MULTIGRADED HILBERT SCHEMES PARAMETRIZING IDEALS
IN THE WEYL ALGEBRA
JEN-CHIEH HSIAO
ABSTRACT. Results of Haiman and Sturmfels [2] on multigraded Hilbert schemes are used to establish a quasi- projective scheme which parametrizes all left homogeneous ideals in the Weyl algebra having a fixed Hilbert function with respect to a given grading by an abelian group.
1. Introduction. Let S = k[x1, . . . , xn] be the polynomial algebra over a commutative ring k. The monomials xu in S are identified with their exponents u ∈ Nn. A grading of S by an abelian group A is a semigroup homomorphism
deg : Nn −→ A.
We may assume that A is generated by deg(xi) for i = 1, . . . , n. For a ∈ A, let Sa be the k-span of the monomial xu with deg(u) = a. We have the decomposition
S = !
a∈A
Sa
which satisfies Sa · Sb ⊆ Sa+b. An admissible ideal in S is a homogen- eous ideal I with the property that (S/I)a = Sa/Ia is a locally free k-module of finite rank (constant on Spec k) for all a ∈ A. The Hilbert function of an admissible ideal I is a map
hI : A −→ N defined by hI(a) := rankk(Sa/Ia).
2010 AMS Mathematics subject classification. Primary 14C05, 16S32.
Keywords and phrases. Hilbert schemes, Weyl algebras.
The author was partially supported by the National Science Foundation of Taiwan.
Received by the editors on November, 25, 2015, and in revised form on May 28, 2016.
DOI:10.1216/RMJ-2017-47-8-2675 Copyright c⃝2017 Rocky Mountain Mathematics Consortium
2675
By fixing any function
h : A −→ N,
Haiman and Sturmfels constructed [2] a scheme HSh over k (called the multigraded Hilbert scheme) which parametrizes all admissible ideals I in S with Hilbert function hI = h. As discussed in [2], their results recover many special cases, including Hilbert schemes of points in affine space, toric Hilbert schemes, Hilbert schemes of abelian groups orbits and Grothendieck Hilbert schemes. It is also mentioned in [2, subsection 6.2] that their results can be applied to the universal enveloping algebra of an A-graded Lie algebra. The purpose of this note is to verify this claim for the special case of the Weyl algebra W = k⟨x1, . . . , xn, ∂1, . . . , ∂n⟩.
In order to have a well-defined degree function on the set B = {xα∂β | α, β ∈ Nn}
of all monomials in W , we assume that our ground ring k is an integral domain of characteristic 0. By Proposition 2.1 of [1, Chapter 1] (the proof works for any integral domain k), the setB forms a k-basis for W . In general, this does not hold in the non-domain case. For example, if k = Z[t]/⟨2t⟩, then t∂2 ∈ k⟨x, ∂⟩ acts as the zero operator on k[x]. On the other hand, in view of the relations ∂ixi − xi∂i = 1 in W , it may be quickly noticed that we must have deg(xi) = − deg(∂i). Therefore, any A-grading
deg :Nn −→ A
on S extends to an A-grading deg : B → A on W by deg(xα∂β) = deg(α)− deg(β). We have the decomposition
W =!
a∈A
Wa
satisfying Wa · Wb ⊆ Wa+b, where Wa is the k-span of the monomials in B with degree a.
Similarly to the case of polynomial algebras, we call a homogeneous left ideal I of W admissible if (W/I)a = Wa/Ia is a locally free k- module of finite rank (constant on Spec k) for all a∈ A. Note that the Hilbert function hI : A → N of an admissible ideal I in W defined by hI(a) = rankk(Wa/Ia) cannot have finite support. This follows from the fact that there is no left ideal of W with finite co-rank over k.
Indeed, if rank(W/I) is finite, then the two k-linear maps φx and φ∂ on W/I induced by multiplications of x and ∂, respectively, would satisfy the equality φ∂φx− φxφ∂ = idW/I, which is not possible by comparing the traces of the linear maps from both sides.
Our goal is to prove the following analog of [2, Theorem 1.1].
