On nilpotent commuting varieties and cohomology of Frobenius kernels

0.2 MB·

by  Nham V. Ngo

ON NILPOTENT COMMUTING VARIETIES AND COHOMOLOGY OF
FROBENIUS KERNELS
NHAMV. NGO
4
1
0 Abstract. Thepaperstudiesthedimensionsofirreduciblecomponentsofcommutingvarietiesof
2 (restricted) nilpotent r-tuples in a classical Lie algebra g defined over an algebraically closed field
c k. As applications, we obtain some new results on the structure of the (even) cohomology ring of
e FrobeniuskernelsGr for each r≥1,whereGisthesimply connected,simplealgebraic group such
D that Lie(G)=g. Explicit calculations for rank two groups are also presented.

6
1 1. Introduction

] 1.1. Let g be a classical Lie algebra defined over an algebraically closed field k of characteristic
T
p > 0. Denote N = {x ∈ g : x[p] = 0}, therestricted nullconeof g. Note thatN coincides with the
R 1 1
nilpotent cone N of g when p ≥ h, the Coxeter number of g. In this paper, we study the dimension
.
h and related properties of the commuting variety
t
a
m Cr(N1) = {(x1,...,xr)∈ N1r :[xi,xj] = 0,1 ≤ i,j ≤ r}.
[ It is well known that for r = 2 and p ≥ h, such commuting variety is completely described by

4 Premet in [Pr]. Explicitly, he showed that C2(N) has pure dimension dimg, each irreducible
v component is of the form G·(x,z(x)∩N) for some distinguished nilpotent element x, where z(x)
0
is the centralizer of x in g. In a part of the Ph.D. dissertation, the author proved that if g is
2
8 either sl2 or sl3, then Cr(N) is irreducible for each r ≥ 1 [Ngo2]. Recently, Sˇivic and the author
6 studied the reducibility of C (N) for type A and all r ≥ 1 [NS]. Among other results, we proved
r
.
1 that Cr(N) is reducible for r ≥ 4 and rank(g) ≥ 3. For r = 3, it was shown to be irreducible for
0 rank(g)≤ 5. Very little is known about C (N ) for p < h. In fact, one could only find in literature
r 1
4
related results for r = 1 [CLNP][UGA] or r = 2,p = 2,g = sl [L].
1 n
: Our motivation for studying such commuting varieties is to investigate cohomological properties
v
ofFrobeniuskernelsofG. Ther-thFrobeniuskernelsG ,forallr ≥ 1,areinfinitesimalsubgroupsof
i r
X Gwhosecoordinatealgebras arefinitedimensionalandlocal. Thesegroupsplay afundamentalrole
r in relating cohomology theory of finite groups to that of reductive group schemes [Jan1]. However,
a
cohomology theory for these objects are not well understood except few special cases. In the case
r = 1, the first Frobenius kernel G is a familiar object and received considerable interest from
1
representation theorists due to the equivalence between the category of G -modules and that of the
1
restricted Lie algebra (g = Lie(G),[p])-modules, see for example [Jan1]. For higher values of r, the
problem on computing the cohomology of G turns out to be complicated. Bendel, Nakano, and
r
Pillen have made some progress in the two papers [BNP1][BNP2] where they explicitly calculated
the first and second degrees of Hi(G ,H0(λ)) with H0(λ) = indG(λ) the induced module of the
r B
highest weight λ. In the special case when G = SL , the author computed Hi(G ,H0(λ)) for each
2 r
i,r ≥ 1 and dominant weight λ [Ngo1]. In general, no conjecture has been made. Geometrically,
Suslin,Friedlander,andBendel,demonstratedthatthespectrumofthecohomology ringH2•(G ,k)
r
can be identified with the commuting variety C (N ) [SFB1]. This groundbreaking result deduces
r 1
the study of cohomology for Frobenius kernels to that of the variety C (N ).
r 1
The paperwas partly supported by EPSRC grant EP/K022997/1.
1
2 NHAMV.NGO
1.2. Thepaperisstructuredasfollows. WefirstprovethatPremet’sresultdoesnotholdforC (N)
r
in general. In other words, we point out that for all r larger than some constant depending on the
type and rank of g, the commuting variety C (N) is not equidimensional. Our main ingredient is
r
the G-saturation variety V = G·wr with w a fixed nilpotent commutative subalgebra of g defined
r
at the beginning of Section 3.2. The dimension of V gives sharp lower bounds for those of C (u )
r r 1
and C (N ). It also allows us to compute, for g of type A and C, the dimension of the commuting
r 1
variety over the square zero set O = {x ∈ g : (i(x))2 = 0}, where the inclusion i : g ֒→ gl is the
2 n
natural representation of g (defined in 2.1). Consequently, we calculate the dimension of C (N )
r 1
for g of rank 2.
The rest of the paper, as applications of the previous part, is devoted to explore the structure of
the cohomology ring H•(G ,k) and complexity of G -module M. In particular, we first show that
r r
if G = SL , then the graded commutative ring H•(G ,k) is Cohen-Macaulay for each r ≥ 1. This
2 r
result significantly strengthens the one in [Ngo1] where the author proved that the commutative
ring H2•(G ,k) is Cohen-Macaulay. However, this special property can not be generalized for
r red
arbitrary G, as we show in the later part that H•(G ,k) is, in general, not equidimensional. We
r
are also able to provide a universal lower bound (depending only on r and rank(g)) for the Krull
dimension of H•(B ,k) and H•(G ,k) and then compute exactly this amount for G of rank 2. Note
r r
that our last results are much stronger than the computations of Kaneda et. al. on the Krull
dimension of B cohomology ring for SL [KSTY, 4.7]. Finally, we obtain some properties of the
r 3
complexity, c (M), of a module over G .
Gr r
2. Notation
2.1. Representation theory. Let k be an algebraically closed field of characteristic p > 0. Let g
be a classical Lie algebra over k, i.e. g is of type A,B,C, or D. Throughout the paper, we always
assume p is a good prime for g, i.e. p is arbitrary for type A and it is greater than 2 for other types
(see details for exceptional types in [Jan2, 2.6]), unless otherwise stated. To be convenient for later
usage, we give an explicit description of these classical Lie algebras as subalgebras of the general
linear algebra gl for some n > 0 as follows. For type A , (i.e. sl ) it is exactly the space of
n ℓ ℓ+1
traceless (ℓ+1)×(ℓ+1)matrices. For othertypes, ourLiealgebras aredefinedbythesamestrategy
as in [Hum, 1.2] but using different nondegenerate (skew)-symmetric bilinear forms. Indeed, let J
ℓ
be the anti-identity ℓ×ℓ matrix (consisting of 1’s on the anti diagonal and 0’s elsewhere). It is
easy to see that the forms
0 0 J
0 J 0 J ℓ
ℓ , ℓ , 0 1 0
(cid:18) −Jℓ 0 (cid:19) (cid:18) Jℓ 0 (cid:19)  Jℓ 0 0 
 
