AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 5, May 2012, Pages 2461–2478 S 0002-9947(2012)05372-4
Article electronically published on January 13, 2012
D-MODULE STRUCTURE OF LOCAL COHOMOLOGY MODULES OF TORIC ALGEBRAS
JEN-CHIEH HSIAO
Abstract. Let S be a toric algebra over a fieldK of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules HiI(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.
1. Introduction
Lyubeznik [Lyu93] introduced an approach of studying local cohomology mod- ules using the theory of D-modules. He obtained many finiteness properties of the local cohomology modules HIi(R) when R is a regular ring containing a field of characteristic 0. For example, taking advantage of holonomicity of HIi(R) as a D-module, he showed that for any maximal ideal m the number of associated prime ideals of HIi(R) contained in m is finite and that all Bass numbers of HIi(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 HIi(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 Rx
and of HIi(R) as D-modules for a regular ring R, among them [Bøg95], [Bøg02], [Lyu97], [ ´AMBL05]. The first D-finiteness result of Rxfor a singular ring R is due to Takagi and Takahashi [TT08] which says Rxis 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 Sf and the local cohomology modules HIi(S) as D-modules, where S is a toric algebra (not necessarily normal) over a fieldK 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] gave a detailed description of D(S). Based on their results, we prove that any localization Sf = S[f−1] of S is generated by f−1 over D(S)
Received by the editors December 17, 2009 and, in revised form, April 8, 2010 and April 14, 2010.
2010 Mathematics Subject Classification. Primary 13D45, 13N10, 14M25.
The author was partially supported by the NSF under grants DMS 0555319 and DMS 0901123.
2012 American Mathematical Societyc Reverts to public domain 28 years from publication 2461
(Theorem 3.1). This implies immediately that HIi(S) is D(S)-finitely generated if D(S) is left Noetherian. Unfortunately, D(S) is not always left Noetherian [ST09].
Nonetheless, we can show that HIi(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 cohomol- ogy modules HIi(S). Characteristic cycles are formal sums of subvarieties (counted with multiplicities) of the characteristic variety of a D-module M . Here, the char- acteristic 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 poly- nomial algebra, one can explicitly compute the Bass numbers and the associated primes of HIi(S) from its characteristic cycles [ ´AM04]. The cohomological dimen- sion of I and the Lyubeznik numbers can also be computed from them [ ´AM00].
Through our finiteness results of HIi(S), we are able to compute the characteris- tic 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 vari- ety Ch(HmdimS(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 notation 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).
2. 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 ΓI(M ) := lim−→HomR(R/Ik, M ). Then ΓI is a left exact R-linear covariant functor and the i-th local cohomology functor HIi is defined to be its i-th right derived functor. We call HIi(M ) the i-th local cohomology module of M supported at the ideal I. If I is generated by f1, . . . , ft, then HIi(M ) is the i-th cohomology of the ˇCech complex
0→ M →
t i=1
Mfi→
1≤i<j≤t
Mfifj → · · · → Mf1···ft→ 0.
2.2. Toric algebras. We introduce some notation for later use. For more infor- mation on toric algebras, the reader is referred to [Ful93], [MS05] or [ILL+07].
Let A be a d× n integer matrix with columns a1, . . . , an. AssumeZA = Zd. For a field K, the semigroup subring SA,K := K[NA] = K[ta1, . . . , tan] of the Laurent polynomial ringK[Zd] =K[t±11 , . . . , t±1d ] is called the toric algebra associated to the matrix A. Denote NA := R≥0A∩ Zdas the saturation ofNA. Then SA,K:=K[NA]
is the normalization of SA,K.
2.3. Rings of differential operators. For a commutative algebra R over a field K, set D0(R;K) := R, and for i > 0,
Di(R;K) := {f ∈ HomK(R, R)| [f, r] ∈ Di−1(R;K) for all r ∈ R}.
Then the ring of differential operators is defined to be D(R;K) :=
j
Dj(R;K).
When R is a polynomial ring over a fieldK 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.
Lemma 2.1. If M is a D(R;K)-module and f ∈ R, then the R-module structure on Mf extends uniquely to a D(R;K)-module structure such that the natural map M → Mf is a D(R;K)-module homomorphism. In particular, via the ˇCech complex HIi(M ) has a natural D(R;K)-module structure.