Theorem 1.1. Given a Hilbert function h : A → N, there exists a quasi-projective scheme over k that represents the Hilbert functor
HWh : k -Alg−→ Set
where, for a k-algebra R, the set HWh (R) consists of homogeneous ideals I ⊆ R ⊗k W such that
(R⊗kWa)/Ia
is a locally free R-module of rank h(a) for every a∈ A.
In Section 4, we will recall the techniques from [2] that are needed in the proof of Theorem 1.1. Roughly speaking, we first show that, for any finite set of degrees D, the Hilbert functor HWh
D is represented by a quasi-projective scheme that is a closed subscheme of a certain relative Grassmann scheme. Here, the k-module
WD = !
a∈D
Wa,
and, by abusing notation, the restriction of the Hilbert function h : A → N to D is also denoted by h. Then, we specify a special finite set D such that HWh is a subfunctor of HWhD represented by a closed subscheme of HWh
D.
Although the strategy of proving Theorem 1.1 is very similar to the polynomial algebra case, there are still several issues that require some modifications. For example, the key feature that makes these mechanisms work for the multigraded Hilbert scheme HSh is the nice behaviors of monomial ideals in S, e.g., the fact that antichains of monomial ideals in S are finite [3] is essential in the construction of HSh. In Section 2, we will see that monomial ideals in W do not have the expected behaviors in general. In particular, the naive generalization of Gr¨obner basis theory to Weyl algebra does not work very well. For example, the ideal⟨∂2, x∂− 1⟩ and its naive initial ideal ⟨∂2, x∂⟩ = ⟨∂⟩
in W = k⟨x, ∂⟩ do not have the same Hilbert function. In order to get around this, we consider the initial ideal of a left ideal in W as a monomial ideal in the associated graded algebra gr W (which is a polynomial algebra) and utilize the Gr¨obner basis theory for the Weyl algebra developed in [4]. Basic facts about the Gr¨obner basis theory for W will be reviewed in Section 3. Finally, the proof of Theorem 1.1 will be elaborated in Section 5.
The same results of this paper applied to right ideals may be achieved similarly.
2. Monomial ideals in the Weyl algebra. Let k be an integral domain of characteristic 0, and let W = k⟨x, ∂⟩ = k⟨x1, . . . , xn, ∂1, . . . ,
∂n⟩ be the nth Weyl algebra. Many classical facts regarding the Weyl algebra are proved under the assumption that k is a field, see e.g., [1].
This has an advantage in that W is left and right Noetherian. For our purposes, we will not make this assumption. Many classical properties of the Weyl algebra extend to this more general setting. For example, by [1, Chapter 1, Proposition 2.1] (the proof of which works for integral domain k) the set
B = {xα∂β | α, β ∈ Nn}
forms a k-basis for W , where xα = xα11· · · xαnn and ∂β = ∂1β1· · · ∂nβn. The unique expression of an element δ of W in terms of this k-basisB is called the canonical form of δ. In this paper, elements inB are the only monomials of the Weyl algebra W , and a product of monomials in W may not be a monomial. Also, the total degree of the monomial xα∂β is |α| + |β|, and the total degree of an element in W is defined as the total degree of its leading monomials. The total degrees of elements of W induce the Bernstein filtration on W , whose associated graded ring gr W = k[x, ξ] = k[x1, . . . , xn, ξ1, . . . , ξn] is the polynomial algebra of 2n variables over k. Moreover, by considering the isomorphism of free k-modules
Ψ : k[x, ξ]−→ W = k⟨x, ∂⟩
that sends xαξβ to xα∂β, we have the following Leibniz formula which is helpful for the multiplication of elements in W . Note that, in the formula of Proposition 2.1, the denominator k1!· · · kn! is used only to obtain a nice expression. We will never need to find the inverse of elements in the domain k.
Proposition 2.1. [4, Theorem 1.1.1]. For any two polynomials f and g in k[x, ξ], we have
Ψ(f )· Ψ(g) = "
k1,...,kn≥0
1
k1!· · · kn! Ψ
#∂kf
∂ξk · ∂kg
∂xk
$ .
In particular, we have a convenient formula for multiplying two monomials.
Corollary 2.2. Let xα∂β, xα′∂β′ be monomials in W . We have (xα∂β)· (xα′∂β′) = "
k1,...,kn≥0
#%n i=1
ki!
#βi ki
$#α′i ki
$
xαi+α′i−ki∂βi+βi′−ki
$ .