are symplectic and orthogonal bilinear forms of sp ,so and so . Now for each matrix m in
2ℓ 2ℓ 2ℓ+1
gl , denote mJ the matrix reflected over the anti-diagonal. Similarly as in [Hum, 1.2], mt is the
ℓ
transposedmatrixofm. Finally classical Liealgebras (otherthantypeA)canbedefinedasfollows:
• Type C or D (i.e. sp or so ): is the space of matrices of the form
ℓ ℓ 2ℓ 2ℓ
m m
11 12
m m
21 22
(cid:18) (cid:19)
satisfying m ∈ gl , m ,m are (skew) symmetric over the anti-diagonal, and m =
ij ℓ 12 21 11
±mJ .
22
• Type B (i.e. so ) is the space of matrices of the form
ℓ 2ℓ+1
m b m
11 1 12
c 0 c
1 2
 
m b m
21 2 22
 
ON NILPOTENT COMMUTING VARIETIES AND COHOMOLOGY OF FROBENIUS KERNELS 3
where m ∈ gl , b ,b and c ,c are column and row vectors in kℓ such that m ,m are
ij ℓ 1 2 1 2 12 21
skew symmetric over the anti-diagonal, m = −mJ ,J b = −ct, and J b = −ct.
11 22 ℓ 1 2 ℓ 2 1
The above construction of classical linear Lie algebras implies an inclusion i : g ֒→ gl for
n
some n > 0. Fix a Borel subalgebra b consisting of upper triangular matrices in g, and a Cartan
subalgebratconsistingofdiagonalmatrices ing. NowletGbeasimplyconnected, simplealgebraic
groupoverk,stabilizingtheaforementionedbilinearforms,suchthatLie(G) = g,(explicitdefinition
of G could be found in [MT, §1.2]). Fix a maximal torus T of G, let B ⊂ G be the Borel subgroup
of G containing T satisfying Lie(B) = b and Lie(T) = t. Let U ⊂ B be the unipotent radical of
B, then Lie(U) = u, consisting of strictly upper triangular matrices. From now on, the symbol
⊗ means the tensor product over the field k, unless otherwise stated. Suppose H is an algebraic
group over k and M is a (rational) module of H. Denote by MH the submodule consisting of all
the fixed points of M under the H-action.
For each positive integer r, let F : G → G(r) be the r-th Frobenius morphism, see for example
r
[Jan1, I.9]. The scheme-theoretic kernel G = ker(F ) is called the r-th Frobenius kernel of G.
r r
Givenaclosedsubgroup(scheme)H ofG,writeH forthescheme-theoretickerneloftherestriction
r
F :H → H(r). In other words, we have
r
H = H ∩G .
r r
Given a rational G-module M, write M(r) for the module obtained by twisting the structure map
for M by F . Note that G acts trivially on M(r). Conversely, if N is a G-module on which G
r r r
acts trivially, then there is a unique G-module M with N = M(r). We denote the module M by
N(−r). Let M be a B-module. Then the induced G-module can be defined as
indGM = (k[G]⊗M)B.
B
2.2. Geometry. Let R be a commutative Noetherian ring with identity. We use R to denote
red
the reduced ring R/NilradR where NilradR is the radical ideal of 0 in R, which consists of all
nilpotent elements of R. Let MaxSpec(R) be the spectrum of all maximal ideals of R. This set is
a topological space under the Zariski topology. The notation dim(−) will be interchangeably used
as the dimension of a variety or the Krull dimension of a ring.
Let N be the nilpotent cone of g, consisting of all nilpotent elements in g. The adjoint action
of G on g stabilizes N and is denoted by “·”. For each x ∈ N, denote O = G·x the orbit of x
x
under the dot action of G. Let x be a fixed regular nilpotent element and z be its centralizer
reg reg
in g. It is well known that z ⊂ N, dimz = rankg =: ℓ, and the regular orbit O = G·x
reg reg reg reg
is dense in N. The restricted nullcone N of g is defined as a subvariety of N satisfying
1
x ∈ N ⇔ x[p] = 0
1
