FINITE GENERATIONS OF RINGS OF DIFFERENTIAL OPERATORS OF SEMIGROUP ALGEBRAS

MUTSUMI SAITO AND WILLIAM N. TRAVES

1. Introduction

Let A := { a_{1}, a_{2}, …, a_{n} } be a finite subset of ℤ^{d}. We denote by ℕA, ℤA, and ℝ_{≥0}A the monoid, the abelian group, and the cone generated by A, respectively.

Let R_{A} denote the semigroup algebra ℂ[ℕA] of ℕA. We consider two rings: the ring D(R_{A}) of differential operators of R_{A} and its graded ring Gr(D(R_{A})) with respect to the order filtration. As a starting point for the study of D(R_{A}), we have examined the finite generations of D(R_{A}) and Gr(D(R_{A})).

While considering the finite generation of Gr(D(R_{A})), we encountered the notion of a scored semigroup; a semigroup ℕA is scored if the difference (ℝ_{≥0}A ∩ ℤA) \ ℕA consists of a finite union of hyperlane sections of ℝ_{≥0}A ∩ ℤA parallel to facets of the cone ℝ_{≥0}A.

We have proved the following.

Theorem 1.1 (Theorem 3.2.12 in [1], Theorem 5.13, Theorem 6.3 in [2]).

1. Gr(D(R_{A})) is finitely generated if and only if R_{A} is a scored semigroup algebra.

2. D(R_{A}) is finitely generated for all semigroup algebras R_{A}.

By the standard argument of filtered rings, we obtain the Noetherian properties of D(R_{A}) from those of Gr(D(R_{A}));

Corollary 1.2 (Corollary 6.4 in [1]). If the semigroup ℕA is scored, then D(R_{A}) is left and right Noetherian.

Here we exhibit the idea of the proof of the theorem, and explain the scored property. The scored property implies Serre’s condition (S_{2}). However neither the scored property nor the Cohen-Macaulay property implies the other. The problem of the Noetherian properties of D(R_{A}) is still open.

2. Motivation

In the theory of A-hypergeometric systems (or GKZ systems), the associative algebra composed of contiguity differential operators plays an important role. We [1] proved that this algebra (called the symmetry algebra) is anti-isomorphic to the algebra D(R_{A}). Hence we expect that the study of D(R_{A}) has some fruitful applications to the theory of A-hypergeometric systems.

We also expect that the study of D(R_{A}) brings new insights into the ring R_{A}.

This report is based on [1] and [2].

3. Definitions

In this section, we briefly recall some fundamental facts about the rings of differential operators of semigroup algebras. Let A := { a_{1}, a_{2}, …, a_{n} } be a finite set of column vectors in ℤ^{d}. Sometimes we identify A with the matrix (a_{1}, a_{2}, …, a_{n}). Throughout this paper, we assume that ℤA = ℤ^{d} for simplicity.

The ring ℂ[t_{1}^{±1}, …, t_{d}^{±1}]⟨∂_{1}, …, ∂_{d}⟩ of differential operators with Laurent polynomial coefficients is the ring of differential operators on the algebraic torus (ℂ^{×})^{d}, where [∂_{i}, t_{j}] = δ_{ij}, [∂_{i}, t_{j}^{−1}] = −δ_{ij}t_{j}^{−2}, and the other pairs of generators commute. Here [ , ] denotes the commutator and δ_{ij} is 1 if i = j and 0 otherwise.

3.1. The Rings R_{A} and D(R_{A}). The semigroup algebra R_{A} := ℂ[ℕA] = ⨁_{a∈ℕA} ℂt^{a} is the ring of regular functions on the affine toric variety defined by A, where t^{a} = t_{1}^{a_{1}}t_{2}^{a_{2}} ⋯ t_{d}^{a_{d}} for a = ^{t}(a_{1}, a_{2}, …, a_{d}). Its ring of differential operators D(R_{A}) can be realized as a subring of the ring ℂ[t_{1}^{±1}, …, t_{d}^{±1}]⟨∂_{1}, …, ∂_{d}⟩ of differential operators on the big torus as follows:

D(R_{A}) = {P ∈ ℂ[t_{1}^{±1}, …, t_{d}^{±1}]⟨∂_{1}, …, ∂_{d}⟩ : P (R_{A}) ⊂ R_{A}}.

