COMPUTATIONAL ASPECTS OF GROTHENDIECK LOCAL RESIDUES

COMPUTATIONAL ASPECTS OF GROTHENDIECK LOCAL RESIDUES

by

Shinichi Tajima & Yayoi Nakamura

Dedicated to Professor Tatsuo Suwa on his sixtieth birthday Abstract. — Grothendieck local residues are studied from a view point of algebraic analysis. The main idea in this approach is the use of regular holonomicD-modules attached to a zero-dimensional algebraic local cohomology class. A new method for computing Grothendieck local residues is developed in the context of Weyl alge- bra. An effective computing algorithm that exploits first order annihilators is also described.

Résumé (Aspects effectifs des résidus locaux de Grothendieck). — On ´etudie le r´esidu local de Grothendieck du point de vue de l’analyse alg´ebrique. L’id´ee principale de cette approche est l’utilisation de D-modules holonomes r´eguliers attach´es `a une classe alg´ebrique de cohomologie locale en dimension z´ero. On d´eveloppe une m´ethode nouvelle pour calculer les r´esidus locaux de Grothendieck dans le cadre de l’alg`ebre de Weyl. Cette m´ethode permet de d´ecrire un algorithme efficace, lequel utilise les annulateurs du premier ordre.

1. Introduction

In this paper, we consider Grothendieck local residues and its duality in the context of holonomic D-modules. Upon using the regular holonomic system associated to a certain zero-dimensional algebraic local cohomology class, we derive a method for computing Grothendieck local residues. We also give an effective algorithm that serves exact computations.

In§2, we study local residues from the viewpoint of the analyticD-module theory.

By using the local residue pairing, we associate to an algebraic local cohomology class attached to a given regular sequence an analytic linear functional acting on the space of germs of holomorphic functions. We apply Kashiwara-Kawai duality theorem on holonomic systems [3] to the residue pairing and show that the kernel of the above analytic functional can be described in terms of partial differential operators. This result ensures in particular the computability of the Grothendieck local residues.

2000 Mathematics Subject Classification. — Primary 32A27; Secondary 32C36, 32C38.

Key words and phrases. — Grothendieck local residues, algebraic local cohomology classes, holonomic D-modules.

In§3, we give a framework in the Weyl algebra, and develop there a method for computing Grothendieck local residues. The key ingredient of the present method is the annihilating ideal in the Weyl algebra of the given zero-dimensional algebraic local cohomology class. We show that the use of generators of the annihilating ideal in the Weyl algebra reduces the computation of the local residues to that of linear equations.

In §4, we derive an algorithm for computing Grothendieck local residues that exploits only first order partial differential operators. The resulting algorithm (Algo- rithm R) is efficient and thus can be available in use for actual computations in many cases. We also present an criterion to the applicability of this algorithm.

In§5, we give an example to illustrate an effectual way of using Algorithm R.

In Appendix, we present an algorithm that outputs the first order partial differen- tial operators which annihilate a direct summand in question of the given algebraic local cohomology class.

2. Local duality theorem

LetOXbe the sheaf of holomorphic functions onX=CⁿandFa regular sequence given by n holomorphic functions f1, . . . , fn on X. Denote by I the ideal of OX

generated byf1, . . . , fn andZ the zero-dimensional variety V(I) ={z∈X |f1(z) =· · ·=fn(z) = 0} of the idealI consisting of finitely many points.

There is a canonical mapping ι from the sheaf of n-th extension groups Extⁿ_OX(OX/I, Ω_Xⁿ) to the sheaf ofn-th algebraic local cohomology groups

Hⁿ[Z](Ω_Xⁿ) with support onZ:

ι:Extⁿ_OX(OX/I, Ω_Xⁿ)−→ H[Z]ⁿ (Ω_Xⁿ)

whereΩ_Xⁿ is the sheaf of holomorphicn-forms onX. We denote byω_F = dz

f1· · ·fn

the image by the mappingιof the Grothendieck symbol dz

f1. . . fn

∈ Extⁿ_OX(OX/I, Ωⁿ_X), i.e.,

(1) ω_F =ι

dz f1. . . fn

∈ Hⁿ[Z](Ωⁿ_X),

wheredz=dz1∧ · · · ∧dzn. Letω_F,β denote the germ atβ∈Z of the algebraic local cohomology classω_F:

ω_F,β ∈ H[β]ⁿ (Ωⁿ_X),

whereHⁿ_[β](Ω_Xⁿ) stands for the algebraic local cohomology supported atβ.

Let Hⁿ_{β}(Ωⁿ_X) be the sheaf of n-th local cohomology groups at β ∈ Z and let Resβ:Hⁿ_{β}(Ω_Xⁿ)→Cbe the local residue map. Recall that the mapping

Hⁿ_{β}(Ω_Xⁿ)× OX,β−→ Hⁿ_{β}(Ωⁿ_X)

composed with the local residue map Resβ defines a natural pairing between two topological vector spacesH_{β}ⁿ (Ω_Xⁿ) andOX,β. Thus, the algebraic local cohomology class ω_F,β ∈ Hⁿ[β](Ωⁿ_X) which also belongs to Hⁿ_{β}(Ωⁿ_X) induces a linear functional Resβ(ω_F) that acts onOX,β. Namely, Resβ(ω_F) is defined to be

Resβ(ω_F)(ϕ(z)) = Resβ(ϕ(z)ω_F,β)

for ϕ(z)∈ OX,β, β ∈Z. We consider the kernel space Ker of the linear functional Resβ(ω_F) defined to be

Ker ={ψ(z)∈ O^X,β |Resβ(ω_F)(ψ(z)) = 0}.