Proof. See [Lyu00], Example (b).
When R is a regular algebra over a field K of characteristic 0, D(R; K) is well understood (see e.g. [Bj¨o79]). In this case, the local cohomology modules HIi(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 HIi(R) via the theory of D-modules. On the bright side, when R = SA,K is a toric algebra over an algebraically closed fieldK 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 SA := SA,K, whereK is an algebraically closed field of charac- teristic 0. Following [ST01], the noncommutative ring DA := D(SA,K) can be described as aZd-graded subring of
D(K[Zd];K) = K[t±11 , . . . , t±1d ]∂1, . . . , ∂d,
where [∂i, tj] = δij, [∂i, t−1j ] = −δijt−2j and the other pairs of variables commute.
More precisely, with the notation θi:= ti∂i, one has DA=
a∈Zd
taI(Ω(a)),
where Ω(a) = NA \ (−a + NA) and I(Ω(a)) is the vanishing ideal of Ω(a) in K[θ1, . . . , θd].
3. Finiteness properties of HIi(SA)
In this section, F will be denoted to be the set of all facets of R≥0A. For a face τ ofR≥0A, we writeN(A ∩ τ) := NA ∩ Rτ and denote Z(A ∩ τ) as the group generated byN(A ∩ τ).
We recall some notation in [ST01], which are crucial to the proofs of Theorems 3.1 and 3.3.
For a∈ Zd and τ a face ofR≥0A, define
Eτ(a) :={l ∈ C(A ∩ τ) | a − l ∈ NA + Z(A ∩ τ)}/Z(A ∩ τ).
Notice that
Eτ(a)⊆ [Zd∩ C(A ∩ τ)]/Z(A ∩ τ) = Z(A ∩ τ)/Z(A ∩ τ).
Here Z(A ∩ τ) := C(A ∩ τ) ∩ Z d is the saturation ofZ(A ∩ τ), so each Eτ(a) is a finite set. Consider the ordering≤ on Zd defined by
[a≤ b] ⇐⇒ [Eτ(a)⊆ Eτ(b)] for all faces τ ofR≥0A.
This ordering induces an equivalence relation onZd by [a∼ b] ⇐⇒ [a ≤ b and b ≤ a].
Now we are ready for the first main theorem.
Theorem 3.1. SA[f−1] = DA·f−1is cyclic as a left DA-module for any monomial f ∈ SA.
Proof. It’s clear that SA[f−1]⊇ DA·f−1. Conversely, write f = tbfor some b∈ NA (we may assume b = 0). Since SA[f−1]⊆ DA· {t−mb | m ∈ N}, it suffices to show that t−mb∈ DA· t−bfor all m∈ N. By Proposition 4.1.5(1) in [ST01], it is enough to prove that−b ∼ −mb. Indeed, we show that, for any face σ, Eσ(−b) = Eσ(−mb) as follows:
(1) Let σ be a facet, and suppose b /∈ σ. Then Fσ(−b) < 0, and hence Fσ(−mb) <
0. So Eσ(−b) = Eσ(−mb) = ∅.
(2) Let τ be a face and suppose that b /∈ τ. Then there exists a facet σ containing τ with b /∈ σ. Hence Eσ(−b) = Eσ(−mb) = ∅ by (1), and thus Eτ(−b) = Eτ(−mb) = ∅.
(3) Let τ be a face and suppose b∈ τ. Then b ∈ NA ∩ τ = N(A ∩ τ). Hence,
±b ∈ Z(A ∩ τ), and thus Eτ(−b) = Eτ(−mb) by definition. Remark 3.2. Via the ˇCech complex, it follows immediately from Theorem 3.1 that HIi(SA) is finitely generated over DA if DA is left Noetherian. The left Noetheri- aness of DA was studied by Saito and Takahashi [ST09]. They proved that DA is left Noetherian if SAsatisfies Serre’s condition (S2). Serre’s condition is, by [Ish88], equivalent to
SA=
τ : facets
K[NA + Z(A ∩ τ)].
Saito and Takahashi also gave a necessary condition (on SA) for DA to be left Noetherian. However, DA is not always left Noetherian.
Nonetheless, we have
Theorem 3.3. For any i and any monomial ideal I, HIi(SA) is of finite length as a DA-module.
