D-module structure of local cohomology modules of toric algebras

Let S be a toric algebra over a field K of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules H^i_I(S) are of finite length over the ring of differential operators D(S;K), generalizing the classical case of a polynomial algebra S. As an application, we compute the characteristic cycles of some local cohomology modules.


Introduction
Lyubeznik [Lyu93] introduced an approach of studying local cohomology modules using the theory of D-modules. He obtained many finiteness properties of the local cohomology modules H i I (R) when R is a regular ring containing a field of characteristic 0. For example, taking advantage of holonomicity of H i I (R) as a D-module, he showed that for any maximal ideal m the number of associated prime ideals of H i I (R) contained in m is finite and that all Bass numbers of H i I (R) are finite. When R is a regular local ring of positive characteristic, analogous results were obtained by Huneke and Sharp using the Frobenius functor [HS93]. When R is not regular, the situation is more subtle. There are characteristic-free examples where H i I (R) have infinitely many associated primes [SS04]. Also, an example by Hartshorne [Har70] shows that in general the Bass numbers can be infinite (see Example 3.9).
After [Lyu93], there have been several studies on the finiteness properties of R x , and H i I (R) as D-modules for a regular ring R, among them [Bøg95], [Bøg02], [Lyu97], [ÁMBL05]. The first D-finiteness result of R x for a singular ring R is due to Takagi and Takahashi [TT08] which says R x is generated by x −1 over D when R is a Noetherian graded ring with finite F-representation type. In particular, their theorem applies to the case where R is a normal toric algebra over a perfect field of positive characteristic.
In the present article, we study finiteness properties of the localizations S f and the local cohomology modules H i I (S) as D-modules where S is a toric algebra (not necessarily normal) over a field K of characteristic 0, f is a monomial element and I is a monomial ideal of S. In this case, the ring of differential operators D(S) := D(S; K) is much more complicated than the case where S is regular. Using the natural grading of D(S) introduced by Jones [Jon94] and Musson [Mus94], Saito and Traves [ST01,ST04] [ST09]. Nonetheless, we can show that H i I (S) is actually of finite length as a D(S)-module (Theorem 3.3). In view of Hartshorne's example (Example 3.9), this result is quite surprising.
As an application, we compute the characteristic cycles of some local cohomology modules H i I (S). Characteristic cycles are formal sums of subvarieties (counted with multiplicities) of the characteristic variety of a D-module M . Here, the characteristic variety Ch(M ) is the support of the associated graded module gr M in the spectrum Spec(gr D(S)) of the associated graded ring gr D(S). When S is a polynomial algebra, one can explicitly compute the Bass numbers and the associated primes of H i I (S) from its characteristic cycles [ÁM04]. The cohomological dimension of I and the Lyubeznik numbers can also be computed from them [ÁM00]. Through our finiteness results of H i I (S), we are able to compute the characteristic cycles of some local cohomology modules. We will show that for normal toric algebras S (in fact, for a more general class of toric algebras) the characteristic variety Ch(H dimS m (S)) of the top local cohomology with maximal support is abstractly isomorphic to the ambient toric variety Spec(S) (Theorem 4.13).
In section 2, we briefly recall the notions of local cohomology, toric algebras and the ring of differential operators of a commutative algebra over a field. We also describe the structure of rings of differential operators over toric algebras following the notations in [ST01] and [ST04]. In section 3, our main results on the finiteness properties mentioned above are presented. Also, we relate our finiteness results to the notion of sector partition introduced in [SS90] and [MM06]. Some discussions on gr D(S) and the computations of characteristic cycles are in section 4. As suggested by the referee, some relations between our results in section 4 and the recent work of Saito [Sai10] are discussed (see Remarks 4.7, 4.14).
Acknowledgment. This work originates from a conversation between Uli Walther and William Traves. The author is grateful to his advisor Uli Walther for introducing this problem and providing many useful discussions and comments. He would also like to thank Ezra Miller for the comments on sector partition and Example 3.9, and Christine Berkesch for carefully reading this paper. Special thanks go to the referee of this paper who helped improve the presentation, simplified the proof of Theorem 3.1, and pointed out the relations between our results and the results in [Sai10].

