COMMUTING VARIETIES OF r-TUPLES OVER
LIE ALGEBRAS

arXiv:1209.1659v2 [math.RT] 7 Nov 2013

NHAM V. NGO

Abstract. Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p and let g be the Lie algebra of G. It is well known that for p large enough the spectrum
of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r-tuples of elements in the nilpotent cone of g [Suslin-Friedlander-Bendel, J. Amer. Math.
Soc, 10 (1997), 693–728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties
Cr (gl2 ), Cr (sl2 ) and Cr (N ) where N is the nilpotent cone of sl2 . Our calculations lead us to state
a conjecture on Cohen-Macaulayness for commuting varieties of r-tuples. Furthermore, in the case
when g = sl2 , we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of sl3 , we are able to verify the aforementioned properties for
Cr (u). Finally, applying our calculations on the commuting variety Cr (Osub ) where Osub is the
closure of the subregular orbit in sl3 , we prove that the nilpotent commuting variety Cr (N ) has
singularities of codimension ≥ 2.

1. Introduction
1.1. Let k be an algebraically closed field of characteristic p (possibly p = 0). For a Lie algebra
g over k and a closed subvariety V of g, the commuting variety of r-tuples over V is defined as
the collection of all r-tuples of pairwise commuting elements in V . In particular, for each positive
integer r, we define
Cr (V ) = {(v1 , . . . , vr ) ∈ V r | [vi , vj ] = 0 for all 1 ≤ i ≤ j ≤ r}.
When r = 2, we call Cr (V ) an ordinary commuting variety (cf. [Vas]). Let N be the nilpotent
cone of g. We then call Cr (N ) the nilpotent commuting variety of r-tuples. For simplicity, from
now on we will call Cr (V ) a commuting variety over V whenever r > 2 in order to distinguish it
from ordinary commuting varieties.
The variety C2 (g) over a reductive Lie algebra g was first proved to be irreducible in characteristic
0 by Richardson in 1979 [R]. In positive characteristics, Levy showed that C2 (g) is irreducible under
certain mild assumptions on G [L]. In 2003, Premet completely determined irreducible components
of the nilpotent commuting variety C2 (N ) for an arbitrary reductive Lie algebra in characteristic 0
or p, a good prime for the root system of G. In particular, he showed that the nilpotent commuting
variety C2 (N ) is equal to the union of irreducible components C(ei ) = G · (ei , Lie(Z[G,G] (ei ))), where
the ei ’s are representatives for distinguished nilpotent orbits in g (which contain elements whose
centralizers are in the nilpotent cone).
Up to now, the properties of being Cohen-Macaulay and normal for ordinary commuting varieties
are in general not verified [K]. There is a long-standing conjecture stating that the commuting
variety C2 (g) is always normal (see [Po] and [Pr]). Artin and Hochster claimed that C2 (gln ) is
a Cohen-Macaulay integral domain (cf. [K],[MS]). This is verified up to n = 4 by the computer
program Macaulay [Hr]. There is not much hope for verifying Cohen-Macaulayness of nilpotent
commuting varieties since their defining ideals are not radical, thus creating great difficulties for
computer calculations. As pointed out by Premet, all components of C2 (N ) share the origin 0, so
if it is reducible then it can never be normal.
1

2

NHAM V. NGO

Not much is known about commuting varieties. In case V = gln , commuting varieties were
studied by Gerstenhaber, Guralnick-Sethuraman, and Kirillov-Neretin. Gerstenhaber proved that
Cr (gln ) is reducible for all n ≥ 4 and r ≥ 5 [G]. In 1987, Kirillov and Neretin lowered the bound
on r by simply showing that C4 (gl4 ) is reducible. Moreover, they proved that when n = 2 or 3 the
commuting variety Cr (gln ) is irreducible for all r ≥ 1 [KN]. In 2000, Guralnick and Sethuraman
studied the case when r = 3 and concluded that C3 (gln ) can be either irreducible (for n ≤ 10) or
reducible (for n ≥ 30) [GS], [S]. In general, the study of Cr (V ) for arbitrary r remains mysterious
even in simple cases.
1.2. Main results. The results in the present paper were motivated by investigating the cohomology for Frobenius kernels of algebraic groups. To be more precise, let G be an algebraic group
defined over k, and let Gr be the r-th Frobenius kernel of G. Then there is a homeomorphism
between the maximal ideal spectrum of the cohomology ring for Gr and the nilpotent commuting
variety over the Lie algebra Lie(G) whenever the characteristic p is large enough [SFB1, SFB2].
In this paper, we tackle the irreducibility, normality and Cohen-Macaulayness of this variety for
simple cases. The main results are summarized in the following.
Theorem 1.2.1. Suppose p > 2. Then for each r ≥ 1, the commuting varieties Cr (gl2 ), Cr (sl2 )
and Cr (N ) are irreducible, normal and Cohen-Macaulay.
For readers’ convenience, we sketch the structure of the paper as follows. We first introduce
notation, terminology and conventions in Sections 2 and 3. We also prove several properties related
to the map m : G ×B Cr (u) → Cr (N ) where u is the Lie algebra of the unipotent radical of a fixed
Borel subgroup B of G, and N is the nilpotent cone of g. For instance, we show that the map is
surjective and satisfies the hypotheses of Zariski’s Main Theorem. These results are analogous to
those for the moment map from G×B u to N . In the next section, we first show a connection between
Cr (gln ) and Cr (sln ) for arbitrary n, r ≥ 1. This link reduces all of the works for Cr (gln ) to that for
Cr (sln ). Then we consider the case n = 2 and prove the properties of being irreducible, normal,
and Cohen-Macaulay for both varieties by exploiting a fact of determinantal rings. In addition, the
analogs are shown for the nilpotent commuting variety over rank 2 matrices in Section 5. It should
be noticed that the nilpotency condition makes the defining ideal of this variety non-radical, thus
creating more difficulties in our task. In the case when g = sl2 , to obtain the Cohen-Macaulayness
of Cr (N ), we first analyze the geometry by intersecting Cr (N ) with a hypersurface and reduce the
problem to showing that a certain class of ideals are radical. Then we prove that this family of
ideals belongs to a principal radical system, a deep concept in commutative algebra introduced by
Hochster [BV]. As a consequence, we show that the moment map m : G ×B ur → Cr (N ) admits
rational singularities (cf. Proposition 5.3.1).
As an application we can compute the characters of the coordinate algebra for this variety (cf.
Theorem 5.4). Combining this with an explicit calculation for the reduced Gr -cohomology ring in
[N], we obtain an alternative proof for the main result in [SFB1] for the case when G = SL2 .
In Section 6, we study commuting varieties with additional restrictions on each tuple. In particular, let V1 , . . . , Vr be closed subvarieties of a Lie algebra g. Define
C(V1 , . . . , Vr ) = {(v1 , . . . , vr ) ∈ V1 × · · · × Vr | [vi , vj ] = 0 , 1 ≤ i ≤ j ≤ r},

a mixed commuting variety over V1 , . . . , Vr . In the case where g = sl2 , we can explicitly describe
the irreducible decomposition for any mixed commuting variety over g and N . This shows that
such varieties are mostly not Cohen-Macaulay or normal.
Section 7 involves results about the geometric structure of nilpotent commuting varieties over
various sets of 3 by 3 matrices. In particular, let N be the nilpotent cone of sl3 . We first study
the variety Cr (u) and then obtain the irreducibility and dimension of Cr (N ). Next we apply our
calculations on Cr (Osub ) to classify singularities of Cr (N ) and show that they are in codimension
greater than or equal to 2 (cf. Theorem 7.2.3), here Osub is the closure of the subregular orbit in

COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

3

N . This result indicates that the variety Cr (N ) satisfies the necessary condition (R1) of Serre’s
criterion for normality.
2. Notation
2.1. Root systems and combinatorics. Let k be an algebraically closed field of characteristic
p. Let G be a simple, simply-connected algebraic group over k, defined and split over the prime
field Fp . Fix a maximal torus T ⊂ G, also split over Fp , and let Φ be the root system of T in G. Fix
a set Π = {α1 , . . . , αn } of simple roots in Φ, and let Φ+ be the corresponding set of positive roots.
Let B ⊆ G be the Borel subgroup of G containing T and corresponding to the set of negative roots
Φ− , and let U ⊆ B be the unipotent radical of B. Write U + ⊆ B + for the opposite subgroups. Set
g = Lie(G), the Lie algebra of G, b = Lie(B), u = Lie(U ).
2.2. Nilpotent orbits. We will follow the same conventions as in [Hum2] and [Jan]. Given a
G-variety V and a point v of V , we denote by Ov the G-orbit of v (i.e., Ov = G · v). For example,
consider the nilpotent cone N of g as a G-variety with the adjoint action. There are well-known
orbits: Oreg = G · vreg , Osubreg = G · vsubreg , (we abbreviate it by Osub ,) and Omin = G · vmin
where vreg , vsubreg , and vmin are representatives for the regular, subregular, and minimal orbits.
Denote by z(v) and Z(v) respectively the centralizers of v in g and G. It is well-known that
dim z(v) = dim Z(v) and dim Ov = dim G − dim z(v). For convenience, we write zreg (zsub and
zmin ) for the centralizers of vreg (vsub or vmin ). It is also useful to keep in mind that every orbit is
a smooth variety. Sometimes, we use OV for the structure sheaf of a variety V (see [CM], [Hum2]
for more details).
2.3. Basic algebraic geometry conventions. Let
commutative Noetherian ring with
√ R be a √
identity. We use Rred to denote the reduced ring R/ 0 where 0 is the radical ideal of the trivial
ideal 0, which consists of all nilpotent elements of R. Let Spec R be the spectrum of all prime
ideals of R. If V is a closed subvariety of an affine space An , we denote by I(V ) the radical ideal
of k[An ] = k[x1 , . . . , xn ] associated to this variety.
Given a G-variety V , B act freely on G × V by setting b · (g, v) = (gb−1 , bv) for all b ∈ B, g ∈ G
and v ∈ V . The notation G ×B V stands for the fiber bundle associated with the projection
π : G ×B V → G/B with fiber V . Topologically, G ×B V is the quotient space of G × V in which
the equivalence relation is given as
(g, v) ∼ (g ′ , v ′ ) ⇔ (g′ , v ′ ) = b · (g, v) for some b ∈ B.

In other words, each equivalence class of G ×B V represents a B-orbit in G × V . The map m :
G ×B V → G · V defined by mapping each equivalence class [g, v] to the element g · v for all
g ∈ G, v ∈ V is called the moment morphism. It is obviously surjective. Let X be an affine variety.
Then we always write k[X] for the coordinate ring of X which is the same as the ring of global
sections OX (X). Although G ×B V is not affine, we still denote k[G ×B V ] for its ring of global
sections. It is sometimes useful to make the identification k[G ×B V ] ∼
= k[G × V ]B .
Let f : X → Y be a morphism of varieties. Denote by f∗ the direct image functor from the
category of sheaves over X to the category of sheaves over Y . One can see that this is a left exact
functor. Hence, we have the right derived functors of this direct image. We call these functors
higher direct images and denote them by Ri f∗ with i > 0. In particular, if Y = Spec A is affine
and F is a quasi-coherent sheaf on X, then we have Ri f∗ (F) ∼
= L(Hi (X, F)) where L is the exact
functor mapping an A-module M to its associated sheaf L(M ). Here we follow conventions in [H].
3. Commutative algebra and Geometry
In this section we introduce concepts in commutative algebra and geometry that play important
roles in the later sections. In particular, we recall the definition of Cohen-Macaulay varieties and
their properties. We also show that the moment map G ×B Cr (u) → Cr (N ) is always surjective

4

NHAM V. NGO

for arbitrary type of G, and that it is a proper birational morphism. We then review a number of
well-known results in algebraic geometry.
3.1. Cohen-Macaulay Rings. We first define regular sequences, which are the key ingredient in
the definition of Cohen-Macaulay rings. Readers can refer to [E] and [H] for more details.
Definition 3.1.1. Let R be a commutative ring and let M be an R-module. A sequence x1 , . . . , xn ∈
R is called a regular sequence on M (or an M -sequence) if it satisfies
(1) (x1 , . . . , xn )M 6= M , and
(2) for each 1 ≤ i ≤ n, xi is not a zero-divisor of M/(x1 , . . . , xi−1 )M .
Consider R as a left R-module. For a given ideal I of R, it is well-known that the length of any
maximal regular sequence in I is unique. It is called the depth of I and denoted by depth(I). The
height or codimension of a prime ideal J of R is the supremum of the lengths of chains of prime
ideals descending from J. Equivalently, it is defined as the Krull dimension of R/J. In particular, if
R is an integral domain that is finitely generated over a field, then codim(J) = dim R − dim(R/J).
We are now ready to define a Cohen-Macaulay ring.
Definition 3.1.2. A ring R is called Cohen-Macaulay if depth(I) = codim(I) for each maximal
ideal I of R. A variety V is called Cohen-Macaulay if its coordinate ring k[V ] is a Cohen-Macaulay
ring.
Example 3.1.3. Smooth varieties are Cohen-Macaulay. The nilpotent cone N of a simple Lie
algebra g over an algebraically closed field of good characteristic is Cohen-Macaulay [Jan, 8.5].
3.2. Minors and Determinantal rings. Let U = (uij ) be an m × n matrix over a ring R. For
indices a1 , . . . , at , b1 , . . . , bt such that 1 ≤ ai ≤ m, 1 ≤ bi ≤ n, i = 1, . . . , t, we put


ua1 b1 · · · ua1 bt

..  .
..
[a1 , . . . , at | b1 , . . . , bt ] = det  ...
.
. 
uat b1 · · · uat bt

We do not require that a1 , . . . , at and b1 , . . . , bt are given in ascending order. Note that
[a1 , . . . , at | b1 , . . . , bt ] = 0

if t > min(m, n). To be convenient, we let [∅ | ∅] = 1. If a1 ≤ · · · ≤ at and b1 ≤ · · · ≤ bt we call
[a1 , . . . , at | b1 , . . . , bt ] a t-minor of U .
Definition 3.2.1. Let B be a commutative ring, and consider an m × n matrix


x11 · · · x1n

.. 
..
X =  ...
.
. 
xm1 · · · xmn

whose entries are independent indeterminates over R. Let R(X) be the polynomial ring over all
the indeterminates of X, and let It (X) be the ideal in R(X) generated by all t-minors of X. For
each t ≥ 1, the ring
R(X)
Rt (X) =
It (X)
is called a determinantal ring.
For readers’ convenience, we recall nice properties of determinantal rings as follows.

Proposition 3.2.2. [BV, 1.11, 2.10, 2.11, 2.12] If R is a reduced ring, then for every 1 ≤ t ≤
min(m, n), Rt (X) is a reduced, Cohen-Macaulay, normal domain of dimension (t−1)(m+n−t+1).

COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

5

3.3. The moment morphism. Suppose G is a simple algebraic group. It is well-known that
N = G · u where u is the Lie algebra of the unipotent radical subgroup U of the Borel subgroup B
of G, and the dot is the adjoint action of G on the Lie algebra g. Note that if u1 , u2 are commuting
in u, then so are g · u1 , g · u2 in N for each g ∈ G. This observation can be generalized to give the
following moment map
m : G ×B Cr (u) → Cr (N )

by setting m[g, (u1 , . . . , ur )] = (g · u1 , . . . , g · ur ) for all g ∈ G, and (u1 , . . . , ur ) ∈ Cr (u). In the
case when r = 1, this is the moment map in the Springer resolution. Therefore, we also call it the
moment morphism for each r ≥ 1. The following proposition shows surjectivity of this morphism.
Theorem 3.3.1.

1

The moment morphism m : G ×B Cr (u) → Cr (N ) is always surjective.

Proof. Suppose (v1 , . . . , vr ) ∈ Cr (N ). Let b′ be the vector subspace of g spanned by the vi . As
[vi , vj ] = 0 for all 1 ≤ i, j ≤ r, b′ is an abelian, hence solvable, Lie subalgebra of g. Thus, there
exists a maximal solvable subalgebra b′′ of g containing b′ . By [Hum1, Theorem 16.4], b′′ and our
Borel subalgebra b are conjugate under some inner automorphism Ad(g) with g ∈ G. So there exist
u1 , . . . , ur ∈ b such that
(v1 , . . . , vr ) = Ad(g −1 )(u1 , . . . , ur ) = g−1 · (u1 , . . . , ur ) = m[g−1 , (u1 , . . . , ur )].

As all the vi are nilpotent and commuting, so are the ui . This shows that m is surjective.



As a corollary, we establish the connection between irreducibility of Cr (u) and Cr (N ).
Theorem 3.3.2. For each r ≥ 1, if Cr (u) is irreducible then so is Cr (N ).
Proof. As the moment morphism G × Cr (u) → Cr (N ) is surjective and G is irreducible, the irreducibility of Cr (N ) follows from that of Cr (u).