A left ideal in W is called a left monomial ideal if it is generated by monomials. Unlike the monomial ideals in a polynomial algebra, it may occur that an element in a left monomial ideal I is a sum of monomials which are not in I. For example, in the first Weyl algebra W = k⟨x, ∂⟩, the element ∂x = x∂ + 1 is in the principal ideal I = W x generated by x, but the identity 1 (hence, x∂) is not in I by considering the total degrees of elements in W . Moreover, there exists an infinite antichain of monomial ideals in W , see Example 2.6. Thus, the direct generalization of the main theorem in [3] does not hold for the Weyl algebra. Nonetheless, we still have the following analog of Dickson’s lemma for monomial ideals in polynomial algebras.
Proposition 2.3. Every left monomial ideal of W is generated by finitely many monomials.
Proof. Let I be a left monomial ideal of W . By passing to the associated graded algebra with respect to the Bernstein filtration, it may be observed that the ideal gr I is a monomial ideal of gr W . By Dickson’s lemma, gr I is finitely generated by monomials of degrees
≤ m for some m. Standard arguments regarding filtered algebras, see, for example, the proof of [1, Theorem 8.2.3], show that I is generated by elements with total degree≤ m, say f1, . . . , ft. We only need finitely many monomials in I to generate f1, . . . , ft; thus, I is, in fact, generated
by finitely many monomials. !
Example 2.4. Every left monomial ideal in the first Weyl algebra W = k⟨x, ∂⟩ is principally generated by one monomial. In order to see this, suppose that I is a left monomial ideal of W and assume that xα∂β ∈ I. Observe that, by Corollary 2.2, we have
(i) (x∂)(xα∂β) = xα+1∂β+1+αxα∂β, so xα+1∂β+1 ∈ I, and hence, xα+s∂β+t ∈ I for all 0 ≤ t ≤ s;
(ii) for α ≥ 1, ∂(xα∂β) = xα∂β+1 + axα−1∂β, xα∂β+1 ∈ I if and only if xα−1∂β ∈ I.
It suffices to show that, for any two monomials xα∂β and xα′∂β′ in I, there exists a monomial xα′′∂β′′ ∈ I such that xα∂β, xα′∂β′ ∈ W xα′′∂β′′. We may assume by symmetry that β′ ≥ β and consider only the following cases.
(a) [0 ≤ β′−β ≤ α′−α]. In this case, xα′∂β′ = xα+(α′−α)∂β+(β′−β) ∈ W xα∂β by (i); thus, we simply take (α, β) = (α′′, β′′).
(b) [α′ − α < β′ − β]. In this case, take β′′ = β and α′′ =
&
0 if α′ − β′+ β ≤ 0;
α′ − β′+ β otherwise.
It follows from (i) that xα∂β, xα′∂β′ ∈ W xα′′∂β′′. On the other hand, the Leibniz formula in (ii) also shows that, if xα′′+1∂β′′ and xα′′+1∂β′′+1 are both in I, then xα′′∂β′′ ∈ I. Hence, if there exists an m ∈ N such that
{xα′′+m∂β′′+i | 0 ≤ i ≤ m} ⊂ I,
then xα′′∂β′′ ∈ I. The proof of this example is completed by taking m = α + β′ − β − α′′. Indeed, for β′− β ≤ i ≤ m,
0≤ (β′′+ i)− β′ ≤ m + (β − β′)≤ m + (α′′− α′) = (α′′+ m)− α′; thus, by (i),
{xα′′+m∂β′′+i | β′ − β ≤ i ≤ m} ⊂ W xα′∂β′ ⊂ I.
Also, for 0≤ i < β′ − β,
0≤ i = (β′′+ i)− β < β′ − β = (α′′ + m)− α;
thus, again by (i),
{xα′′+m∂β′′+i | 0 ≤ i < β′− β} ⊂ W xα∂β ⊂ I.
The observations in Example 2.4 also imply the following lemma.
Lemma 2.5. In the first Weyl algebra W , if xα′∂β′ ∈ W xα∂β with α ≥ 1, then
[(α′, β′)− (α, β)] ∈ Σ = {(s, t) ∈ N2 | 0 ≤ t ≤ s}.
