Operations with regular holonomic D-modules with support a normal crossing

OPERATIONS WITH REGULAR HOLONOMIC D-MODULES WITH SUPPORT A NORMAL CROSSING JOSEP ` ALVAREZ MONTANER Abstract. The aim of this work is to describe some operations in the category of regular holonomic D-modules with support a normal crossing and variation zero introduced in [2]. These operations will allow us to compute the characteristic cycle of the local cohomology supported on homogeneous prime ideals of these modules. In particular, we will be able to describe their Bass and dual Bass numbers. 1. Introduction Let X=Cn,OXthe sheaf of holomorphic functions in Cn,andDXthe sheaf of differential operators in Cnwith holomorphic coefficients. Galligo, Granger and Maisonobe [8], described in terms of linear algebra the category Mod(DX)T hr of regular holonomic DX-modules such that their solution complex RHomDX(M,OX) are perverse sheaves relatively to the stratification given by the union Tof the coordinate hyperplanes in Cn. In Section 2 we recall the definition and the basic properties of the category DT v=0 of modules with variation zero introduced in [2] (see also [3]). Moreover, we define the category of modules with unipotent monodromy that is also a full abelian subcategory of Mod(DX)T hr. This category is closed under extensions and includes DT v=0. In Section 3 we describe some operations in the category DT v=0. We have to point out that we will consider the case of R=k[x1,...,x n] being the polynomial ring in nindependent variables over any field kof characteristic zero and Dbeing the ring of differential operators over R. We can consider this case due to the good behavior of this category with respect to flat base change (see [3]). First, we describe the restriction to an homogeneous prime ideal of a module with variation zero. However, the main result of this section is a description of the kernel, the cokernel and the image of the homomorphism λi:M−→ M[1 xi]of localization of a module with variation zero Mby the variable xi. In Section 4, by using the results of the previous section and Brodmann’s exact sequence, we give an algorithm that allows us to compute the characteristic cycle of the local cohomology modules Hp pα(M), where pα⊆Ris an homogeneous prime ideal. In particular, we give a different approach to the computation of the Bass numbers µp(pα,M):=dim k(pα)Extp Rpα(k(pα),M pα) given by K. Yanagawa in [11]. 2000 Mathematics Subject Classification. Primary 32C38, Secondary 16D40. Key words and phrases. D-modules, Bass numbers. 1 2J. ` ALVAREZ MONTANER Finally, in Section 5 we define Matlis duality in the category DT v=0. This duality theory is nothing but the duality in the lattice {0,1}n. The Matlis dual of an injective DT v=0-module is projective so, by using the results of the previous section, we describe projective resolutions in DT v=0. In the sequel we will denote 1=ε1+···+εnwhere ε1,...,ε nis the natural basis of Zn. For all α∈{0,1}n,Xαwill be the linear subvariety of Xdefined by the homogeneous prime ideal pα=<x i|αi=1>. For unexplained terminology on the theory of algebraic D-modules we refer to [4], [6]. 