Now we are going to give an alternative description of the kernel space Ker in terms of partial differential operators.

LetD^Xbe the sheaf onX of linear partial differential operators. Then the sheaves Ωⁿ_X,Hⁿ[β](Ω_Xⁿ) andHⁿ[Z](Ωⁿ_X) arerightDX-modules. Note also thatOXandHⁿ[β](OX) have a structure of left DX-modules. We denote by Ann_DX(ω_F) the right ideal of DXconsisting of linear partial differential operators which annihilate the cohomology classω_F:

Ann_DX(ω_F) ={P ∈ DX|ω_FP = 0}.

Note that, if we setω_F =σ_Fdzwithσ_F ∈ H[Z]ⁿ (OX), the right idealAnn_DX(ω_F) can be rewritten as

Ann_DX(ω_F) ={P ∈ DX|P^∗σ_F = 0}, whereP^∗ stands for the formal adjoint operator ofP.

TheDX-moduleDX/Ann_DX(ω_F) is isomorphic to Hⁿ_[Z](Ω_Xⁿ). We thus in partic- ular have the following theorem (cf.[2], [3], [7]);

Theorem 2.1. — LetFbe a regular sequence given bynholomorphic functions andω_F an algebraic local cohomology class defined by (1)whose support contains a pointβ.

(i) D^X/Ann_DX(ω_F)is a regular singular holonomic system.

(ii) DX/Ann_DX(ω_F)is simple at each point β∈Z.

The theorem implies the folloiwng result on the local cohomology solution space of the holonomic systemDX/Ann_DX(ω_F);

Corollary 2.2. — Let β ∈Z. Then

Hom_DX(DX/Ann_DX(ω_F),Hⁿ_{β}(Ω_Xⁿ)) = Hom_DX(DX/Ann_DX(ω_F),Hⁿ[β](Ω_Xⁿ))

=Cω_F,β

holds.

The above result means that the holonomic systemDX/Ann_DX(ω_F) completely characterize the algebraic local cohomology classω_F as its solution.

Example 2.3 (cf. [1]). — LetF={f1, f2}be a regular sequence andIbe the ideal in C[x, y] generated by functionsf1 andf2given below. Letj_F(x, y) = det

∂(f1, f2)

∂(x, y)

be the Jacobian off1 andf2. We fix the lexicographical orderingxy and use the term orderingin computations of Gr¨obner basis ofI.

(i) Let f1 =x(x²−y³−y⁴), f2 = x²−y³. We haveI =hx²−y³, xy⁴, y⁷i and V(I) ={(0,0)}with the multiplicity 11. The algebraic local cohomology classω_F= hdx∧dy

f1f2

i is supported only at the origin (0,0). The annihilating idealAnn_DX(ω_F) of ω_F is generated by multiplication operatorsx(x²−y³−y⁴),x²−y³and a first order differential operator P = 3x_∂x^∂ + 2y_∂y^∂ −12. By solving the system of differential equations ω_Fy⁷ = ω_Fxy⁴ = ω_F(x² −y³) = ω_FP = 0 together with the formula j_F(x, y)ω_F= 11δ(0,0)dx∧dy whereδ(0.0)=h

1 xy

i∈ H²_[(0,0)](OX) is the delta function with support at the origin, we have the following representation ofω_F;

ω_F= 1

x⁵y + 1 x³y⁴+ 1

xy⁷

dx∧dy

.

(ii) Letf1=xandf2= (x²−y³)(x²−y³−y⁴). We haveI=hx, y⁷+y⁶iand its primary decomposition I =hx, y+ 1i ∩ hx, y⁶i. The annihilating ideal Ann_DX(ω_F) of the algebraic local cohomology classω_F=h

dx∧dy f1f2

iis generated by x, y⁷+y⁶ and P = (y²+y)_∂y^∂ −5y−5. We have a representation

1 xy− 1

xy² + 1 xy³ − 1

xy⁴ + 1 xy⁵ − 1

xy⁶

dx∧dy

+

dx∧dy x(y+ 1)

ofω_F by solving the system of differential equationsω_Fx=ω_F(y⁶+y⁷) =ω_FP= 0 together with the formula j_F(x, y)ω_F = (6δ(0,0)+δ(0,−1))dx∧dy where δ(0,−1) = h 1

x(y+1)

iis the delta function with support at (0,−1).

(iii) Let f1 = x² −y³−y⁴ and f2 = x(x² −y³). We have I = hx² −y⁴− y³, xy⁴, y⁸+y⁷iand its primary decompositionI=hx, y+ 1i ∩ hx²−y⁴−y³, xy⁴, y⁷i. The variety {(0,−1)} is simple and {(0,0)} is of multiplicity 11. The annihilating idealAnn_DX(ω_F) of the algebraic local cohomology classω_F=h

dx∧dy f1f2

iis generated byx²−y⁴−y³, xy⁴, y⁸+y⁷ and P = (4xy+ 3x)_∂x^∂ + (2y²+ 2y)_∂y^∂ −12y−12. We have a representation ofω_F as

1 xy + 1

x⁵y − 1 xy² + 1

xy³ − 1 xy⁴ + 1

x³y⁴ + 1 xy⁵ − 1

xy⁶ + 1 xy⁷

dx∧dy

+

−dx∧dy x(y+ 1)

by solvingω_F(x²−y³−y⁴) =ω_Fxy⁴ =ω_F(y⁸+y⁷) =ω_FP = 0 together with the formulaj_F(x, y)ω_F = (11δ(0,0)+δ(0,−1))dx∧dy.