3.2. The Ring Gr(D(R_{A})). Next we explain the order filtration. A differential operator

P = ∑_{a∈ℕ^{d}} c_{a}(t)∂^{a} ∈ ℂ[t_{1}^{±1}, …, t_{d}^{±1}]⟨∂_{1}, …, ∂_{d}⟩

is said to be of order k if c_{a} ≠ 0 for some a with |a| = k and c_{a} = 0 for all a with |a| > k, where |a| = a_{1} + a_{2} + ⋯ + a_{d}. Let D_{k}(R_{A}) denote the set of differential operators in D(R_{A}) of order at most k. Then {D_{k}(R_{A})}_{k∈ℕ} is called the order filtration of D(R_{A}). We consider the graded ring Gr(D(R_{A})) of D(R_{A}) with respect to the order filtration,

Gr(D(R_{A})) := ⨁_{k∈ℕ}D_{k}(R_{A})/D_{k−1}(R_{A}),

where D_{−1}(R_{A}) = 0. The graded ring Gr(D(R_{A})) is a subring of the commutative ring

Gr(ℂ[t_{1}^{±1}, …, t_{d}^{±1}]⟨∂_{1}, …, ∂_{d}⟩) = ℂ[t_{1}^{±1}, t_{2}^{±1}, …, t_{d}^{±1}, ξ_{1}, ξ_{2}, …, ξ_{d}],

where ξ_{j} is the element represented by ∂_{j}.

4. Scored Semigroups

We recall the definition of scored semigroups. To this end, let us define the primitive integral support function of a facet (maximal face) of the cone ℝ_{≥0}A. We denote by F the set of facets of the cone ℝ_{≥0}A. Given σ ∈ F, we denote by F_{σ} the primitive integral support function of σ, i.e., F_{σ} is the uniquely determined linear form on ℝ^{d} satisfying F_{σ}(ℝ_{≥0}A) ≥ 0, F_{σ}(σ) = 0, and F_{σ}(ℤ^{d}) = ℤ.

Definition 4.1. The semigroup ℕA is said to be scored if

ℕA = ⋂_{σ∈F}{ a ∈ ℤ^{d} : F_{σ}(a) ∈ F_{σ}(ℕA) }. (4.1)

Example 1. Let

A_{1} = (a_{1}, a_{2}, a_{3}) = ( \begin{matrix}1 & 1 & 1 \\ 0 & 2 & 3 \end{matrix} ) .

Then

F = { σ_{1} = ℝ_{≥0}a_{1}, σ_{3} = ℝ_{≥0}a_{3} },

F_{σ_{1}}(θ_{1}, θ_{2}) = θ_{2}, F_{σ_{3}}(θ_{1}, θ_{2}) = 3θ_{1} − θ_{2}, and

ℕ \ F_{σ_{1}}(ℕA_{1}) = { 1 }, ℕ \ F_{σ_{3}}(ℕA_{1}) = ∅.

The semigroup ℕA_{1} is scored.

[figure: σ_{3}; σ_{1}]

Remark 4.1. 1. By the definition of F_{σ}, the difference ℕ \ F_{σ}(ℕA) is finite for any

σ ∈ F.

2. The semigroup ℕA is scored if and only if

(ℝ_{≥0}A ∩ ℤ^{d}) \ ℕA = ⋃_{σ∈F} ⋃_{m∈ℕ\F_{σ}(ℕA)}F_{σ}^{−1}(m) ∩ ℝ_{≥0}A ∩ ℤ^{d}.

Thus ℕA is scored if and only if the difference (ℝ_{≥0}A ∩ ℤA) \ ℕA consists of a finite union of hyperlane sections of ℝ_{≥0}A ∩ ℤA parallel to facets of the cone ℝ_{≥0}A.