2. Preliminaries Let Cnbe the category whose objects are families {Mα}α∈{0,1}nof finitely dimensional complex vector spaces, endowed with linear maps Mα ui −→ M α+εi,Mα vi ←− M α+εi for each α∈{0,1}nsuch that αi= 0. These maps are called canonical (resp., variation) maps, and they are required to satisfy the conditions: uiuj=ujui,v ivj=vjvi,u ivj=vjuiand viui+id is invertible. Such an object will be called an n-hypercube. A morphism between two n-hypercubes {Mα}αand {Nα}αis a set of linear maps {fα:Mα→N α}α, commuting with the canonical and variation maps. In [9], an equivalence of categories between Mod(DX)T hr and Cnis described. The functor Mod(DX)T hr −→ C nis a contravariant exact functor. The construction of the n-hypercube corresponding to an object Mof Mod(DX)T hr is given in [9]. From the construction one can describe the composition vi◦uiin terms of the partial monodromy around the hyperplane xi= 0. We also want to point out that the n-hypercube describes the characteristic cycle of the corresponding DX-module M. Namely, let CC(M)=mαT∗ XαCnbe the characteristic cycle of M. Then, one has the equality dimCMα=mα. The papers [2] and [3] study objects in the category Mod(DX)T hr having the following property: Definition 2.1. We say that an object Mof Mod(DX)T hr has variation zero if the morphisms viare zero for all 1≤i≤nand all α∈{0,1}nwith αi=0. Modules with variation zero form a full abelian subcategory of Mod(DX)T hr that will be denoted DT v=0. The simple objects of DT v=0 are of the form: DX DX({xi|αi=1},{∂j|αj=0}). This module is isomorphic to the local cohomology module H|α| Xα(OX). Every holonomic module has finite length, so if M∈D T v=0 then, there exists a finite increasing filtration {Fj}j≥0of Mby objects of DT v=0 such that for all j≥1 one has DX-module isomorphisms Fj/Fj−1≃H |α| Xα(OX),α∈{0,1}n. OPERATIONS 3 The category DT v=0, regarded as a subcategory of Mod(DX)T hr, is not closed under extensions. However, its objects can be characterized by the following particular filtration: Proposition 2.2. ([2]) An object Mof Mod(DX)T hr has variation zero if and only if there is a increasing filtration {Fj}0≤j≤nof Mby objects of Mod(DX)T hr and there are integers mα≥0for α∈{0,1}nsuch that for all 1≤j≤none has D-module isomorphisms Fj/Fj−1≃|α|=j(H|α| Xα(OX))⊕mα. 2.1. Closing the category DT v=0 for extensions. In this section we will find the minimal full abelian subcategory of Mod(DX)T hr containing DT v=0 that is closed under extensions. Definition 2.3. We say that an object Mof Mod(DX)T hr has m-trivial monodromy if the composition of morphisms (vi◦ui)mis zero for all 1≤i≤nand all α∈ {0,1}nwith αi=0. Modules with m-trivial monodromy form a full abelian subcategory of Mod(DX)T hr that will be denoted DT (vu)m=0. Notice that we have DT v=0 ⊆D T vu=0 ⊆D T (vu)2=0 ⊆···⊆DT (vu)m=0 ⊆··· In particular, we get an increasing filtration of the following category: Definition 2.4. We say that an object Mof Mod(DX)T hr has unipotent monodromy if the composition of morphisms vi◦uiis nilpotent for all 1≤i≤nand all α∈{0,1}nwith αi=0. Modules with unipotent monodromy form a full abelian subcategory of Mod(DX)T hr that will be denoted DT uni. It is easy to see that the simple objects of the category DT uni are isomorphic to the local cohomology modules H|α| Xα(OX), for α∈{0,1}n. The categories DT (vu)m=0, regarded as a subcategories of Mod(DX)T hr, are not closed under extensions for all m, but DT uni is. Proposition 2.5. The category DT uni, regarded as a subcategory of Mod(DX)T hr,is closed under extensions. Proof: Let 0//Mi//Mπ//M //0be an exact sequence in Mod(DX)T hr such that M,M ∈D T uni. Consider, for all α∈{0,1}nsuch that 4J. ` ALVAREZ MONTANER αi= 0, the commutative diagram in the category Cnof n-hypercubes: . . .. . .. . . 