Example 2.4 ([4]). — Letf =x³+y⁷+xy⁵. We consider the regular sequence given by partial derivarivesf1= 3x²+y⁵andf2= 5xy⁴+7y⁶off. The primary decomposition of the ideal I = hf1, f2i is given by h3125x+ 151263,25y+ 147i ∩I0 where I0 = h3x²+y⁵,5xy⁴+ 7y⁶, y⁸i.

For a direct summandω1 with support at {(−¹⁵¹²⁶³₃₁₂₅ ,−¹⁴⁷₂₅)}of the algebraic local cohomology classω_F=h

dx∧dy f1f2

i, the annihilating idealAnn_DX(ω1) is given byh25y+ 147,3125x+ 151263iDX.

For the other direct summandω0 with support at the origin (0,0), its annihilating ideal AnnAn(ω0) is generated by the idealI0and the second order differential operator y ∂²

∂y² +

−43

18y⁴+84 5 xy ∂²

∂x²+ 50

147y+ 9 ∂

∂y +

6250

1361367y⁴+ 125 9261y³+

− 78125 3176523x− 5

63

y²+8125

64827x+252 5

y− 25 441x

∂

∂x

− 762939453125

218041257467152161y⁷+ 6103515625

494424620106921y⁶− 8300781250 30270895108587y⁵ + 156250000

205924456521y⁴+

− 37841796875

211896265760109x+ 781250 1400846643

y³

+ 927734375

1441471195647x− 78125 1361367

y²+

− 1953125

1400846643x+21250 64827

y

− 390625

66706983x+650 441.

Kashiwara-Kawai duality theory on holonomic systems ([3]) together with Theorem 2.1 implies the following result which gives a characterization of the space Ker.

Theorem 2.5. — Let Kerbe the kernel space of the residue mapping Resβ(ω_F). Then Ker ={Rϕ(z)|ϕ(z)∈ OX,β, R∈ Ann_DX(ω_F)}

holds.

Observe that the stalk at β ∈ Z of O^X/I is a finite dimensional vector space, the quotient space Ker/I ⊂ OX/I is a one codimensional vector subspace. Hence, if generators of the ideal Ann_DX(ω_F) are given, the determination of Ker can be reduced to a problem in the finite dimensional vector space.

Example 2.6. — Letf1=x³andf2=y²+2x²+3x. The varietyV(I) of the idealI= hf1, f2iis the origin{(0,0)}with the multiplicity 6. Letω_F =h

dx∧dy f1f2

i∈ H²[(0,0)](Ω_Xⁿ).

Then the right idealAnn_DX(ω_F) is generated byf1,f2and the first order differential operator

P = 6x ∂

∂x + (3y+ 2xy) ∂

∂y+ (−2x−15).

It is easy to verify that P enjoys the property P(I) ⊆ I. Under the identification O^X/I ∼= SpanC{1, y, x, xy, x², x²y},

P1 =−2x−15, P x=−2x²−9x, P x²=−3x²,

P y=−9y, P xy =−6xy, P x²y = 0.

Thus, by Theorem 2.5, we have Ker/I∼= SpanC{1, y, x, xy, x²}. Note that, the relative ˇCech representation

ω_F = 1 x³y²

−2 1

xy⁴

−3 1

x²y⁴

+ 9 1

xy⁶

dx∧dy

of the cohomology classω_F implies the following formula;

Res(0,0)

ϕ(x, y)dx∧dy x³(y²+ 2x²+ 3x)

=1 2

∂³ϕ

∂x²∂y(0,0)−1 3

∂³ϕ

∂y³(0,0)−1 2

∂⁴ϕ

∂x∂y³(0,0) + 3 40

∂⁵ϕ

∂y⁵(0,0).

3. A method for computing the local residues

LetK be the fieldQ of rational numbers. LetF ={f1, . . . , fn} be a regular se- quence ofnpolynomialsfi∈K[z] =K[z1, . . . , zn],i= 1, . . . , nandIthe ideal inK[z]

generated by thesenpolynomials. LetI=I1∩ · · · ∩I`be the primary decomposition of the ideal I. Put Z = V(I), Zλ = V(Iλ) and let H<sub>[Z]</sub><sup>n</sup> (Ω<sub>X</sub><sup>n</sup>) = Γ(X,H<sup>n</sup>[Z](Ω<sup>n</sup><sub>X</sub>)), H<sub>[Z</sub><sup>n</sup><sub>λ]</sub>(Ω<sub>X</sub><sup>n</sup>) = Γ(X,H<sup>n</sup>[Zλ](Ω<sub>X</sub><sup>n</sup>)) for λ = 1, . . . , `. We have the following direct sum decomposition;

H_[Z]ⁿ (Ωⁿ_X) =H_[Zⁿ_1](Ω_Xⁿ)⊕ · · · ⊕H_[Zⁿ`](Ω_Xⁿ).