Preliminaries
2.1. Local Cohomology. General facts regarding local cohomology can be found in [ILL+07] or [BS98]. Here, we only recall some basics.
Let R be a Noetherian commutative ring, M an R-module, and I an ideal of R. Define Γ I (M ) := lim − → Hom R (R/I k , M ). Then Γ I is a left exact R-linear covariant functor and the i-th local cohomology functor H i I is defined to be its i-th right derived functor. We call H i I (M ) the i-th local cohomology module of M supported at the ideal I. If I is generated by f 1 , . . . , f t , then H i I (M ) is the i-th cohomology of theČech complex 2.2. Toric Algebras. We introduce some notations for later use. For more information on toric algebras, the reader is referred to [Ful93], [MS05] or [ILL+07]. Let A be a d × n integer matrix with columns a 1 , . . . , a n . Assume ZA = Z d . 2.3. Rings of Differential Operators. For a commutative algebra R over a field K, set D 0 (R; K) := R and for i > 0 Then the ring of differential operators is defined to be When R is a polynomial ring over a field K of characteristic 0, D(R; K) is the usual Weyl algebra. In this paper, a module over D(R; K) means a left D(R; K)-module. When R is a regular algebra over a field K of characteristic 0, D(R; K) is well understood (see e.g. [Bjö79]). In this case, the local cohomology modules H i I (R) are holonomic as D(R; K)-modules and hence are of finite length (see [Lyu93]). This essential property enables Lyubeznik to achieve many finiteness results of the local cohomology modules.
Unfortunately, D(R; K) does not behave well when R is singular; we don't have a notion of holonomicity in this case. This complicates the study of H i I (R) via the theory of D-modules. On the bright side, when R = S A,K is a toric algebra over an algebraically closed field K of characteristic 0, there is a nice combinatorial structure for D(R; K) which we will present in the next subsection. Our finiteness results about local cohomology modules substantially rely on this structure.
2.4. Rings of Differential Operators over Toric Algebras. In the rest of this paper, we denote S A := S A,K where K is an algebraically closed field of characteristic 0. Following [ST01], the noncommutative ring D A := D(S A , K) can be described as a Z d -graded subring of , and the other pairs of variables commute. More precisely, with the notation θ i := t i ∂ i , one has where Ω(a) = NA \ (−a + NA) and I(Ω(a)) is the vanishing ideal of Ω(a) in 3. Finiteness properties of H i I (S A ) In this section, F will be denoted to be the set of all facets of R ≥0 A. For a face τ of R ≥0 A, we write N(A ∩ τ ) := NA ∩ Rτ and denote Z(A ∩ τ ) as the group generated by N(A ∩ τ ).
We recall some notations in [ST01], which are crucial to the proofs of Theorems 3.1 and 3.3. For a ∈ Z d and τ a face of R ≥0 A, define This ordering induces an equivalence relation on Z d by Now, we are ready for the first main theorem.
(2) Let τ be a face and suppose that b / ∈ τ . Then there exists a facet σ containing τ Remark 3.2. Via theČech complex, it follows immediately from Theorem 3.1 that The left Noetherianess of D A was studied by Saito and Takahashi [ST09]. They proved that D A is left Noetherian if S A satisfies Serre's condition (S 2 ). Serre's condition is, by [Ish88], equivalent to Saito and Takahashi also gave a necessary condition (on S A ) for D A to be left Noetherian. However, D A is not always left Noetherian. Proof. In view of theČech complex, since any localization

Nonetheless, we have
Consider the notations in the beginning of this section. For a, b ∈ Z d , we will write Since there are only finitely many faces and since each E τ (a) is contained in , there are finitely many equivalence classes determined by ∼; we denote them [α 1 ], [α 2 ], . . . , [α k ]. We may rearrange the order so that, for any pair i < j, either α i > α j or α i and α j are incomparable. Denote T a := b≥a Kt b . Then the filtration Example 3.4. For 1-dimensional S A , the composition series of K[Z] is easy to describe. In this case, R ≥0 A has two faces, 0 and σ = R ≥0 A. For a ∈ Z, for all a ∈ Z. Therefore, [0] and [−1] are the two equivalence classes determined by ∼ and we have the composition series Remark 3.5. Theorem 3.1 and Theorem 3.3 also hold for any field K with characteristic 0 by the isomorphism Remark 3.6. Suppose I = m, the maximal graded ideal of S A . Here we assume that the semigroup NA is pointed, so that 0 is the only invertible element.
(1) Recall that H i m (S A ) can be computed as the i-th cohomology of the Ishida complex [Ish88] or [ILL+07]. Therefore, H 1 m (S A ) is finitely generated as an S A -module. Indeed, it suffices to observe that (2) In general, Schäfer and Schenzel [SS90] showed that there is a partition of Z d with respect to which H i m (S A ) can be written as a finite direct sum of K-vector spaces. This decomposition coincides with the sector partition appearing in [MM06] (see also [HM05] for a more general notion of sector partition). More precisely, let Conv(A) be the set of all faces of R ≥0 A and for any filter (cocomplex) ∇ of Conv(A), denote Then the P ∇ 's form a partition (sector partition) of Z d and On the other hand, for a ∈ Z d denote and consider the equivalence relation a ≡ a ′ ⇔ ∇(a) = ∇(a ′ ). Then P ∇(a) is the equivalence class containing a. Notice that P ∇ could be empty and that [a ∈ P ∇ ] ⇔ [P ∇ = P ∇(a) ]. Theorem 6 in [MM06] shows that the partition determined by the equivalence relation ∼ in the proof of Theorem 3.3 is finer than the sector partition determined by ≡.
Theorem 3.7. If S A is normal and I is a monomial ideal in S A , then every simple subquotient of H i I (S A ) is of the form K[P ∇ ] coming from the sector partition. Proof. By Theorem 3.3 and Remark 3.6(2), we only have to show that ∼ and ≡ define the same equivalence relation on Z d . Note that the normality of Example 3.8. Consider A = 1 1 1 0 1 2 . Then S A = K[t, ts, ts 2 ] is a 2-dimensional normal toric algebra. Let I = (ts) be the ideal of S A generated by ts. We shall describe a composition series of H 1 Following the notations in the proof of Theorem 3.3 and Remark 3.6, let where σ 1 = R ≥0 1 0 and σ 2 = R ≥0 1 2 . Then P ∇0 = P ∇(a0) = NA, where a 0 = 0 0 and a i , i = 1, 2, 3, is the ith column of A. In terms of notation in Theorem 3.3, Quotienting out S A , we obtain a composition series of H 1 Example 3.9. This example is essentially due to Hartshorne [Har70]. We adopt its combinatorial description which can be found in [ILL+07] or [MS05].
Consider A = . This equality can be verified by the following data: In what follows, we will use the description: When R is a regular algebra over K, Spec(gr D(R; K)) can be identified as the cotangent bundle of the variety Spec R with the projection π : Spec(gr D(R; K)) → Spec R induced by the embedding R ֒→ gr D(R; K). The fiber of π over a closed point of Spec R is the cotangent space over that point which is isomorphic to the affine space K dim R .