Proof. Suppose otherwise that α′ − α < β′ − β. Applying Example 2.4 (i) to xα∂β ∈ W xα∂β, we obtain
{xβ′−β+α−1∂β+i | 0 ≤ i ≤ β′− β − 1} ⊂ W xα∂β. Since xα′∂β′ ∈ W xα∂β, we have
xβ′−β+α−1∂β′ = x(β′−β)−(α′−α)−1(xα′∂β′)∈ W xα∂β. Therefore,
{xβ′−β+α−1∂β+i | 0 ≤ i ≤ β′ − β} ⊂ W xα∂β.
Through repeated use of Example 2.4 (ii), we obtain xα−1∂β ∈ W xα∂β, which is impossible in view of the Bernstein filtration on W . ! We remark that the same argument in the proof of Lemma 2.5 generalizes to the nth Weyl algebra W , namely, if xα′∂β′ ∈ W xα∂β with αi ≥ 1 for some i ∈ {1, . . . , n}, then
[(α′i, βi′)− (αi, βi)]∈ Σ = {(s, t) ∈ N2 | 0 ≤ t ≤ s}.
Example 2.6. Using Lemma 2.5, it may readily be verified that {W x∂β | β ∈ N}
is an infinite antichain of monomial ideals in the first Weyl algebra W . 3. Gr¨obner bases in the Weyl algebra. In this section, the ground ring k is a field of characteristic 0. We recall the Gr¨obner bases theory for Weyl algebra over k developed in [4, subsection 1.1].
A total order ≺ on the set B of monomials in W is called a term order for W if the following two conditions hold:
(1) 1 = x0∂0 is the ≺-smallest element;
(2) xα∂β ≺ xa∂b implies xα+s∂β+t ≺ xa+s∂b+t for all (s, t)∈ N2n.
The initial monomial in≺(δ) of an element δ ∈ W is the commutative monomial xαξβ ∈ k[x, ξ] such that xα∂β is the ≺ - largest monomial appearing in the canonical form of δ. For a W -ideal I, its initial ideal is the ideal in k[x, ξ], generated by {in≺δ | δ ∈ I}. A finite set G of W is said to be a Gr¨obner basis for a W -left ideal I with respect to ≺ if I is generated by G and the initial ideal in≺I is generated by {in≺g | g ∈ G}. From [4, Theorem 1.1.10], every left ideal I of W admits a Gr¨obner basis G with respect to any given term order≺. Note that not every finite monomial generating set of a monomial ideal forms a Gr¨obner basis. For example, the initial ideal of I = W x + W ∂ = W is k[x, ξ], which is not generated by x and ξ. Nonetheless, we have the following analog of the normal form algorithm: every element δ ∈ W has a unique normal form δG ∈ W with respect to G such that δ ≡ δG modulo I and that every monomial appearing in the canonical form of δG is not divisible by Ψ(in≺g) for any g ∈ G. Here, a monomial xα∂β is said to be divisible by xa∂b in W if αi ≥ ai and βi ≥ bi for all i. A monomial of W is called a standard monomial of I with respect to ≺ if it is not divisible by Ψ(in≺g) for any g in a Gr¨obner basis G for I.
The next lemma is an immediate consequence of the normal form algorithm.
Lemma 3.1. Let ≺ be a term order on W , and let I be a left ideal of W .
(1) The images of the standard monomials of I in W/I form a k- basis.
(2) The map
Ψ : gr W = k[x, ξ] −→ W
induces an isomorphism between the k-vector spaces gr W/ in≺I and W/I, which sends the standard monomials of in≺I in gr W to the standard monomials of I in W .
(3) If I is homogeneous with respect to an A-grading of W , then in≺I is homogeneous with respect to the induced A-grading on gr W , and the map Ψ restricts to an isomorphism between (gr W/ in≺I)a and (W/I)a for each a ∈ A. In particular, the Hilbert functions of I and in≺I are identical.
4. Main techniques. For the reader’s convenience, we recall the general framework for the construction of the multigraded Hilbert scheme in [2].
