Saturation rank for finite group schemes: Finite groups and Infinitesimal group schemes

0.27 MB·

by  Yang Pan

SATURATION RANK FOR FINITE GROUP SCHEMES: FINITE
GROUPS AND INFINITESIMAL GROUP SCHEMES
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1
YANGPAN
0
2

n Abstract. We investigate the saturation rank of a finite group scheme, de-
a fined over an algebraically closed field k of positive characteristic p. We be-
J
gin byexploringthe saturation rankfor finitegroups andinfinitesimal group

1 schemes. Special attention is given to reductive Lie algebras and the second
1 FrobeniuskernelofthealgebraicgroupSLn.

]
T
R
h. 1. Introduction
t
a
m ThispaperisconcernedwiththesaturationrankoffinitegroupschemesG thatare
defined over an algebraically closed field k of characteristic p>0. The Paper [FS]
[
writtenbyFriedlanderandSuslinenablesustoconsiderthecohomologicalsupport

1 variety V , defined as the maximal ideal spectrum of the even cohomological ring.
G
v
The classical contexts of V concern a recent study on the representation theory
1 G
of finite groups G and finite dimensional restricted Lie algebras g. In the field of
5
9 finite groups, by virtue of Quillen’s work, it was shown that the dimension of VG
2 is the maximal rank of an elementary abelian p-subgroup. When it turns to g,
0 by setting g = Spec(U (g)∗) where U (g) is the restricted enveloping algebra, we
0 0
.
1 areinformed that Vg and the restrictednullcone V(g) are naturally homeomorphic
0 varieties[SFB2, 1.6,5.11]. Inspiredby the approachofelementaryabelianp-groups
7
to a finite group and the generalized definition of elementary abelian subgroup
1
schemes given in [Far4], we now consider
:
v
i rp(G):=max{cxE(k); E is an elementary abelian subgroup ofG}
X
wherecx (k)isthecomplexityofE. ByconceringtheirreduciblecomponentsofV
r E G
a inconjunctionwiththeirreducibilityofV ,wederivetheinequalityr (G)≤dimV .
E p G
We probablyhave foundthat, the behavioroffinite groupsandinfinitesimal group
schemes are rather different when applying them to the aforementioned formula.
Incidentally it is quite clear if we look at the case of g = sl (k), indicating that it
2
may be of no hope to investigate the variety V by looking at pieces coming from
G
elementary abelian subgroups when G is an infinitesimal group scheme.
We are now drawn the attention to the saturation rank defined for finite group
schemes. In [Far3], the saturation rank srk(G) is exploited for conditioning the
indecomposibility of Carlson modules. Though it is defined using the theory of
p-points, expounded by Friedlander and Pevtsova in their recent paper [FP], the
homeomorphismbetweenthe space P(G)ofp-points andthe projectivizationof V
G
2010 Mathematics Subject Classification. 17B45.
1
gives us an interchangeable interpretation. From its definition, it is readily seen
that
srk(G)≤r (G)≤dimV .
p G
The question now was, how the number srk(G) reveals the properties of V or G.
G
For instance, when srk(G) = dimV we have V is equi-dimensional and there are
G G
only finitely many elementary subgroups of G with complexity equaling srk(G).
The problem was turned into an investigation of the space P(G). We have already
known that, the consideration of P(G) for a finite group scheme generalizes the
earlier version of the rank variety for a finite group and the variety of 1-parameter
subgroups for an infinitesimal group scheme. It is worth detecting the rank srk(G)
via these two precursorsas our preliminary exploration,even though it is typically
difficult to capture those infinitesimal group schemes of higher height.
The purpose of this paper is to study the rank srk(G) for finite groups, finite
dimensional restricted Lie algebras and a specific example SL arising from the
n(2)
secondFrobeniuskernelofthealgebraicgroupSL . Themethodsweuserangefrom
n
Quillen Stratification for finite groups, nilpotent orbits for reductive Lie algebras
and nilpotent commuting variety for SL . We achieve this by translating from
n(2)
the homeomorphism
//
Ψ :P(G) ProjV .