where(−)[p] is thep-power operation of therestricted Liealgebra g. Sinceour classical Liealgebras
could be embedded into gl for some n > 0, one may identify x[p] = i(x)p for x ∈ g. Hence, for
n
p ≥ h,theCoxeternumberofg,wehaveN = N,seeforinstance[CLNP,§1]. Completedescription
1
of N is referred to the paper of Carlson, Lin, Nakano, and Parshall [CLNP]. Set u = N ∩u. It
1 1 1
follows that u = u whenever p ≥ h.
1
3. Commuting varieties
Suppose V is a closed affine subvariety of g. We define the commuting variety of r-tuples over
V as follows
C (V) = {(x ,...,x )∈ Vr | [x ,x ] = 0, 1 ≤ i≤ j ≤ r}.
r 1 r i j
We will just call it the commuting variety over V for short. In case when V = N (or N ), we call
1
C (V) the (restricted) nilpotent commuting variety of g. For more details of such varieties, one can
r
refer to [Ngo2].
4 NHAMV.NGO
3.1. Irreducible component associated to regular nilpotent elements. One has seen such
an irreducible component in the case when r = 2, see for example [Pr]. We show here that its
generalized version for arbitrary r is still an irreducible of C (N). Similarly, we point out an
r
irreducible component for C (u). The dimension of these components gives some criterion for
r
irreducibility and equidimensionality.
Proposition 3.1.1. For each r ≥ 1, the closed subvariety V := G·(x ,z ,...,z ) is an
reg reg reg reg
irreducible component of C (N) whose dimension is dimN +(r−1)ℓ.
r
Proof. First note that V is irreducible as it is the image of the surjective morphism
reg
m :G×zr−1 → V
reg reg
(g,x ,...,x )7−→ g·(x ,x ,...,x )
1 r−1 reg 1 r−1
for all g ∈ G and x ∈ z . Now consider the projection from C (N) to its first component
i reg r
ρ:C (N) → N,
r
(x ,...,x )7→ x .
1 r 1
Since the orbit O is open in N, so is its preimage ρ−1(O ) = G·(x ,z ,...,z ) (here we
reg reg reg reg reg
use the fact that z is commutative). So the closure ρ−1(O )= V is an irreducible component
reg reg reg
of C (N).
r
Applying the theorem on the dimension of fibers to the restriction of ρ :V → N, we have
reg
dimV = dimN +dimρ−1(x ) = dimN +(r−1)ℓ,
reg reg
which completes our proof. (cid:3)
From now on, we call V the irreducible component of C (N) associated to regular nilpotent
reg r
elements. ReplacingGbyB intheaboveargument,oneobtainsasimilarresultforC (u)asfollows.
r
Proposition 3.1.2. For each r ≥ 1, the closed subvariety B ·(x ,z ,...,z ) is an irreducible
reg reg reg
component of C (u) whose dimension is dimu+(r−1)ℓ.
r
An easy corollary immediately follows.
Corollary 3.1.3. For each r ≥ 1, if the commuting variety C (N) (or C (u)) is irreducible or
r r
equidimensional then its dimension is dimN +(r−1)ℓ (or dimu+(r−1)ℓ).
3.2. Dimension of commuting varieties. We introduce here a special square zero vector space
which plays the main role in our investigations on the dimensions of commuting varieties. Our
construction below is based on the definition of classical linear Lie algebras in the beginning of
Section 2. Explicitly, the space w is defined by matrices of the form
0 m
(1)
0 0
(cid:18) (cid:19)
where m satisfies the following
• Type A : If n = 2ℓ then m is an ℓ×(ℓ+1) matrix, otherwise, if n = 2ℓ−1, then m ∈ gl ,
n ℓ
• Type C : m ∈ gl and m = mJ,
ℓ ℓ
• Type B or D : m ∈ gl and m = −mJ.
ℓ ℓ ℓ
It is not hard to see that w is a nilpotent Lie subalgebra of g (as it is square zero). Moreover, it is
commutative and
ℓ2 if g = sl ,
2ℓ
ℓ(ℓ+1) if g = sl ,
(2) dimw =  2ℓ+1
ℓℓ222+−ℓℓ ieflsge.= sp2ℓ,
2