Accordingly, the algebraic local cohomology class ω_F = dz

f1· · ·fn

can be decom- posed into

ω_F=ω1+· · ·+ωλ+· · ·+ω`

withωλ∈H_[Zⁿ_λ](Ω_Xⁿ), λ= 1, . . . , `. Letϕ(z)∈K[z] and letβ ∈Zλ. Sinceω_F =ωλ

onZλ, we have

Resβ(ω_F)(ϕ(z)) = Resβ(ωλ)(ϕ(z)).

To compute the Grothendieck local residue Resβ(ω_F)(ϕ(z)) at β ∈ Zλ, it suffices to consider the linear functional Resβ(ωλ) associated to the direct summand ωλ of ω_F. SinceωλIλ = 0 holds, Resβ(ωλ) defines a linear functional acting on the space K[z]/Iλ.

Taking these facts in account, we introduce vector spaces EI^λ = K[z]/Iλ and E^√_Iλ =K[z]/√

Iλ. Letj_F(z) = det

∂(f1, . . . , fn)

∂(z1, . . . , zn)

be the Jacobian of f1, . . . , fn. Let us consider the correspondenceγ which assignesj_F(z)g(z) modIλ tog(z).

Lemma 3.1. — Let γ(g) =j_F(z)g(z) modIλ. Then (i) γ:E^√_Iλ →EIλ is a well-defined linear map.

(ii) γ:E^√_Iλ →EIλ is injective.

Proof. — LetJ_F =hj_F(z)i ⊂K[z] be the ideal generated byj_F(z). Then the ideal quotientIλ :J_F is equal to the radical√

Iλ. (i) Letg∈√

Iλ. Thenj_F(z)g(z) is in Iλ which means the well-definedness of the mapγ:E^√_Iλ →EI^λ.

(ii) Let g ∈ E^√_Iλ and assumeγ(g) = 0 in EIλ. Then, j_F(z)g(z) ∈Iλ and thus g(z)∈√

Iλ,i.e.,g= 0 inE^√_Iλ.

Letj_F,λ(z) =j_F(z) modIλ ∈EIλ. Then, γ(g) =j_F,λ(z)g(z) modIλ. We intro- duceEJ,λ to be

EJ,λ= Imγ={j_F,λ(z)g(z) modIλ|g∈E^√_Iλ}. LetEK,λdenote the subspace of EIλ defined to be

EK,λ={h(z)∈EIλ |Resβ(ω_F)(h(z)) = 0, β∈Zλ}. Proposition 3.2. — EI^λ =EJ,λ⊕EK,λ.

Proof. — It follows from dimEIλ = #Zλ+ dimEK,λand #Zλ= dimE^√_Iλ that dimEIλ = dimE^√_Iλ+ dimEK,λ.

Thus, Lemma 3.1 implies dimE^√_Iλ = dimEJ,λ, which gives dimEIλ = dimEJ,λ+ dimEK,λ.

The proof of Lemma 3.1 also yieldsEJ,λ∩EK,λ={0}, which completes the proof.

By Proposition 3.2, we see that, for any polynomial ϕ(z) ∈ K[z], there exist polynomialsgλ(z)∈E^√_Iλ andhλ(z)∈EK,λ such that

(2) ϕ(z) =j_F,λ(z)gλ(z) +hλ(z) modIλ. Letµλ= dimEI^λ/dimE^√_Iλ.

Lemma 3.3. — Let ϕ(z)∈ K[z] and ϕλ(z) = ϕ(z) modIλ ∈EIλ. Assume ϕλ(z) = γ(g(z)) +h(z). Then, Resβ(ω_F)(ϕ(z)) =µλg(β).

Proof. — Sinceβ∈Zλ,

Resβ(ω_F)(ϕ(z)) = Resβ(ωλ)(ϕ(z))

= Resβ(ωλ)(ϕλ(z))

= Resβ(ωλ)(γ(g(z)) +h(z))

= Resβ(ωλ)(j_F,λ(z)gλ(z)).

LetδZλ ∈H_[Zⁿ_λ](Ω_Xⁿ) be the delta function supported on Zλ. Since the multiplicity atβ ∈Zλ of the idealIλ is equal toµλ, we havej_F,λ(z)ωλ=µλδZλ. Thus we have

Resβ(ωλ)(j_F,λ(z)g(z)) = Resβ(j_F,λ(z)ωλ)(g(z))

=µλResβ(δZλ)(g(z))

=µλg(β), which implies

Resβ(ω_F)(ϕ(z)) =µλg(β).

Let An := K[z1, . . . , zn]h∂/∂z1, . . . , ∂/∂zni be the Weyl algebra on n variables z = (z1, . . . , zn)∈X. Let AnnAn(ωλ) be the right ideal ofAn given by annihilators of the cohomology class ωλ. The right module An/AnnAⁿ(ωλ) is a simple holo- nomic system at each point β ∈Zλ. And thus the dimension of the solution space HomAn(An/AnnAn(ωλ), ωλAn) is equal to #Zλ= dimE^√_Iλ. Reasoning on the dual- ity for the holonomic systemAn/AnnAn(ωλ) yields the following result, which is the counterpart in the Weyl algebra of the Theorem 1.