3.4. Zariski’s Main Theorem. Zariski’s Main Theorem is one of the powerful tools to study
structure sheaves of two schemes. In this subsection, we state the version for varieties and show that
the moment map in the preceding subsection satisfies the hypotheses of Zariski’s Main Theorem.
We first look at proper morphisms. As defining properness requires terminology from algebraic
geometry, we refer readers to [H, Section II.4] for the details. We only introduce some important
characterizations of proper morphisms which will be useful later.
Proposition 3.4.1. In the following properties, all the morphisms are taken over Noetherian
schemes.
(a) A closed immersion is proper.
(b) The composition of two proper morphims is proper.
(c) A projection X × Y → X is proper if and only if Y is projective.
We also recall that a rational map ϕ : X → Y (which is a morphism only defined on some open
subset) is called birational if it has an inverse rational map.
There are many versions of Zariski’s Main Theorem. Here we state a “pre-version” of the
theorem since the Main Theorem immediately follows from this result (cf. [H, Corollary III.11.3
and III.11.4]).
Theorem 3.4.2. Let f : X → Y be a birational proper morphism of varieties and suppose Y is
normal. Then f∗ OX = OY .
We now verify that the morphism m : G ×B Cr (u) → Cr (N ) satisfies the hypotheses of Zariski’s
Main Theorem. In other words, we have
1The author would like to thank Christopher M. Drupieski for the main idea in this proof

6

NHAM V. NGO

Proposition 3.4.3. For each r ≥ 1, the moment morphism m : G ×B Cr (u) → Cr (N ) is birational
proper.
Proof. We generalize the proofs of Lemmas 1 and 2 in [Jan, 6.10]. For the properness, we consider
the map
ǫ : G ×B Cr (u) ֒→ G/B × Cr (N )

with ǫ[g, (u1 , . . . , ur )] = (gB, g · (u1 , . . . , ur )) for all g ∈ G and (u1 , . . . , ur ) ∈ Cr (u). By the same
argument as in [Jan, 6.4], we can show that this map is a closed embedding; hence a proper
morphism by Proposition 3.4.1(a). Next, as G/B is projective, the projection map p : G/B ×
Cr (N ) → Cr (N ) is proper by Proposition 3.4.1(c). Therefore, part (b) of Proposition 3.4.1 implies
that m = p ◦ ǫ is also proper.
Consider the projection of Cr (N ) onto the first factor p1 : Cr (N ) → N . Recall that zreg is the
centralizer of a fixed regular element vreg in N . From Lemma 35.6.7 in [TY], zreg is a commutative
Lie algebra. Then we have
p−1
1 (Oreg ) = C(Oreg , N , . . . , N ) = G · (vreg , zreg , . . . , zreg ).

−1
As Oreg is an open subset in N , the preimage p−1
1 (Oreg ) is open in Cr (N ). Let V = p1 (Oreg ).
Since ZG (vreg ) ⊆ B, we have m induces an isomorphism from m−1 (V ) onto V . It follows that m is
a birational morphism.


Remark 3.4.4. We have not shown that m satisfies all the hypotheses of Zariski’s Theorem since
the normality for Cr (N ) is still unkown. As mentioned in Section 1, the variety Cr (N ) is normal
only when it is irreducible. In particular, Premet already proved that C2 (N ) is irreducible if and
only if G is of type A [Pr]. By considering the natural projection map Cr (N ) → C2 (N ), it follows
that if G is not of type A then Cr (N ) is reducible for each r ≥ 2. In this paper we prove for
arbitrary r ≥ 1 that the variety Cr (N ) is irreducible for types A1 and A2 . The result for type An
with arbitrary n, r > 2 remains an open problem.
Conjecture 3.4.5. If G is of type A, then Cr (N ) is irreducible. Moreover, it is normal. In other
words, the morphism
m : G ×B Cr (u) → Cr (N )
satifies all the hypotheses of Zariski’s Theorem.

3.5. Singularities and Resolutions. Here we state an observation on determinning the singular
points of an affine variety defined by homogeneous polynomials, and then define a resolution of
singularities.
Proposition 3.5.1. Let V be an affine variety whose defining radical ideal is generated by a nonempty set of homogeneous polynomials of degree at least 2. Then 0 is always a singular point of
V.
Proof. Suppose V is an affine subvariety of the affine space Am associated with the coordinate ring
k[x1 , . . . , xm ]. Let f1 , . . . , fn be the set of polynomials defining V . Note that dim V < m since
n ≥ 1. Consider the Jacobian matrix

 df1
· · · xdfm1
x1
 ..
.. 
..
 .
.
. 
dfn
x1

···

dfn
xm

i
As all the fi are homogeneous of degree ≥ 2, we have df
xj (0, . . . , 0) = 0 for all 1 ≤ i ≤ n, 1 ≤ j ≤ m.
It follows that the tangent space at 0 has dimension m which is greater than dim V . Thus 0 is a
singular point of V .


COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

7

Definition 3.5.2. A variety X has a resolution of singularities if there exists a non-singular variety
Y such that there is a proper birational morphism from Y to X.
Definition 3.5.3. A variety X has rational singularities if it is normal and has a resolution of
singularities
f :Y →X
i
such that the higher direct image R f∗ OY vanishes for i ≥ 1. (Sometimes one calls f a rational
resolution.)
In Lie theory, the nullcone N admits a resolution of singularites, the Springer resolution, and
also has rational singularities. One of our goals in the present paper is to generalize this resolution
for the nilpotent commuting variety over rank two Lie algebra.
4. Commuting Varieties over 2 by 2 matrices
4.1. Recall that the problem of showing that the commuting variety over n × n matrices is CohenMacaulay and normal is very difficult to verify. With ordinary commuting varieties, computer
verification works up to n = 4 [Hr]. There are also some studies on the Cohen-Macaulayness of
other structures closely similar to ordinary commuting varieties by Knutson, Mueller, ZolbaninSnapp and Zoque (cf. [K],[Mu], [MS], [Z]). Very little appears to be known for commuting varieties
in general. In this section, we confirm the properties of being Cohen-Macaulay and normal for
Cr (gl2 ) and Cr (sl2 ) with arbitrary r ≥ 1.
4.2. Nice properties of Cr (gl2 ) and Cr (sl2 ). We first show a general result connecting the
commuting varieties over gln and sln .
Theorem 4.2.1. 2 For each n and r ≥ 1, if p does not divide n, then there is an isomorphism of
varieties from Cr (gln ) to Cr (sln ) × Ar defined by setting


Tr(vr )
Tr(v1 )
(1)
I n , . . . , vr −
In × (Tr(v1 ), . . . , Tr(vr ))
ϕ : (v1 , . . . , vr ) 7→ v1 −
n
n
for vi ∈ gln .
Proof. It is easy to see that adding or subtracting cIn from the vi does not change the commuting
conditions on the vi . So the morphism ϕ in (1) is well-defined and its inverse is

a1
ar 
ϕ−1 : (u1 , . . . , ur ) × (a1 , . . . , ar ) 7→ u1 + In , . . . , ur + In .
n
n
This completes the proof.


This result implies that our work for Cr (gl2 ) will be done if we can prove that Cr (sl2 ) is CohenMacaulay. Notice that Popov proved the normality of this variety in the case r = 2 [Po, 1.10].
However, his proof depends on computer calculations to verify that the defining ideal is radical.
Here we propose another approach that completely solves the problem for arbitrary r. Let slr2 be
the affine space defined as





x1 y 1
xr y r
,...,
| xi , yi , zi ∈ k, 1 ≤ i ≤ r .
z1 −x1
zr −xr

Then the variety Cr (sl2 ) can be defined as the zero locus of the following ideal
(2)

J = hxi yj − xj yi , yi zj − yj zi , xi zj − xj zi | 1 ≤ i ≤ j ≤ ri.

Proposition 4.2.2. For each r ≥ 1, the variety Cr (sl2 ) is:
(a) irreducible of dimension r + 2,
(b) Cohen-Macaulay and normal.
2The author would like to thank William Graham for his assistance in generalizing the result to gl .
n

8

NHAM V. NGO

Proof. It was shown by Kirillov and Neretin that Cr (gl2 ) and Cr (gl3 ) are irreducible for all r in
characteristic 0, [KN, Theorem 4]. The irreducibility of Cr (sl2 ) in characteristic 0 then follows by
Theorem 4.2.1. Here we provide a general proof for any characteristic p 6= 2.
Consider the ideal J above as the ideal I2 (X ) generated by all 2-minors of the following matrix


x1 x2 · · · xr


X =  y1 y2 . . . yr  .
z1 z2 · · · zr

)
. It follows immediately from PropoThen we can identify k[Cr (sl2 )] with the ring R2 (X ) = Ik(X
2 (X )
sition 3.2.2 that R2 (X ) is a Cohen-Macaulay and normal domain, hence completing the proof. 