0//Mα u i OO πα//Mα ui OO iα//Mα u i OO //0 0//Mα+εi v i OO πα+εi//Mα+εi vi OO iα+εi//Mα+εi v i OO //0 0//Mα u i OO πα//Mα ui OO iα//Mα u i OO //0 where (v i◦u i)m=0and(v i◦u i)m = 0. It is easy to check that for m max{m,m },(vi◦ui)m=0soM∈D T uni. Since every holonomic module has finite length, it follows that the objects of DT uni are characterized as follows: Proposition 2.6. An object Mof DT hr has unipotent monodromy if and only if there is a finite increasing filtration {Fj}j≥0of Mby objects of DT hr such that for all j≥1one has D-module isomorphisms Fj/Fj−1≃H |α| Xα(OX),α∈{0,1}n. However, notice that the modules M∈D T uni are not characterized by a filtration given by the height (as in Proposition 2.2), unless they are modules with variation zero, i.e. we can not give a filtration {Fj}j≥0of Mwhere the submodules Fj correspond to the n-hypercubes: (Fj)β=   Mβif |β|≤j 0 otherwise, the canonical and variation maps being either zero or equal to those in M. 3. Operations in the category DT v=0 By using flat base change, we can define the category DT v=0 of modules with variation zero, as well the corresponding category of n-hypercubes, for the case of Dbeing the ring of differential operators over R, where Ris the polynomial or the formal power series ring in nindependent variables, x1,...,x n, over any field kof characteristic zero (see [3, Remark 4.1]). From now on, we will only consider the case R=k[x1,...,x n] in order to take advantage of the Zn-graded structure of these modules given by the equivalence of categories (see [2]), between the category of modules with variation zero and the category of straight modules introduced by K. Yanagawa [11]. 3.1. Restriction to a face ideal. Let Zα⊆Znbe the coordinate space spanned by {εi|αi=1},α∈{0,1}n. The restriction of Rto the homogeneous prime ideal pα⊆Ris the Zα-graded k-subalgebra of R R[pα]:= k[xi|αi=1]. OPERATIONS 5 The restriction to pαof a Zn-graded module Mis the R[pα]-module M[pα]:=  β∈Zα Mβ Restriction gives us a functor that plays in some cases the role of the localization functor. For details on the description of the morphisms and further considerations we refer to [10]. Let I=Iα1∩···∩Iαmbe the minimal primary decomposition of a squarefree monomial ideal I⊆R. Then, the restriction of Ito the face ideal pαis the squarefree monomial ideal I[pα]⊆R[pα]whose face ideals in the minimal primary decomposition are those face ideals Iαjcontained in pα. Namely I[pα]= αj≤α Iαj. Notice that the restriction to pαof the local cohomology module H|β| pβ(R) supported on a face ideal pβ⊆Rvanishes if and only if pβpα. The restriction to pαof a module M∈D T v=0 is again a module with variation zero. This can be seen considering the restriction of a fixed increasing filtration of M,0=F0⊆F1⊆···⊆Fn=Mgiven by Proposition 2.2. Proposition 3.1. i) The vertices of the |α|-hypercube corresponding to M[pα] are the vector spaces (M[pα])γ=Mγfor γ≤α. ii) The map uj:(M[pα])γ→(M[pα])γ+εjis the same map as uj:Mγ→ Mγ+εj. Proof: Let CC(M)=mγT∗ XγXbe the characteristic cycle of M. Then, the characteristic cycle of M[pα]is CC(M[pα])= γ≤α mγT∗ XγX so we get the vertices of the n-hypercube. In order to get the linear maps ui’s we only have to point out that the filtration of the module M[pα]is determined by the filtration of M, in particular they have the same extension problems. Then we are done by [2, Section 3] and [3, Theorem 4.1].  3.2. Localization by a variable. Let Mbe a module in DT v=0 and xi∈Ra variable. We will describe the n-hypercube of the localization M[1 xi]. First of all, it is worthwhile to point out that localization by a variable is an exact functor in the category of modules with variation zero due to the fact that R[1 xi] is a flat module in DT v=0. It also follows that localization commutes with restriction to homogeneous prime ideals. Proposition 3.2. i) The vertices of the n-hypercube corresponding to M[1 xi] are the vector spaces M[1 xi]γ=Mγif γi=0. In this case we also have M[1 xi]γ+εi=Mγ. 6J. ` ALVAREZ MONTANER