zε,−Zk−4(ε2−ε)−1v)=X (Zkv).
1 2 2A1+A2
If the root A∈Φ is short, we can assume that A=A +A , hence
P 1 2
g (Z,Zk−1v)·X (−Zk+1v)=X (Zkv).
1 2A1+A2 A1+A2
This means that (2) holds for these roots for any k ≥5.
Consider the case where Φ = Φ = G , so G is split. Let Π = {A ,A }, Φ+ =
P 2 1 2
{A ,A ,A +A ,2A +A ,3A +A ,3A +2A }. BythehypothesisofTheorem1, R/M 6∼=
1 2 1 2 1 2 1 2 1 2
F , hence we can take ε∈R\M such that ε2−ε∈R\M =R∗. If the root A is long, we
2
can assume that A=3A +2A . Then
1 2
[X (Zv),X (Zk−1)]=X (Zkv).
A2 3A1+A2 3A1+2A2
Therefore,(2) holds for long roots for any k ≥2. If the rootA is short,we can assume that
A=2A +A . Then
1 2
[X (s),X (t)]=X (st)·X (s2t)·X (s3t)·X (s3t2).
A1 A2 A1+A2 2A1+A2 3A1+A2 3A1+2A2
Hence,
[X (Zε),X (−(ε2−ε)−1Zk−2v)]−1·[X (Z),X (−ε(ε2−ε)−1Zk−2v)]=
A1 A2 A1 A2
X (Zkv)X ((ε+1)Zk+1v)X (ε(ε2−ε)−1Z2k+1v), (3)
2A1+A2 3A1+A2 3A1+2A2
and the roots 3A +A and 3A +2A are long. This means that (2) holds for short roots
1 2 1 2
for any k≥3.
We areleft with the casewhere Φ is of type B ,C , or F . Recallthat by the hypothesis
l l 4
ofTheorem1 Gcontainsa splittorus ofrank≥2. Hence inthe F casethe classificationof
4
Tits indices over localrings [8] says that G is a split group. Hence we can assume that P is
a BorelsubgroupofG, andΦ =Φ is arootsystemoftype F . Then if the rootA∈Φ is
P 4 P
short, by Lemma 2 the map N is surjective for any non-collinear B,C ∈Φ such that
BC11 P
A=B+C. If this rootis long, then it belongs to the long rootsubsystem of F , whichhas
4
typeD . Then(forexample,byLemma1)wecanfindtwolongrootsB,C ∈Φ , suchthat
4 P
B+C = A, and necessarily, iB+jC is not a root for any i,j > 0 distict from i = j = 1.
Then
X (Zkv)=[X (Zu ),X (Zk−1u )]
A B 10 C 01
7
for some u ∈V ⊗ R′[Z] and u ∈V ⊗ R′[Z].
10 B R 01 C R
Consider the case where Φ is of type B , l≥3. By the classification of Tits indices over
l
local rings [8], we can assume that P is a parabolic subgroup of type Π\J, where Π is a
set of simple roots of Φ and J = {α ,α }. Then Φ can be identified with a root system
1 2 P
of type B . One readily sees, using the fact that l ≥ 3, that for any relative root A ∈ Φ
2 P
andany pair B,C ∈Φ satisfyingA=B+C, we canfind a pair oflong roots β ∈π−1(B),
P
γ ∈π−1(C) such that β+γ is a root (one can assume that A is one of two simple roots of
Φ , due to the lifting of the relative Weyl group [6, Exp. XXVI Th. 7.13 (ii)]). Then by
P
Lemma 2 the map N is surjective.
BC11
It remains to consider the case where Φ is of type C , l ≥ 3. First assume that P is a
l
parabolic subgroup of type Π\J, where J = {α ,α } for two short simple roots α ,α of
i j i j
Φ. Then Φ can be identified with a root system of type BC . One readily sees that for
P 2
all extra-short and short relative roots A ∈ Φ the set π−1(A) consists of short roots, and
P
hence by Lemma 2 the map N is surjective for any decomposition A = B+C. Let A
BC11
be a long root. Let A and A be a short and an extra-short simple roots of Φ . We can
1 2 P
assume without loss of generality that A = 2A +2A . Take k ≥ 4. Then by Lemma 2
1 2
(2) and by the generalized Chevalley commutator formula, for any R-algebra R′, and any
v ∈V ⊗ R′[Z], we have
A R
X (Zkv)=[X (Zu ),X (Zk−2u )]·X (Zk−1u )
A A1 1 2A2 2 A1+2A2 3
for some u ∈V ⊗ R′[Z], u ∈V ⊗ R′[Z], and u ∈V ⊗ R′[Z]. Further, by
1 A1 R 2 2A2 R 3 A1+2A2 R
Lemma 2 (1) and by the generalized Chevalley commutator formula, since π−1(A +2A )
1 2
consists of short roots, we have
X (Zk−1u )=[X (Zu ),X (Zk−3u )]
A1+2A2 3 A1+A2 4 A2 5
forsomeu ∈V ⊗ R′[Z]andu ∈V ⊗ R′[Z]. Hence(2)holdsforAforanyk ≥4.