G G
WhenGisafinitegroup(orequivalentlyaconstantfinitegroupscheme),theproofof
Theorem2.2,following[Qui1][Qui2]showssrk(G)istheminimalrankofamaximal
elementaryabelianp-subgroup. RecallthatthedimensionofV isthemaximalrank
G
r (G) ofanelementaryabelianp-subgroup. Itis straightforwardlyseenthatsrk(G)
p
equals the dimension of V when V is of equi-dimension, and vice versa. When G
G G
isaninfinitesimalgroupschemeofheight≤r,invirtueoftheupper-semicontinuity
ofa mapdefined onthe supportvariety V (G), we find srk(G) is determined by the
r
local data. When it applies to reductive Lie algebras (infinitesimal group schemes
of height ≤ 1), the regular nilpotent element will be involved in. We prove in
Theorem 3.1, under certain mild restriction on the reductive algebraic group G,
that srk(g) coincides with the semisimple rank rk (G) of G. With being curious,
ss
we also consider the higher height case, i.e. the second Frobinus kernel SL . As
n(2)
being shown in Theorem 4.4, the saturation rank srk(G ), the height of G and
(r) (r)
the semisimple rank rk (G) might constitute an equality, giving a generalisation
ss
to all r-th Frobenius kernel G of the result of Theorem 3.1.
(r)
This paper is organised as follows. In Section 2, we introduce the definition of
saturation rank for all finite group schemes, specializing to finite groups and infin-
itesimal group schemes. The characterisation of the saturation rank for reductive
Lie algebras, and the geomertric realization using nilpotent orbits for an open set
is found in Section 3. Section 4 deals with the saturation rank for the second
Frobenius kernel of SL , endeavoring to approacha general result.
n
Acknowledgement. The results of this paper are part of the author’s doctoral
thesis, which he was writing at the University of Kiel. He would like to thank his
advisor, Rolf Farnsteiner, for his continuous support. Furthermore, he thanks the
members of his working group for proofreading the paper.
2
2. Precursors for finite group schemes
2.1. Notations. In this section, we are to introduce and investigate the satura-
tion rank for all finite group schemes, with an emphasis on constant finite group
schemesandinfinitesimalgroupschemes. Intermsofconstantfinitegroupschemes,
we show that the saturation rank is determined by their maximal elementary
abelian subgroup schemes; see Theorem 2.2. In the context of infinitesimal group
schemes, we reveal that the saturation rank is controlled by the local data; see
Theorem 2.10. The techniques we use range from Quillen stratification for finite
groups([Qui1],[Qui2])to supportvarietiesforinfinitesimalgroupschemes([SFB1],
[SFB2]). Before we start, we first recall the definition of the saturation rank pro-
posed by Farnsteiner ([Far3, Sect. 6.4]).
Let G be a finite group scheme. For a subgroup H ⊆ G, the canonical inclu-
(cid:31)(cid:127) //
sion map ι :H G induces a continuous yet not necessarily injective map
H
//
ι∗,kH :P(H) P(G). The definition of p-points ensures that
P(G)= ι∗,kU(P(U)).
U⊆G
[
unipotentabelian
Motivated by this, we consider the set Max (G) of maximal abelian unipotent
au
subgroups of G as well as the subsets
Max (G) :={U ∈Max (G); cx (k)≥ℓ}
au ℓ au U
for every ℓ≥1. Setting P(G)ℓ := U∈Maxau(G)ℓι∗,kU(P(U)), the number
srk(G):=mSax{ℓ≥1; P(G)=P(G)ℓ}
is referred to as the saturation rank of G.
Remark2.1. In[Far4,Section6.2.1],theauthorhasprovedforanyabelianunipo-
tentgroupschemeU ⊆G,thereexistsauniqueelementaryabeliansubgroupscheme
//
EU ⊆ U such that ι∗,kEU :P(EU) P(U) is a homeomorphism. We then con-
siderthesetMax (G)ofmaximalelementaryabeliansubgroupsofG togetherwith
ea
the subsets
Max (G) :={E ∈Max (G); cx (k)≥ℓ}
ea ℓ ea E
for every ℓ ≥ 1. Replacing Max (G) with Max (G) in P(G) and redefining
au ℓ ea ℓ ℓ
srk(G), wefindthat,there isnodifference onthe numbersrk(G)betweenthesetwo
settings assured by the homeomorphism ι∗,kEU.