Corollary 4.2.3. Suppose p 6= 2. Then for each r ≥ 1, the variety Cr (gl2 ) is
(a) irreducible of dimension 2r + 2,
(b) Cohen-Macaulay and normal.
Proof. Follows immediately from Theorem 4.2.1 and Proposition 4.2.2.



This computation allows us to state a conjecture about Cohen-Macaulayness for commuting
varieties which is a generalization of that for ordinary commuting varieties.
Conjecture 4.2.4. Suppose p ∤ n. Then both commuting varieties Cr (gln ) and Cr (sln ) are CohenMacaulay.
5. Nilpotent Commuting Varieties over sl2
With the nilpotency condition, problems involving commuting varieties turn out to be more
difficult. The irreducibility of ordinary nilpotent commuting varieties was studied by Baranovsky,
Premet, Basili and Iarrobino (cf. [Ba], [Pr], [B], [BI]). However, there has not been any successful
work on normality and Cohen-Macaulayness even in simple cases. In this section, we completely
prove this conjecture for the nilpotent commuting variety over sl2 .
5.1. Irreducibility. Let G = SL2 , g = sl2 , and k be an algebraically closed field of characteristic
p 6= 2. The nilpotent cone of g then can be written as follows.



x y
2
| x + yz = 0 with x, y, z ∈ k
N =
z −x
Note that for each r ≥ 1, Cr (u) = ur , so that the moment map in Section 3.3 can be rewritten as
(3)

m : G ×B ur → Cr (N ).

As G/B and ur are smooth varieties, so is the vector bundle G ×B ur . In addition, the moment map
m is known to be proper birational from Proposition 3.4.3. It follows that G ×B ur is a resolution
of singularities for Cr (N ) through the morphism m. We will see later in this section that m is in
fact a rational resolution. First we study geometric properties of Cr (N ).
Proposition 5.1.1. For every r ≥ 1, we have
(a) Cr (N ) is an irreducible variety of dimension r + 1.
(b) The only singular point in Cr (N ) is the origin 0.
Proof.
(a) The first part follows immediately from the surjectivity of m in (3) and the fact that G ×B ur
is irreducible. Now since G ×B ur is a vector bundle over the base G/B with each fiber isomorphic
to ur , we have
dim G ×B ur = dim G/B + dim ur = r + 1.
As m is birational, Cr (N ) has the same dimension as G ×B ur .

COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

9

(b) By Proposition 3.5.1 we immediately get that 0 is a singular point of the variety. It is enough
to show that every non-zero element in Cr (N ) belongs to a smooth open subset of dimension r + 1.
Let 0 6= v = (v1 , . . . , vr ) ∈ Cr (N ). We can assume that v1 6= 0 ∈ N . Then considering the
projection on the first factor p : Cr (N ) → N , we see that v ∈ G · (v1 , ur−1 ) = p−1 (G · v1 ) which is
open in Cr (N ) as G · v1 is the regular orbit in N . Now we define an action of the reductive group
G × kr−1 on Cr (N ) as follows:

G × kr−1 × Cr (N ) → Cr (N )
(g , a1 , . . . , ar−1 ) • (v1 , . . . , vr ) 7−→ g · (v1 , a1 v2 , . . . , ar−1 vr ).

It is easy to see that p−1 (G · v1 ) = G × kr−1 • (v1 , v1 , . . . , v1 ). As every orbit is itself a smooth
variety, we obtain p−1 (G · v1 ) is smooth of dimension r + 1.

5.2. Cohen-Macaulayness. We denote by ∩∗ the scheme-theoretic intersection in order to distinguish with the regular intersection of varieties. Before showing that Cr (N ) is Cohen-Macaulay, we
need some lemmas related to Cohen-Macaulay varieties. The first one is an exercise in [E, Exercise
18.13] (see also [BV, Lemma 5.15]).
Lemma 5.2.1. 3 Let X, Y be two Cohen-Macaulay varieties of the same dimension. Suppose the
scheme-theoretic intersection X ∩∗ Y is of codimension 1 in both X and Y . Then X ∩∗ Y is
Cohen-Macaulay if and only if X ∪ Y is Cohen-Macaulay.
The following lemmas involve properties about radical ideals that determine whether a schemetheoretic intersection ∩∗ coincides with the intersection of varieties ∩.

Lemma 5.2.2. Let I ⊳ k[x1 , . . . , xm ] be the radical ideal associated to a variety V in Am . Then the
variety V × 0 ⊂ Am × An is represented by the ideal I + hy1 , . . . , yn i ⊳ k[x1 , . . . , xr , y1 , . . . , yn ].
Proof. It is easy to see that we have an isomorphism of rings
k[x1 , . . . , xr , y1 , . . . , yn ] ∼ k[x1 , . . . , xr ]
=
I + hy1 , . . . , yn i
I

where the latter ring is reduced. The result immediately follows.



Lemma 5.2.3. Let I1 , I2 be radical ideals of k[x1 , . . . , xm ]. Let J1 , J2 be ideals of R = k[x1 , . . . , xm , y1 , . . . , yn ].
If I2 ⊆ I1 and J1 ⊆ J2 , then we have
p
p
I1 + J1 + I2 + J2 = I1 + J2
provided I1 + J2 is a radical ideal of R.
In particular, suppose V1 ⊆ V2 are varieties of Am and W is a variety of An . Then we have

(V1 × W ) ∩∗ (V2 × 0) = (V1 × W ) ∩ (V2 × 0) = V1 × 0.
√
√
√
Proof. It is
√ that I1 + J1 + I2 + J2 ⊆ I1 + J2 = I1 + J2 . On the other hand,
√ well-known
I1 + J2 ⊆ I1 + J1 + I2 + J2 . This shows
√ the first part of the lemma.
I1 + J1 with I1 = I(V1 ) and J1 is an ideal of R. Let
For the remainder,
let
I(V
×
W
)
=
1
√
I(V2 × 0) = I2 + J2 where I2 = I(V2 ), J2 = hy1 , . . . , yn i by the preceding lemma. Then we have
I1 + J2 is radical again by the previous lemma. It can be seen that J1 ⊆ J2 as an ideal of R. Hence
the first statement of the lemma implies that
R
= V1 × 0.
V1 × W ∩∗ V2 × 0 = Spec
I1 + J2

3In [E], although the author did not state the word “scheme-theoretic intersection”, we implicitly understand from
the exercise that the intersection needs to be scheme-theoretic.

10

NHAM V. NGO

For each r ≥ 1, the variety Cr (N ) is defined as the zero locus of the family of polynomials
{x2i + yi zi , xi yj − xj yi , xi zj − xj zi , yi zj − yj zi | 1 ≤ i ≤ j ≤ r}

in R = k[slr2 ] = k[xi , yi , zi | 1 ≤ i ≤ r]. It is easy to check by computer that this ideal is not
radical. This causes some difficulties for us to investigate the algebraic properties of this variety
like Cohen-Macaulayness. Let Ir be the radical ideal generated by the family of polynomials above
in R. We first reduce the problem to checking a certain condition in commutative algebra.
Lemma 5.2.4. If Is + hy1 + z1 i is a radical ideal for each s ≥ 1, then Cr (N ) is Cohen-Macaulay
for each r ≥ 1.

Proof. We argue by induction on r. When r = 1, C1 (N ) = N , which is a well-known CohenMacaulay variety. Suppose that Cr−1 (N ) is Cohen-Macaulay for some r ≥ 2. As we have seen
earlier Cr (N ) is irreducible of dimension r+1 and (y1 +z1 ) is not zero in the coordinate ring IRr . The
hypothesis and Lemma 5.15 in [BV] imply that it suffices to show the variety Cr (N ) ∩ V (y1 + z1 )
is Cohen-Macaulay of dimension r.
Let V = Cr (N ) ∩ V (y1 + z1 ). Solving from the constraint x21 + y1 z1 = 0, we have either
x1 = y1 = −z1 or x1 = −y1 = z1 . Then we decompose V = V1 ∪ V2 ∪ V3 where the Vi are irreducible
algebraic varieties defined by
V1 = V ∩ V (x1 = y1 = z1 = 0),
V2 = V ∩ V (x1 = y1 = −z1 6= 0),
V3 = V ∩ V (x1 = −y1 = z1 6= 0).