Fix a commutative ring k and an arbitrary index set A. Consider the pair (T, F ) of graded k-modules
T =!
a∈A
Ta with a collection of operators
F = '
a,b∈A
Fa,b,
where Fa,b ⊆ Homk(Ta, Tb) satisfies Fb,c◦ Fa,b ⊆ Fa,c and idTa ∈ Fa,a. In fact, (T, F ) is a small category of k-modules with objects Ta and arrows, the elements of which are in F .
For a commutative k-algebra R, we denote by R ⊗ T the graded
R-module !
a
(R⊗ Ta)
with operators (Fa,b = (1R⊗ −)(Fa,b). A homogeneous submodule L =!
a
La ⊆ R ⊗ T
is an F -submodule if (Fa,b(La)⊆ Lb for all a, b∈ A. Fix a function h : A −→ N.
Let HTh(R) be the set of F -submodules L⊆ R⊗T such that (R⊗Ta)/La is a locally free R-module of rank h(a) for each a ∈ A. We have the Hilbert functor
HTh : k -Alg −→ Set.
For any subset D of A, denote by (TD, FD) the full subcategory of (T, F ) with objects Ta and the set of arrows FD,a,b = Fa,b for a, b∈ A.
We have a natural transformation of Hilbert functors HTh −→ HThD
given by restriction of degrees.
Theorem 4.1. [2, Theorem 2.2]. Let (T, F ) be a graded k-module with operators as above. Suppose that there exist homogeneous k-submodules M ⊆ N ⊆ T and a subset F′ ⊆ F satisfying the following conditions:
(i) N is a finitely generated k-module;
(ii) N generates T as an F′-module;
(iii) for every field K ∈ k -Alg and every L ∈ HTh(K), M spans (K⊗ T )/L;
(iv) there is a subset G ⊆ F′, generating F′ as a category, such that GM ⊆ N.
Then, HTh is represented by a quasi-projective scheme over k.
We remark that the statement of Theorem 4.1 is slightly stronger than that of [2, Theorem 2.2]. However, the same proof works in this setting. Indeed, in the proof of [2, Theorem 2.2], only Step 6 applies to conditions (ii) and (iv) where any element in T needs to be produced using elements in N and operators in F . However, this does not require the full set of F . Any subset F′ ⊆ F satisfying conditions (ii) and (iv) will suffice.
Moreover, hypothesis (iii) implies that dimK(K⊗ T )/L = "
a∈A
h(a)
is finite; thus, Theorem 4.1 works only for h having finite support. For the general case, we need the following theorem.
Theorem 4.2. [2, Theorem 2.3]. Let (T, F ) be graded k-modules with operators, and let D ⊆ A be such that HThD is represented by a scheme over k. Assume that, for each degree a ∈ A:
(v) there is a finite subset
E ⊆ '
b∈D
Fb,a such that
Ta/"
b∈D
Eb,a(Tb) is a finitely generated k-module;
(vi) for every field K ∈ k -Alg and every LD ∈ HThD(K), if L′ denotes the F -submodule of K ⊗ T generated by LD, then dim(K ⊗ Ta)/L′a ≤ h(a).
Then, the natural transformation
HTh −→ HThD
makes HTh a subfunctor of HThD, represented by a closed subscheme of the Hilbert scheme HTh
D.
In order to find a suitable finite set D of degrees satisfying hypotheses (v) and (vi) in Theorem 4.2, we also need the following facts.
Proposition 4.3. [2, Proposition 3.2]. Let S be an A-graded polyno- mial ring. Given a degree function
deg :Nn −→ A and a Hilbert function
h : A −→ N,
there is a finite set of degrees D ⊆ A that satisfies the following two properties:
(g) every monomial ideal of S with Hilbert function h is generated by monomials of degrees in D, and
(h′) every monomial ideal I of S generated in degrees D satisfies:
if hI(a) = h(a) for all a ∈ D, then hI(a)≤ h(a) for all a ∈ A.
Such a set D in Proposition 4.3 is called a supportive set of degrees in [2]. There is also a so-called very supportive set E which is used to define equations of HSh in the positive grading case. Since the A- grading on W is never positive, we will not pursue here the analogous results on very supportive sets.
5. Proof of Theorem 1.1. Fix any Hilbert function h : A → N, and let D be a finite subset of degrees in A. Our first task is to construct k-submodules M, N and a subset FD′ ⊆ FD satisfying the hypotheses
in Theorem 4.1 for the graded k-modules WD = !
a∈D
Wa
with the set of operators FD to be defined later.