2.2. Constantfinitegroupschemes.LetGbe afinite group. Thenitdefines a
constantfunctorG whichassigntoeachfinitelygeneratedconnectedcommutative
G
k-algebra the group G itself. This functor is represented by k×|G|, indexed by the
elements of G, with its k-linear dual kG. We call this finite group scheme G
G
retrieved from G a constant finite group scheme.
Let G be a constantfinite groupscheme with G=G (k). We denote by H·(G,k)
G G
the cohomology ring H∗(G,k) of G if char(k) = 2 and the subring Hev(G,k) of
elements of even degree if char(k) > 2. Evens and Venkov have proved indepen-
dently that H·(G,k) is a finitely generated commutative k-algebra. We denote by
3
V = max(H·(G,k)) the maximal ideal spectrum, an affine variety corresponding
G
to H·(G,k). Let E ≤G be an elementary abelian p-subgroup of G. Then there is
a restriction map
res :H·(G,k) // H·(E,k)
G,E
which gives rise to a map of affine varieties res∗ :V // V . Quillen has
G,E E G
investigated the variety V and shown that it is stratified by pieces coming from
G
elementary abelian subgroups of G, which is known as Quillen Stratification; see
[Qui1] and [Qui2] for details. A weak version of his result is the following
V = res∗ V ,
G G,E E
E≤G
[
elemab
where res∗ V =V(ker(res )) is an irreducible closed subvariety of V .
G,E E G,E G
Notation 2.2. Keep the notation for H·(G,k) and V . We denote by H·(G,k)†
G
the augmentation ideal of H·(G,k). Let I be an ideal of H·(G,k), then gr(I) is
defined to be the unique maximal homogeneous ideal inside of I. We denote by
ProjV the set of homogeneousideals of H·(G,k) which are maximal among those
G
homogeneous ideals other than the augmentation ideal H·(G,k)†. Then ProjV
G
can be identified with the set of gr(m) for m ∈ V \ H·(G,k)† . Let E be an
G
elementaryabeliansubgroupofG. Observethatres∗ (gr(m))=gr(res∗ (m))for
G,E(cid:8) (cid:9) G,E
any m∈V \ H·(E,k)† . Then the map res∗ :V // V induces a map
E G,E E G
(cid:8) (cid:9)res∗ :ProjV //ProjV .
G,E E G
AnelementaryabeliansubgroupofG isisomorphictoG whereEisanelementary
G E
abelian p-subgroup of G; see [Far4, 6.2]. Without any real ambiguity, we will use
G and E alternatively for the sake of convenience. By denoting
E
V (ℓ)= res∗ V ,
G G,E E
GE∈Ma[xea(GG)ℓ
we have the following Lemma:
Lemma 2.1. Suppose G is a constant finite group scheme with G (k)=G. Then
G G
srk(G )=max{ℓ≥1; V =V (ℓ)}.
G G G
Proof. We utilize the homeomorphism Ψ :P(G ) //ProjV presented in
GG G G
[FP, Sect. 4] to verify this. First assume that V = V (ℓ). Let [α] ∈ P(G ) be
G G G
anequivalence class, then Ψ ([α])∈ProjV . Thus there is m∈V \ H·(G,k)†
GG G G
such that Ψ ([α]) = gr(m). By our assumption, there exists a maximal ele-
GG (cid:8) (cid:9)
mentary abelian subgroup scheme G of G with cx (k) ≥ ℓ such that m ∈
E G GE
res∗ (V \ H·(E,k)† ). Then gr(m) ∈ res∗ ProjV . The bijective map Ψ
G,E E G,E E GE
ensures gr(m(cid:8)) = res∗G,E(cid:9)(ΨGE([β])) = ΨGG([ι∗,kE ◦β]) for some [β] ∈ P(GE). As
a result, ΨGG([α]) = ΨGG([ι∗,kE ◦β]), which gives [α] = ι∗,kE([β]) since ΨGG is
bijective. Hence, we have P(G )=P(G ) .
