Local cohomology, arrangements of subspaces and monomial ideals

Advances in Mathematics 174 (2003) 35–56

http://www.elsevier.com/locate/aim

Local cohomology, arrangements of subspaces and monomial ideals Josep A`lvarez Montaner,a,*,1 Ricardo Garcı´ a Lo´pez,b,2 and Santiago Zarzuela Armengoub,3 a

Departament de Matema`tica Aplicada I, Universitat Polite`cnica de Catalunya, Avinguda Diagonal 647, Barcelona 08028, Spain b Departament d’A`lgebra i Geometria, Universitat de Barcelona, Gran Via 585, Barcelona 08007, Spain Received 1 December 2000; accepted 8 March 2002 Communicated by Anders Bjo¨rner

0. Introduction Let Ank denote the affine space of dimension n over a field k; let X CAnk be an arrangement of linear subvarieties. Set R ¼ k½x1 ; y; xn  and let ICR denote an ideal which defines X : In this paper we study the local cohomology modules HIi ðRÞ :¼ indlimj ExtiR ðR=I j ; RÞ; with special regard of the case where the ideal I is generated by monomials. If k is the field of complex numbers (or, more generally, a field of characteristic zero), the module HIi ðRÞ is known to have a module structure over the Weyl algebra An ðkÞ; and one can therefore consider its characteristic cycle, denoted CCðHIi ðRÞÞ in this paper (see e.g. [3, I.1.8.5]). On the other hand, the arrangement X defines a partially ordered set PðX Þ whose elements correspond to the intersections of irreducible components of X and where the order is given by inclusion. Our first result is the determination of the characteristic cycles CCðHIi ðRÞÞ in terms of the cohomology of some simplicial complexes attached to the poset PðX Þ: It follows from the formulas obtained that, in either the complex or the real case, these characteristic cycles determine the Betti numbers of the complement of the arrangement in Ank : In fact, it was proved by Goresky and MacPherson that the *Corresponding author. E-mail addresses: [email protected] (J. A`lvarez Montaner), [email protected] (R. Garcı´ a Lo´pez), [email protected] (S. Zarzuela Armengou). 1 Partially supported by the University of Nice. 2 Partially supported by the DGCYT PB98-1185 and by INTAS 97 1644. 3 Partially supported by the DGCYT PB97-0893. 0001-8708/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0001-8708(02)00050-6

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Betti numbers of the complement of an arrangement X can be computed as a sum of non-negative integers, one for each non-empty intersection of irreducible components of X : These integers are dimensions of certain Morse groups (cf. [9, Part III, Theorems 1.3 and 3.5]). We will see that, over a field of characteristic zero, one can give a purely algebraic interpretation of them in terms of local cohomology. More precisely, in Section 1, and following closely Bjo¨rner–Ekedahl’s proof of the c-adic version of Goresky–MacPherson’ s formula, we will establish the existence of a Mayer–Vietoris spectral sequence ðiÞ

E2i;j ¼ indlimPðX Þ HIjp ðRÞ ) HIji ðRÞ; where p runs over PðX Þ; Ip is the (radical) ideal of definition of the irreducible ðiÞ

variety corresponding to p; and indlimPðX Þ is the ith left-derived functor of the inductive limit functor in the category of inductive systems of R-modules indexed by PðX Þ: The main ingredient in the proof is the Matlis–Gabriel theorem on structure of injective modules. Our formula for the characteristic cycle of local cohomology follows essentially from the fact, proved in Section 1 as well, that this spectral sequence degenerates at the E2 -term. In case the arrangement X is defined by a monomial ideal I; the local cohomology modules HIi ðRÞ have a natural Zn -grading. In Section 2, we relate the multiplicities of the characteristic cycle of HIi ðRÞ to its graded structure (Proposition 2.1). Using results of Mustat-a˘, we can conclude that the multiplicities of the characteristic cycle of HIi ðRÞ determine and are determined by the graded Betti numbers of the Alexander dual ideal of I: The relation between the Zn -graded structure of HIi ðRÞ and the multiplicities of its characteristic cycle has also been established by Yanagawa in a preprint [17]. The degeneration of the Mayer–Vietoris spectral sequence provides a filtration of each local cohomology module HIi ðRÞ; where the successive quotients are given by the E2 -term. In general, not all the extension problems attached to this filtration have a trivial solution (this happens for example for the arrangement given by all coordinate hyperplanes in Ank ; see Remark 1.4(i)). This is a major difference between the case we consider here and the cases considered by Bjo¨rner and Ekedahl. Namely, in the analogous situation for the c-adic cohomology of an arrangement defined over a finite field the extensions appearing are trivial not only as extensions of Qc -vector spaces but also as Galois representations, and for the singular cohomology of a complex arrangement the extensions appearing are trivial as extensions of mixed Hodge structures (cf. [4, p. 179]). This contrasts with the fact that the proof of the degeneration of the Mayer–Vietoris spectral sequence for c-adic or singular cohomology uses deeper facts than the proof of the degeneration for local cohomology (it relies on the strictness of Deligne’s weight filtration). In Section 3, we solve these extensions problems in case the ideal I is monomial (the result stated is actually more general, in that we work in the category of estraight modules, which is a slight variation of a category introduced by Yanagawa [16], and includes as objects the local cohomology modules considered above). It