ON NILPOTENT COMMUTING VARIETIES AND COHOMOLOGY OF FROBENIUS KERNELS 5
Remark 3.2.1. One could realizes w as the Lie algebra of a unipotent radical for some para-
bolic subgroup P of G. Explicitly, suppose V is the natural representation of g with the basis
w
{v ,...,v }wheren = ℓ+1,(2ℓ+1, or 2ℓ)if g is of typeA (B , or C ,D ). Let W bethe subspace
1 n ℓ ℓ ℓ ℓ
of V generated by the first ⌊n⌋ basis vectors v′s. Under the bilinear forms defined in Section 2, we
2 i
have 0 ⊂ W ⊂ V is a totally isotropic flag, so Proposition 12.13 in [MT] states that the stabilizer
of this flag in G is a parabolic subgroup, denoted by P . Simple calculations would show that
w
w = Lie(U) where U is the unipotent radical of P . (The reader should refer to Examples 12.4 and
w
17.9 in [MT] for the details).
Proposition 3.2.2. The commuting variety C (u) is not equidimensional for
r
2+ 1 if g = sl or sl ,
ℓ−1 2ℓ 2ℓ+1
2 if g = sp ,
r >  2ℓ
2+ ℓ−23 if g = so2ℓ,
2+ 4 if g = so .
ℓ−3 2ℓ+1

Proof. As w is commutative, wr isa subvariety of Cr(u). Easy computation shows that dimwr
(= rdimw) is greater than dimu+(r −1)ℓ for all r satisfying the hypothesis. Hence the result
follows by Corollary 3.1.3. (cid:3)
Since w ⊂ u for all p ≥ 2, we further have wr ⊂ C (u ). Thus, we obtain the following
1 r 1
Corollary 3.2.3. For each r ≥ 1 and for all prime p ≥ 2 (not necessarily good prime), we always
have
dimC (u ) ≥ rdimw.
r 1
Before getting similar results for C (N), we set V = G·wr for each r ≥ 1. It’s easy to see that
r r
V is a closed subvariety of C (N), see for example [Jan2, §8.7]. We now compute the dimension
r r
of V .
r
Proposition 3.2.4. For each r ≥ 1, one has
(r+1)ℓ2 if g = sl ,
2ℓ
(r+1)ℓ(ℓ+1) if g = sl ,
dimV = (r+1)dimw=  2ℓ+1
r ((rr++11))ℓℓ222+−ℓℓ ieflsge.= sp2ℓ,
2