ii) The map uj:M[1 xi]γ→M[1 xi]γ+εj,whereγi=0,is: Id if j=i uj:Mγ→M γ+εjif j=i. In this case, this map is also equal to uj:M[1 xi ]γ+εi→M[1 xi ]γ+εi+εj. Proof: The vertices of the n-hypercube corresponding to M[1 xi] can be easily described by means of a formula given in [5]. Namely, let CC(M)=mαT∗ XαX be the characteristic cycle of M. Then, the characteristic cycle of M[1 xi]is CC(M[1 xi ]) =  αi=0 mα(T∗ XαX+T∗ Xα+εiX) In order to get the linear maps ui’s we will use induction on the length lof M.Letl= 1, i.e. M=H|α| pα(R) is a local cohomology module. If αi= 0 then M[1 xi]=0.Ifαi= 0, we just have to prove that the map ui:M[1 xi]α→M[1 xi]α+εi is the identity. For simplicity we will use the restriction to pα+εi. Then localizing by xithe minimal graded injective resolution: 0−→ H|α| pα(R)−→ ∗E(R/pα)(1)−→ ∗E(R/pα+εi)(1)−→ 0 we get an isomorphism H|α| pα(R)[ 1 xi]∼ =∗E(R/pα)(1), so we get the desired result by the description of injective modules given in the proof of [2, Theorem 4.3]. Notice that the same argument can be used to describe the n-hypercube of the localization H|α| pα(R)[ 1 xβ] of a local cohomology module by any squarefree monomial xβ:= xβ1 1···xβn n,β∈{0,1}n. The case l= 2 is proved as well since a module of length two has to be, for convenient α, β ∈{0,1}nand j∈{1,...,n}, isomorphic to one of the following modules H|α| pα(R)⊕H|α| pα(R),H |α| pα(R)⊕H|β| pβ(R),H |α| pα(R)[ 1 xj ] that correspond to the n-hypercubes with non vanishing part k2,k, k k 1  k If l>2 we consider the submodule M⊆Mwhose corresponding n-hypercube only has the vertices and linear map we want to study. Namely, uj:Mγ−→ M γ+εj, where γi=0. Ifthen-hypercube of Mhas another vertex different from zero then length M<length Mso we are done by induction. If the n-hypercube of Monly has the vertices Mγand Mγ+εj, i.e. M=M, we can give a precise description of this module in terms of the rank of uj. Namely, let m=rk(uj), mγ=dim kMγ and mγ+εj=dim kMγ+εj. Then, Mis the direct sum of mcopies of H|γ| pγ(R)[ 1 xj], mγ−mcopies of H|γ| pγ(R), OPERATIONS 7 mγ+εj−mcopies of H|γ+εj| pγ+εj(R). So we are done since we can reduce to the cases of length l=1andl=2.  3.3. Image, kernel and cokernel of the localization by a variable. Let M be a module in DT v=0 and xi∈Ra variable. We will describe the n-hypercubes of the image, kernel and cokernel of the morphism λi:M−→ M[1 xi] of localization of Mby the variable xi. Proposition 3.3. i) The vertices of the n-hypercube corresponding to Im λi are computed from the characteristic cycle CC(Im λi)=  αi=0 (mαT∗ XαX+rk(ui)T∗ Xα+εiX). ii) The map uj:(Imλi)γ→(Imλi)γ+εjare the same as the corresponding for the module M. Proof: We will use induction on the length lof M.Letl= 1, i.e. M=H|α| pα(R) is a local cohomology module. If αi= 0 then Im λi=0. Ifαi= 0 we have the exact sequence: 0//H0 (xi)(M)//M//M[1 xi]//H1 (xi)(M)//0. Notice that H0 (xi)(M)=0andH1 (xi)(M)=H|α+εi| pα+εi(R) so we are done. The case l= 2 is easy to compute using the description of a module of length two. If l>2, we consider the submodule M⊆Mwhose corresponding n-hypercube only has the vertices and linear map we want to study. Namely, uj:Mγ−→ M γ+εj, where γi= 0. It follows as in the proof of Proposition 3.2.  