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turns out that these extensions can be described by a finite set of linear maps, which for local cohomology modules supported at monomial ideals can be effectively computed in combinatorial terms from certain Stanley–Reisner simplicial complexes (using the results in [13]). If the base field is the field of complex numbers, the category of e-straight modules considered in Section 3 is a full subcategory of the category of regular holonomic An ðCÞ-modules. Then, by the Riemann–Hilbert correspondence, the category of e-straight modules is equivalent to a full subcategory of the category of perverse sheaves in Cn (with respect to the stratification given by the coordinate hyperplanes). This category has been described in terms of linear algebra by Galligo et al. [6]. Given such a perverse sheaf, one can attach it a set of partial variation maps. In Section 4, and using the description in [6], we prove that the category of e-straight modules is equivalent to the category of those perverse sheaves where all partial variations vanish (and then all partial monodromies are the identity map). If ðP; pÞ is a poset, we will denote by KðPÞ the simplicial complex which has as vertexes the elements of P and where a set of vertexes p0 ; y; pr determines an rdimensional simplex if p0 o?opr : If K is a simplicial complex and E is a k-vector space, we will denote by Simp * ðK; EÞ the complex of simplicial chains of K with coefficients in E: On dealing with arrangements defined over fields of characteristic zero we will use some notions from D-module theory, we refer to [5] or [3] for unexplained terminology. All modules over a non-commutative ring (or over a sheaf of noncommutative rings) will be assumed to be left modules. On dealing with arrangements defined over fields of positive characteristic we will refer to the notion of F -module introduced in [11, Definition 1.1]. If A is a ring, we denote by ModA the category of A-modules. For any ring A; if M is an A-module endowed with a filtration fFk gkX0 ; it will be always assumed that the filtration is exhaustive, i.e. S M ¼ k Fk ; and we will agree that F1 ¼ f0g: We denote Vectk the category of kvector spaces. For arrangements of subspaces over a field of characteristic zero defined by monomial ideals, an algorithm to compute the characteristic cycles of the local cohomology modules considered in this paper was given by Alvarez Montaner [1]. In the same case, some partial results were obtained by Barkats in her thesis [2].

1. Filtrations on local cohomology modules Let S be an inductive system of R-modules. Roos [14] introduced a complex which has as ith cohomology the ith left-derived functor of the inductive limit functor evaluated at S (and the dual notion for projective systems as well, this is actually the case treated by Roos in more detail). We recall his definition in the case of interest for us.

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Let ðP; pÞ be a partially ordered set, let C be an abelian category with enough projectives and such that the direct sum functors are exact (usually, C will be a category of modules, sometimes with enhanced structure: a D-module structure, a F -module structure or a grading). We will regard P as a small category which has as objects the elements of P and, given p; qAP; there is one morphism p-q if ppq: A diagram over P of objects of the category C is by definition a covariant functor F : P-C: Note that the image of F is an inductive system of objects of C indexed by P: The category which has as objects the diagrams of objects of C and as functors the natural transformations is abelian and will be denoted DiagðP; CÞ: Definition. The Roos complex of F is the homological complex of objects of C defined by Roosk ðF Þ :¼

"

p0 o?opk

Fp0 ?pk ;

where Fp0 ypk ¼ F ðp0 Þ and, if i40 and we denote by pp0 ypˆi ypk the projection from "p0 o?opk Fp0 ypk onto Fp0 ypˆi ypk ; the differential on Fp0 ypk is given by F ðp0 -p1 Þ þ

k X

ð1Þi pp0 ypˆi ypk :

i¼1

This construction defines a functor Roos * ðÞ : DiagðP; CÞ-CðCÞ; where CðCÞ denotes the category of chain complexes of objects of C: It is easy to see that this functor is exact and commutes with direct sums. Let X CAnk be an arrangement defined by an ideal ICR: Given pAPðX Þ; we will denote by Xp the linear affine variety in Ank corresponding to p and by Ip CR the radical ideal which defines Xp in Ank : Note that the poset PðX Þ is isomorphic to the poset of ideals fIp gp ; ordered by reverse inclusion. We denote by hðpÞ the kcodimension of Xp in Ank (that is, hðpÞ equals the height of the ideal Ip ). The height of an ideal JCR will be denoted by hðJÞ: Let M be a R-module, iX0 an integer. Then one can define a diagram of Rmodules H½i  ðRÞ on the poset PðX Þ by *

H½i  ðMÞ : p/HIip ðMÞ: *

This defines a functor H½i  ðÞ : ModR -DiagðPðX Þ; ModR Þ: *

Lemma. If E is an injective R-module, then the augmented Roos complex Roos * ðH½0  ðEÞÞ-HI0 ðEÞ-0 *

is exact.

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Proof. Since both Roos * ðÞ and H½0  ðÞ commute with direct sums, by the Matlis– *

Gabriel theorem we can assume that there is a prime ideal pCR such that E ¼ ER ðR=pÞ; the injective envelope of R=p in the category of R-modules. Note also that for any ideal JCR; HJ0 ðER ðR=pÞÞ ¼ ER ðR=pÞ if pDJ and is zero otherwise. It will be enough to prove that if mCR is a maximal ideal, then the complex ðRoos * ðH½0  ðEÞÞÞm -ðHI0 ðEÞÞm -0 *

is exact. If Igm; this complex is zero. Otherwise, it equals the augmented complex Simp * ðK; ERm ðRm =pRm ÞÞ-ERm ðRm =pRm Þ-0; where K is the simplicial complex attached to the subposet of PðX Þ which has as vertexes those linear subspaces Xp such that Ip Dm: As K has a unique maximal element, it is contractible and then the lemma follows. & Fix now an injective resolution 0-R-E n of R in the category of R-modules. Each of the modules E j ðjX0Þ defines a diagram H½0  ðE j Þ over PðX Þ and one obtains *

a double complex Roosi ðH½0  ðE j ÞÞ; *

ip0; jX0

(the change of sign on the indexing of the Roos complex is because we prefer to work with a double complex which is cohomological in both degrees). This is a second quadrant double complex with only a finite number of non-zero columns, so it gives rise to a spectral sequence that converges to HIn ðRÞ (because of the lemma above). More precisely, we have E1i;j ¼ Roosi ðH½0  ðE j ÞÞ ) HIji ðRÞ: *

The differential d1 is that of the Roos complex, and since this complex computes the ith left-derived functor of the inductive limit, the E2 term will be ðiÞ