The semigroup ring ℂ[ℕA] is Cohen-Macaulay if and only if it satisfies Serre’s condition (S_{2}) and the reduced homology modules of certain simplicial complexes vanish ([3, Theorem 4.1]). In our case, Serre’s (S_{2}) condition can be stated as follows:

ℕA = ⋂_{σ∈F}(ℕA + ℤ(A ∩ σ)). (S_{2})

Proposition 4.1 (Proposition 2.6 in [2]). Any scored semigroup satisfies (S_{2}).

Proof. Let ℕA be a scored semigroup. It is enough to show that for any facet σ ∈ F we have

ℕA + ℤ(A ∩ σ) = { a ∈ ℤ^{d} : F_{σ}(a) ∈ F_{σ}(ℕA) }. (4.2)

The inclusion ‘⊂’ is clear from the definition of F_{σ}. To prove the other inclusion ‘⊃’, let a ∈ ℤ^{d} satisfy F_{σ}(a) ∈ F_{σ}(ℕA). For every σ′ ∈ F different from σ, there exists a_{i} ∈ A such that a_{i} ∉ σ′ and a_{i} ∈ σ. Since F_{σ′}(a_{i}) > 0 and ℕ \ F_{σ′}(ℕA) is finite, there exists m_{i} ∈ ℕ such that F_{σ′}(a + m_{i}a_{i}) ∈ F_{σ′}(ℕA).

Doing this argument for every σ′ ∈ F different from σ, we find b ∈ ℕ(A ∩ σ) such that

F_{σ′}(a + b) ∈ F_{σ′}(ℕA) (∀σ′ ∈ F \ {σ}).

Since

F_{σ}(a + b) = F_{σ}(a) ∈ F_{σ}(ℕA)

and ℕA is scored, we see a + b ∈ ℕA. Hence a ∈ ℕA + ℤ(A ∩ σ).

[figure: σ_{2}; σ_{1}]

Example 2.

A_{2} = ( \begin{matrix}0 & 1 & 2 \\ 1 & 1 & 0 \end{matrix} ) .

The semigroup ℕA_{2} satisfies (S_{2}). Hence the semigroup ring ℂ[ℕA_{2}] is Cohen-Macaulay as well since the cone ℝ_{≥0}A is simplicial. Thus ℂ[ℕA_{2}] is Cohen-Macaulay but not scored.

Example 3. Let

A_{3} = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 2 & 3 & 0 & 2 & 3 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{matrix} ) .

[figure: θ_{1} = 1; θ_{3}; θ_{2}]

Then the semigroup ℕA_{3} is clearly scored. However the semigroup ring ℂ[ℕA_{3}] is not Cohen-Macaulay ([3, Example 3.9]).

5. Graded Structure

Put θ_{j} := t_{j}∂_{j} for j = 1, 2, …, d. Then it is easy to see that θ_{j} ∈ D(R_{A}) for all j. We introduce a ℤ^{d}-grading on the ring D(R_{A}) as follows: For a = ^{t}(a_{1}, a_{2}, …, a_{d}) ∈ ℤ^{d}, set

D(R_{A})_{a} := {P ∈ D(R_{A}) : [θ_{j}, P] = a_{j}P for j = 1, 2, …, d}.

Then D(R_{A}) is ℤ^{d}-graded; D(R_{A}) = ⨁_{a∈ℤ^{d}} D(R_{A})_{a} = ⨁_{a∈ℤ^{d}} t^{a}𝕀(Ω(a)), where 𝕀(Ω(a)) is a certain ideal of ℂ[θ].

Since each D_{k}(R_{A}) is ℤ^{d}-graded – D_{k}(R_{A}) = ⨁_{d∈ℤ^{d}} D_{k}(R_{A}) ∩ D(R_{A})_{d} – the graded ring Gr(D(R_{A})) inherits the grading; Gr(D(R_{A})) = ⨁_{d∈ℤ^{d}} Gr(D(R_{A}))_{d}.

The following decomposition is a key to the finite generations of D(R_{A}) and Gr(D(R_{A})).