Moreover, we can explicitly describe these varieties as follows:
V1 = 0 × 0 × 0 × Cr−1 (N )
x2
x2
y1 ,
z1 , . . . , xr ,
V2 = {(x1 , y1 , z1 , x2 ,
x1
x1
= {(x1 , x1 , −x1 , x2 , x2 , −x2 , . . . , xr ,
x2
x2
V3 = {(x1 , y1 , z1 , x2 , y1 , z1 , . . . , xr+1
x1
x1
= {(x1 , −x1 , x1 , x2 , −x2 , x2 , . . . , xr ,

xr
xr
y1 , z1 ) | 0 6= x1 = y1 = −z1 ∈ k}
x1
x1
xr , −xr ) | xi ∈ k},
xr
xr
, y1 , z1 ) | 0 6= x1 = −y1 = z1 ∈ k}
x1
x1
−xr , xr ) | xi ∈ k}.

Observe that V1 is Cohen-Macaulay of dimension r by the inductive hypothesis, and V2 and V3
are affine r-spaces so they are Cohen-Macaulay. From Lemma 5.2.3, we have the scheme-theoretic
intersection
V1 ∩∗ V2 = {(0, 0, 0, x2 , x2 , −x2 , . . . , xr , xr , −xr ) | xi ∈ k, 2 ≤ i ≤ r}

which is an affine (r − 1)-space. Hence by Lemma 5.2.1 the union V1 ∪ V2 is a Cohen-Macaulay
variety of dimension r. Next we consider the scheme-theoretic intersection (V1 ∪ V2 ) ∩∗ V3 . Note
that V2 ∩∗ V3 = {0}, we have
(V1 ∪ V2 ) ∩∗ V3 = V1 ∩∗ V3 = {(0, 0, 0, x2 , −x2 , x2 , . . . , xr , −xr , xr ) | xi ∈ k}

for the same reason as earlier. Then again Lemma 5.2.1 implies that V1 ∪ V2 ∪ V3 is Cohen-Macaulay
of dimension r.

We are now interested in the conditions under which the sum of two radical ideals is again
radical. One of the well-known concepts in commutative algebra related to this problem is that of
a principal radical system introduced by Hochster, which shows certain class of ideals are radical.
Theorem 5.2.5. [BV, Theorem 12.1] Let A be a Noetherian ring, and let F be a family of ideals
in A, partially ordered by inclusion. Suppose that for every member I ∈ F one of the following
assumptions is fulfilled:

COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

11

(a) I is a radical ideal; or
(b) There exists an element x ∈ A such T
that I + Ax ∈ F and
A
i
√
• x is not a zero-divisor I and ∞
i=0 (I + Ax )/I = 0, or
• there exists an ideal J ∈ F with I ( J, such that xJ ⊆ I and x is not a zero-divisor of
A
√
.
J
Then all the ideals in F are radical ideals.
Such a family of ideals is called a principal radical system. This concept plays an important role
in the proof of Hochster and Eagon showing that determinantal rings are Cohen-Macaulay [HE].
Before applying this theorem, we need to set up some notations.
Fix r ≥ 1. For each 1 ≤ m ≤ r, let Im be the radical ideal associated to the variety 0 × · · · × 0 ×
Cm (N ) ⊆ Cr (N ) with 0 ∈ N . Each ideal Im is prime by the irreducibility of Cm (N ) and it is easy
to see that
r−m
X
hxj , yj , zj i .
Im = Ir +
j=1

We also let, for each 1 ≤ m ≤ r,
r
m
X
X
hxj − yj , yj + zj i .
hxi , yi , zi i +
Pm =
j=m+1

i=1

Note that each Pm is a prime ideal since R/Pm is isomorphic to k[xm+1 , . . . , xr ]. Now we consider
the following family of ideals in R
m
X
hyi + zi i}rm=1
F ={Ij }rj=1 ∪ {Pj }rj=1 ∪ {Ir +
i=1

∪ {Ir +

r
X
i=1

hyi + zi i +

n
X
i=1

hxj + yj i}rn=1 ∪ m = hx1 , y1 , z1 , . . . , xr , yr , zr i .

Proposition 5.2.6. The family F is a principal radical system.

Proof. It is obvious that {Ij }rj=1 , {Pj }rj=1 , and m are radical. So we just have to consider the two
following cases:
P
(a) I = Ir + m
i=1 hyi + zi i for some 1 ≤ m ≤ r − 1. Observe that I + hym+1 + zm+1 i is an
element in F. Let J = Ir−m . It√
is easy to see ym+1 +zm+1 ∈
/ Ir−m so that ym+1 +zm+1 is not
a zero-divisor in the domain
P R/ J = R/Ir−m . It remains to show that (ym+1 +zm+1 )J ⊆ I.
Recall that Ir−m = Ir + m
j=1 hxj , yj , zj i. Then it suffices to prove that
(ym+1 + zm+1 ) xj ∈ I,
(ym+1 + zm+1 ) yj ∈ I,
(ym+1 + zm+1 ) zj ∈ I

for all 1 ≤Pj ≤ m. This is done
P in the Appendix 8.1.
(b) I = Ir + P ri=1 hyi + zi i + ni=1 hxj + yj i for some 0 ≤ n ≤ r − 1, where if n = 0, we set
I = Ir + ri=1 hyi + zi i. It is clear that I + hxn+1 + yn+1 i ∈ F. Choose J = Pn , then√the
same argument as in the previous case gives us xn+1 + yn+1 is not a zero-divisor of R/ J.
We also refer the reader to the Appendix 8.2 for the proof of (xn+1 + yn+1 )J ⊆ I.

Here is the main result of this section.
Theorem 5.2.7. For each r ≥ 1, the variety Cr (N ) is Cohen-Macaulay and therefore normal.

12

NHAM V. NGO

Proof. It immediately follows from Lemma 5.2.4 and Proposition 5.2.6.



Now we can summarize our results into a theorem as we stated in Section 1.2.
Theorem 5.2.8. Let gl2 and sl2 be Lie algebras defined over k of characteristic p 6= 2. Then for
each r ≥ 1, we have the commuting varieties Cr (gl2 ), Cr (sl2 ), and Cr (N ) are irreducible, normal,
and Cohen-Macaulay.
Remark 5.2.9. We claim that the theorem above holds even when the field k is not algebraically
closed. Indeed, all the proofs in the last section do not depend on the algebraic closedness of
k since the theory of determinantal rings does not require it. There should be a way to avoid
Nullstellensatz’s Theorem (hence algebraic closedness is not necessary) in the arguments in this
section.
5.3. Rational singularities. We prove in this section that the moment map
m : G ×B ur → Cr (N )

admits rational singularities. Since m is already a resolution of singularities, it is equivalent to
show the following.
Proposition 5.3.1.
(a) OCr (N ) = m∗ OG×B ur .
(b) The higher direct image Ri m∗ (OG×B ur ) = 0 for i > 0. Hence G×B ur is a rational resolution
of Cr (N ) via m.
Proof.
(a) This follows from Theorem 5.2.7, and Zariski’s Main Theorem 3.4.2.
(b) By [H, Proposition 8.5], we have Ri m∗ (OG×B ur ) ∼
= L Hi (G ×B ur , OG×B ur ) for each i ≥ 0
(as we pointed out in the end of Section 2.3). Note that L(k) = OG×B ur , so we have
H (G × u , OG×B ur ) ∼
=
i

B

r

∞
M
j=0

i

j

r∗

H (G/B, LG/B S (u )) =

∞
M

j ∗r
Ri indG
B (S (u )).

j=0

As we are assuming that u is a one-dimensional space of weight corresponding to the negative
root −α, S j (u∗r ) can be considered as the direct sum (kjα )⊕Pr (j) , where Pr (j) is the number
of ways getting r as a sum of j non-negative numbers. This weight is dominant, so by
j ∗r
Kempf’s vanishing theorem we obtain Ri indG
B (S (u )) = 0 for all i > 0 and j ≥ 0. It
follows that
Ri m∗ (OG×B ur ) = 0
for all i ≥ 1.

Now we state the result of this section.
Theorem 5.3.2. The moment map m : G ×B ur → Cr (N ) is a rational resolution of singularities.
5.4. Applications. As a corollary, we establish the connection with the reduced cohomology ring
of Gr . This gives an alternative proof for the main result in [SFB1] for the special case G = SL2 .
Theorem 5.4.1. For each r > 0, there is a G-equivariant isomorphism of algebras
k[G ×B ur ] ∼
= k[Cr (N )].
Consequently, there is a homeomorphism between Spec H• (Gr , k)red and Cr (N ).