E2i;j ¼ indlimP H½j  ðRÞ ) HIji ðRÞ *

ð1Þ

(we write P for PðX Þ in order to simplify the writing of our formulas). Hereafter this sequence will be called Mayer–Vietoris spectral sequence. Remark 1.1. (i) Instead of an injective resolution of R in the category of R-modules, one could take as well an acyclic resolution with respect to the functors GJ ðÞ; JCR an ideal (recall that GJ ðMÞ :¼ fmAM j (rX0 such that J r m ¼ 0g). This fact will be used in the next sections. (ii) If the base field k is of characteristic zero, we can choose an injective resolution of R in the category of modules over the Weyl algebra An ðkÞ: Since An ðkÞ is free as an R-module, it follows (see e.g. [3, II.2.1.2]) that this is also an injective resolution of R in the category of R-modules. If the base field k is of characteristic p40; the ring R

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has a natural F -module structure and its minimal injective resolution is a complex of F -modules and F -module homomorphisms (see [11, (1:2:b00 )]). Therefore, the spectral sequence above may be regarded as a spectral sequence in the category of An ðkÞ-modules (respectively, of F -modules). The main result of this section is the following: Theorem 1.2. Let X CAnk be an arrangement of linear varieties. Let Kð4pÞ be the simplicial complex attached to the subposet fqAPðX Þ j q4pg of PðX Þ: Then: (i) There are R-module isomorphisms ðiÞ indlimP H½j  ðRÞC " ½HIjp ðRÞ#k H˜ i1 ðKð4pÞ; kÞ; *

hðpÞ¼j

where H˜ denotes reduced simplicial homology. We agree that the reduced homology with coefficients in k of the empty simplicial complex is k in degree 1 and zero otherwise. (ii) The Mayer–Vietoris spectral sequence ðiÞ

E2i;j ¼ indlimP H½j  ðRÞ ) HIji ðRÞ *

degenerates at the E2 -term. (iii) If k is a field of characteristic zero, the isomorphisms in ðiÞ are also isomorphisms of An ðkÞ-modules and the spectral sequence in ðiiÞ is a spectral sequence of An ðkÞmodules. If k is a field of positive characteristic, the isomorphisms in ðiÞ are also isomorphisms of F -modules and the spectral sequence in ðiiÞ is a spectral sequence of F -modules. Proof (cf. [4, Proposition 4.5]). (i) Given pAPðX Þ and a R-module M we consider the following three diagrams: FM;Xp ; defined by FM;Xp ðqÞ ¼ M if qXp and FM;Xp ðqÞ ¼ 0 otherwise, FM;4p ; defined by FM;4p ðqÞ ¼ M if q4p and FM;4p ðqÞ ¼ 0 otherwise, FM; p ; defined by FM; p ðqÞ ¼ M if q ¼ p and FM; p ðqÞ ¼ 0 otherwise (in all three cases F ðp-qÞ ¼ id if F ðpÞ ¼ F ðqÞ and it is zero otherwise). In the category of diagrams of R-modules over PðX Þ we have an exact sequence 0-FM;4p -FM;Xp -FM;p -0: Let KðXpÞ be the simplicial complex attached to the subposet fqAPðX Þ j qXpg of PðX Þ: Then one has Roos * ðFM;Xp Þ ¼ Simp * ðKðXpÞ; MÞ

and

Roos * ðFM;4p Þ ¼ Simp * ðKð4pÞ; MÞ:

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Since the complex KðXpÞ is contractible (to the vertex corresponding to p), the long exact homology sequence obtained from the sequence of complexes 0-Roos * ðFM;4p Þ-Roos * ðFM;Xp Þ-Roos * ðFM;p Þ-0 gives ðiÞ

indlimP FM;p DH˜ i1 ðKð4pÞ; MÞ; where the tilde denotes reduced homology and we agree that the reduced homology of the empty simplicial complex is M in degree 1 and zero otherwise. Notice that for any pAPðX Þ the module HIjp ðRÞ vanishes unless hðIp Þ ¼ j; so one has an isomorphism of diagrams H½j  ðRÞC"hðpÞ¼j FH j *

ðiÞ

Ip

ðiÞ

indlimP H½j  ðRÞD " indlimP FH j *

hðpÞ¼j

Ip

ðRÞ;p D

ðRÞ;p :

Thus,

" H˜ i1 ðKð4pÞ; HIjp ðRÞÞ:

hðpÞ¼j

By the universal coefficient theorem, for i40 H˜ i1 ðKð4pÞ; HIjp ðRÞÞDHIjp ðRÞ#k H˜ i1 ðKð4pÞ; kÞ: Although the isomorphism given by the universal coefficient theorem is a priori only an isomorphism of k-vector spaces, it is easy to check that in our case it is also an isomorphism of An ðkÞ-modules (if charðkÞ ¼ 0) or of F -modules (if charðkÞ40). In particular, it is always an isomorphism of R-modules. (ii) and (iii) Observe first that if ICR is an ideal and we set h ¼ hðIÞ; then all associated primes of HIh ðRÞ are minimal primes of I (this is due to the structure of the minimal injective resolution of R; in fact it holds for any Gorenstein ring, see e.g. [12, Theorem 18.8]). It follows that if p; qCR are prime ideals such that pgq and we set i ¼ hðpÞ; j ¼ hðqÞ; then HomR ðHpi ðRÞ; Hqj ðRÞÞ ¼ 0: From this last fact and (i) above, it follows that the Mayer–Vietoris sequence degenerates at the E2 -term. Part (iii) follows from Remark 1.1(ii) above and the observation at the end of the proof of part (i). & Corollary 1.3. Let X CAnk be an arrangement of linear varieties, let ICR be an ideal defining X : Then, for all rX0 there is a filtration fFjr grpjpn of HIr ðRÞ by R-submodules such that r Fjr =Fj1 D " ½HIjp ðRÞ#k H˜ hðpÞr1 ðKð4pÞ; kÞ hðpÞ¼j

r ¼ 0). This filtration is functorial with respect to affine for all jXr (we agree that Fr1 transformations. Moreover, if charðkÞ ¼ 0 it is a filtration by holonomic An ðkÞmodules and if charðkÞ40 it is a filtration by F -modules. Set

mr;p ¼ dimk H˜ hðpÞr1 ðKð4pÞ; kÞ:

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If charðkÞ ¼ 0; the characteristic cycle of the holonomic An ðkÞ-module HIr ðRÞ is CCðHIr ðRÞÞ ¼

X

mr;p TXn p Ank ;

where TXn p Ank denotes the relative conormal subspace of T n Ank attached to Xp : If k ¼ R is the field of real numbers, the Betti numbers of the complement of the arrangement X in AnR can be computed in terms of the multiplicities fmi;p g as dimQ H˜ i ðAnR  X ; QÞ ¼

X

miþ1;p :

p

If k ¼ C is the field of complex numbers, then one has dimQ H˜ i ðAnC  X ; QÞ ¼

X

miþ1hðpÞ;p :

p

Proof. The filtration fFjr g is the one given by the degeneration of the Mayer–Vietoris spectral sequence. It is a filtration by An ðkÞ-modules (respectively, F -modules) by Theorem 1.2(iii). The formula for the characteristic cycle follows from the fact that if hðIp Þ ¼ h; then CCðHIhp ðRÞÞ ¼ TXnp Ank and the additivity of the characteristic cycle with respect to short exact sequences. The formula for the Betti numbers of the complement AnR  X follows from a theorem of Goresky–MacPherson [9, III.1.3. Theorem A], which states (slightly reformulated) that H˜ i ðAnR  X ; ZÞD " H hðpÞi1 ðKðXpÞ; Kð4pÞ; ZÞ: p

Regarding a complex arrangement in AnC as a real arrangement in A2n R ; the formula for the Betti numbers of the complement of a complex arrangement follows from the formula for real arrangements. & Remark 1.4. (i) Set R ¼ k½x; y; consider the ideal I ¼ ðx  yÞCk½x; y and denote I1 ¼ ðxÞ; I2 ¼ ðyÞ; m ¼ ðx; yÞ: For the filtration of HI1 ðRÞ introduced above one has F1 CHI11 ðRÞ"HI12 ðRÞ; F2 ¼ HI2 ðRÞ; and the sequence 0-F1 -F2 -F2 =F1 -0 is nothing but the Mayer–Vietoris exact sequence 2 ðRÞ-0: 0-HI11 ðRÞ"HI12 ðRÞ-HI1 ðRÞ-Hm

This sequence is not split, e.g. because the maximal ideal m is not a minimal prime of I and so cannot be an associated prime of HI1 ðRÞ: Therefore, the extension problems attached to the filtration introduced in Corollary 1.3 are non-trivial in general. This question will be studied in Section 3. (ii) The formalism of Mayer–Vietoris sequences can be applied to functors other than H½i  ðÞ (and other than those considered in [4]). For example, one can consider *

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the diagrams ExtiR ðR=½ * ; RÞ : p/ExtiR ðR=Ip ; RÞ and, similarly as for the local cohomology modules, one has a spectral sequence ðiÞ

E2i;j ¼ indlimP ExtjR ðR=Ip ; RÞ ) Extji R ðR=I; RÞ; which degenerates at the E2 -term. Therefore, one can endow the module ExtrR ðR=I; RÞ with a filtration fGjr grpjpn such that Gjr =Gjr1 D " ½ExtjR ðR=Ip ; RÞ#k H˜ hðpÞr1 ðKð4pÞ; kÞ: hðpÞ¼j

The functoriality of the construction gives that the natural morphism ExtrR ðR=I; RÞ-HIr ðRÞ is filtered. In case ICR is a monomial ideal, it would be interesting to compare this filtration with the one defined in [13, Theorem 3.3]. 2. Betti numbers vs. multiplicities The ring R ¼ k½x1 ; y; xn  has a natural Zn -graduation given by degðxi Þ ¼ ei ; where e1 ; y; en denotes the canonical basis of Zn : If M ¼ "aAZn Ma and N ¼ "aAZn Na are graded R-modules, a morphism f : M-N is said to be graded if f ðMa ÞDNa for all aAZn : Henceforth, the term graded will always mean Zn -graded. We denote by * ModR the category which has as objects the graded R-modules and as morphisms the graded morphisms. If M; N are graded R-modules, we denote by * HomR ðM; NÞ the group of graded morphisms from M to N (this group should not be confused with the internal Hom in the category * ModR ; in particular it is usually not a graded R-module). Its derived functors will be denoted * Exti ðM; NÞ; iX0: R We recall some facts about * ModR which will be relevant for us (see [10] for proofs and related results). If M is a graded module one can define its * -injective envelope * EðMÞ (in particular, * ModR is a category with enough injectives). A graded version of the Matlis–Gabriel theorem holds: the indecomposable injective objects of * ModR are the shifted injective envelopes * EðR=pÞðaÞ; where p is an homogeneous prime ideal of R and aAZn ; and every graded injective module is isomorphic to a unique (up to order) direct sum of indecomposable injectives. Let IDR be a monomial ideal. Then, R=I j is a finitely generated graded R-module for any jX0; and * ExtiR ðR=I j ; NÞCExtiR ðR=I j ; NÞ for any graded R-module N: It follows that injective objects of * ModR ; which usually are not injective as objects of ModR ; are acyclic with respect to the functor GI ðÞ (see [10]). It also follows that the local cohomology modules HIi ðRÞ are objects of * ModR : Our aim is to relate the dimensions of its homogeneous components to the multiplicities of its characteristic cycle.