Proof. In view of the ˇCech complex, since any localization SA[f−1] (with monomial f ) is a DA-submodule of K[Zd], it suffices to show that K[Zd] has a composition series.
Consider the notation in the beginning of this section. For a, b ∈ Zd, we will write a < b if a≤ b but a b. Then, for each a ∈ Zd,
b≥aKtb is generated by ta as a DA-module. Moreover,
b≥aKtb
b>aKtb ∼=
b∈[a]
Ktb
is, by Theorem 4.1.6 in [ST01], a simple DA-module where [a] = {b ∈ Zd | b ∼ a}. Since there are only finitely many faces and since each Eτ(a) is contained in Z(A ∩ τ)/Z(A ∩ τ), there are finitely many equivalence classes determined by ∼; we denote them [α1], [α2], . . . , [αk]. We may rearrange the order so that, for any pair i < j, either αi > αj or αiand αj are incomparable. Denote Ta:=
b≥aKtb. Then the filtration
0 Tα1 · · · Σil=1Tαl · · · Σkl=1Tαl=K[Zd]
is a composition series of DA-submodules ofK[Zd]. Example 3.4. For 1-dimensional SA, the composition series of K[Z] is easy to describe. In this case,R≥0A has two faces, 0 and σ =R≥0A. For a∈ Z,
E0(a) ={ ∈ {0} | a − ∈ NA}/{0} and Eσ(a) ={ ∈ Z | a − ∈ Z}/Z.
Thus E0(a) ={0} if a ∈ NA, E0(a) =∅ if a /∈ NA, and Eσ(a) ={0} for all a ∈ Z.
Therefore, [0] and [−1] are the two equivalence classes determined by ∼, and we have the composition series
0 K[NA] = T0 T−1=K[Z].
Remark 3.5. Theorem 3.1 and Theorem 3.3 also hold for any fieldK with charac- teristic 0 by the isomorphism
HIi(SA,K)⊗ K ∼= HIi(SA,K).
Remark 3.6. Suppose I = m, the maximal graded ideal of SA. Here we assume that the semigroupNA is pointed, so that 0 is the only invertible element.
(1) Recall that Hmi(SA) can be computed as the i-th cohomology of the Ishida complex [Ish88] or [ILL+07]. Therefore, Hm1(SA) is finitely generated as an SA-module. Indeed, it suffices to observe that
Hm1(SA)
σ: rays
K[NA + Z(A ∩ σ)] ⊆
τ : facets
K[NA + Z(A ∩ τ)] ⊆ SA
and the fact that SA is finite over SA. Moreover, Hmd(SA) is cyclic as a left DA-module. This is because the d-th module in the Ishida complex is K[Zd], which is cyclic by Theorem 3.1.
(2) In general, Sch¨afer and Schenzel [SS90] showed that there is a partition ofZd with respect to which Hmi(SA) can be written as a finite direct sum ofK-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 ofR≥0A, and for any filter (cocomplex)∇ of Conv(A), denote
P∇=
A∩τ∈∇
[NA + Z(A ∩ τ)] \
A∩τ /∈∇
[NA + Z(A ∩ τ)].
Then the P∇’s form a partition (sector partition) ofZd and Hmi(SA) =
∇
K[P∇]⊗KHi(Conv(A), Conv(A)\ ∇; K).
On the other hand, for a∈ Zd denote
∇(a) := {face τ of R≥0A| a ∈ NA + Z(A ∩ τ)}
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 equiva- lence relation∼ in the proof of Theorem 3.3 is finer than the sector partition determined by≡.
Notice that eachK[P∇] is naturally a left Zd-graded DA-module because each K[NA + Z(A ∩ τ)] is as well. If SA is normal, the DA-moduleK[P∇] is simple. In fact, we have
Theorem 3.7. If SA is normal and I is a monomial ideal in SA, then every simple subquotient of HIi(SA) 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 onZd. Note that the normality of SAimplies that
Eτ(a) ={0} if a ∈ NA + Z(A ∩ τ) and Eτ(a) =∅ if a /∈ NA + Z(A ∩ τ).
So we have
a∼ b
⇔Eτ(a) = Eτ(b) for all τ
⇔Eτ(a) ={0} if and only if Eτ(b) ={0}
⇔a ∈ NA + Z(A ∩ τ) if and only if b ∈ NA + Z(A ∩ τ)
⇔a ≡ b.