Theorem 3.4. — Let R1, . . . , Rs be generators of AnnAn(ωλ). For h(z) ∈ EI,λ, the following two conditions are equivalent;

(i) h(z)∈EK,λ, i.e.,Resβ(ω_F)(h(z)) = 0holds for^∀β∈Zλ. (ii) There exist u1(z), . . . , us(z)∈K[z]such that h(z) =Ps

k=1Rkuk(z) modIλ. Example 3.5. — Letf1= 144y⁴+(288x²+2304x+952)y²+144x⁴−768x³+952x²−343 andf2= 36y²+ 36x²−49.

The primary decomposition of the idealI=hf1, f2iis given byI=I1∩I2 where I1 = h144x² + 168x+ 49,144y²−168x−245i and I2 = h6x−7, y²i. Let ω_F = hdx∧dy

f1f2

i =ω1+ω2 where ω1 ∈H_[Z²_1](Ω_X²) andω2 ∈H_[Z²_2](Ω_X²). A monomial basis ofK[x, y]/I1is{1, x, y, xy}. The annihilating ideal of the algebraic local cohomology classesω1 is generated byI1 andP1 = (84x+ 49)_∂x^∂ + (48xy+ 28y)_∂y^∂ −84. By the computation

P11 =−84, P1x= 49,

P1y = 48xy−56y, P1xy=−28xy+⁹⁸₃y

in EI,1, we find Span_K{P11, P1x, P1y, P1xy}= Span_K{1,6xy−7y}.

We findK[x, y]/I2∼= Span_K{1, y}. The annihilating ideal ofω2is generated by I2

andy_∂y^∂ −1. We have Span_K{P11, P1y}= Span_K{1, y}.

Remark 3.6. — There is an algorithm, due to T.Oaku ([6], see also [10]), for comput- ing the Gr¨obner basis of the left ideal AnnAⁿ(σ_F) of the algebraic local cohomology class σ_F =

1 f1. . . fn

∈ H_[Z]ⁿ (O^X). The annihilating ideal AnnAn(σλ) of the di- rect summand σλ ∈ H_[Zⁿ_λ](O^X) of σ_F is equal to the left ideal in An generated by AnnAn(σ_F)∪Iλ, i.e.,

AnnAn(σλ) =AnhAnnAn(σ_F), Iλi.

Thus generators of the right ideal AnnAn(ωλ) ={R∈An |R^∗∈AnnAn(σλ)}can be explicitly constructed by using Gr¨obner basis computation inAn. For an alternative approach, we refer the reader to [5].

Theorem 3 ensures that the decomposition (2) ofϕ(z) can be rewritten in the form ϕ(z) =j_F,λ(z)gλ(z) +

s

X

k=1

Rkuk(z) mod Iλ

wheregλ(z)∈E^√_Iλ,uk(z)∈K[z],k= 1, . . . , sandRk ∈AnnAn(ωλ),k= 1, . . . , sare generators of AnnAn(ωλ). Then, the formulaρ−µλgλ(β) = 0 represents the relation between residues and the variety. The final step of the computation of residues is achieved in the following manner; Find a generatorrλ(ρ) of the intersection ofK[ρ]

and the ideal inK[ρ, z] given by√

Iλandρ−µλgλ(z),i.e.,hrλ(ρ)i=K[ρ]∩ h√ Iλ, ρ− µλgλ(z)i. Then the roots ofrλ(ρ) = 0 are exactly the residue Resβ(ω_F)(ϕ(z)),β ∈Zλ. This computation can be done, for instance, through Gr¨obner basis computations in the polynomial ringK[ρ, z].

4. Algorithm for computing residues with first order differential operators

We use the notation as in the preceding section and we recall properties of the first order partial differential operator that annihilates the algebraic local cohomology class ωλ. Upon using first order annihilators, we derive an algorithm for computing Grothendieck local residues which works for almost every case.

4.1. Use of first order annihilators. — LetWλ be the vector space defined by Wλ={h(z)ω_F |h(z)∈K[z],supp(h(z)ω_F)⊆Zλ},

the image byιof the extension groupExtⁿ_K[z](K[z]/Iλ, K[z]dz). SinceWλ=K[z]ωλ

and AnnK[z](ωλ) =Iλ hold, we have the following proposition;

Proposition 4.1 ([8]). — Let P ∈An be a first order differential operator which anni- hilates ωλ. Then

(i) Wλ is closed under the right action of P, i.e., ωP ∈Wλ,^∀ω∈Wλ. (ii) Iλ is closed under the left action ofP, i.e.,P(f)∈Iλ,^∀f ∈Iλ.

Let Ann⁽¹⁾_An(ωλ) be the right ideal inAngenerated by differential operators of order at most one that annihilate the algebraic local cohomology classωλ. Let EL,λ be a subspace ofEK,λ defined to be

EL,λ={RhmodIλ|R∈Ann⁽¹⁾_An(ωλ), h∈K[z]}, which is equal to

(3) {RhmodIλ|ωλR= 0,ord(R) = 1, h∈K[z]}. Proposition 4.1 yields

EL,λ={RhmodIλ|ωλR= 0,ord(R) = 1, h∈EIλ}.