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A homogeneous prime ideal of R is of the form p ¼ ðxi1 ; y; xik Þ 1pi1 o?oik pn: We will denote by * P the set of homogeneous prime ideals of R: Putting O ¼ f1; 0gn ; there is a bijection O - * P; a /pa ¼ /xik j aik ¼ 1S: For aAO; we will set jaj ¼ ja1 j þ ? þ jan j: Let ICR be a monomial ideal, rX0 an integer, let X CAnk be the arrangement defined by I: It will be convenient to reindex the multiplicities introduced in Section 1 as follows: ( mr;p if there is a pAPðX Þ with pa ¼ Ip ; mr; a :¼ 0 otherwise: Then we have: Proposition 2.1. In the situation and with the notations described above, mr;a ¼ dimk HIr ðRÞa

for all rX0; aAO:

Proof. Since * -injective modules are GI -acyclic, by Remark 1.1(i) one can assume that the filtration fFjr grpjpn of HIr ðRÞ introduced in Corollary 1.3 has been constructed from an injective resolution of R in * ModR : From this fact, it follows that we can assume that the R-modules Fjr are graded, the r inclusion maps Fj1 +Fjr are graded morphisms and one has isomorphisms of graded modules r Fjr =Fj1 D " ðHpj a ðRÞÞ"mr;a : jaj¼j

Note that for a; bAO; one has ( ðHpjaja ðRÞÞb

¼

0

if baa;

k

if b ¼ a:

Using these facts and the exactness of the functors desired result follows. &

* ModR -Vectk ;

M/Ma the

n Given a reduced monomial ideal ICR; the modules TorR i ðI; kÞ are Z -graded and the graded Betti numbers of I are defined as

bi;a ðIÞ :¼ dimk TorR i ðI; kÞa ;

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where aAZn : The Alexander dual ideal of I is the ideal * + Y Y xi j SDf1; y; ng; xi eI : I3 ¼ iAS

ieS

Mustat-a˘ [13] showed that if aAO; then one has bi;a ðI 3 Þ ¼ dimk HI

jaji

ðRÞa ;

and all other graded Betti numbers are zero. So, we have: Corollary 2.2. If I is a reduced monomial ideal and aAO; then bi;a ðI 3 Þ ¼ mjaji;a : In particular, if charðkÞ ¼ 0; the graded Betti numbers of a monomial ideal I can be obtained from the (D-module theoretic) characteristic cycles of the local cohomology of R supported at its Alexander dual I 3 : Also, if k ¼ C or k ¼ R; it follows from the corollary above and Corollary 1.3 that the (topological) Betti numbers of the complement in Ank of the arrangement defined by a monomial ideal I can be obtained from the (algebraic) graded Betti numbers of I 3 : This fact was already proved using a different approach in [8].

3. Extension problems If M is a graded R-module and aAZn ; as usual we denote by MðaÞ the graded Rmodule whose underlying R-module structure is the same as that of M and where the grading is given by ðMðaÞÞb ¼ Maþb : If aAZn ; we set xa ¼ xa11 ?xann and suppþ ðaÞ ¼ fi j ai 40 g: We recall the following definition of Yanagawa: Definition (Yanagawa [16, 2.7]). A Zn -graded module is said to be straight if the following two conditions are satisfied: (i) dimk Ma oN for all aAZn : (ii) The multiplication map Ma {y/xb yAMaþb is bijective for all a; bAZn with suppþ ða þ bÞ ¼ suppþ ðaÞ:

* ModR which has as objects the straight The full subcategory of the category P modules will be denoted Str. Let e ¼  ni¼1 ei ¼ ð1; y; 1Þ: In order to avoid shiftings in local cohomology modules, we will consider instead the following (equivalent) category:

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Definition. We will say that a graded module M is e-straight if MðeÞ is straight in the above sense. We denote e-Str the full subcategory of * ModR which has as objects the e-straight modules. It follows from [13,16, Theorem 2.13] that if ICR is a monomial ideal and rX0 is an integer, HIr ðRÞ is an e-straight module. Proposition 3.1. Let M be a e-straight module. There is a finite increasing filtration fFj g0pjpn of M by e-straight submodules such that for all 0pjpn one has graded isomorphisms Fj =Fj1 C " ðHpj a ðRÞÞ"ma : aAO jaj¼j

where ma ¼ dimk Ma : Proof. The existence of an increasing filtration fGj gj of M by e-straight submodules such that all quotients Gj =Gj1 are isomorphic to local cohomology modules supported at homogeneous prime ideals is an immediate transposition to e-straight modules of [16, 2.12] (which relies on [15, 2.5]). Inspection of Yanagawa’s proof shows that, in order to prove the existence of a filtration fFj gj satisfying the condition of the proposition, it is enough to show that if pa ; pb are homogeneous prime ideals with jaj ¼ jbj ¼ l; then * Ext1R ðHpl a ðRÞ; Hpl b ðRÞÞ ¼ 0: The minimal * injective resolution of Hpl b ðRÞ is 0-Hpl b ðRÞ- * ER ðR=pb ÞðeÞ- "

ijbi ¼0

* ER ðR=p bei ÞðeÞ-?:

(see e.g. [16, 3.12]). Thus the vanishing of the above * Ext module follows from the following: Claim. For all homogeneous prime ideals q*pb with hðqÞ  hðpb Þ ¼ 1; one has * HomR ðH l ðRÞ; * ER ðR=qÞðeÞÞ ¼ 0: pa Proof. If q is such an homogeneous prime ideal, we will assume that q*pa (otherwise, the statement follows easily from the fact that q is the only associated prime of * ER ðR=qÞ). Set ac :¼ e  aAO: It suffices to prove the vanishing of * HomR ðH l ðRÞðeÞ; * EðR=qÞÞ: By the equivalence of categories proved in [16, 2.8], pa there is a bijection between this group and * HomR ðR=pa ðac Þ; R=qÞ: A graded morphism j : R=pa ðac Þ-R=q is determined by jð1ÞAðR=qÞac : But ðR=qÞac ¼ 0; so we are done. The equality ma ¼ dimk Ma is proved as in the proof of Proposition 2.1. & Remark. Even if M is a local cohomology module supported at a monomial ideal, in general the submodules Fj introduced in the proposition above are not. This is one of the reasons to consider the category of e-straight modules.