Therefore, the simple subquotients ofK[Zd] are precisely the DA-modules K[P∇] =
b∈[a]
Ktb∼=
b≥aKtb
b>aKtb.
Example 3.8. Consider A = 1 1 1
0 1 2
. Then SA =K[t, ts, ts2] is a 2-dimensional normal toric algebra. Let I = (ts) be the ideal of SA generated by ts. We shall describe a composition series of HI1(SA). By ˇCech complex, HI1(SA) =K[Zd]/SA. Following the notation in the proof of Theorem 3.3 and Remark 3.6, let
∇0={0, σ1, σ2,R≥0A}, ∇1={σ1,R≥0A}, ∇2={σ2,R≥0A},
∇12={σ1, σ2,R≥0A}, ∇A={R≥0A},
where σ1=R≥01
0
and σ2=R≥01
2
. Then
P∇0 = P∇(a0)=NA,
P∇1 = P∇(−a1)= [NA + Z(A ∩ σ1)]\ [NA + Z(A ∩ σ2)], P∇2 = P∇(−a2)= [NA + Z(A ∩ σ2)]\ [NA + Z(A ∩ σ1)], P∇12 =∅, and
P∇A = P∇(−a3)=Z2\
(NA + Z(A ∩ σ1))∪ (NA + Z(A ∩ σ2)) , where a0=0
0
and ai, i = 1, 2, 3, is the i-th column of A. In terms of notation in Theorem 3.3,
Ta0 =K[P∇0] = SA, T−a1 =K[P∇0]⊕ K[P∇1], T−a2 =K[P∇0]⊕ K[P∇2],
T−a3 =K[Z2] =K[P∇0]⊕ K[P∇1]⊕ K[P∇2]⊕ K[P∇A].
So 0⊂ Ta0 ⊂ T−a1 ⊂ T−a1+ T−a2⊂ T−a3=K[Z2] is a composition series ofK[Z2].
Quotienting out SA, we obtain a composition series of HI1(SA):
0⊂ K[P∇1]⊂ K[P∇1]⊕ K[P∇2]⊂ K[P∇1]⊕ K[P∇2]⊕ K[P∇A].
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 =
1 1 1 1
0 1 0 1 0 0 1 1
. Then SA =K[r, rs, rt, rst] is a normal toric algebra.
Consider the ideal I = (r, rs) of SA. Then the socle HomSA(SA/m, HI2(SA)) is infinite dimensional, where m = (r, rs, rt, rst) is the maximal graded ideal of SA. However, according to Theorem 3.3 HI2(SA) is of finite length over DA.
In fact, HI2(SA) is DA-simple. To see this, using the notation in Remark 3.6 we consider the filter ∇ = {σ12,R≥0A}, where σ12 =R≥0 1 1
0 1 0 0
is a facet of R≥0A.
We claim that
P∇= [NA + Z(A ∩ σ12)]\ [(NA + Za1)∪ (NA + Za2)], where a1=
1
0 0
and a2 = 1
1 0
. Therefore, in view of the ˇCech complex we have the isomorphism
HI2(SA) ∼=K[P∇], which is DA-simple by Theorem 3.7.
The claim is equivalent to the equality NA + Z(A ∩ σ12)
∩
σ=σ13,σ24,σ34
(NA + Z(A ∩ σ))
=
NA + Z(A ∩ σ12)
∩
(NA + Za1)∪ (NA + Za2) ,
where σ13=R≥0 1 1
0 0 0 1
, σ24=R≥0 1 1
1 1 0 1
, and σ34=R≥0 1 1
0 1 1 1
. This equality can be verified by the following data:
NA + Za1={t(x, y, z)∈ Z3| y ≥ 0 and z ≥ 0}, NA + Za2={t(x, y, z)∈ Z3| x ≥ y and z ≥ 0}, NA + Z(A ∩ σ12) ={t(x, y, z)∈ Z3| z ≥ 0}, NA + Z(A ∩ σ13) ={t(x, y, z)∈ Z3| y ≥ 0}, NA + Z(A ∩ σ24) ={t(x, y, z)∈ Z3| x ≥ y}, NA + Z(A ∩ σ34) ={t(x, y, z)∈ Z3| x ≥ z}.