LetMIλ be a finite set of monomials ofK[z] satisfying the condition that the space BI^λ = Span_K{m(z)∈MI^λ}generated by these monomials is isomorphic toEλ as a vector space. Such monomialsMIλ can be obtained by Gr¨obner basis computations with respect to a term order inK[z].

Definition 4.2. — Lλ ={P =p1(z) ∂

∂z1

+· · ·+pn(z) ∂

∂zn

+q(z)∈An | pi(z), q(z)∈ BIλ, i= 1, . . . , n, ωλP= 0}.

We have the following lemma;

Lemma 4.3. — LetR∈An be a linear partial differential operator of order one which annihilates ωλ. Then there exists a linear partial differential operator P inLλ such that R−P ∈IλAn.

We thus have the following results;

Proposition 4.4 ([5]). — Iλ and Lλ generate the ideal Ann⁽¹⁾_An(ωλ) over An, i.e., Ann⁽¹⁾_An(ωλ) = (Iλ∪Lλ)An holds.

Proposition 4.5

EL,λ∼={q(z)∈BI^λ |P =p1(z) ∂

∂z1

+· · ·+pn(z) ∂

∂zn

+q(z)∈Lλ}. Proof. — By (3),

EL,λ={RhmodIλ|ωλR= 0,ord(R) = 1, h∈K[z]}.

Since the first order partial differential operator R◦h, a composition of R and the multiplication operatorh, annihilatesωλ, it follows from the above lemma that there exists aP ∈Lλ such thatRh−P ∈IλAn. We thus have

Rh= (R◦h)1 = (R◦h−P)1 +P1 =P1 modIλ, which completes the proof.

In Appendix of the present paper, we give an algorithm that computes the vector spaceLλ.

We arrive at the following result;

Theorem 4.6. — The following conditions are equivalent;

(i) EK,λ=EL,λ.

(ii) AnnAⁿ(ωλ) = Ann⁽¹⁾_An(ωλ).

Corollary 4.7. — If dimEL,λ= dimEIλ−dimE^√_Iλ, thenEK,λ=EL,λ.

4.2. Algorithm for computing residues. — When conditions in Theorem 4.6 are satisfied, or equivalently dimEL,λ= dimEIλ−dimE^√_Iλ holds, one can compute the residues Resβ(ω_F)(ϕ(z)) forβ∈Zλin the following manner; Letg1(z), . . . , gdλ(z) be a canonical monomial basis of the vector spaceE^√_Iλ, wheredλ = dimE^√_Iλ. Let q1(z), . . . , qκ^λ(z) (κλ = dimEI^λ −dλ) be a basis ofEL,λ. A polynomialϕ(z)∈K[z]

can be represented inK[z]/Iλ as

ϕ(z) =j_F,λ(z)g(z) + (b1q1(z) +· · ·+bκλqκλ(z)) modIλ

withg(z) =Pdλ

j=1cjgj(z),cj∈K,j= 1, . . . , dλandbk ∈K,k= 1, . . . , κλ. Thus, by Lemma 3.3, we have

Resβ(ω_F)(ϕ(z)) =µλg(β).

The output of the following algorithm is the desired univariate polynomial rλ(ρ) such that

{ρ∈C|rλ(ρ) = 0}={ρ∈C|ρ= Resβ([ ϕ(z)dz f1· · ·fn

]), β∈Zλ}.

Algorithm R (Computation of the Grothendieck local residue)

Input : a regular sequencef1(z), . . . , fn(z), a holomorphicn-formϕ(z)dz, the Gr¨obner bases of primary idealsIλ,√

Iλ.

(i) Choose a basis q1(z), . . . , qκλ(z) of EL,λ from the output of Algorithm A (in Appendix).

(ii) Choose the basisg1(z), . . . , gdλ(z)∈E^√_Iλand computeej:=j_F,λ(z)gj(z) mod Iλ, j= 1, . . . , dλ, wherej_F,λ(z) =j_F(z) modIλwithj_F(z) = det_∂(f

1,...,fn)

∂(z1,...,zn)

. (iii) Computeϕλ(z) =ϕ(z) modIλ∈EIλ.

(iv) Determine the coefficients cj, j = 1, . . . , dλ by solving the following linear equation;

ϕλ=c1e1+· · ·+cdλedλ+b1q1+· · ·+btqt. (v) Putg(z) =Pdλ

j=1cjgj(z).

(vi) Compute a generatorrλ(ρ) of the idealK[ρ]∩ h√

Iλ, ρ−µλg(z)iwhereµλ= dimEIλ/dimE^√_Iλ is the multiplicity of the pointβ.

Output : the polynomialrλ(ρ).

The above algorithm may admit several extension. One of the most natural gen- eralizations is probably the use of higher order annihilators. Such a generalization, which involves construction of higher order annihilators, will be treated in elsewhere ([9]). In the rest of this section, we give an example for illustration.

Example 4.8. — Letf1= (x²+y²−1)² andf2= (x²+y²)²+ 3x²y−y³. The primary decomposition of the idealI=hf1, f2iis given byI1∩I2 where

I1=h16y⁴+ 32y³+ 24y²+ 8y+ 1,80y³+ 107y²+ 48y−x²+ 8i and

I2=hy²−2y+ 1, y+ 5x²−1i with the radical √

I1 = h4x²−3,2y+ 1i and √

I2 = hx, y−1i. The varieties are Z1=V(I1) ={(^√₂³,−¹2),(^−√₂³,−¹2)}andZ2=V(I2) ={(0,1)}. Letω_F=ω1+ω2

with ω1 ∈ H_[Z²

1](Ω²_X) and ω2 ∈ H_[Z²

2](Ω_X²). The Jacobian j_F(x, y) is −36xy⁴+ (−24x³+ 36x)y²+ 12x⁵−12x³.