Proof. By (2), it suffices to prove the first equality. From the Remark 3.2.1, we have the moment
map m :G×Pw wr → V . It follows that
r
dimV ≤ dim(G/P )+rdimw = (r+1)dimw.
r w
Ontheotherhand,letx beanelementinwwhichalsobelongstoamaximalorbitinN intersecting
w
with w. (The best candidate for such x is the matrix form (1) where m is a diagonal matrix
w
consisting of 1’s or −1’s.) In particular, the corresponding partition of x is [2s,1t] where s and t
w
satisfy the following condition
• Type A : 2s+t = ℓ+1 with t =0, or 1.
ℓ
• Type C : s = ℓ and t = 0.
ℓ
• Type B : 2s+t = 2ℓ+1 with s even and t =1, or 3.
ℓ
• Type D : 2s+t = 2ℓ with s even and t = 0, or 2.
ℓ
Using the Corollary 6.1.4 in [CM], one easily verifies that dimO ≥ 2dimw. So
xw
dimV ≥ dim G·(x ,wr−1) = dim(G·x )+(r−1)dimw ≥ (r+1)dimw.
r w w
Finally, we have obtained(cid:0) the equality.(cid:1) (cid:3)
6 NHAMV.NGO
Using the same argument as for C (u) and C (u ), one easily obtains the below properties.
r r 1
Corollary 3.2.5. The nilpotent commuting variety C (N) is not equidimensional for
r
3 if g = sl ,
2ℓ+1
3+ ℓ−21 if g = sl2ℓ,
r > 3+ 4 if g = sp ,
 ℓ−1 2ℓ

3+ ℓ−43 if g = so2ℓ,
3+ 8 if g = so .
 ℓ−3 2ℓ+1


Proof. It is not hard to see that for these values of r, dimVr > dimVreg, so the result follows by
Corollary 3.1.3. (cid:3)
Remark 3.2.6. The above result shows that Premet’s result on equidimensionality of C (N) [Pr]
2
isnotvalid forr ≥3. Ourresultalsoimplies thatthestructureof C (N)could beverycomplicated
r
when r is large. Hence, the task of describing irreducible components of these varieties becomes
challenging.
We next obtain a lower bound for the dimension of C (N ). This bound depends on r,ℓ, and
r 1
the type of g, not depend on p. Recall that we have been assuming that p is a good prime for G.
Corollary 3.2.7. For each r ≥ 1, one has
(r+1)ℓ2 if g = sl ,
2ℓ
(r+1)ℓ(ℓ+1) if g = sl ,
dimC (N ) ≥ dimV =  2ℓ+1
r 1 r ((rr++11))ℓℓ222−+ℓℓ ieflsge.= sp2ℓ,
2