D-MODULE STRUCTURE OF LOCAL COHOMOLOGY MODULES OF TORIC ALGEBRAS 9
In this subsection, we discuss the the map π for a certain class of toric algebras. We shall see that in some cases the fibers of π behave nicely (Theorems 4.5, 4.8). We also give an example (Example 4.6) of more complicated nature.
To begin with, consider the natural order filtration of D A inherited from that of D(K[Z d ]). With respect to this filtration, one can regard grD A as a commutative subalgebra of grD( When gr D A is finitely generated over K, Musson showed that it has dimension 2d [Mus87]. Saito and Traves proved that gr D A is finitely generated over K if and only if the semigroup NA is scored [ST04]. By definition, a semigroup NA is scored if or equivalently, NA \ NA is a union of finitely many hyperplane sections parallel to some facets of R ≥0 A. The scored condition implies Serre's condition (S 2 ) (see Remark 3.2).
We should remark that if S A is normal, then gr D A is Gorenstein ([Mus87] Theorem D). In general, even for 1-dimensional semigroup ring (which is always scored), the associated graded ring can have bad singularities. In fact, we have the following Proposition 4.1. If S A is a 1-dimensional toric algebra which is not normal, then gr D A is not Cohen-Macaulay.
Therefore gr D A is a two dimensional toric algebra over K.
We claim that Take ℓ to be the maximal number of 2|Ω(−w)| for w ∈ Hole(NA). To prove (4.1), it is enough to show that t u ξ v ∈ gr D A for all pairs u, v ∈ N satisfying u+v ≥ ℓ. Since [t u ξ v ∈ gr D A ⇔ t v ξ u ∈ gr D A ], by symmetry we may assume So the claim is proved. Now, applying the criterion in Remark 3.2 to the toric algebra gr D A , we see that gr D A doesn't satisfy Serre's condition (S 2 ). Hence, gr D A is not Cohen-Macaulay.
Remark 4.2. The claim (4.1) holds true for any affine curve with injective normalization (see the proof of Theorem 3.12 in [SS88]). In fact, this codimension is known to be the Letzter-Makar-Limanov invariant, which plays an important role in the theory of Calogero-Moser space. For more information, see the work by Berest and Wilson [BW99]. Now, let m be the maximal graded ideal of S A corresponding to the closed point 0 of the toric variety Spec(S A ). Let I = √ mgrD A be the radical of the extended ideal of m under the embedding S A ֒→ grD A . We are going to show that if NA is simplicial and scored then grD A /I is isomorphic to S A as K-algebras. This implies that the reduced induced structure of the fiber of π : Spec(grD A ) → Spec(S A ) over the point 0 is isomorphic to the ambient toric variety.
The following two lemmas are needed: For scored NA, Lemma 4.4. Let NA be scored. Then for any a ∈ Z d and σ ∈ F , n σ,−a = n σ,a + F σ (a).