Proof. Note that the moment map m is G-equivariant. This implies that the comorphism m∗
is compatible with the G-action. The isomorphism follows from part (a) of Proposition 5.3.1.
Combining this observation with [N, Proposition 5.2.4], we have the homeomorphism between the
spectrum of the reduced Gr -cohomology ring and Cr (N ).


COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

13

Now we want to describe the characters for the coordinate algebra k[Cr (N )]. Before doing that,
we need to introduce some notation. Here we follow L
the convention in [Jan].
Let V be a graded vector space over k, i.e., V = i≥0 Vn , where each Vn is finite dimensional.
Set
X
cht V =
ch(Vn )tn
n≥0

the character series of V .
From the isomorphism in Theorem 5.4.1, the character series for the coordinate algebra of Cr (N )
can be computed via that of k[G ×B ur ]:

(4)
cht (k[Cr (N )]) = cht k[G ×B ur ] .

Theorem 5.4.2. For each r > 0, we have
(5)

cht k[Cr (N )] =

X

X

χ(nα)tn .

n≥0 a1 +···+ar =n

where the latter sum is taken over all r-tuple (a1 , . . . , ar ) of non-negative integers satisfying
a1 + · · · + ar = n.

Proof. From [Jan, 8.11(4)], k[G ×B ur ] is a graded G-algebra and for each n ≥ 0,
ch k[G ×B ur ]n = ch H0 (G/B, S n (ur∗ )).

The argument in Proposition 5.3.1 gives us

Hi (G/B, S n (u∗r )) = 0
for all i > 0, n ≥ 0. Hence, it follows by [Jan, 8.14(6)] that

χ(S n (u∗r )) = ch H0 (G/B, S n (u∗r ))

for each n ≥ 0. Here we recall that for each finite-dimensional B-module M , the Euler characteristic
of M is defined as
X
χ(M ) =
(−1)i ch Hi (G/B, M ).
i≥0

Then we write

ch(S n (u∗ ))

= e(nα); hence for each n we have

ch(S n (ur∗ )) = ch(S n (u∗ )⊗r )
= ch

X

a1 +···+ar =n

X

=

a1 +···+ar =n

X

=

a1

∗

ar

∗

!

S (u ) ⊗ · · · ⊗ S (u )

ch (S a1 (u∗ ) ⊗ · · · ⊗ S ar (u∗ ))

e(nα).

a1 +···+ar =n

Thus by the linear property of χ, we obtain
X
χ (S n (u∗r )) =

χ(e(nα)) =

a1 +···+ar =n

X

Combining all the formulas, we get
X
X

cht k[G ×B ur ] =
ch k[G ×B ur ]n tn =
n≥0

Therefore, the identity (4) completes our proof.

χ(nα).

a1 +···+ar =n

X

χ(nα)tn .

n≥0 a1 +···+ar =n



14

NHAM V. NGO

Remark 5.4.3. The above formula can be made more specific by making use of partition functions.
Let S be the set of r numbers a1 , . . . , ar ∈ N. Set PS : N → N with PS (m) the number of ways
getting m as a linear combination of elements in S with non negative coefficients. This is called a
partition function. Kostant used this notation with S = Π = {α1 , . . . , αn }, the set of simple roots,
to express the number of ways of writing a weight λ as a linear combination of simple roots, see
[SB] for more details. In our case, set S = {α1 = · · · = αr = 1} and denote Pr = PS . Then from
(5) we can have
ch k[Cr (N )]n = Pr (n)χ(nα).
As a consequence, we obtain a result on the multiplicity of H0 (λ) in k[Cr (N )].
Corollary 5.4.4. For each λ = mα ∈ X + , we have


k[Cr (N )] : H0 (λ) = Pr (m).

This result also shows that the coordinate algebra of Cr (N ) has a good filtration as a G-module.
6. The Mixed Cases

6.1. Now we turn our concerns into more complicated commuting varieties. Let V1 , . . . , Vr be
closed subvarieties of a Lie algebra g. Define
C(V1 , . . . , Vr ) = {(v1 , . . . , vr ) ∈ V1 × · · · × Vr | [vi , vj ] = 0 , 1 ≤ i ≤ j ≤ r},

a mixed commuting variety over V1 × · · · × Vr . It is obvious that when V1 = · · · = Vr , this variety
coincides with the commuting variety over V1 . We apply in this section our calculations from
previous sections to study mixed commuting varieties over sl2 and its nilpotent cone N .
We assume k is an algebraically closed field of characteristic p 6= 2. Set
Ci,j = C(N , . . . , N , sl2 , . . . , sl2 )
| {z } | {z }
i times

j times

with i, j ≥ 1, a mixed commuting variety over sl2 and N . Note that if j = 0, then Ci,0 = Ci (N ).
Otherwise, one gets Cj (sl2 ) if i = 0. By permuting the tuples, this variety is isomorphic to any
mixed commuting variety in which N appears i times and sl2 appears j times. Hence we consider
Ci,j a representative of such varieties. The following are some basic properties.
Proposition 6.1.1. For each i, j ≥ 1, we have:
(a) The variety Ci,j is reducible. Moreover, we have Ci,j = Ci+j (N ) ∪ 0 × . . . × 0 × Cj (sl2 ).
(b) dim Ci,j = i + j + 1.
(c) Ci,j is not normal. It is Cohen-Macaulay if and only if i = 1.
Proof. Part (a) and (b) immediately follow from the decomposition
Ci,j = Ci+j (N ) ∪ 0 × . . . × 0 × Cj (sl2 ).

Indeed, the inclusion ⊇ is obvious. For the other inclusion, we consider v = (v1 , . . . , vi+j ) ∈ Ci,j . If
vm 6= 0 for some 1 ≤ m ≤ i, then it is distinguished. Hence by [TY, Corollary 35.2.7] the centralizer
of vm is in N . This implies that all the vn with i + 1 ≤ n ≤ i + j must be in N so that v ∈ Ci+j (N ).
Otherwise, we have v1 = · · · = vi = 0; hence v ∈ 0 × . . . × 0 × Cj (sl2 ). Also note from earlier that
dim Ci+j (N ) = i + j + 1 and dim Cj (sl2 ) = j + 2.
(c) The mixed commuting variety Ci,j is never normal for all i, j ≥ 1 as it is reducible and
the irreducible components overlap each other (they all share the origin). Recall that a variety is
Cohen-Macaulay only if its irreducible components are equidimensional. In our case, this happens
only when i = 1. The remainder is to show that C1,j = C(N , sl2 , . . . , sl2 ) is Cohen-Macaulay. Let
q
I(Cj+1 (N )) = I(Cj (N )) + J1

COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

15

where J1 = x21 + y1 z1 , x1 yi − xi y1 , x1 zi − xi z1 , y1 zi − yi z1 | i = 2, . . . , j + 1 , an ideal of k[slj+1
2 ].
Note also by Lemma 5.2.2 that
I(0 × Cj (sl2 )) = hx1 , y1 , z1 i + I(Cj (sl2 ))
and that I(Cj (sl2 )) ⊆ I(Cj (N )), J1 ⊆ hx1 , y1 , z1 i. Hence, by Lemma 5.2.3, we have
Cj+1 (N ) ∩∗ 0 × Cj (sl2 ) = Cj+1 (N ) ∩ 0 × Cj (sl2 ) = 0 × Cj (N ).

Observe, in addition, that each variety is Cohen-Macaulay. So Lemma 5.2.1 implies that C1,j is
Cohen-Macaulay.

Remark 6.1.2. One can see that mixed commuting varieties are the generalization of commuting
varieties. Strategies in studying the latter objects can be applied to that of the first objects.
However, we have not seen any work in the literature related to this concept. Our results in this
section show that mixed commuting varieties are usually not irreducible, hence fail to be normal or
Cohen-Macaulay. The reducibility also causes big dificulties in computing their dimensions. If V1
and Vn are the minimal and maximal element of the family V1 , . . . , Vn ordered by inclusion, then
for each r ≥ 1 one can easily see the following
dim Cr (V1 ) ≤ dim C(V1 , . . . , Vn ) ≤ dim Cr (Vn ).
These bounds are rough and depend on the dimensions of the commuting varieties. So new methods
are required to investigate a mixed commuting variety over various closed sets in a higher rank Lie
algebra.
7. Commuting Variety of 3 by 3 matrices
We turn our assumption to G = SL3 and g = sl3 defined over an algebraically closed field k.
Recall that N denotes the nilpotent cone of g. Not much is known about commuting varieties in
this case. In the present section, we only focus on the calculations on the nilpotent commuting
variety. In particular, we apply determinantal theory to study the smaller variety Cr (u), and then
use the moment morphism to obtain results on Cr (N ). We next show that for each r ≥ 1 all of the
singular points of Cr (N ) are in the commuting variety Cr (Osub ).
7.1. Irreducibility. First, we identify ur with the



0 0
0 0 0

ur =  x 1 0 0  , . . . ,  x r 0

yr zr
y1 z1 0

affine space as follows.