Proof. Thefactthatwissquarezeroimpliesthatitisalways contained inN1,andsoVr ⊂ Cr(N1).
Therefore, the inequality follows. (cid:3)
3.3. Recall from the Section 2.1 that we have the embedding i : g ֒→ gl for some n > 0. Now
n
let O = {x ∈ g : (i(x))2 = 0}. We next compute the dimensions of C (O ∩u) and C (O ). This
2 r 2 r 2
deduces the dimensions of C (u ) and C (N ) when g is of type A and p = 2. We assume for the
r 1 r 1
rest of this section that g is of type A or C.
Suppose O is the maximal orbit in O , i.e. O =O . (Note that such orbit is not unique if g is
2 2 2 2
of type D). In fact, O = O defined in the proof of Proposition 3.2.4. Hence, we recall that the
2 xw
partition of O is of the form [2s,1t] for some non-negative integers s,t. In particular, their values
2
are following
• g = sl : t = 1 if n is odd, otherwise t = 0; hence s = n−t,
n 2
• g = sp : t = 0 and s = ℓ.
2ℓ
We review the dimension of O .
2
Lemma 3.3.1. We have
n2−t2 if g = sl ,
dimO = dimO = 2 n
2 2 (ℓ2+ℓ if g = sp2ℓ.
Proof. It easily follows from [CM, Corollary 6.1.4]. (cid:3)
We first compute the dimension of C (O ∩u).
r 2
Lemma 3.3.2. For each r ≥ 1, we have
dimC (O ∩u)= rdimw
r 2
and wr is an irreducible of maximal dimension in C (O ∩u).
r 2
ON NILPOTENT COMMUTING VARIETIES AND COHOMOLOGY OF FROBENIUS KERNELS 7
Proof. By [Jan2, Theorem 10.11], one has
1 1
dim(O ∩u) = max{dim(O∩u)} = max{ dimO} = dimO .
2 2
O⊂O2 O⊂O2 2 2
It follows that dimC (O ∩u) ≤ r dimO . On the other hand, note that dimO = 2dimw (from
r 2 2 2 2
(2)). Hence the fact that wr is a subset of C (O ∩u) implies the lemma. (cid:3)
r 2
Now we prove the main result of this subsection.
Theorem 3.3.3. For each r ≥ 1, one has
(r+1)⌊n2⌋ if g = sl ,
dimC (O ) = (r+1)dimw= 4 n
r 2 ((r+1)ℓ22+ℓ if g = sp2ℓ.
Consequently, V is an irreducible component of maximal dimension in C (O ).
r r 2
Proof. It is noted that C (O ) = G·C (O ∩u) for each r ≥ 1. It follows that
r 2 r 2
dimC (O ) =dimG−dimN (C (O ∩u))+dimC (O ∩u)
r 2 G r 2 r 2
=dimG−dimN ((O ∩u)r)+rdimw
G 2
=dimG−dimN (O ∩u)+rdimw
G 2
=dim[G·(O ∩u)]−dim(O ∩u)+rdimw
2 2
=dimO −dimw+rdimw
2
=(r+1)dimw
where N (S) = {g ∈ G : g · S ⊆ S}, the normalizer of S ⊆ g. Thus, we have proved that
G
dimC (O )= dimV for each r ≥ 1, hence the theorem follows from Proposition 3.2.4. (cid:3)
r 2 r
As an application, we are now able to compute the dimension of C (N ) when p = 2 and g is of
r 1
type A. Indeed, since N = O when p = 2, we immediately have
1 2
Proposition 3.3.4. Suppose g = sl and p =2. Then for each r ≥ 1,
n
n2
dimC (N )= dimC (O )= (r+1)⌊ ⌋.
r 1 r 2
4
Remark 3.3.5. The last result not only points out the case when the equality in Corollary 3.2.7
occurs but also generalize a result in [L] on the dimension of C (N ).
2 1
3.4. Rank 2 cases. Apply Theorem 3.3.3, we explicitly calculate the dimension of C (N ) for g
r 1
of rank 2, i.e. g is of type A or C . We keep assuming that the characteristic of k is a good prime.
2 2
We first present the result for type A .
2
Corollary 3.4.1. Suppose g is of type A and p is any prime. For each r ≥ 1, one has
2
2r+1 if p 6= 2,
dimC(u )=
1
(2r if p = 2,
and
2r+4 if p 6= 2,
dimC (N ) =
r 1
(2r+2 if p = 2.
Proof. Suppose p > 2. Then from Proposition 7.1.1 and Theorem 7.1.2 in [Ngo2], we have for each
r ≥ 1
dimC (u ) = dimC (u) = 2r+1 , dimC (N ) = dimC (N) = 2r+4.
r 1 r r 1 r
Now if p = 2, then Lemma 3.3.2 and Proposition 3.3.4 give the desired results. (cid:3)
8 NHAMV.NGO
Corollary 3.4.2. Suppose g is of type C and p ≥ 3. For r ≥ 1, one has
2
2r+2 if r = 1,p 6= 3,
dimC (u ) =
r 1
(3r else,
and
2r+6 if r ≤ 2,p 6=3,
dimC (N )=
r 1
(3r+3 else.
Proof. First we consider the case when p = 3. By [UGA, Theorem 5.1], N = O so that
1 2
dimC (u ) =3r , dimC (N )= 3(r+1)
r 1 r 1
by Lemma 3.3.2 and Theorem 3.3.3.
Now we assume p > 3. Then N = N and u = u. Recall that u is the space of all matrices of
1 1
the form
0 x y z
0 0 t y
 