Proposition 5.1 (Proposition 3.4 in [2]). There exist m ∈ ℕ, b_{i} ∈ ℤ^{d}, and faces τ_{i} of ℝ_{≥0}A with i = 1, 2, …, m such that

(ℝ_{≥0}A ∩ ℤ^{d}) \ ℕA = ∐_{i=1}^{m}(b_{i} + ℕ(A ∩ τ_{i})). (5.1)

The decomposition (5.1) is called a Stanley decomposition of (ℝ_{≥0}A ∩ ℤ^{d}) \ ℕA. Note that it is not unique.

6. Finite Generation of Gr(D(R))

The Serre’s condition (S_{2}) means that every graded component D(R_{A})_{d} is a singly generated ℂ[θ]-module.

Proposition 6.1. The ℂ[θ]-modules D(R_{A})_{a} are singly generated for all a if and only if the semigroup ℕA satisfies (S_{2}).

Next we consider the scored property. Let c ∈ ℕA be a vector such that c+ℝ_{≥0}A∩ℤ^{d} ⊂ ℕA. It is well known that such a vector always exists.

Proposition 6.2. There exists L ∈ ℕ independent of d ∈ c + ℝ_{≥0}A ∩ ℤ^{d} such that the minimal degree of polynomials in 𝕀(Ω(−d)) is equal to ∑_{σ∈F} F_{σ}(d) + L.

Furthermore, L = 0 if and only if ℕA is scored.

Thus the scored property means that the degree of generators of graded components far from the hyperplanes grows linearly without a constant term. This fact intuitively explains Theorem 1.1 (1).

The proof of the finite generation of Gr(D(R_{A})) for a scored semigroup ring R_{A} goes as follows:

1. Fix a Stanley decomposition (5.1).

2. Let M be one plus the maximum of F_{σ}(b_{i}) (σ ∈ F, i = 1, …, m).

3. Decompose ℤ^{d} into pieces;

ℤ^{d} = ⋃_{μ} S_{μ},

where μ runs over all maps from F to {−∞, ∞} ∪ {k ∈ ℤ : |k| < M}, and

S_{μ} := { d ∈ ℤ^{d} : F_{σ}(d) = μ(σ) for all σ ∈ F }.

Here we agree that F_{σ}(d) = +∞ (−∞, respectively) mean F_{σ}(d) ≥ M (≤ −M, respectively).

4. Let Ray(A) denote the set of rays of the hyperplane arrangement determined by A, {(F_{σ} = 0) : σ ∈ F}. For each ray ρ ∈ Ray(A), take d_{ρ} from ρ ∩ ℤ^{d} so that |F_{σ}(d_{ρ})| ≥ M whenever it is not zero.

5. Put F_{μ} := {ρ ∈ Ray(A) : ρ ⊂ σ if μ(σ) ≠ ±∞}.

6. Then there exists a finite subset S_{μ,fin} ⊂ S_{μ} such that S_{μ} = S_{μ,fin} + ∑_{ρ∈F_{μ}} ℕd_{ρ}.

7. Put D(R_{A})_{S_{μ}} = ⨁_{a∈S_{μ}} D(R_{A})_{a}. Then Gr(D(R_{A}))_{S_{μ}} is generated by Gr(D(R_{A}))_{a} (a ∈ S_{μ,fin}) and Gr(D(R_{A}))_{ρ} (ρ ∈ F_{μ}).

8. Gr(D(R_{A})) is finitely generated.

Example 4. (Continuation of Example 2)

Since ℕA_{2} satisfies (S_{2}), each D(R_{A_{2}})_{a} is singly generated. For a = ^{t}(a_{1}, a_{2}), put

Q_{a} := \{ \begin{matrix}t_{1}^{a_{1}}t_{2}^{a_{2}} & (a_{1} ≥ 0, a_{2} ≥ 1, or a_{1} ≥ 0 even, a_{2} = 0) \\ t_{1}^{a_{1}}t_{2}∂_{2}^{|a_{2}|+1} & (a_{1} ≥ 0, a_{2} < 0, or a_{1} ≥ 0 odd, a_{2} = 0) \\ t_{2}^{a_{2}}∂_{1}^{|a_{1}|} & (a_{1} < 0, a_{2} ≥ 1, or a_{1} < 0 even, a_{2} = 0) \\ t_{2}∂_{1}^{|a_{1}|}∂_{2}^{|a_{2}|+1} & (a_{1}, a_{2} < 0, or a_{1} < 0 odd, a_{2} = 0). \end{matrix}