G G ℓ
On the other hand, we assume P(G ) = P(G ) . Let m ∈ V . If m = H·(G,k)†,
G G ℓ G
then m∈res∗ V for any maximal elementary abelian subgroup G of G . So it
G,E E E G
4
suffices to consider m6=H·(G,k)†. Then gr(m)=kerα· for some [α] ∈P(G ). By
G
our assumption, there is a maximal elementary abelian subgroup G of G with
E G
cxGE(k) ≥ ℓ such that [α] = ι∗,kE([β]) where [β] ∈ P(GE). This gives kerα· =
ker(β· ◦res ), and consequently ker(res ) ⊂ gr(m) ⊂ m. As a result, m ∈
G,E G,E
V(ker(res ))=res∗ V . Therefore, we have V =V (ℓ).
G,E G,E E G G
Combining these two, it has been shown that P(G )=P(G ) if and only if V =
G G ℓ G
V (ℓ) for ℓ≥1. This ultimately gives srk(G )=max{ℓ≥1; V =V (ℓ)}. (cid:3)
G G G G
With the finiteness property of the number of elementary abelian subgroups of a
finite group behind us, Lemma 2.1 tells us that the saturation rank srk(G ) of a
G
constantfinitegroupschemeisclear,i.e. istheminimaldimensionofanirreducible
component of V . Recall that a maximal elementary abelian subgroup E of G has
G
the property:
• E is not conjugate to a proper subgroup of any other elementary abelian
subgroups.
Let M(G) be the set of representatives from each conjugacy class of a maximal
elementary abeliansubgroup. The following theoremis dedicated to establishing a
relation between the set M(G) and the set of irreducible components of V .
G
Theorem 2.2. Let G be a constant finite group scheme with G (k) = G. Then
G G
the assignment
E 7−→res∗ V
G,E E
induces a bijection
elementary abelian subgroups irreducible components of the
∼
−→ .
E in the set M(G) affine variety V
( ) (cid:26) G (cid:27)
Proof. We first show that the assignment is well-defined. Let E be a maximal
elementaryabeliansubgroup. Weadoptthenotionsfrom[Ben,Sect. 5.6]asfollows:
V+ :=V − res∗ V , V+ :=res∗ V+.
E E E,F F G,E G,E E
F<E
[
elemab
Then by [Ben, Lemma 5.6.2], there exists an element ̺ of H·(G,k) with the
E
property V+ = res∗ V −V(̺ ). Suppose additionally that E′ is an elemen-
G,E G,E E E
tary abelian subgroup which is not conjugated to E. Since E is maximal, we
have resG,E′(̺E) = 0 according to [Ben, Lemma 5.6.2]. This subsequently implies
res∗G,E′ VE′ ⊆ V(̺E), ensuring res∗G,EVE * resG,E′ VE′. Thus, res∗G,EVE is maxi-
malin V , i.e. isanirreduciblecomponent. The bijection followsimmediately. (cid:3)
G
Corollary 2.3. Let G be a constant finite group scheme with G (k) = G. Then
G G
srk(G ) is the minimal rank of a maximal elementary abelian subgroup.
G
Proof. Since dimres∗ V =dimH·(E,k)=rk(E), the result is readily seen. (cid:3)
G,E E
Corollary 2.4. LetG beaconstantfinitegroupschemewithG (k)=G. Suppose
G G
additionally V is equidimensional, then srk(G )=dimV , and vice versa.
G G G
5
Example 2.3. We consider the dihedral group D and suppose p=2. It has two
8
generators a and b with relations:
a4 =1=b2, and aba=b.
There are two maximal elementary abelian 2-subgroups:
e,a2,b,a2b , and e,a2,ab,a3b
which are isomorphic to(cid:8)Z/2Z×Z/2(cid:9)Z. As a(cid:8)result, srk(G(cid:9) )=dimV =2.