4 A1+A2 R 5 A2 R
By the classification of Tits indices over local rings [8], the only remaining case is when
P is a parabolic subgroup of type Π\J for J ={α ,α}, where l =2i. Now α is short, α
i l i l
is long, and Φ can be identified with a root system of type B = C . As in Lemma 3, we
P 2 2
put A =π(α ), A =π(α ). Then if the root A∈Φ is short, by the lifting of the relative
1 i 2 l P
Weyl group [6, Exp. XXVI Th. 7.13 (ii)]), we can assume that A=A +A . By Lemma 3
1 2
for any R-algebra R′, and any v ∈V ⊗ R′[Z], we have
A R
X (Zkv)= [X (Zv ),X (Zk−1u )]
A Y A1 i A2 1
i
for some u ∈V ⊗ R′[Z], v ∈V ⊗ R′[Z]. If the root A∈Φ is long, we can assume
1 A2 R i A1 R P
that C =2A +A . By Lemma 3 for any R-algebraR′, and any v ∈V ⊗ R′[Z], we
1 2 2A1+A2 R
have
X (Zkv)=[X (Zu ),X (Zk−1u )]· [X (Zv ),X (Zk−2u )]
A A1 1 A1+A2 2 Y A1 i A2 3
i
for some u ∈V ⊗ R′[Z], u ∈V ⊗ R′[Z], u ∈V ⊗ R′[Z], v ∈V ⊗ R′[Z].
1 A1 R 2 A1+A2 R 3 A2 R i A1 R
Hence (2) holds for A for any k ≥3.
Proof of Theorem 1. RecallthatwecanrepresentSpec(R)asafinitedisjointunionSpec(R)=
n n
Spec(R ), so that E(R)= E(R ) and for any 1≤i≤n we have
i i
` Q
i=1 i=1
E(R )= X (V ), A∈Φ ,
i (cid:10) A A PRi(cid:11)
for a set of root subschemes X , A∈Φ , over R . Hence we can assume that
A PRi i
E(R)=hX (V ), A∈Φ i
A A P
8
from the very beginning.
We show that [E(R),E(R)] contains any X (v), A ∈ Φ , v ∈ V , by induction on
A P A
|levA|. Take
I ={s∈R | X (tsv)∈[E(R),E(R)] ∀ t∈R}.
A
By [7, Th. 2] for any u,u′ ∈V we have
A
X (u+u′)=X (u)X (u′) X (u )
A A A Y iA i
i>0
for some u ∈ V . Hence by the inductive hypothesis I is an ideal of R. If I 6= R, let M
i iA
be a maximal ideal of R containing I. Let F denote both the localizationhomomorphism
M
R →R and the induced homomorphism G(R[Y,Z])→G(R [Y,Z]). By Lemma 5 there
M M
is an m > 0 such that we can represent the element F (X (ZmYv)) of G(R [Y,Z]) as a
M A M
product
n
F (X (ZmYv))= [h (Y,Z),g (Y,Z)],
M A Y i i
i=1
for some h (Y,Z),g (Y,Z)∈E(ZR [Y,Z]), 1≤i≤n. By [7, Lemma 15] there exist
i i M
h′(Y,Z), g′(Y,Z)∈E (R[Y,Z],ZR[Y,Z])≤E (R[Y,Z])=E(R[Y,Z])
i i P P
and s ∈ R\M such that F (h′(Y,Z)) = h (Y,sZ) and F (g′(Y,Z)) = g (Y,sZ) for all i.
M i i M i i
Hence
n
F X ((sZ)mYv) =F [h′(Y,Z),g′(Y,Z)] .
M(cid:0) A (cid:1) M(cid:16)Y i i (cid:17)
i=1
Then by [7, Lemma 14] there exists t∈R\M such that
n
X ((tsZ)mYv)= [h′(Y,tZ),g′(Y,tZ)].
A Y i i
i=1
Substituting Z by 1 and Y by an arbitrary element of R, we see that (ts)m ∈ I. But
(ts)m ∈R\M, a contradiction.
TheauthorsaresincerelygratefultoVictorPetrovandNikolaiVavilovfortheirinspiring
comments on the subject of this paper.
References
[1] H.Azad,M.Barry,G.Seitz,Onthestructureof parabolic subgroups,Comm.inAlgebra
18 (1990), 551–562.
[2] H. Bass, K-theory and stable algebra, Publ. Math. I.H.E´.S. 22 (1964), 5–60.
[3] A. Bak, N. Vavilov, Structure of hyperbolic unitary groups I. Elementary subgroups,
Algebra Colloquium 7 (2000), 159–196.
[4] A. Borel, J. Tits, Groupes r´eductifs, Publ. Math. I.H.E´.S. 27 (1965), 55–151.
[5] N. Bourbaki, Groupes et alg`ebres de Lie. Chapitres 4–6, Hermann, Paris, 1968.
[6] M. Demazure, A. Grothendieck, Sch´emas en groupes, Lecture Notes in Mathematics,
Vol. 151–153,Springer-Verlag,Berlin-Heidelberg-New York, 1970.
9
[7] V. Petrov, A. Stavrova,Elementary subgroups of isotropic reductive groups, St. Peters-
burg Math. J. 20 (2009), 625–644.
[8] V. Petrov, A. Stavrova, Tits indices over semilocal rings, Preprint POMI, N 2 (2009),
1–22.
[9] A. Stavrova,Stroenije isotropnyhreduktivnyhgrupp,PhD thesis, St. Petersburg State
University, 2009.
[10] M.R. Stein, Generators,relations,andcoveringsofChevalleygroupsovercommutative
rings, Amer. J. Math. 93 (1971), 965–1004.
[11] A.A. Suslin, On the structure of the special linear group over polynomial rings, Math.
USSR Izv. 11 (1977), 221–238.
[12] J. Tits, Algebraic and abstract simple groups, Ann. of Math. 80 (1964), 313–329.
[13] L.N.Vaserstein,Normal subgroups of orthogonal groups over commutative rings,Amer.
J. Math. 110 (1988), 955–973.
[14] L.N.Vaserstein,Normal subgroups of symplectic groups over rings,K-Theory2(1989),
647–673.
10