Once the n-hypercube for Im λiis determined we can easily compute the nhypercube for Ker λiand Coker λi. Proposition 3.4. i) The vertices of the n-hypercube corresponding to Ker λi are computed from the characteristic cycle CC(Ker λi)=CC(M)−CC(Im λi). ii) The map uj:(Kerλi)γ→(Kerλi)γ+εjare the same as the corresponding for the module M. Proposition 3.5. i) The vertices of the n-hypercube corresponding to Coker λi are computed from the characteristic cycle CC(Coker λi)=CC(M[1 xi ]) −CC(Im λi). ii) The map uj:(Cokerλi)γ→(Cokerλi)γ+εjare the same as the corresponding for the module M[1 xi]. Example: Let R=k[x1,x 2,x 3]. Given the 3-hypercube of a module M∈D T v=0 (see below), consider the morphism λ3:M−→ M[1 x3]. Then we have the following exact sequences of 3-hypercubes: 0Kerλ3 ooM ooImλ3 oo0 oo 8J. ` ALVAREZ MONTANER 0         > > > > > > > > 0  > > > > > > >0       > > >  > > > 0         0  > > > > > > > >k 1  k 1        k 0         > > > > > > > > k 1  0  > > > > > > >0       > > >  > > > 0          k 0 > > > > > > >k 1  k 1        k 0         > > > > > > > > k 1  > > > > > > > >0       = = =  = = = 0         k  > > > > > > >0  0         0 0Imλ3 ooM[1 x3] ooCokerλ3 oo0 oo 0         > > > > > > > > k 1  > > > > > > > >0       = = =  = = = 0         k  > > > > > > >0  0         0 0         > > > > > > > > k 1  1  > > > > > > >0       > > >  > > > 0          k 1 > > > > > > >k 1  0         k 0         > > > > > > > 0  > > > > > > >0       > > >  > > > 0         0  > > > > > > > >k 1  0         k It is not difficult to check out that: M∼ =H1 (x1)(R)[ 1 x2]⊕H2 (x1x2,x3)(R) M[1 x3]∼ =H1 (x1)(R)[ 1 x2x3]∼ =∗E(R/(x1))(1) Kerλ3∼ =H2 (x1x2,x3)(R) Im λ3∼ =H1 (x1)(R)[ 1 x2] Cokerλ3∼ =H2 (x1,x3)(R)[ 1 x2]∼ =∗E(R/(x1,x 3))(1) 4. Bass numbers of modules with variation zero It is easy to prove that the injective objects of DT v=0 are of the form: D D({xi|αi=1},{xj∂j+1|αj=0}),α∈{0,1}n. These modules are isomorphic to the shifted graded injective hulls ∗E(R/pα)(1) of the quotients R/pα. In particular, the minimal injective resolution of a module with variation zero Mis in the form: I•(M): 0//I0d0//I1d1//··· dm−1//Imdm//···, where the j-th term is Ij= α∈{0,1}n ∗E(R/pα)(1)µj(pα,M). OPERATIONS 9 The Bass numbers of Mwith respect to the face ideal pα⊆Rare the invariants defined by µj(pα,M). In general, the Bass numbers µp(p,M):=dim k(p)Extp Rp(k(p),M p) with respect to any prime ideal p⊆Rcan be described as the multiplicities of the characteristic cycle of Hp p(M). Namely, by using the same arguments as in [1, Proposition 2.1] we have: Proposition 4.1. Let p⊆Rbe a prime ideal and CC(Hp p(M))=λp,p,α T∗ XαX be the characteristic cycle of the local cohomology module Hp p(M). Then, the Bass numbers with respect to pof Mare µp(p,M)=λp,p,αp, where Xαpis the subvariety of X=Spec(R)defined by p. Our aim in this section is to compute the characteristic cycle of the local cohomology modules Hp pα(M), where pα∈Ris a face ideal. In particular, we give a different approach to the computation of the Bass numbers of these modules given by K. Yanagawa in [11]. We have to point out that, by means of [7, Theorem 1.2.3], one may compute the Bass numbers of Mwith respect to any prime ideal. To this purpose we will use the Brodmann sequence ···−→Hp pα+εi(M)−→ Hp pα(M)−→ Hp pα(M)[ 1 xi ]−→ Hp+1 pα+εi(M)−→··· in an iterated way. By using the additivity of the characteristic cycle with respect to short exact sequences it will be enough to compute the characteristic cycle of the kernel and cokernel of the localization morphism. This will be done by using the description given in Propositions 3.4 and 3.5. We present the following: Algorithm: INPUT: A module with variation zero M∈D T v=0 and the face ideal pα⊆R. Denote pαk:= (xi|αi=1,i≤k). OUTPUT: The characteristic cycle of Hp pα(M)∀p. •For i=1,...,n, while αi=1: •Consider the Brodmann exact sequence: ··· //Hp pαi(M)//Hp pαi−1(M)λi,p //Hp pαi−1(M)[ 1 xi]//Hp+1 pαi(M)//··· •Compute the characteristic cycle of Ker λi,p and Coker λi,p,∀p by using Propositions 3.4 and 3.5. •CC(Hp pαi(M)) = CC(Ker λi,p)+CC(Coker λi,p−1),∀p.