D8 D8
2.3. Infinitesimalgroupschemes. We say a finite group scheme G is infinites-
imal if its coordinate algebra k[G] is a local algebra. Then the augmentation ideal
k[G]† of k[G] is its unique maximal ideal. Associated to G, it is of height ≤r ∈N
0
if xpr =0 for all x∈k[G]†. A class of infinitesimal groupschemes that have served
as prototypical examples arise from reduced algebraic group schemes G by taking
their r-th Frobenius kernel G via the Frobenius map Fr. The representing alge-
(r)
bra of G is then k[G ]=k[G]/I, where I is an ideal generatedby elements xpr
(r) (r)
for x ∈ k[G ]†. In particular, when r = 1, we find that k[G ] is the dual of the
(r) (1)
restricted enveloping algebra of algebraic Lie algebra g = Lie(G), a sepecial case
of restricted Lie algebras which carries a canonical [p]-structure equivariant under
the adjoint action of G(k). In general, there is a categorical equivalence between
the category of finite dimensional p-restricted Lie algebras and category of infini-
tesimal group schemes of height ≤ 1. Henceforth, for any given finite dimensional
restricted Lie algebra (g,[p]), we denote the associated infinitesimal group scheme
G :=Spec((U (g))∗) by g.
g 0
LetG be aninfinitesimalgroupschemeofheight≤r. Aninfinitesimal1-parameter
subgroup of G over a commutative k-algebra R is a homomorphism of R-group
R
//
schemes G G . Let V (G) be the functor, which sends every commu-
a(r),R R r
tative k-algebra A to the group V (G)(A) = Hom (G ,G ). The functor
r Gr/A a(r),A A
V (G)isrepresentedbyanaffineschemeoffinitetypeoverk,[SFB1,Theorem1.5].
r
Thecoordinatealgebrak[V (G)]ofV (G)isthenagradedconnectedalgebragener-
r r
ated by homogeneous elements of degree pi,0≤i≤r−1. In what follows, we will
onlyconcentrateonthek-rationalpointsoftheschemeV (G),whichisstilldenoted
r
//
byV (G). Note thataninfinitesimal1-parametersubgroup v :G G over
r a(r)
k may be factored as
u // (cid:31)(cid:127) //
G G G
a(r) a(s)
forsome1≤s≤r,whereG isanelementaryabeliansubgroup(see[Far4,6.2]).
a(s)
Let C be the category whose objects are elementary abelian subgroups of G, and
G
whose morphisms are inclusions. Similarly, define CkG to be the category hav-
ing commutative Hopf subalgebras of kG whose underlying associative algbera is
isomorphic to some truncated polynomial ring k[x ,...,x ]/(xp,...,xp) as its ob-
1 n 1 n
jects,andmorphismsarealsogivenbyinclusions. Thereisacategoricalequivalence
//
between CG and CkG via the functor F :CG CkG by sending E to kE.
//
Let Grd(kG):Comk Ens be the Grassmannscheme, i.e., the k-functor that
assigntoeverycommutativek-algebraRthesetGr (kG)(R)ofR-directsummands
d
6
ofkG⊗kR of rank d; see [Jan1, Sect I.1.9]. We begin with the considerationof the
subfunctor Sub (kG)⊆Gr (kG) which is given by
d d
Sub (kG)(R):={V ∈Gr (kG)(R)|V ·V ⊆V,∆(V)⊆V ⊗ V}
d d R
for every commutative k-algebra R. Recall that the base change kG ⊗k R of the
Hopf k-algebra kG is then a Hopf R-algebra.
Proposition 2.5. Keep the notations for Gr (kG),Sub (kG) as above. Then the
d d
functor Sub (kG) is a closed subfunctor of Gr (kG).
d d
Proof. Let ψ :R //S be a k-algebra homomorphism. Then it induces a Hopf
S-algebra homomorphism id⊗(ψ⊗ˆ id):(kG⊗kR)⊗RS //kG⊗kS by send-
ing x ⊗ r ⊗ s to x ⊗ ψ(r)s, it follows that Gr (kG)(ψ) sends Sub (kG)(R) to
d d
Sub (kG)(S). As as result, Sub (kG) is a subfunctor of the k-functor Gr (kG).