0 0 0 −x
 0 0 0 0 
 
Forr ≥ 2,thecommutatoronthesematricesimpliesthatthevarietyC (u)isdefinedbypolynomials
r
x t −x t , x y −x y
i j j i i j j i
for 1 ≤ i < j ≤ r. It is then easy to see that C (u) is a closed subvariety of the one defined by
r
polynomials {x t −x t } in the affine space ur, which is the product of an affine space of
i j j i 1≤i,j≤r
dimension 2r (corresponding to free parameters y ,z ) and the determinantal variety of all 2×2
i j
minors over the matrix
x x ··· x
1 2 r .
t t ··· t
1 2 r
(cid:18) (cid:19)
Denote this product by P, we then have by [Ngo2, Proposition 3.2.2], dim(P) = 3r+1. Now as P
is irreducible and C (u) is a proper subvariety of P, we obtain dimC (u) < 3r+1. On the other
r r
hand, dimC (u) ≥ dimC (O ∩u)= 3r. Thus, we have shown that dimC (u) = 3r.
r r 2 r
Next, as N contains finitely many orbits we decompose
C (N) = V ∪G· x ,C (z(x )∩N) ∪G· x ,C (z(x )∩N)
r reg [2,2] r−1 [2,2] [2,1,1] r−1 [2,1,1]
where x[2,2] (or x[2,1,1]) is a(cid:0)representative in the orbi(cid:1)t of pa(cid:0)rtition [2,2] (or [2,1,1]). He(cid:1)nce, any
irreducible component of C (N), other than V , lies in one of the last two subvarieties. On the
r reg
other hand, by similar argument as in [Pr, Proposition 2.1], we have GL (k) acting on C (N) as
r r
follows
a a ··· a
11 12 1r
r r
a a ··· a
 2:1 2:2 : 2:r •(x1,...,xr) = a1ixi,..., arixi
!
i=1 i=1
 ar1 ar2 ··· arr  X X
 