0

0  | xi , yi , zi ∈ k, 1 ≤ i ≤ r .

0

For each pair of elements in u, the commutator equations are xi zj − xj zi with 1 ≤ i ≤ j ≤ r. It
follows that
k[x1 , y1 , z1 , . . . , xr , yr , zr ]
k[Cr (u)] = p
.
hxi zj − xj zi | 1 ≤ i ≤ j ≤ ri
Proposition 7.1.1. For each r ≥ 1, the variety Cr (u) is
(a) irreducible of dimension 2r + 1,
(b) singular at the point




0 0 0
0 0 0
X =  0 0 0  , · · · ,  0 0 0  ,
y1 0 0
yr 0 0
with y1 , . . . , yr ∈ k,
(c) Cohen-Macaulay and normal.

16

NHAM V. NGO

Proof. We first consider the following isomorphism of rings
k[Cr (u)] ∼
= k[y1 , . . . , yr ] ⊗ p

(6)

k[x1 , z1 , . . . , xr , zr ]
.
hxi zj − xj zi | 1 ≤ i ≤ j ≤ ri

Let V be the variety associated to the second ring in the tensor product. Then parts (a)
and (c) follow if we are able to show that V is irreducible, Cohen-Macaulay and normal.
Indeed, these properties can be obtained from determinantal varieties as we argued earlier.
Consider the matrix


x1 x2 · · · xr
X =
.
z1 z2 · · · zr
It is easy to see that
k(X )
k[V ] ∼
=
I2 (X )
where I2 (X ) is the ideal generated by 2-minors of X . Hence by Theorem 3.2.2, k[V ] is a
Cohen-Macaulay, normal domain of Krull dimension r + 1. As tensoring with a polynomial
ring does not change the properties except the dimension, we have k[Cr (u)] is a CohenMacaulay, normal domain of Krull dimension 2r + 1 [BK]. This proves parts (a) and (c).
(b) Note that the group H = GLr × GL2 acts on V under the following action:

 

r
!
a11 · · · a1r


r
r
0 0 0
X
X
a
b


 ..
.
..
..  ,
×   xi 0 0  
a1i vj
a1i vi , . . . ,
7→
 .
.
c d 
0
z
0
i=1
i=1
i
ar1 · · · arr
i=1


0
0
0
0
0  for each 1 ≤ i ≤ r. Now consider a nonzero
where vi =  axi + bzi
0
cxi + dzi 0
element v in V . Without loss of generality, we can assume that the entry x1 6= 0. Then v
belongs to the open set W = V ∩ V (x1 6= 0). On the other hand, it is not hard to see that




0 0 0
0 0 0
W = H ·  1 0 0  , . . . ,  1 0 0 
0 0 0
0 0 0
which is a smooth orbit of dimension r+1. It follows that v is non-singular. From the isomorphism (6), we have the set of all singular points of the right-hand side is V (k[y1 , . . . , yr ]) ×
{0}. This determines the singular locus as desired.


Now we use properties of Cr (u) as ingredients to study the irreducibility and dimension of Cr (N ).
Theorem 7.1.2. The variety Cr (N ) is an irreducible variety of dimension 2r + 4.
Proof. Irreducibility follows from Theorem 3.3.2. Then, the birational properness of the moment
morphism
m : G ×B Cr (u) → Cr (N )
implies that
dim Cr (N ) = dim G ×B Cr (u) = dim Cr (u) + dim G/B = 2r + 4.


COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

17

7.2. Singularities. Serre’s Criterion states that a variety V is normal if and only if the set of
singularities has codimension ≥ 2 and the depth of V at every point is ≥ min(2, dim V ). This
makes the task of determining the dimension of the singular locus of a variety necessary in order
to verify normality.
Note that the problem on the singular locus of ordinary commuting varieties was studied by
Popov [Po]. He showed that the codimension of singularities for C2 (g) is greater than or equal to
2 for an arbitrary reductive Lie algebra g. In other words, the variety C2 (g) holds the necessary
condition for being normal. This author has not seen any analogous work in the literature for
arbitrary commuting varieties.
We prove in this subsection that the set of all singularities for Cr (N ) has codimension ≥ 2. Let
α, β be the simple roots of the underlying root system Φ of G. Then we have the set of positive
roots Φ+ = {α, β, α + β}. Recall that zreg and zsub are respectively the centralizers of vreg and vsub
in g. We have shown in Proposition 7.1.1(b) that the vector space ur−α−β includes all singularities
of Cr (u). Before locating the singularities of Cr (N ), we need some lemmas.
Lemma 7.2.1. Suppose r ≥ 2. For each 1 ≤ i ≤ r, the subset V [i] = G · (zreg , . . . , vreg , . . . , zreg )
with vreg in the i-th of Cr (N ) is a smooth open subvariety of dimension 2r + 4.
Proof. As a linear combination of commuting nilpotents is also nilpotent, we define an action of
the algebraic group G × Mr−1,r on Cr (N ) where Mr−1,r is the set of all r − 1 × r-matrices and can
be identified with (Gr )r−1 :
(G × Mr−1,r ) × Cr (N ) → Cr (N )
r
r
X
X
ar−1,j vj ).
a1j vj , . . . , vi , . . . ,
(g, (aij )) • (v1 , . . . , vr ) 7−→ g · (
j=1

j=1

for every element (v1 , . . . , vr ) ∈ Cr (N ). Now observe that zreg is a vector space of dimension 2,
choose {vreg , w} a basis of this space. It is easy to check that for r ≥ 2 we have
V [i] = (G × Mr−1,r ) • (w , 0 , . . . , vreg , . . . , 0).
In other words, our original variety is an orbit under the bullet action. So it is smooth.
Now for each 1 ≤ i ≤ r, consider the projection map from Cr (N ) to the i-th factor, p : Cr (N ) →
N . We then have
V [i] = p−1 (Oreg )

which is an open subset in Cr (N ). As the variety Cr (N ) is irreducible, the dimension of V [i] is the
same as dim Cr (N ).

Lemma 7.2.2. The intersection of zsub and Osub is exactly the union of u−α ×u−α−β and u−α ×uβ .
Hence, we have dim (zsub ∩ Osub ) = 2.
Proof. It can be computed that zsub consists of all matrices of the form


x1 0
0
 x2 x1
t2 
x3 0 −2x1

where x1 , x2 , x3 , t2 are in k. As the determinant of a nilpotent matrix is always 0, we obtain



0 0 0


zsub ∩ N =  x2 0 t2  | x2 , x3 , t2 ∈ k .


x3 0 0

18

NHAM V. NGO

On the other hand, it is well-known that Osub consists of all matrices of rank one and Osub =
Osub ∪ {0}. Then

 

0 0 0  
0 0 0 

zsub ∩ Osub =  x2 0 0  ∪  x2 0 t2 

 

x3 0 0
0 0 0
= u−α × u−α−β ∪ u−α × uβ .
It immediately follows that dim (zsub ∩ Osub ) = 2.



For each r ≥ 1, let Crsing be the singular locus of Cr (N ). The following theorem determines the
location of Crsing .
Theorem 7.2.3. For each r ≥ 1, we have Crsing ⊆ Cr (Osub ). Moreover, codim Crsing ≥ 2.
Proof. It is obvious that our result is true for r = 1. Assume now that r ≥ 2. It suffices to prove
that any element in the complement of Cr (Osub ) in Cr (N ) is smooth. Let V = Cr (Osub ) and
suppose w ∈ Cr (N )\V . Say w = (w1 , . . . , wr ) with some wn ∈
/ Osub , i.e., wn is a regular element.
Consider the projection onto the n-th factor pn : Cr (N ) → N . As G · wn = Oreg is open in N , the
preimage p−1
n (G · wn ) is an open set of Cr (N ). Note that
−1
p−1
n (G · wn ) = G · pn (wn ) = G · (zreg , . . . , vreg , . . . , zreg ).