d d d
The closedness of Sub (kG) is verified in this fashion: for every commutative k-
d
algebra A and every morphism f :Speck(A) // Grd(kG) , f−1(Subd(kG)) is a
closed subfunctor of Spec (A); see [Jan1, I.1.12]. By Yoneda’s Lemma, the mor-
k
phism corresponds to an A-point W ∈Gr (kG)(A).
d
Fix a basis {v ,...,v } of kG. Let α be the elements of A that are given by
1 n ijℓ
n
v ·v = α v , 1≤i,j ≤n
i j ijℓ ℓ
ℓ=1
X
Denote by {w ,...,w } a set of generatorsof the locally free A-module W of rank
1 d
d, and define elements a ,b ,c ∈A via
ri ri ri
n
w = v ⊗a ,1≤r ≤d,
r i ri
i=1
X
n n
∆(w )= (v ⊗b )⊗(v ⊗c ),1≤r ≤d.
r i ri j rj
i=1j=1
XX
′
BydefinitionofGrd(kG)(A), there existsanA-submoduleW ⊆kG⊗kAsuchthat
′
kG⊗kA=W ⊕W
// ′
and we denote by pr:kG⊗kA W the corresponding projection. This A-
linear map is given by
n
pr(v ⊗1)= v ⊗κ , 1≤j ≤n.
j i ij
i=1
X
We let I ⊆A be the ideal generatedby the elements for 1≤i,j,t≤n,1≤r,s≤d
n n n n n
h = α a a κ ; g = b κ c , ; γ = b κ c .
rst ijℓ ri sj tℓ trj rq tq rj tri ri tq rq
ℓ=1i=1j=1 q=1 q=1
XXX X X
//
Now let ψ :A R be a homomorphism of k-algebras. Then we have
′
kG⊗kR=WR⊕WR
7
′
where X =X⊗ R for X ∈ W,W . The corresponding projection
R A
n o
// ′
prR :kG⊗kR WR
is given by
n
pr (v ⊗1)= v ⊗ψ(κ ), 1≤j ≤n.
R j i ij
i=1
X
In view of
n n n
(w ⊗1)·(w ⊗1)= v ⊗ α ψ(a )ψ(a )
r s ℓ ijℓ ri sj
ℓ=1 i=1j=1
X XX
this gives rise to
n
pr ((w ⊗1)·(w ⊗1))= v ⊗ψ(h ).
R r s t rst
t=1
X
By the same token, we have
n n
(pr ⊗id )◦∆ (w ⊗1)= (v ⊗1)⊗(v ⊗ψ(g )),
R R R r t j trj
t=1j=1
XX
n n
(id ⊗pr )◦∆ (w ⊗1)= (v ⊗ψ(γ ))⊗(v ⊗1).
R R R r i tri t
t=1 i=1
XX
Now suppose that ψ(I) = 0. Then the three forgoing identities imply W ·W ⊆
R R
kerpr = W , as well as ∆ (W ) ⊆ ker(pr ⊗id )∩ker(id ⊗pr ) = (W ⊗
R R R R R R R R R R
(kG⊗kR))∩((kG⊗kR)⊗RWR)=WR⊗RWR,thusWR ∈Subd(kG)(R). Conversely,
ifwehaveW ∈Sub (kG)(R),thenI ⊆ker(ψ). Observethatf−1(Sub (kG))(R)=
R d d
{ψ ∈Speck(A)(R)|WR =Grd(kG)(ψ)(W)∈Subd(kG)(R)}. This shows
f−1(Subd(kG))(R)={ψ ∈Speck(A)(R)|ψ(I)=0}.
Consequently, f−1(Subd(kG)) is a closed subfunctor of Speck(A), as desired. (cid:3)
Remark 2.4. LetAb (kG)⊆Gr (kG) be the subfunctor ofcommutative subalge-
d d
bras (contains identity element of kG) of kG. It is a closed subfunctor, which can
be proved similar to Proposition 2.5. The above proposition shows that the sets
Sub (kG)(k),Ab (kG)(k) of rational k-points of these functors are closed subsets
d d
of the Grassmann variety Gr (kG)(k).
d
Proposition2.6. LetC(ℓ,G)bethesetconsistingofobjectsofC havingcomplexity
G
ℓ. Then C(ℓ,G) is a projective variety.