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47

Henceforth we will assume that a e-straight module M together with an increasing filtration fFj gj of M as in Proposition 3.1 have been fixed. For each 0pjpn; one has an exact sequence ðsj Þ: 0-Fj1 -Fj -Fj =Fj1 -0; which defines an element of * Ext1R ðFj =Fj1 ; Fj1 Þ: In this section, our aim is to show that this element is determined, in a sense that will be made precise below, by the klinear maps xi : Ma -Maþei ; where jaj ¼ j and i is such that ai ¼ 1: In particular, the sequence ðsj Þ splits if and only if xi Ma ¼ 0 for a; i in this range. We will prove first the following lemma: Lemma. The natural maps * Ext1 ðFj =Fj1 ; Fj1 Þ- * Ext1 ðFj =Fj1 ; Fj1 =Fj2 Þ R R

are injective for all jX2: Proof. From the short exact sequence ðsj1 Þ; applying * HomðFj =Fj1 ; Þ we obtain the exact sequence * Ext1 ðFj =Fj1 ; Fj2 Þ- * Ext1 ðFj =Fj1 ; Fj1 Þ- * Ext1 ðFj =Fj1 ; Fj1 =Fj2 Þ; R R R

and we have to prove that * Ext1R ðFj =Fj1 ; Fj2 Þ ¼ 0: Applying again * HomðFj =Fj1 ; Þ to the exact sequences ðsl Þ for lpj  2; and descending induction, the assertion reduces to the statement * Ext1R ðFj =Fj1 ; Fl =Fl1 Þ ¼ 0 for lpj  2: By Proposition 3.1, it suffices to prove that if p; qCR are homogeneous prime ideals of heights hðpÞ ¼ j and hðqÞ ¼ l and lpj  2; then * Ext1R ðHpj ðRÞ; Hql ðRÞÞ ¼ 0: We will have q ¼ pb ; l ¼ jbj; for some bAO: As observed in the proof of Proposition 3.1, the jbj

minimal * -injective resolution of Hpb ðRÞ is 0-Hpjbjb ðRÞ- * ER ðR=pb ÞðeÞ- "

ijbi ¼0

* ER ðR=p bei ÞðeÞ-?:

Thus, again as in the proof of Proposition 3.1, it suffices to prove that * HomR ðH j ðRÞ; * p

ER ðR=pbei ÞðeÞÞ ¼ 0:

This follows as in the proof of Theorem 1.2(ii), because pbei is the only associated prime of ER ðR=pbei Þ: & Remark 3.2. If ICR is an ideal defining an arbitrary arrangement of linear varieties in Ank ; one can prove an analogous lemma for the filtration of the local cohomology module HIr ðRÞ introduced in Section 1.

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48

The extension class of ðsj Þ maps, via the morphism in the above lemma, to the extension class of the sequence ðs0j Þ: 0-Fj1 =Fj2 -Fj =Fj2 -Fj =Fj1 -0: Let d0

0-Fj1 =Fj2 - * E0 - * E1 -? be the minimal * -injective resolution of Fj1 =Fj2 : Given a graded morphism Fj =Fj1 -Im d 0 ; one obtains an extension of Fj1 =Fj2 by Fj =Fj1 taking the following pull-back:

and all extensions of Fj1 =Fj2 by Fj =Fj1 are obtained in this way. Take aAO with jaj ¼ j: Applying the functor Hpna ðÞ to this diagram, we obtain a commutative square

where (i) The upper horizontal arrow is an isomorphism because * E 0 C"jbj¼j1 * ðER ðR=p ÞÞ"mb ðeÞ; b

and then Hpi a ð * E 0 Þ ¼ 0 for all iX0:

(ii) The morphism da is the connecting homomorphism of the given extension. Since Fj =Fj1 ¼ "jaj¼j Hp0a ðFj =Fj1 Þ; the morphism j is determined by the morphisms ja for jaj ¼ j; and these are in turn determined by the connecting homomorphisms da via the commutative square above. Observe also that the module Hp1a ðFj =Fj1 Þ is isomorphic to a direct sum of local cohomology modules of the form Hpj a ðRÞ: We will show next that, because of this fact, it suffices to consider the restriction of da to the k-vector space of homogeneous elements of multidegree a:

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49

Lemma (Linearization). Let k1 ; k2 X0 be integers, let M1 ¼ Hpl a ðRÞ"k1 ; M2 ¼ Hpl a ðRÞ"k2 : The restriction map

* HomR ðM1 ; M2 Þ-HomVect

k

ððM1 Þa ; ðM2 Þa Þ

is a bijection. Moreover, these bijections are compatible with composition of graded maps. Proof. Taking the components of a graded map M1 -M2 ; it will be enough to prove that all graded endomorphisms of N :¼ Hpl a ðRÞ are multiplications by constants of the base field k: Let j : N-N be a graded endomorphism. Note that j can be regarded also as an endomorphism of the graded module NðeÞ: By Yanagawa [16], NðeÞ is a straight module and j is determined by its restriction to the Nn -graded part of NðeÞ; which is R=pa ðac Þ ðac ¼ e  aÞ: A graded R-module map R=pa ðac Þ-R=pa ðac Þ is determined by the image of 1AðR=pa ðac ÞÞac : Since ðR=pa ðac ÞÞac ¼ ðR=pa Þ0 ¼ k; j must be the multiplication by some constant, as was to be proved.

&

Thus, any extension of Fj1 =Fj2 by Fj =Fj1 is determined by the k-linear maps daa : Hp0a ðFj =Fj1 Þa -Hp1a ðFj1 =Fj2 Þa : One can easily check that Hp0a ðFj =Fj1 Þa CMa ; and using a C˘ech complex one obtains that " # Hp1a ðFj1 =Fj2 Þa ¼ Ker