Let us compute residues Resβ

h_ϕ(x,y)

f1f2 dx∧dyi

onZ1 and atZ2. (a) Computation onZ1={(^√₂³,−¹2),(−^√2³,−¹2)}. — We identify

K[x, y]/I1∼= Span_K{1, x, xy, xy², xy³, y, y², y³} andK[x, y]/√

I1∼= Span_K{1, x}. Algorithm A outputs following six first order anni- hilators that form a basis of the vector spaceL1;

−197

62y³−927

248y²−699 496y− 83

496 ∂

∂x +

−11

31xy³−81

62xy²−285

248xy−145 496x ∂

∂y +xy³+14 31x, −941

93y³−3501

248y²−407

62y−3029 2976

∂

∂x +607

186xy³+159

31xy²+647

248xy+40 93x ∂

∂y +xy²− 1 31x, 861

31y³+2333

62 y²+526 31y+79

31 ∂

∂x +

−200

31xy³−277

31 xy²−233

62 xy−14 31x∂

∂y +xy−16 31x, 5045

558xy³+401

31 xy²+13775

2232xy+274 279x ∂

∂x +

−151

279y³−2351

2232y²−347

558y− 343 2976

∂

∂y+y³− 3 31y, −1012

279xy³−165

31 xy²−1451

558xy−118 279x ∂

∂x +175

279y³+665

558y²+202 279y+ 53

372 ∂

∂y +y²+16 31y,

−416

9 xy³−64xy²−266

9 xy−41 9 x ∂

∂x +

−16

9 y³−34

9 y²−25 9 y−2

3 ∂

∂y + 1.

Thus, by Proposition 4.5, we have EL,1= Span_K

1, y²+16

31y, y³− 3

31y, xy−16

31x, xy²− 1

31x, xy³+14 31x

.

Put q1(x, y) = 1, q2(x, y) = y²+ ¹⁶₃₁y, q3(x, y) = y³− ₃₁³y, q4(x, y) = xy− ¹⁶₃₁x, q5(x, y) =xy²−31¹x,q6(x, y) =xy³+¹⁴₃₁xandg1(x, y) = 1,g2(x, y) =x. We find that e1(x, y) =−576xy³−864xy²−432xy−72xande2(x, y) =−432y³−648y²−324y−54.

Forϕ1(x, y) =−159y³−216y²+ (−4x−96)y−14, we have ϕ1= 8

243e1− 1

324e2−85

6q1−218q2−481 3 q3+92

9 q4+256

9 q5+512 27 q6, which implies

ϕ(x, y) = 8 243+

− 1 324x

j_F(x, y)−85

6 −218 y²+16

31y

−481 3

y³− 3 31y +92

9

xy−16 31x

+256 9

xy²− 1 31x

+512 27

xy³+14 31x

modI1. Thus we findg(x, y) =₂₄₃⁸ −₃₂₄¹ x. The Gr¨obner basis ofh4x²−3,2y+ 1, ρ−2g(x, y)i with respect to the lexicographical ordering x y ρ is {236196ρ²−62208ρ+ 4069,2y+ 1,3x+ 243ρ−32}. rλ(ρ) = 236196ρ²−62208ρ+ 4069 is the desired polynomial.

(b) Computation at Z2 = {(0,1)}. — We identify K[x, y]/I2 ∼= Span_K{1, x, y, xy} and K[x, y]/√

I2 ∼= SpanK{1}. The following three first order annihilators are the output of the Algorithm A ;

3 62y− 3

62 ∂

∂x+

−15

31xy+15 31x∂

∂y +xy−16 31x, 17

45xy−32 45x ∂

∂x+

−2 3y+2

3 ∂

∂y+y, 62

45xy−77 45x ∂

∂x+

−2 3y+2

3 ∂

∂y+ 1.

Thus, we have

EL,2= Span_K

1, y, xy−16 31x

.

Putq1(x, y) = 1,q2(x, y) =y,q3(x, y) =xy−¹⁶₃₁x. We find e1(x, y) =j_F(x, y) modI2=−324

5 xy+324 5 x.

Then,ϕ2(x, y) =ϕ(x, y) modI2 is rewritten in the following form;

ϕ2(x, y) =

−4x−3 5

y+8 5

=−16

243e1(x, y) +8

5q1(x, y)−3

5q2(x, y)−124

15 q3(x, y).

We findg(x, y) =− 16

243. Thus we havehx, y−1, ρ−4g(x, y)i=h243ρ+ 64, x, y−1i. We arrive at Res(0,0)

hϕ(x,y)dx∧dy f1f2

i=−64/243.

5. Example

Let ϕλ(z) be a polynomial in K[z] and let ϕλ(z) = ϕ(z) modIλ ∈ EIλ. It is obvious that if the condition

(4) ϕλ(z)∈EJ,λ⊕EL,λ

holds, one can apply Algorithm R to compute the Grothendieck local residue Resβ

h_ϕ(z)dz

f1...fⁿ

i, β ∈Zλ. This fact does not imply that the range of application of Algorithm R is the condition (4). In this section, we present an example to show the usage of the Algorithm R and illustrate a method to extend the range of application of Algorithm R.