Let F be the monoid of operators on W generated by multiplication of monomials in W . Note that this is slightly different from the poly- nomial case due to the non-commutativity of W . Denote the set of all operators in F that send Wa into Wb by Fa,b. Then,
F = '
a,b∈A
Fa,b.
Moreover, we have Fb,c◦Fa,b ⊆ Fa,c for all a, b, c∈ A, and Fa,a contains the identity map on Wa for all a∈ A. Thus, (W, F ) is a small category of k-modules with the components Wa of W as objects and elements of F as arrows. Note also that, for a k-algebra R, an admissible left ideal in R⊗kW is equivalent to a left F -submodule L of R⊗kW such that (R ⊗k Wa)/La is a locally free R-module of rank h(a) for each a∈ A.
Define
FD := '
a,b∈D
Fa,b.
Then, (WD, FD) is a full subcategory of (W, F ). Consider, for each k- algebra R, the set HWhD(R) of all admissible FD-submodules of R⊗kWD and, for each k-algebra homomorphism
φ : R −→ S, the map
HWhD(φ) : HWhD(R)−→ HWhD(S).
There is a natural transformation of Hilbert functors HWh −→ HWhD,
given by sending L∈ HWh (R) to LD := !
a∈D
La ∈ HWhD(R).
For each a ∈ A, let Ba be the set of monomials (excluding 1 = x0∂0) with degree a. Denote the set of minimal elements in Ba with respect to the partial ordering
xα∂β ≤ xα′∂β′ ⇐⇒ (α, β) ≤ (α′, β′)
by G′a. Recall that xα′∂β′ is said to be divisible by xα∂β if (α, β) ≤ (α′, β′). By Dickson’s lemma, we have G′a is finite for each a ∈ A.
For a, b ∈ A, let Ga,b be the set of operators on W consisting of left multiplication by elements in G′b−a. Denote by FD′ the monoid (category) generated by
GD := '
a,b∈D
Ga,b.
For a, b∈ D, denote the set of all operators in FD′ that send Wa into Wb by Fa,b′ . The following example shows that strict inequality FD′ ! FD
can occur.
Example 5.1. Consider the Z-grading on the first Weyl algebra W = k⟨x, ∂⟩ with deg(x) = − deg(∂) = 1. Let D = {0, 2} ⊆ Z.
Then, GD = {x∂, x2, ∂2}. Observe that the element ∂x3 ∈ F0,2 ⊆ FD
does not lie in the monoid FD′ generated by elements in GD.
The A-grading on W induces an A-grading on gr W by setting deg ξ := deg ∂. The Hilbert function h : A → N can be viewed as a Hilbert function for ideals in the polynomial algebra gr W with this induced A-grading. Let CD be the set of ideals of gr W generated by monomials in degrees D with Hilbert functions agreeing with h on D.
Denote the union over all I ∈ CD of the Ψ-images of the standard monomials of I in (gr W )D by M′, i.e.,
M′ ={Ψ(xαξβ)| xαξβ ∈ (gr W )D\ I, for some I ∈ CD}.
Since CD is finite by [3], the set M′ is also finite.
Let
N′ = GDM′∪# '
a∈D
G′a
$ , M = kM′ and N = kN′.
We verify that (WD, FD′ , FD), N, M and GD satisfy the hypotheses of Theorem 4.1 which is rewritten below.
(i) N is a finitely generated k-module;
(ii) N generates WD as an FD′ -module;
(iii) for every field K ∈ k -Alg and every LD ∈ HWhD(K), M spans (K ⊗ WD)/LD;
(iv) there is a subset GD ⊆ FD′ , generating FD′ as a category, such that GDM ⊆ N.