Remark 3.10. Helm and Miller [HM03] studied the Bass numbers of local coho- mology modules over toric algebras. As a generalization of Hartshorne’s example, their main result ([HM03], Theorem 7.1) implies that for a Gorenstein normal toric algebra SA,NA is not simplicial if and only if there exists an NA-graded prime p of dimension 2 such that Hpd−1(SA) has infinite-dimensional scole.
4. Associated graded rings gr DA and characteristic cycles 4.1. Associated graded rings gr DA. Let R be aK-algebra as in subsection 2.3.
The definition of D(R;K) gives an order filtration of D(R; K):
0⊆ R = D0(R;K) ⊆ D1(R;K) ⊆ D2(R;K) ⊆ · · · . Define the associated graded ring of D(R;K) to be
gr D(R;K) := D0⊕ (D1/D0)⊕ (D2/D1)⊕ · · · ,
where Di := Di(R;K). From the definition of D(R; K), gr D(R; K) is a com- mutative R-algebra and we have the natural embedding R → gr D(R; K). For example, if R = K[t1, . . . , td] is a polynomial algebra over K, then D(R; K) = K[t1, . . . , td]∂1, . . . , ∂d is the Weyl algebra. The associated graded ring gr D(R; K)
=K[t1, . . . , td, ξ1, . . . , ξd] is a 2d-dimensional polynomial algebra overK, where ξi
is the image of ∂iin the associated graded ring gr D(R;K). In what follows, we will use the description
gr D(K[t±11 , . . . , t±1d ];K) = K[t±11 , . . . , t±1d , Θ1, . . . , Θd],
where Θi= tiξi is the image of θi= ti∂i in the associated graded ring gr D(R;K).
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 spaceKdim R.
In this subsection, we discuss 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 a more complicated nature.
To begin with, consider the natural order filtration of DAinherited from that of D(K[Zd]). With respect to this filtration, one can regard grDA as a commutative subalgebra of grD(K[Zd]) = K[t±11 , . . . , t±1d , Θ1, . . . , Θd]. When gr DA is finitely generated over K, Musson showed that it has dimension 2d [Mus87]. Saito and
Traves proved that gr DA is finitely generated overK if and only if the semigroup NA is scored [ST04]. By definition, a semigroup NA is scored if
NA =
σ: facet
{a ∈ Zd | Fσ(a)∈ Fσ(NA)}
or, equivalently, NA \ NA is a union of finitely many hyperplane sections parallel to some facets of R≥0A. The scored condition implies Serre’s condition (S2) (see Remark 3.2).
We should remark that if SA is normal, then gr DA is Gorenstein ([Mus87], Theorem D). In general, even for a 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 SAis a 1-dimensional toric algebra which is not normal, then gr DA is not Cohen-Macaulay.
Proof. Using the formula in Lemma 4.3, we see that gr DA is again a toric algebra overK. Indeed, notice that 0 is the only facet and that n0,w =|Ω(w)|. Note also that|Ω(−w)| = w + |Ω(w)| by Lemma 4.4. So by Lemma 4.3,
gr DA=K
tξ, t|Ω(w)|ξ|Ω(−w)|:|w| ∈ {a1, . . . , an} ∪ Hole(NA) . Therefore gr DA is a two-dimensional toric algebra overK.
We claim that
(4.1) dimKK[t, ξ]
gr DA
<∞.
Take to be the maximal number of 2|Ω(−w)| for w ∈ Hole(NA). To prove (4.1), it is enough to show that tuξv∈ gr DAfor all pairs u, v∈ N satisfying u+v ≥ . Since [tuξv∈ gr DA⇔ tvξu∈ gr DA], by symmetry we may assume w0:= u− v ≥ 0.
• If w0∈ Hole(NA), 2u ≥ u + v ≥ ≥ 2|Ω(−w0)|. So (tξ)u−|Ω(−w0)|∈ gr DA, and hence tuξv= (tξ)u−|Ω(−w0)|t|Ω(−w0)|ξ|Ω(w0)|∈ gr DA.
• If w0∈ NA we have tuξv= (tξ)vtw0 = (tξ)vt|Ω(−w0)|∈ gr DA.