Proof. Let X := H ∈Grpℓ(kG)(k)|H is a commutative Hopf subalgebra ofkG .
Then by Lemma 1.2(1) of [Far2] on Hopf algebras, we have
(cid:8) (cid:9)
X =Subpℓ(kG)(k)∩Abpℓ(kG)(k).
According to our Remark 2.4, X is a closed subvariety of Gr (kG)(k). Consider
pℓ
thesetY :=C(pℓ,kG),consistingofobjectsofCkG withdimensionpℓ. ThenY ⊂X
is a subset of X. By the equivalence of categories between C and C , it suffices
G kG
to endow Y with a projective variety structure, i.e. to show Y is closed.
8
Let kG := {u ∈ kG | up = 0} be the set of p-nilpotent elements in kG, then
p
it is a closed conical subvariety. By setting H := H ∩kG for H ∈ X, we are
p p
going to verify that: The underlying associative algebra of H is isomorphic to
k[x1,...,xℓ]/(xp1,...,xpℓ) if and only if dimkHp ≥ pℓ −1. First if we have such
algebraicisomorphismfor H, then Hp =RadH and dimkHp ≥pℓ−1. Conversely,
if dimkHp ≥ pℓ −1 with dimkH = pℓ, then H = Hp⊕k1H and H must be local
since the identity element is the unique non-zeroidempotent element in H. Notice
thatH is commutative,then it representsaninfinitesimal groupscheme. Theorem
in [Wat, Sect 14.4]ensures that H has to be isomorphic to a truncated polynomial
ring, say k[x ,...,x ]/(xpe1,...,xpet). By the definition of H, the p-th power of
1 t 1 t
x for 1 ≤ i ≤ t should be zero, i.e. xp = 0. This implies all e = 1 and further
i i i
t= t e =ℓ by dimension. Thus for such H, its underlying associative algebra
i=1 i
is isomorphic to k[x ,...,x ]/(xp,...,xp). Finally, recall the following map
P 1 ℓ 1 ℓ
//
X N0 ; H 7→dimkH ∩kGp
is upper semicontinuous [Far1, Lemma 7.3]. Thus Y = {H ∈ X | dimkH ∩kGp ≥
pℓ−1} is closed. (cid:3)
(cid:31)(cid:127) //
Suppose ι :E G is the canonical inclusion of an elementary abelian sub-
E
//
group. Then there is a morphism ι :V (E) V (G) of support varieties.
∗,E r r
We set
C(ℓ↑,G):= C(r,G)
r≥ℓ
[
and
V = ι (V (E)).
C(ℓ↑,G) ∗,E r
E∈C[(ℓ↑,G)
Theorem 2.7. Suppose G is an infinitesimal group scheme of height ≤r. Then
srk(G)=max ℓ|V (G)=V .
r C(ℓ↑,G)
(cid:8) (cid:9)
Proof. Let{v0,...,vpr−1}bethedualbasisofthestandardbasis{T0,T1,...,Tpr−1}
of k[Ga(r)] = k[T]/(Tpr). Denote by ui = vpi for 0 ≤ i < r, this gives kGa(r) =
k[u ,...,u ]. Now weturntoverifyourstatement. Proposition3.8of[FP]gives
0 r−1
a bijection
//
Θ :Proj(V (G)) P(G) ; [α]7→[α ◦ǫ]
G r ∗
(cid:31)(cid:127) // //
where ǫ:k[u ]≃kZ/pZ kG ,and α :kG kG . Ifα∈Pt(G),
r−1 a(r) ∗ a(r)
then it represents an equivalence class given by the map Θ , say [α]=Θ ([β]) for
G G
some β ∈ V (G). Further if we assume that V (G) = V , then there exists a
r r C(ℓ↑,G)
maximalelementaryabeliansubgroupE ≤G withcx (k)≥ℓsuchthatβ =ι (γ)
E ∗,E
forsomeγ ∈Vr(E). Therefore,[α]=[β∗◦ǫ]=[ι∗,kE◦γ∗◦ǫ]=ι∗,kE([γ∗◦ǫ])whichlies
in ι∗,kE(P(E)), and this gives P(G)=P(G)ℓ. On the other hand, if α∈Vr(G)\{0}
and suppose P(G) = P(G) , then Θ ([α]) is an equivalence class of P(G). By our
ℓ G
assumption there exists a maximal abelian unipotent subgroup U along with an
elementary abelian subgroup E with cx (k) ≥ ℓ and a p-point β ∈ Pt(E ) such
U EU U
that ΘG([α]) = ι∗,kEU([β]); see [Far4, Lemma 6.2.1]. Again by the bijective map
9
Θ , we have [β] =[γ ◦ǫ] for some γ ∈V (E ), thus Θ ([α]) =Θ ([ι ◦γ]) and
EU ∗ r U G G EU
α=ι ◦γ ∈ι (V (E )). Therefore, V (G)=V , as desirable. (cid:3)
EU ∗,EU r U r C(ℓ↑,G)
Corollary 2.8. In Theorem 2.7, if G is of height ≤1, then
srk(G)=max ℓ|V (G)=V .