Conditions (i) and (iv) obviously hold by our construction. For condi- tion (ii), given a ∈ D and xα∂β ∈ Wa, we want to show that xα∂β is generated by N over FD′ by induction on the total degree |α| + |β| of xα∂β. If xα∂β ∈ G′a ⊆ N, the statement is automatically true. For
xα∂β ∈ Wa \ G′a,
there exists an xα′∂β′ ∈ G′a ⊆ N such that xα∂β is divisible by xα′∂β′. Note that the total degree of the element
(xα∂β− xα−α′∂β−β′ · xα′∂β′)∈ Wa
is strictly less than that of xα∂β; thus, by inductive hypothesis, it is generated by N over FD′ . Therefore, it suffices to show that xα−α′∂β−β′ ∈ W0 acts on Wa as a sum of operators in FD′ . In fact, we will show that every element xα¯∂β¯ ∈ W0 acts as a sum of operators in FD′ by induction on the total degree of xα¯∂β¯. Recall that FD′ is the monoid generated by '
a,b∈D
G′a,b.
In particular, if xα¯∂β¯ ∈ G′0, it acts as an operator in Ga,a ⊆ FD′ for any a∈ D. For
xα¯∂β¯ ∈ W0\ G′0,
there exists an xα¯′∂β¯′ ∈ G′0 such that xα¯∂β¯is divisible by xα¯′∂β¯′. Since xα− ¯¯ α′∂β− ¯¯ β′ and (xα¯∂β¯− xα−¯¯ α′∂β− ¯¯ β′· xα¯′∂β¯′) are in W0 and have total degree strictly less than xα¯∂β¯, they both act as a sum of operators in FD′ . We conclude that
xα¯∂β¯= (xα¯∂β¯− xα¯− ¯α′∂β¯− ¯β′ · xα¯′∂β¯′) + xα¯− ¯α′∂β¯− ¯β′ · xα¯′∂β¯′ also acts as a sum of operators in FD′ . This establishes condition (ii).
We need the next lemma to verify condition (iii).
Lemma 5.2. Let R ∈ k -Alg, LD ∈ HWhD(R) and L ⊆ R ⊗k W be the left ideal generated by LD. Then, La = LD,a for all a∈ D.
Proof. Observe that, for a∈ D, La = "
b∈D
Fb,a(LD,b)⊇ Fa,a(LD,a)⊇ LD,a.
Conversely, we have Fb,a(LD,b) ⊆ LD,a for any a, b ∈ D, since LD ∈ HWh
D(R) is an FD-submodule of R⊗k WD. ! For condition (iii), fix a field K ∈ k -Alg and an FD-submodule LD ∈ HWhD(K). Let L ⊆ K ⊗k W be the ideal generated by LD. Fix any term order≺ on W , and let I = in≺L ⊂ gr W be the initial ideal of L with respect to ≺. By Lemma 3.1 (3), the Hilbert functions hI and hL coincide. Hence, hI(a) = hLD(a) for all a ∈ D by Lemma 5.2.
In particular, the ideal I ∈ CD. From Lemma 3.1 (2), M′ spans (K ⊗k WD)/LD, which is exactly the statement of condition (iii).
At this point, we have shown, by using Theorem 4.1, that HWh
D is represented by a quasi-projective scheme for any finite set D of degrees in A. In order to complete the proof of Theorem 1.1, it remains to verify conditions (v) and (vi) (for each degree a ∈ A) in Theorem 4.2 for some suitable finite subset D of A.
(v) There is a finite subset E ⊆ )
b∈DFb,a such that Wa/"
b∈D
Eb,a(Wb) is a finitely generated k-module;
(vi) for every field K ∈ k -Alg and every LD ∈ HWhD(K), if L denotes the F -submodule of K ⊗ W generated by LD, then dim(K ⊗ Wa)/La ≤ h(a).
Applying Proposition 4.3 to the case where S = gr W with the induced degree function
deg :N2n −→ A
given by deg(ξ) = deg(∂), we can find a finite subset D of A that satisfies conditions (g) and (h′) for gr W with respect to the same Hilbert function h. From now on, fix such a finite set D. The goal is to use Theorem 4.2 to show that the natural transformation
HWh −→ HWhD
makes HWh a subfunctor of HWh
D, represented by a closed subscheme of the Hilbert scheme HWh
D. We may assume that there exists an admis- sible function L of W , whose Hilbert function is hL = h, for otherwise, the statement of Theorem 4.2 is null. Choose any term order ≺ on W . By Lemma 3.1 (3), the Hilbert function hin≺L of the initial ideal in≺L of L in gr W coincides with h, and hence, in≺L is generated in degrees D by condition (g) in Proposition 4.3. Since the s-pair of two homogeneous elements in the Weyl algebra is still homogeneous, there exists a Gr¨obner basis for L consisting of homogeneous elements in degrees D. In particular, the ideal L of W is also generated in degrees D. Therefore, for each a∈ A, the component
La = "
b∈D
Fb,a(Lb), and it has finite k-codimension h(a) in Wa.