Example 5.1. — Let I be an ideal generated by f1 = 3x⁴−6x³+ 3x² +y⁵, f2 = 5y⁴x+ 7y⁶. The Gr¨obner basis of the idealI =hf1, f2i with respect to the graded total lexicographical ordering (withxy) is given by

{−3087x⁸+ 12348x⁷−18522x⁶+ 12348x⁵−3087x⁴−125x³y⁴,

−21x⁴y+ 42x³y−21x²y+ 5xy⁴,3x⁴−6x³+ 3x²+y⁵}. Its primary decomposition isI=I1∩I2∩I3 where

I1=h147x²−294x+ 25y+ 147,5x+ 7y²i, pI1=h147x²−294x+ 25y+ 147,5x+ 7y²i, I2=hy⁴, x²−2x+ 1i,p

I2=hx−1, yi,

I3=h21x²y−5xy⁴,6x³−3x²−y⁵, x⁴, x³y,5x³+ 7x²y²i,p

I3=hx, yi. We find Z1 =V(I1) consists of four simple points, Z2 =V(I2) = {(0,1)} with the multiplicity µ2 = 8 and Z3 = V(I3) ={(0,0)} with the multiplicity µ3 = 12. Let ω_F =ω1+ω2+ω3withωλ∈H_[Z²_λ](Ω_X²),λ= 1,2,3.

(a) Since µ1 = 1, EI1 = EJ,1 and thus EK,1 = EL,1 = {0}, one can apply Al- gorithm R to compute Resβ

ϕ(x, y)dx∧dy f1f2

for any ϕ(x, y) ∈ K[x, y] without computingEL,1

(b) We use the following identification; EI2 ∼= Span_K{1, y, x, y², xy, y³, xy², xy³}. Algorithm A outputs the following 10 operators which form a basis of the vector space L2;

(₁₄⁵xy−₁₄⁵y)_∂x^∂ + (−¹⁵₂₈xy²+₂₈⁵y²)_∂y^∂ +xy³+₁₄⁵y, (−xy³)_∂y^∂ +xy²,

(−¹4xy+¹₄y)_∂x^∂ + (−¹8xy²+¹₈y²)_∂y^∂ +xy−³4y, (−¹₃xy+¹⁴₁₅y³)_∂y^∂ +x,

(₂₈⁵xy−28⁵y)_∂x^∂ + (−¹⁵56xy²−56⁵y²)_∂y^∂ +y³+¹⁵₂₈y,

−y³_∂y^∂ +y²,

(−¹⁴15xy³+²⁸₁₅y³−¹3y)_∂y^∂ + 1,

(−xy³−⁵₇xy+y³+⁵₇y)_∂x^∂ + (¹⁵₁₄xy²−⁵₇y²)_∂y^∂ , (−xy²+y²)_∂x^∂ + (3xy³−2y³)_∂y^∂ ,

(−x+ 1)_∂x^∂ + (−⁴⁹15xy³+xy+⁷₃y³−²3y)_∂y^∂ .

Taking the zeroth order parts of these operators, we have EL,2= Span_K{1, y², y³+15

28y, x, xy−3

4y, xy², xy³+ 5 14y}.

Since dimEL,2= 7 which is equal toκ2= 8−1, we haveEK,2=EL,2. Thus, one can compute local residues Resβ

h_ϕ(x,y)

f1f2 dx∧dyi

atβ= (1,0) by using the Algorithm R.

(c) We identify EI3 ∼= SpanK{1, x, y, x², xy, y², x³, x²y, xy², y³, xy³, y⁴}. Algo- rithm A outputs the following 14 operators which form a basis of the vector space L3;

(−51584680665

475773195424x²y− 27238684725

1903092781696xy−1903092781696^661775625 x²)_∂x^∂

+ (−1903092781696^94539375 y⁴+237886597712^2924301625 xy³+1903092781696^308828625 y³− 27238684725 3806185563392xy²

− 38134158615

7612371126784y²−10014110299529

4757731954240x²y−1903092781696^44118375 xy+ 44628681825 475773195424x³

− 159058523925

3806185563392x²+7612371126784^3891240675 x)_∂y^∂ +xy³+ 16343210835

475773195424y, (−791242092165

475773195424x²y+ 637344986607

1903092781696xy+ 15484572075 1903092781696x²)_∂x^∂ + (1903092781696^2212081725 y⁴− 11784672235

237886597712xy³−1903092781696^7226133635 y³+ 637344986607 3806185563392xy² + 4461414906249

38061855633920y²+ 26125989295

951546390848x²y+1903092781696^1032304805 xy−1044245230899 475773195424x³ +3721734504591

3806185563392x²− 91049283801 7612371126784x)_∂y^∂ +y⁴−1912034959821

2378865977120y.

(74312890965999

59471649428000x²y+ 7848009842967

47577319542400xy+1903092781696^7626831723 x²)_∂x^∂ + (1903092781696^1089547389 y⁴− 168509956839

1189432988560xy³− 17795940687

9515463908480y³+ 7848009842967 95154639084800xy²