for all (x ,...,x) ∈ C (N). In particular, suppose V is an irreducible component of C (N). Then
1 r r r
any permutationof (x ,...,x )∈ V isalso inV. Thisindicates thatif V ⊆ G·(x,C (z(x)∩N))
1 r r−1
then V must be in C (O ). Hence, the dimension of C (N) is in fact the maximum of the set
r x r
{dimV ,dimC (O ), dimC (O )}. We already know that the first amount is 2r +6 (by
reg r [2,2] r [2,1,1]
Proposition 3.1.1). Since C (O ) ⊂ C (O ), and dimC (O )= dimC (O )= 3(r+1), we
r [2,1,1] r [2,2] r [2,2] r 2
finally obtain
dimC (N) = max{2r+6,3r+3}.
r
This completes our proof. (cid:3)
ON NILPOTENT COMMUTING VARIETIES AND COHOMOLOGY OF FROBENIUS KERNELS 9
4. Structure of the cohomology ring of Frobenius kernels
We keep assuming that p is a good prime for G. For each r ≥ 1, let
H•(G ,k) = Hi(G ,k) , H2•(G ,k) = H2i(G ,k)
r r r r
i≥0 i≥0
M M
where the latter is usually called the even cohomology ring of G . It is well known that H•(G ,k)
r r
is a graded commutative k-algebra. This section is aimed to answer the question on whether or
not this G -cohomology ring is Cohen-Macaulay. This question is motivated from the conjectures
r
in [Ngo1, §7.2]. Explicitly, we show that H•((SL ) ,k) is Cohen-Macaulay for all r ≥ 1. Some
2 r
relation about Cohen-Macaulayness of U - and B -cohomology rings is also obtained. Finally, we
r r
apply some properties of commuting varieties in the previous section to point out the values of r
for which the ring H•(G ,k) is not Cohen-Macaulay.
r
We begin by recalling some distinguished features of a Cohen-Macaulay graded commutative
ring.
Proposition 4.0.3. [Ben, Proposition 2.5.1] Let R = R be a finitely generated graded com-
i≥0 i
mutative k-algebra. Then the following are equivalent.
L
(a) R is Cohen-Macaulay
(b) There existsahomogeneous polynomial subring k[x ,...,x ]suchthat R isfinitelygenerated
1 r
free module over k[x ,...,x ].
1 r
(c) Ifk[x ,...,x ]isahomogeneous polynomial subringofR overwhichRisafinitelygenerated
1 r
module then R a free module over it.
We use these equivalances as a main tool to prove Cohen-Macaulayness of a cohomology ring.
4.1. Cohomology ring of (SL ) . Assume only in this part that G = SL . We prove that the
2 r 2
cohomology ringH•(G ,k)isCohen-Macaulay foreachr ≥ 1. Thissignificantlyimprovesaresultof
r
the author in [Ngo1] where he showed that the commutative ring H2•(G ,k) is Cohen-Macaulay.
r red
We first need a lemma.
Lemma 4.1.1. Let S be a k-algebra on which B acts as algebra automorphisms1. Suppose that U
acts trivially on S, under the action of B, and S is regular as a commutative ring. Then the ring
indGS is Cohen-Macaulay.
B
Proof. We have
indGS = (k[G]⊗S)B
B
∼= (k[G]⊗S)U B/U
∼= (cid:2)k[G]U ⊗S T(cid:3)
Now it is not hard to compute that k[G]U is(cid:0)in fact a p(cid:1)olynomial ring over 2 variables, see [Po,
2.1], so that it is a regular ring. As tensoring preserves regularity, we get k[G]U⊗S is regular. Now
since T is linearly reductive, the main result of Hochster-Robert in [HR] implies that the invariant
ring [k[G]U ⊗S]T is Cohen-Macaulay; hence completing our proof. (cid:3)
Remark 4.1.2. The Lemma 4.1.1 would not hold if the ring S was just Cohen-Macaulay. In fact,
Hochster gave an explicit example in [Ho, p. 900] for a more general fact, that is, the invariant
subring of a Cohen-Macaulay ring under a torus action is not Cohen-Macaulay. As a consequence,
it is not true in general that the ring indGR is Cohen-Macaulay provided that R is so. In other
B
words,thisprovidesacounterexampleforConjecture7.2.1 inanearlier paperoftheauthor[Ngo1].
It remains interesting to know under what conditions the conjecture holds.
1Weusually call such S a B-algebra
10 NHAMV.NGO
Now we can tackle the Cohen-Macaulayness of the G -cohomology ring.
r
Theorem 4.1.3. For each r ≥ 1, the ring H•(G ,k) is Cohen-Macaulay, when G= SL .
r 2
Proof. First recall from [Ngo1, Corollary 4.1.2] that the cohomology ring H•(U ,k) can be consid-
r
ered as a free module over the polynomial ring k[x ,...,x ] where each x is of degree 2. Then
1 r i
a slight modification for Theorem 6.1.2 in [Ngo1] would give us that H•(B ,k) is a free module
r
pr−1 pr−2
over R = k[x ,x ,...,x ]. In other words, there is an isomorphism of R-modules as well as
1 2 r
B-modules as follows
H•(B ,k) ∼= v⊗R
r
v∈B
M
where B is the set of independent generators of H•(B ,k) as an R-module.
r
Following the strategy in [Ngo1, Theorem 4.3.1], one obtains the following isomorphism of alge-
bras as well as G-modules
H•(G ,k)(−r) ∼= indG(v⊗R)(−r).
r B
v∈B
M
Nowitiseasytoseethateach gradedk-algebraindG(v⊗R)(−r) isthegradedk-algebraindG(R(−r))
B B
shifted by deg(v). Hence it suffices to prove that the latter is Cohen-Macaulay.
As U trivially acts on u∗, it does the same on H•(U ,k). Since R ⊂ H•(U ,k), U acts trivially on
r r
R and so on R(−r). Now applying the lemma above, we get indG(R(−r)) is Cohen-Macaulay. Thus
B
the theorem follows from the fact that it is the direct sum of copies of indG(R(−r)). (cid:3)
B
The Poincar´e series associated to the cohomology ring H•(G ,k) is denoted by
r
p (t) = dimHi(G ,k)ti.
Gr r
i≥0
X
Next as a consequence of Theorem 4.1.3, one immediately has
Corollary 4.1.4. Suppose G = SL . For each r ≥ 1, the Poincar´e series p (t) associated to
2 Gr
H•(G ,k) satisfies the Poincar´e duality, i.e.
r
p (1/t) = (−t)dp (t).
Gr Gr
Proof. Follows immediately from [ES, Theorem 12]. (cid:3)
4.2. Cohen-Macaulayness of U and B -cohomology. We are back to the assumption that
r r
G is a classical simple algebraic group. We show here that the Cohen-Macaulayness of H•(U ,k)
r
implies the same for H•(B ,k).
r
Theorem 4.2.1. Let G be a connected, reductive group and r ≥ 1. If the algebra H•(U ,k) is a
r
Cohen-Macaulay ring, then so is H•(B ,k).
r
Proof. By Proposition 4.0.3, suppose H•(U ,k) is a free module over a polynomial ring R. In
r
particular, let B be a basis of H•(U ,k) over R, we then have
r
H•(U ,k) = b∪R.
r
b∈B
M
Now for each r ≥ 1, we have
H•(B ,k) ∼= H•(U ,k)Tr
r r
∼= (b∪R)Tr
b∈B
M
Each direct summand in the last item can be considered as a graded algebra RTr shifted by deg(b).
On the other hand, RTr is Cohen-Macaulay by the main theorem of Hochster-Roberts in [HR]. So
H•(B ,k) is a direct sum of Cohen-Macaulay rings; hence it is so. (cid:3)
r