In order to verify condition (v), it suffices to find a finite subset E ⊆ '
b∈D
Fb,a such that, for any b∈ D, a ∈ A,
Fb,a(Lb)⊆ "
b′∈D
Eb′,a(Wb).
Take Eb,a = Gb,a, and let
E = '
b∈D
Eb,a.
We claim that, in fact,
Fb,a(Wb)⊆ Eb,a(Wb) = Gb,a(Wb).
Since each operator in Fb,a, which is a product of monomials, can be written as a sum of monomials in degree a − b by Corollary 2.2, we verify only that, if deg(xα∂β) = a− b, then xα∂β(Wb) ⊆ Gb,a(Wb). It is certainly true that
xα∂β(Wb)⊆ Gb,a(Wb)
when xα∂β ∈ G′a−b. In general, we have xα∂β is divisible by some element xα′∂β′ ∈ G′a−b and, by inductive hypothesis,
[xα∂β(Wb)− xα′∂β′ · xα−α′∂β−β′(Wb)]⊆ Gb,a(Wb).
Since deg(xα−α′∂β−β′) = 0, we have xα−α′∂β−β′(Wb) ⊆ Wb, and hence,
xα∂β(Wb)⊆ Gb,a(Wb), as desired.
For condition (vi), fix a field K ∈ k -Alg, an element LD ∈ HWhD(K), and let L⊆ K ⊗k W be the ideal generated by LD. From Lemma 5.2, La = LD,a, and hence, hL(a) = h(a) for all a ∈ D. Also, we have hL(a) = hin≺L(a) for all a ∈ A by Lemma 3.1 (3). Let I be the monomial ideal in gr W generated by (in≺L)D. Then, Ia = (in≺L)a for all a∈ D by the same argument of Lemma 5.2, and hence,
hI(a) = hin≺L(a) = hL(a) = h(a) for all a∈ D.
Therefore, by condition (h′) of Proposition 4.3,
hL(a) = hin≺L(a) ≤ hI(a) ≤ h(a) for all a ∈ A.
This establishes condition (vi).
Example 5.3. Let k be a field. Consider the finest possible A-grading on W , where A =Zn and
deg(xi) = − deg(∂i) = ei.
Under this A-grading, any homogeneous ideals are generated by ele- ments in W of the form xap(θ)∂b, a, b ∈ Nn. From [4, Lemma 2.3.1],
such ideals are the torus-fixed ideals of W , which are used in the algo- rithms for solving systems of linear partial differential equations.
Fixing a Hilbert function h : A → N, we remark that, if I, J ∈ HWh (k) and if I is holonomic, then J is also holonomic. Indeed, using the notation in [4], the Hilbert functions of in≺(0,e) I and in≺(0,e) J coincide by Lemma 3.1. Therefore, from [4, Theorem 1.1.6], the ideals in(0,e)I and in(0,e)J in gr W also have the same Hilbert functions under the A-grading inherited from that of W . In particular, dim in(0,e)I = dim in(0,e)J, and the holonomicity of J follows.
REFERENCES
1. S.C. Coutinho, A primer of algebraic D-modules, Lond. Math. Soc. 33, Cambridge University Press, Cambridge, 1995.
2. Mark Haiman and Bernd Sturmfels, Multigraded Hilbert schemes, J. Alg.
Geom. 13 (2004), 725–769.
3. Diane Maclagan, Antichains of monomial ideals are finite, Proc. Amer. Math.
Soc. 129 (2001), 1609–1615 (electronic).
4. Mutsumi Saito, Bernd Sturmfels and Nobuki Takayama, Gr¨obner deforma- tions of hypergeometric differential equations, in Algorithms and computation in mathematics, Volume 6, Springer-Verlag, Berlin, 2000.
National Cheng Kung University, Department of Mathematics, Tainan City, 70101 Taiwan
Email address: jhsiao@mail.ncku.edu.tw
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