" ððFj1 =Fj2 Þxi Þa -

ai ¼1

"

ai ¼1;aj ¼1

ððFj1 =Fj2 Þxi xj Þa

B

- " Maþei ; ai ¼1

where the last arrow is an isomorphism given by multiplication by xi on ððFj1 =Fj2 Þxi Þa : The connecting homomorphism obtained applying Hpna ðÞ to the exact sequence ðs0j Þ can be computed using C˘ech complexes as well. It turns out that, via the isomorphisms above, the corresponding map daa is in this case precisely the map Ma - " Maþei ; ai ¼1

m /"ðxi  mÞ: In conclusion,

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Proposition. The extension class ðsj Þ is uniquely determined by the k-linear maps xi : Ma -Maþei where jaj ¼ j and ai ¼ 1: Remarks. (i) Mustat-a˘ [13] has proved that for local cohomology modules supported at monomial ideals, the linear maps xi : HIj ðRÞa -HIj ðRÞaþei can be explicitly computed in terms of the simplicial cohomology of certain Stanley–Reisner complexes attached to I: (ii) The very definition of e-straight modules shows that they are determined as graded modules by the vector spaces Ma ; aAO and the multiplication maps xi : Ma -Maþei ; ai ¼ 1: However, this fact alone is not very enlightening if one wishes to know how the extension problems arising from the Mayer– Vietoris sequence are related to these data. Regarding e-straight modules as representations of a boolean lattice one can obtain an alternative proof of the results in this section. We have chosen a more algebraic approach because it seems us that it might be better suited to extend our results to local cohomology modules supported at more general types of arrangements (cf. Remark 3.2). 4. Local cohomology and perverse sheaves In this section we will use the following notations: *

* *

*

*

R ¼ C½x1 ; y; xn  (by a slight abuse of notation, we will denote by R as well the sheaf of regular algebraic functions in Cn ). O denotes the sheaf of holomorphic functions in Cn : D denotes the sheaf of differential operators in Cn with holomorphic coefficients. T denotes the union of the coordinate hyperplanes in Cn ; endowed with the stratification given by the intersections of its irreducible components. Xa denotes the linear subvariety of Cn defined by the ideal pa CR; aAO:

We denote PervT ðCn Þ the category of complexes of sheaves of finitely dimensional vector spaces on Cn which are perverse relatively to the given stratification of T [6, I.1]. We denote DThr the full abelian subcategory of the category of regular holonomic modules M in Cn such that their solution complex RHomD ðM; OÞ is an object of PervT ðCn Þ: By the Riemann–Hilbert correspondence, the functor of solutions is then an equivalence of categories between DThr and PervT ðCn Þ: In [6], the category PervT ðCn Þ has been linearized as follows: Let Cn be the category whose objects are families fMa gaAO of finitely dimensional complex vector spaces indexed by O :¼ f1; 0gn ; endowed with linear maps ui

Ma - Maei ;

vi

Ma ’ Maei

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51

for each aAO such that ai ¼ 0: These linear maps are called canonical (resp., variation) maps, and they are required to satisfy the conditions: ui uj ¼ uj ui ;

vi vj ¼ vj vi ;

ui v j ¼ v j ui

and

vi ui þ id is invertible:

Such an object will be called an n-hypercube, the vector spaces Ma will be called its vertices. A morphism between two n-hypercubes fMa ga and fNa ga is a set of linear maps ffa : Ma -Na ga ; commuting with the canonical and variation maps (see [6]). It is proved in [6] that there is an equivalence of categories between PervT ðCn Þ and Cn : Given an object M of DThr ; the n-hypercube corresponding to RHomD ðM; OÞ is constructed as follows (see [7]): Q Consider Cn ¼ ni¼1 Ci ; with Ci ¼ C for 1pipn; let Ki ¼ Rþ CCi and set Vi ¼ Ci WKi : For any a ¼ ða1 ; y; an ÞAO denote Sa :¼ P

GQn i¼1

ak ¼1

O Q

Vi

GCk 

:

VO iak i

Denoting with a subscript 0 the stalk at the origin, one has: (i) The vertices of the n-hypercube associated to M are the vector spaces Ma :¼ HomD0 ðM0 ; Sa;0 Þ: (ii) The linear maps ui are those induced by the natural quotient maps Sa -Saei : (iii) The linear maps vi are the partial variation maps around the coordinate hyperplanes, i.e. for any jAHomD0 ðM0 ; Sa;0 Þ one has ðvi 3 ui ÞðjÞ ¼ Fi ðjÞ  j; where Fi is the partial monodromy around the hyperplane xi ¼ 0: The following is proved as well in [6,7]: P (iv) If CCðMÞ ¼ ma TXna Cn is the characteristic cycle of M; then for all aAO one has the equality dimC Ma ¼ ma : (v) Let a; bAO be such that ai bi ¼ 0 for 1pipn: For each j with bj ¼ 1 choose any lj ACWZ; set lb ¼ flj gj and let Ia;b;lb denote the left ideal in D generated by ðfxi j ai ¼ 1g; f@k j ak ¼ bk ¼ 0g; fxj @j  lj j bj ¼ 1gÞ: Then the simple D : objects of the category DThr are the quotients Ia;b;l b

Definition. We say that an object M of DThr has variation zero if the morphisms vi : Maei -Ma are zero for all 1pipn and all aAO with ai ¼ 0: It is easy to prove that modules with variation zero form a full abelian subcategory of DThr that will be denoted DTv¼0 : Remark 4.1. (i) Let f ¼ xa11 ?xann ; 0pai p1; be a squarefree monomial in C½x1 ; y; xn : From the presentation O½1=f  ¼

D Dðfxi @i þ 1 j ai ¼ 1g; f@j j aj ¼ 0gÞ

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one can check that the D-module O½1=f  has variation zero. Using a Cˇech complex, it follows that if X is a subvariety of Cn defined by the vanishing of monomials, then the local cohomology modules HnX ðOÞ are also modules of variation zero. D of the category DThr has variation zero if and only if bk ¼ 0 (ii) A simple object Ia;b;l b

for 1pkpn: Thus, the simple objects of DTv¼0 are of the form D : Dðfxi j ai ¼ 1g; f@j j aj ¼ 0gÞ This module We have:

is

isomorphic

to

the

local

cohomology

module

jaj

HXa ðOÞ:

Proposition 4.2. An object M of DThr has variation zero if and only if there is an increasing filtration fFj g0pjpn of M by objects of DThr and there are integers ma X0 for aAO; such that for all 0pjpn one has D-module isomorphisms Fj =Fj1 C " ðHjXa ðOÞÞ"ma : aAO jaj¼j