So the claim is proved. Now, applying the criterion in Remark 3.2 to the toric algebra gr DA, we see that gr DA doesn’t satisfy Serre’s condition (S2). Hence,
gr DA is not Cohen-Macaulay.
Remark 4.2. The claim (4.1) holds true for any affine curve with injective normal- ization (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 SAcorresponding to the closed point 0 of the toric variety Spec(SA). Let I =√
mgrDA be the radical of the extended ideal of m under the embedding SA → grDA. We are going to show that if NA is simplicial and scored, then grDA/I is isomorphic to SAasK-algebras. This implies that the reduced induced structure of the fiber of π : Spec(grDA)→ Spec(SA) over the point 0 is isomorphic to the ambient toric variety.
The following two lemmas are needed:
Lemma 4.3 ([ST04]). For scoredNA, gr DA=
a∈Zd
taK[Θ1, . . . , Θd]· Pa where
Pa=
σ∈F
Fσ(Θ)nσ,a and nσ,a= #{Fσ(NA) \ [−Fσ(a) + Fσ(NA)]}.
Lemma 4.4. LetNA be scored. Then for any a ∈ Zd and σ∈ F , nσ,−a= nσ,a+ Fσ(a).
In particular,
(1) if σ is a facet with the property that Fσ(a)≤ 0, then nσ,ka ≤ k · nσ,a for large k∈ N, and furthermore Pak = Pka· P for some P ∈ K[Θ];
(2) if Fσ(a)≤ 0 for all σ ∈ F and −a /∈ NA, then nσ,ka < k· nσ,a for some σ∈ F and large k ∈ N;
(3) if a∈ NA, then nσ,−a= Fσ(a);
(4) if Fσ(a) > 0, then nσ,ka= 0 for large k∈ N.
Proof. To prove nσ,−a = nσ,a+ Fσ(a) for any a∈ Zd and σ ∈ F , it’s enough to show the case where Fσ(a) > 0. Set N = Fσ(NA) and n = Fσ(a) > 0. Then
nσ,−a= #{N \ (n + N)}
= #{b ∈ N | b − n /∈ N}
= n + #{b ∈ N | b − n /∈ N but b − kn ∈ N for some k ≥ 2}
= n + #{c ∈ N | c + n /∈ N}
= n + #{N \ (−n + N)} = n + nσ,a.
Note that for the second equality we need the assumption thatNA is scored.
Now, we prove the four additional statements:
(1) Notice that nσ,ka = kFσ(−a) for large k and that nσ,a = nσ,−a+ Fσ(−a) where nσ,−a≥ 0.
(2) By assumption, −a lies on a hyperplane parallel to some facet, say σ0. Then nσ0,−a> 0 and hence nσ0,ka< k· nσ0,aby (1).
(3) a∈ NA implies nσ,a= 0.
(4) This follows from the definition. Indeed, sinceNA is scored, (Fσ(ka)+N0)⊆ Fσ(NA) for large k. Then Fσ(ka)+Fσ(NA) ⊆ Fσ(NA), and hence nσ,ka = 0.
Theorem 4.5. IfNA is a simplicial scored semigroup, then
gr DA/I =K
t−ai· P−ai; i = 1, . . . , n
∼= SA,
where I =√
m gr DA and t−ai· P−ai is the image of t−ai· P−ai in gr DA/I.
Proof. We sketch how the proof of the left equality goes. LetF = {σ1, . . . , σd} be the set of all facets ofR≥0A, and let
C =−NA = {a ∈ Zd| Fσ(a)≤ 0 for all σ ∈ F }.
We will prove the left equality in three steps. The first step shows Θi ∈ I for i = 1, . . . , d. The second step shows ta· Pa∈ I for all a ∈ Zd\ C. Finally, the third step shows that ta· Pa∈ I if a ∈ C \ (−NA) and that ta· Pa 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 ofZd: {Fσi(Θ) =−1} ∩
⎡
⎣
j=i
{Fσj(Θ) = 0}
⎤
⎦ .
Since NA is simplicial, this is a one point set for each i, say {ui}. Notice that since t−ui ∈ I, Fσniσi,ui = Pui = t−ui· tuiPui ∈ I, where nσi,ui > 0.
Therefore, Fσi ∈ I for each i. Since Fσ1, . . . , Fσd are linearly independent, we conclude that Θi ∈ I for i = 1, . . . , d.