In particular,
(1) if σ is a facet with the property that F σ (a) ≤ 0, then n σ,ka ≤ k · n σ,a for large k ∈ N and furthermore P k a = P ka · P for some P ∈ K[Θ]; (2) if F σ (a) ≤ 0 for all σ ∈ F and −a / ∈ NA, then n σ,ka < k · n σ,a for some σ ∈ F and large k ∈ N.
Proof. To prove n σ,−a = n σ,a + F σ (a) for any a ∈ Z d and σ ∈ F , it's enough to show the case where F σ (a) > 0. Set N = F σ (NA) and n = F σ (a) > 0. Then Note that for the second equality we need the assumption that NA is scored. Now, we prove the four additional statements: (1) Notice that n σ,ka = kF σ (−a) for large k and that n σ,a = n σ,−a + F σ (−a) where n σ,−a ≥ 0.

D-MODULE STRUCTURE OF LOCAL COHOMOLOGY MODULES OF TORIC ALGEBRAS11
Proof. We sketch how the proof of the left equality goes. Let F = {σ 1 , . . . , σ d } be the set of all facets of R ≥0 A, and let We will prove the left equality in three steps. The first step shows Θ i ∈ I for i = 1, . . . , d. The second step shows t a · P a ∈ I for all a ∈ Z d \ C. Finally, the third step shows that t a · P a ∈ I if a ∈ C \ (−NA), and t a · P a is a product of some t −ai · P −ai , i = 1, . . . , n, if a ∈ C ∩ (−NA \ {0}).
(1) For each i = 1, . . . , d, consider the following subset of Z d : Since NA is simplicial, this is a one point set for each i, say {u i }. Notice that since t −ui ∈ I, F nσ i ,u i σi = P ui = t −ui · t ui P ui ∈ I where n σi,ui > 0. Therefore, F σi ∈ I for each i. Since F σ1 , . . . , F σ d are linearly independent, we conclude that Θ i ∈ I for i = 1, . . . , d.
(2) For a ∈ Z d \ C, F σ (a) > 0 for some σ ∈ F . By Lemma 4.4(4), choose k large so that We have P b = P ka and t ka−b ∈ I. By Lemma 4.4(1) (t a P a ) k = t ka P k a = t ka P ka · P = (t b P b ) · P · t ka−b ∈ I. Therefore, t a P a ∈ I as desired.
If −a / ∈ NA, by Lemma 4.4(2) (t a P a ) k = t ka P ka ·P for some nonconstant P ∈ K[Θ]. Since P is a product of some F σ 's, P ∈ I by (1) and hence t a P a ∈ I.
If −a ∈ NA \ {0}, write −a = m i a i . By Lemma 4.4(3), To complete the proof, we establish the right isomorphism. First, recall that if R → S is a homomorphism of commutative rings and Q is a prime ideal in S lying over a prime ideal q of R, then dim(S Q /qS Q ) ≥ htQ − htq. On the other hand, since gr D A is finitely generated as a K-algebra which is also a domain, each maximal ideal of gr D A has height 2d (Here, we use the fact that dim gr D A = 2d). Therefore, dim(grD A /I) ≥ d. Now, consider the surjection from the polynomial ring K[x 1 , . . . , x n ] to K t −ai · P −ai ; i = 1, . . . , n . By Lemma 4.4(3), P −ai = σ∈F F Fσ(ai) σ . Observe that t −ai · P −ai , i = 1, . . . , n, satisfy the relations in the 1 · · · x un n ). Hence we have a surjection S A ∼ = K[x 1 , . . . , x n ]/I A −→ K t −ai · P −ai ; i = 1, . . . , n which is an isomorphism by comparing the dimensions. S A = K[s, t, u, stu −1 ] is a 3-dimensional normal toric algebra which is isomorphic to the toric algebra appearing in Example 3.9. By 4.1, 4.6, and 6.3 in [ST04], Consider the surjection φ : K[a, . . . , o] → gr D A . Using Macaulay 2, we see that a primary decomposition of √ m gr D A is the intersection of the two ideals (o, n, d, a, c, f + g, e, b, f j − il, f i + jm, f 2 + lm, f k + hm, hi − jk, f h − kl) and (m, l, d, a, c, f, e+g, b, gk −io, gi+kn, g 2 +no, gj +hn, hi−jk, gh−jo) modulo Kerφ. Therefore, the fiber π −1 (0) has two components each of which is 4-dimensional.
As a corollary of Theorem 4.5, we can describe the fibers π −1 (p) for every nonzero closed point p in Spec S A . Let p be in the torus orbit O τ for some e-dimensional face τ of R ≥0 A, so p corresponds to a semigroup homomorphism f p : Then p corresponds to the maximal ideal m p = (t a1 − c 1 , . . . , t an − c n ) of S A . The following theorem gives the reduced induced structure of π −1 (p).