By computing 𝕀(Ω(a)), we see that D(R_{A_{2}})_{a} is generated by Q_{a}. Hence L = 1. Moreover we have

Gr(D(R_{A_{2}}))/⟨t_{1}, ξ_{1}⟩ = ℂ[t_{2}, t_{2}ξ_{2}, t_{2}ξ_{2}^{2}, t_{2}ξ_{2}^{3}, ⋯].

Since this is not a finitely generated algebra, neither is Gr(D(R_{A_{2}})).

7. Finite Generation of D(R)

The proof of the finite generation of D(R_{A}) for a general affine semigroup ring R_{A} follows the same steps as that of Gr(D(R_{A})) for a scored semigroup ring made until Step 6. Steps 7 and 8 are the following:

7 Put D(R_{A})_{F_{μ},ℝ} = ⨁_{a∈∑_{ρ∈Fμ} ℕd_{ρ}} D(R_{A})_{a}. Then D(R_{A})_{S_{μ}} is generated by D(R_{A})_{a} (a ∈ S_{μ,fin} + ∑_{ρ∈F_{μ}} ℕ_{<m+2}d_{ρ}) as a right D(R_{A})_{F_{μ},ℝ}-module, where ℕ_{<m+2} denotes the set of natural numbers less than m + 2.

8 D(R_{A}) is finitely generated. Let us explain Step 7 in some more details. Let d = (d − d_{ρ}) + d_{ρ} (d − d_{ρ} ∈ S_{μ},

ρ ∈ F_{μ}). Then there exists an ideal I(d − d_{ρ}, d_{ρ}) of ℂ[θ] such that

D(R_{A})_{d−d_{ρ}}D(R_{A})_{d_{ρ}} = D(R_{A})_{d}I(d − d_{ρ}, d_{ρ}).

When d − (m + 1)d_{ρ} ∈ S_{μ}, we can prove that ∑_{k=1}^{m+1} I(d − kd_{ρ}, kd_{ρ}) = ⟨1⟩. Hence we see that D(R_{A})_{S_{μ}} is generated by D(R_{A})_{a} (a ∈ S_{μ,fin} + ∑_{ρ∈F_{μ}} ℕ_{<m+2}d_{ρ}) as a right D(R_{A})_{F_{μ},ℝ}-module.

Example 5. (Continuation of Example 4)

Recall that Q_{^{t}(0,−p)} = t_{2}∂^{p+1} for p ≥ 1. Then for p, q ≥ 1

Q_{^{t}(0,−p)}Q_{^{t}(0,−q)} = Q_{^{t}(0,−p−q)}(t_{2}∂_{2} − q).

Hence for p ≥ 3

Q_{^{t}(0,−p)} = Q_{^{t}(0,−(p−1))}Q_{^{t}(0,−1)} − Q_{^{t}(0,−(p−2))}Q_{^{t}(0,−2)}.

Thus Q_{^{t}(0,−p)} (p ≥ 3) are generated by Q_{^{t}(0,−1)} and Q_{^{t}(0,−2)}.

References

[1] M. Saito and W.N. Traves, Differential algebras on semigroup algebras, AMS Contemporary Math. 286 (2001), 207–226.

[2] and , Finite generations of rings of differential operators of semigroup algebras, preprint.

[3] N.V. Trung and L.T. Hoa, Affine semigroups and Cohen-Macaulay rings generated by monomials, Trans. of AMS 298 (1986), 145–167.

Department of Mathematics Graduate School of Science Hokkaido University Sapporo, 060-0810 JAPAN

E-mail address: saito@math.sci.hokudai.ac.jp

Department of Mathematics U.S. Naval Academy 572C Holloway Road Annapolis MD, 21402, USA

E-mail address: traves@usna.edu