r C(ℓ,G)
(cid:8) (cid:9)
Proof. Itsufficestoshowthatanyγ ∈V (E)withcx (k)=smayfactorthroughan
1 E
elementarysubgroupE′ ofE fors′ ≤s. SinceG isofheight≤1,wehaveE ∼=G×s .
a(1)
Thehomomorphism γ :G // E isequivalentto dγ :g //Lie(E) where
a(1) a
g = kδ = Lie(G ). The image dγ(δ ) is contained in an elementary subalge-
a a a(1) a
bra es′ of Lie(E) since Lie(E) is elementary abelian. Thus, the image γ(Ga(1)) is
contained in an elementary group scheme es′ of E, as desirable. (cid:3)
Suppose α∈V (G). Consider the following set
r
C(ℓ↑,G) :={E ∈C(ℓ↑,G)|α∈ι (V (E))}
α ∗,E r
together with
rG :=max{ℓ|C(ℓ↑,G) 6=∅}.
α α
Write rG :=min rG |α∈V (G) and OG :={α∈V (G)|rG =rG }.
min α r rmin r α min
Remark 2.5. Let(cid:8)G = G be the(cid:9)Frobenius kernel of a smooth group scheme G.
r
Then G acts on G via the adjoint representation. There results an action of G on
r
V (G) and on C(ℓ,G ). Let α∈V (G) and g ∈G. Based on these facts, we obtain
r r r
the relation rGr =rGr.
α g.α
Lemma2.9. SupposethatG is an infinitesimalgroupschemes of height ≤r. Then
V is a closed subvariety of V (G).
C(ℓ,G) r
Proof. We denote by pr the projection onto the first coordinate:
1
pr :V (G)×C(ℓ,G) // V (G) .
1 r r
Write kG = k[u ,...,u ] as we did in Theorem 2.7. Consider the set Z =
a(r) 0 r−1
{(α ,kE)∈Hom(kG ,kG)×C(pℓ,kG)|α (u )∈kE,0≤i<r}, whichis closed.
∗ a(r) ∗ i
′
Then by categorical equivalence the set Z := {(α,E) ∈ V (G) × C(ℓ,G) | α ∈
r
ι (V (E))} is closed. Proposition 2.6 shows that C(ℓ,G) is complete. Therefore,
∗,G r
the image V =pr (Z′) of Z′ is closed in V (G) by general theory. (cid:3)
C(ℓ,G) 1 r
Theorem 2.10. Suppose that G is an infinitesimal group scheme of height ≤ r.
Then srk(G)=rG and OG is an open subset of V (G).
min rmin r
Proof. Let s = srk(G). Then V (G) = V and C(s ↑,G) 6= ∅ for any α ∈
r C(s↑,G) α
V (G). Thus rG ≥ s, resulting rG ≥ s. On the other hand, rG ≥ rG gives
r α min α min
C(rmGin ↑,G)α 6=∅ for any α∈Vr(G). Therefore, Vr(G)=VC(rG ↑,G) and s≥rmGin.
min
We now consider the function
rG :V (G) // N; α7→ rG.
r α
10