By Lemma 7.2.1, w is non-singular. Therefore, V contains all the singularities of Cr (N ).
We are now computing the dimension of dim Cr (Osub ). It is observed for each r ≥ 1 that
Cr (Osub ) = G · (vsub , Cr−1 (zsub ∩ Osub )).
Now let V1 = uα × uα+β and V2 = uα × u−β . By the preceding lemma, we have zsub ∩ Osub = V1 ∪ V2 .
Also note that for u, v ∈ V1 ∪ V2 we have
[u, v] = 0

⇔

u, v ∈ V1

or

u, v ∈ V2 .

It follows that Cr−1 (zsub ∩ Osub ) = V1r−1 ∪ V2r−1 . So we obtain
Cr (Osub ) = G · (vsub , V1 , . . . , V1 ) ∪ G · (vsub , V2 , . . . , V2 ).
This implies that Cr (Osub ) is reducible. Using theorem of dimension on fibers, we can further
compute that G · (vsub , Vj , . . . , Vj ) has dimension 2r + 2 for each j = 1, 2. Thus dim Cr (Osub ) =
2r + 2 so that codim Cr (Osub ) = 2.

Acknowledgments
This paper is based on part of the author’s Ph.D. thesis. The author gratefully acknowledges the
guidance of his thesis advisor Daniel K. Nakano. Thanks to Christopher Drupieski for his useful
comments. We also thank William Graham for discussions about Cohen-Macaulay rings. Finally,
we are grateful to Alexander Premet and Robert Guralnick for the information about ordinary
commuting varieties.
8. Appendix
We verify in this section the condition of Theorem 5.2.5(b) for I and J are in the context of
Proposition 5.2.6.

COMMUTING VARIETIES OF r-TUPLES OVER LIE ALGEBRAS

19

8.1. Case 1. We check the following
(1) (ym+1 + zm+1 ) xj ∈ I,
(2) (ym+1 + zm+1 ) yj ∈ I,
(3) (ym+1 + zm+1 ) zj ∈ I.

For the first one, we consider for each 1 ≤ j ≤ m,

(ym+1 + zm+1 )xj + I = ym+1 xj + zm+1 xj + I
= xm+1 yj + xm+1 zj + I
= xm+1 hyj + zj i + I
=I

where the second identity is provided by xm+1 yj − xj ym+1 , xm+1 zj − xj zm+1 ∈ Ir ⊆ Im ; and the
last identity is provided by yj + zj ∈ I.
The same technique is applied for showing (2) and (3) as following:
(ym+1 + zm+1 )yj + I = ym+1 yj + zm+1 yj + I
= ym+1 yj + ym+1 zj + I
= ym+1 (yj + zj ) + I
= I,
(ym+1 + zm+1 )zj + I = ym+1 zj + zm+1 zj + I
= zm+1 yj + zm+1 zj + I
= zm+1 (yj + zj ) + I
= I.
8.2. Case 2. We need to check the following
(1) (xn+1 + yn+1 ) xj ∈ I,
(2) (xn+1 + yn+1 ) yj ∈ I,

(3) (xn+1 + yn+1 ) zj ∈ I,

(4) (xn+1 + yn+1 ) (xh − yh ) ∈ I,

for all 1 ≤ j ≤ n and n + 1 ≤ h ≤ r. Verifying (1), (2), and (3) is similar to our work in Case 1.
Lastly, we look at
(xn+1 + yn+1 )(xh − yh ) + I = xn+1 xh + yn+1 xh − xn+1 yh − yn+1 yh + I
= xn+1 xh − yn+1 yh + I
= xn+1 xh + yn+1 zh + I.
Now in order to complete our verification, we will show that xn+1 xh + yn+1 zh ∈ Ir . Indeed, we
have
2
(xn+1 xh + yn+1 zh )2 + Ir = x2n+1 x2h + 2xn+1 xh yn+1 zh + yn+1
zh2 + Ir
2
= −x2n+1 yh zh + 2x2n+1 yh zh + yn+1
zh2 + Ir

2
= x2n+1 yh zh + yn+1
zh2 + Ir

2
= −yn+1 zn+1 yh zh + yn+1
zh2 + Ir

= yn+1 zh (−zn+1 yh + yn+1 zh ) + Ir
= Ir .
As Ir is radical, we obtain xn+1 xh + yn+1 zh ∈ Ir as desired.

20

NHAM V. NGO

References
[B]
[Ba]
[BI]
[BK]
[BV]
[CM]
[E]
[G]
[GS]
[H]
[HE]
[Hum1]
[Hum2]
[Hr]
[Jan]
[K]
[KN]
[L]
[MS]
[Mu]
[N]
[Po]
[Pr]
[R]
[S]
[SB]
[SFB1]
[SFB2]
[TY]
[Vas]
[Z]

R. Basili, On the irreducibility of commuting varieties of nilpotent matrices, J. Pure Appl. Algebra, 149
(2000), 107–120.
V. Baranovsky, The varieties of pairs of commuting nilpotent matrices is irreducible, Transform. Groups, 6
(2001), 3–8.
R. Basili and A. Iarrobino, Pairs of commuting nilpotent matrices, and Hilbert function, J. Algebra, 320
(2008), 1235–1254.
S. Bouchiba and S. Kabbaj, Tensor products of Cohen-Macaulay rings: solution to a problem of Grothendieck,
J. Algebra, 252 (2002), 65–73.
W. Bruns and U. Vetter, Determinantal Rings, Lectures Notes in Math., Springer-Verlag, 1988.
D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand
Reinhold Mathematics Series, 1993.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer, 1995.
M. Gerstenhaber, On dominance and varieties of commuting Matrices, Ann. Math., 73 (1961), 324–348.
R. M. Guralnick and B. A. Sethuraman,Commuting pairs and triples of matrices and related varieties ,
Linear Algebra Appl., 310 (2000), 139–148.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag, 1977.
M. Hochster and A. J. Eagon, Cohen-Macaulay rings, invariant theory, generic perfection of determinantal
loci, Amer. J. Math., 93 (1971), 1020–1058.
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics,
Springer-Verlag, 1978.
, Linear Algebraic Groups, Graduate Texts in Mathematics, Springer-Verlag, 1995.
F. Hreinsdottir, Miscellaneous results and conjectures on the ring of commuting matrices, An. St. Univ.
Ovidius Constanta, 14 (2006), 45–60.
J. C. Jantzen and K-H. Neeb, Lie Theory: Lie Algebras and Representations, Progress in Mathematics,
Birkhauser, 228 2004.
A. Knutson, Some schemes related to the commuting variety, J. Algebraic Geom., 14 (2005), 283–294.
A. A. Kirillov and Y. A. Neretin, The variety An of n-dimensional Lie algebra structures, Amer. Math. Soc.
Transl., 137 (1987), 21–30.
P. Levy, Commuting varieties of Lie algebras over fields of prime characteristic, J. Algebra, 250 (2002),
473–484.
M. Majidi-Zolbanin and B. Snapp, A Note on the variety of pairs of matrices whose product is symmetric,
Cont. Math., 555 (2011).
C. C. Mueller, On the varieties of pairs of matrices whose product is symmetric, Ph.D. thesis, The University of Michigan, 2007.
N. V. Ngo, Cohomology for Frobenius kernels of SL2 , to appear in Journal of Algebra.
V. L. Popov, Irregular and singular loci of commuting varieties, Transform. Groups, 13 (2008), 819–837.
A. Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math., 154 (2003), 653–683.
R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Comp. Math., 38
(1979), 311–327.
K. Sivic, On varieties of commuting triples III, Lin. Alg. and App., 437 (2012), 393–460.
J. R. Schmidt and A. M. Bincer, The Kostant partition function for simple Lie algebras, J. Math. Phys., 25
(1984), 2367–2374.
A. Suslin, E. M. Friedlander, and C. P. Bendel, Infinitesimal 1-parameter subgroups and cohomology, J.
Amer. Math. Soc, 10 (1997), 693–728.
, Support varieties for infinitesimal group scheme, J. Amer. Math. Soc, 10 (1997), 729–759.
P. Tauvel and R. Yu, Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics, SpringerVerlag, 2005.
W. V. Vasconcelos, Arithmetic of Blowup Algebras, Cambridge University Press, 1994.
E. Zoque, On the variety of almost commuting nilpotent matrices, Transform. Groups, 15 (2010), 483–501.

Department of Mathematics, Statistics, and Computer Science, University of Wisconsin-Stout,
Menomonie, WI 54751, USA
Current address: Department of Mathematics and Statistics, Lancaster University, Lancaster,
LA1 4YW, UK
E-mail address: n.ngo@lancaster.ac.uk