(2) For a ∈ Zd\ C, Fσ(a) > 0 for some σ∈ F . By Lemma 4.4(4), choose k large so that
Pka=
Fσ(a)<0
Fσnσ,ka. Now, consider as in (1) the one-point set
⎡
⎣
Fσ(a)<0
{Fσ(Θ) = Fσ(ka)}
⎤
⎦ ∩
⎡
⎣
Fσ(a)≥0
{Fσ(Θ) = 0}
⎤
⎦ = {b}.
We have Pb= Pka and tka−b∈ I. By Lemma 4.4(1)
(taPa)k = tkaPak= tkaPka· P = (tbPb)· P · tka−b∈ I.
Therefore, taPa∈ I as desired.
(3) Let a∈ C.
If−a /∈ NA, by Lemma 4.4(2) (taPa)k= tkaPka·P for some nonconstant P ∈ K[Θ]. Since P is a product of some Fσ’s, P ∈ I by (1), and hence taPa ∈ I.
If−a ∈ NA \ {0}, write −a =
miai. By Lemma 4.4(3), nσ,a= Fσ(−a) =
miFσ(ai) =
minσ,−ai, and therefore
taPa= tmi(−ai)·
σ∈F
F
minσ,−ai
σ =
(t−aiP−ai)mi.
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(SQ/qSQ) ≥ htQ − htq. On the other hand, since gr DA is finitely generated as a K-algebra which is also a do- main, each maximal ideal of gr DA has height 2d. (Here, we use the fact that dim gr DA= 2d.) Therefore, dim(grDA/I)≥ d. Now, consider the surjection from the polynomial ringK[x1, . . . , xn] toK
t−ai· P−ai; i = 1, . . . , n
. By Lemma 4.4(3), P−ai =
σ∈FFσFσ(ai). Observe that t−ai· P−ai, i = 1, . . . , n, satisfy the relations in the toric ideal IA ={xu− xv|Au = Av} (where for u ∈ Zn, xu := xu11· · · xunn).
Hence we have a surjection
SA∼=K[x1, . . . , xn]/IA−→ K
t−ai· P−ai; i = 1, . . . , n
,
which is an isomorphism by comparing the dimensions. Example 4.6. Consider
A =
⎛
⎝1 0 0 1
0 1 0 1
0 0 1 −1
⎞
⎠ .
SA =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],
gr DA=K[s, t, u, stu−1, Θs, Θt, Θu, s−1Θs(Θs+Θu), t−1Θt(Θt+Θu), s−1t−1uΘsΘt
u−1(Θs+ Θu)(Θt+ Θu), tu−1(Θs+ Θu), t−1uΘt, su−1(Θt+ Θu), s−1uΘs].
Set
a = s, b = t, c = u, d = stu−1, e = Θs, f = Θt, g = Θu, h = s−1Θs(Θs+ Θu), i = t−1Θt(Θt+ Θu), j = u−1(Θs+ Θu)(Θt+ Θu), k = s−1t−1uΘs· Θt, l = tu−1(Θs+ Θu), m = t−1uΘt, n = su−1(Θt+ Θu), o = s−1uΘs.
Consider the surjection φ : K[a, . . . , o] → gr DA. Using Macaulay 2, we see that a primary decomposition of √
m gr DA is the intersection of the two ideals (o, n, d, a, c, f + g, e, b, f j− il, fi + jm, f2 + lm, f k + hm, hi− jk, fh − kl) and (m, l, d, a, c, f, e+g, b, gk−io, gi+kn, g2+no, gj +hn, hi−jk, gh−jo) modulo Kerφ.
Therefore, the fiber π−1(0) has two components, each of which is 4-dimensional.
Remark 4.7. The left equality of Theorem 4.5 and Example 4.6 can be achieved alternatively by a result of Saito.
By Proposition 4.14 in [Sai10], π−1(m) = {P(q, ν) | ν ∩ R≥0A = {0}}. If NA is simplicial, then π−1(m) ={P(m0,−R≥0A)}, which is the left equality of Theo- rem 4.5. On the other hand, consider Example 4.6. If P(q, ν)∈ π−1(m), then by [Sai10], Proposition 4.14, q⊇ (Θs, Θt+ Θu) or (Θt, Θs+ Θu). If q = (Θs, Θt+ Θu), then ν ={Θs≤ 0, Θt+Θu≤ 0}. If q = (Θt, Θs+Θu), then ν ={Θt≤ 0, Θs+Θu≤ 0}. They are the two minimal primes mentioned in Example 4.6.