Proof. If M is an object of DTv¼0 ; then the submodules Fj of M corresponding to the hypercube: ( ðFj Þb ¼

Mb

if jbj pj;

0

otherwise:

(the canonical maps being either zero or equal to those in M), give a filtration which satisfies the conditions of the theorem, this follows easily from the fact that the jaj hypercube corresponding to HXa ðOÞ is ( jaj HXa ðOÞd

¼

C

if d ¼ a;

0

otherwise:

Conversely, assume M is an object of DThr endowed with such a filtration fFj g0pjpn : For all 1pjpn; we have exact sequences ðsj Þ: 0-Fj1 -Fj -Fj =Fj1 -0:

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And from them we get corresponding exact sequences in the category of nhypercubes, which in turn give sequences of C-vector spaces:

for dAO with di ¼ 0 (recall the functor of solutions is contravariant). From the jaj description of the n-hypercube corresponding to HXa ðOÞ given above, it follows that we only have to consider those vertices dAO with jd  ei j ¼ j: We have ½Fj =Fj1 d ¼ ½Fj1 dei ¼ 0; so by induction all variations vanish, as was to be proved. & If M is a An ðCÞ-module, then Man :¼ O#R M has a natural D-module structure. This allows to define a functor: ðÞan : ModAn ðCÞ -ModD ; M-Man ; f -id#f : If M is an e-straight module, it can be endowed with a functorial An ðCÞ ¼ R/@1 ; y; @n S—module structure extending its R-module structure as follows: If bAZn and mAMb ; then @i  m :¼ bi x1 i m (see [16, Remark 2.14]). Since the morphism R-O is flat, one has isomorphisms O#R Hpl a ðRÞDHlXa ðOÞ for all aAO; lX0: From this fact, together with Propositions 3.1 and 4.2, it follows that if M is a e-straight module then Man is an object of DTv¼0 : The main result of this section is the following: Theorem 4.3. The functor ðÞan : e  Str-DTv¼0 is an equivalence of categories. We will prove first the following lemma (which in particular gives the fully faithfulness of ðÞan ): Lemma 4.4. Let M; N be e-straight modules. For all iX0; we have functorial isomorphisms * Exti ðM; NÞDExti T R Dv¼0

ðMan ; Nan Þ:

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Proof. It has been proved by Yanagawa that the category of straight modules has enough injectives. It will follow from our proof that the category DTv¼0 has enough injectives as well, so that both Ext functors are defined and can be computed using resolutions. By induction on the length we can suppose that M and N are simple objects, i.e. jaj jbj M ¼ Hpa ðRÞ and N ¼ Hpb ðRÞ: Recall that the minimal * -injective resolution of N is 0-Hpjbjb ðRÞ- * ER ðR=pb ÞðeÞ- "

ijbi ¼0

* ER ðR=p bei ÞðeÞ-?:

ð2Þ

If aAO; set Ea :¼ ð * ER ðR=pa ÞðeÞÞan : We claim that for all aAO; Ea is an injective object of DTv¼0 : Using the description of * -injective envelopes in [10, 3.1.5] (or, alternatively, the equivalence of categories proved in [16, 2.8]) one can see that, for all aAO; there are isomorphisms * ER ðR=p ÞðeÞDP a

1 R½x1 ?x  n ai ¼1

R½x1 ?xˆ1i ?xn 

:

From these isomorphisms it is easy to compute the n-hypercube corresponding to Ea ; namely one has ( C if gi pai for all 1pipn; a Eg ¼ 0 otherwise: The map ui : Eag -Eagei is the identity if Eag DCDEagei and it is zero otherwise. The exactness of the functor HomD ð; Ea Þ on DTv¼0 can now be proved by passing to the category Cn of n-hypercubes, where the assertion reduces to a simple question of linear algebra that is left to the reader. From flatness of R-O and the injectivity of the Ea proved above, it follows that jbj one has the following injective resolution of Nan ¼ HXb ðOÞ in DTv¼0 : jbj

0-HXb ðOÞ-Eb - " Ebei -?:

ð3Þ

ijbi ¼0

Let K1 be the complex obtained applying * HomR ðM; Þ to resolution (2) and let K2 jaj

be the one obtained applying HomD ðHXa ðOÞ; Þ to (3). We have an injection K1 +K2 and we want to show that it is an isomorphism. We have ( C if a ¼ g; jaj * HomR ðH ðRÞ; * ER ðR=p ÞðeÞÞ ¼ ð4Þ g pa 0 otherwise (this can be seen taking the positively graded parts and using [16, 2.8], as done before in similar situations). The same equality holds replacing the left-hand side in (4) by

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55

jaj

HomD ðHXa ðOÞ; Eg Þ; this is easily proved considering the corresponding nhypercubes. It follows that K1 DK2 ; and then we are done. & Proof of Theorem 4.3. By Lemma 4.4 the functor ðÞan is fully faithful, so it remains to prove that it is dense. Let N be an object of DTv¼0 ; let N0 DN be a submodule such that N00 :¼ N=N0 is simple. By induction on the length, there are e-straight Rmodules M 0 and M 00 such that N0 DðM0 Þan and N00 DðM00 Þan : The extension 0-N0 -N-N00 -0 corresponds to an element x of Ext1DT ðN00 ; N0 Þ: Let v¼0

0-M 0 -M-M 00 -0 be an extension such that its class in * Ext1R ðM 00 ; M 0 Þ maps to x via the isomorphism of Lemma 4.4. One can check that NDMan ; and then the theorem is proved. & Remark. The category of e-straight modules, regarded as a subcategory of the category of Zn -graded modules, is closed under extensions [16, Lemma 2.10]. However, note that the category DTv¼0 ; regarded as a subcategory of DThr ; is not. Acknowledgments The authors thank the referee for pointing out a mistake in a previous version of this paper and for a number of valuable comments which have notably improved its readability.

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