As a corollary of Theorem 4.5, we can describe the fibers π−1(p) for every nonzero closed point p in Spec SA. Let p be in the torus orbit Oτ for some e-dimensional face τ ofR≥0A, so p corresponds to a semigroup homomorphism fp:NA → K with fp(ai) = ci, where ci = 0 if and only if ai∈ N(A ∩ τ). Then p corresponds to the/ maximal ideal mp = (ta1− c1, . . . , tan− cn) of SA. The following theorem gives the reduced induced structure of π−1(p).
Theorem 4.8. Under the hypotheses of Theorem 4.5, we have gr DA
mpgr DA
∼= SB⊗ K[δ1, . . . , δe],
where SB is the toric algebra generated by a simplicial scored semigroup NB ∼=NA + Z(A ∩ τ)
Z(A ∩ τ)
andK[δ1, . . . , δe] is a polynomial ring in e(= dim τ ) variables.
Proof. First, notice that ta, a ∈ N(A ∩ τ), acts as a unit on √gr DA
mpgr DA because p∈ Oτ. So, by abusing the notation
gr DA
mpgr DA
∼= gr DA[τ−1]
mpgr DA[τ−1],
where τ−1 means we invert ta for all a∈ N(A ∩ τ). Note also that gr DA[τ−1] ∼= gr D(SA[τ−1]) = gr D(K[NA + Z(A ∩ τ)]).
Next, choose a simplicial scored semigroupNB so that NA + Z(A ∩ τ) = NB ⊕ Z(A ∩ τ).
To do this, let’s first assumeNA is normal. By an exercise of section 1.2 in [Ful93],
R≥0A+Rτ
Rτ is a simplicial rational polyhedral cone with facets γ+RτRτ, where the γ’s are the facets ofR≥0A containing τ . So NA+Z(A∩τ)Z(A∩τ) is a simplicial normal semigroup inZd/Z(A ∩ τ). NB can be obtained by choosing suitable elements in [NA + Z(A ∩ τ )]\ Z(A ∩ τ). For the general simplicial scored semigroup NA, we just have to notice that NA \ NA is a union of hyperplane sections parallel to some facets of R≥0A. SoZ(A ∩ τ) = Z(A ∩ τ) and [NA + Z(A ∩ τ)] \ [NA + Z(A ∩ τ)] is a union of hyperplane sections parallel to some facets ofR≥0A containing τ .
Now,
gr DA[τ−1] ∼= gr D(K[NB ⊕ Z(A ∩ τ)]) ∼= gr DB⊗ gr DZ(A∩τ). Therefore,
gr DA
mpgr DA
∼= gr DB⊗ gr DZ(A∩τ)
mpgr DB⊗ gr DZ(A∩τ)
∼=√ gr DB
mBgr DB ⊗ K[δ1, . . . , δe]
∼= SB⊗ K[δ1, . . . , δe]
by Theorem 4.5, where δ1, . . . , δeare the standard derivations ofK[Z(A ∩ τ)]. 4.2. Characteristic cycles. Let D := D(R;K) as defined in section 2. Let M be a D-module with a filtration{Mi} such that DiMj⊆ Mi+j. The associated graded module gr M :=
Mi/Mi−1 has the natural gr D-module structure. We call{Mi} a good filtration if gr M is finitely generated over gr D. If M is finitely generated over D by x1, . . . , xn, then the filtration{n
j=1Dixj} is good.
From now on, we assume that gr D is Noetherian. This is always the case when R is regular.
For a D-module M with a good filtration{Mi}, define the characteristic variety Ch(M ) of M to be the support of the gr D-module gr M ,
i.e. Ch(M ) = Var(anngr Dgr M )⊆ Spec(gr D).
The characteristic cycle CC(M ) of M is the formal sum of the irreducible compo- nents Vi of Ch(M ) counted with multiplicity. More precisely,
CC(M ) = miVi,
where the multiplicity mi is the length of the (gr D)pi-module (gr M )pi and pi is the prime ideal corresponding to Vi.
