Characteristic cycles of local cohomology modules of monomial ideals II

Transcription

1 Journal of Pure and Applied Algebra 192 (2004) Characteristic cycles of local cohomology modules of monomial ideals Josep Alvarez Montaner Departament de Matematica Aplicada, Universitat Politecnica de Catalunya, Avinguda Diagonal 647, Barcelona 08028, Spain Received 29 November 2003 Communicated by C.A. Weibel Abstract Let R = k[x 1;:::;x n] be the polynomial ring in n independentvariables, where k is a eld of characteristic zero. n this work, we will describe the multiplicities of the characteristic cycle of the local cohomology modules H r (R) supported on a squarefree monomial ideal R in terms of the Betti numbers of the Alexander dual ideal. From this description we deduce a Gorensteinness criterion for the quotient ring R=. On the other side, we give a formula for the characteristic cycle of the local cohomology modules Hp p (H r (R)), where p is any homogeneous prime ideal of R. This allows us to compute the Bass numbers of H r (R) with respect to any prime ideal and describe its associated primes. c 2004 Elsevier B.V. All rights reserved. MSC: Primary: 13D45; 13N10 1. ntroduction Let R = k[x 1 ;:::;x n ] be the polynomial ring in n independentvariables, where k is a eld of characteristic zero. n [1] we studied the local cohomology modules of R supported on a squarefree monomial ideal R using the fact that they have a natural D-module structure (see also [12,19]). More precisely, we computed the characteristic cycle of H r(r) and H m(h p r (R)), where m is the homogeneous maximal ideal of R. As a consequence we gave a criterion to decide when the local cohomology module H r (R) vanishes and to compute the cohomological dimension cd(r; ) in terms of the minimal primary decomposition of the monomial ideal. We also gave a address: josep.alvarez@upc.es (J. A. Montaner) /$ - see front matter c 2004 Elsevier B.V. All rights reserved. doi: /j.jpaa

2 2 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 Cohen Macaulayness criterion for the local ring R= and computed the Lyubeznik numbers p;i (R=) = dim k Ext p n i R (k; H (R)). Our rst aim in this work is to give some new consequences of the main result in [1]. We also want to generalize the formula of the characteristic cycle of Hm(H p r(r)) [1, Theorem 4.4] to the case of any homogeneous prime ideal. This new formula will allow us to study the Bass numbers of local cohomology modules supported on squarefree monomial ideals and describe its associated primes. n Section 2, we introduce the notation we will use throughout this work. n Section 3, we rstrecall the main resultof [1], i.e. a closed formula for the characteristic cycle of any local cohomology module supported on a squarefree monomial ideal in terms of sums of face ideals appearing in the minimal primary decomposition. Let R be the Alexander dual ideal of the ideal R. The algorithm we use to compute the characteristic cycle of H r (R) may be interpreted as an algorithm that allows to minimize the free resolution of given by the Taylor complex. n particular, we can describe the multiplicities of the characteristic cycle in terms of the Betti numbers of. Then, we can easily recover a fundamental result of Eagon and Reiner [8] on the Cohen Macaulayness of R= as well, a generalization of this result given by Terai [18]. By using this description of the multiplicities we can also give a criteria for the Gorensteinness of R= in terms of the minimal primary decomposition of. Moreover, we can describe the type of a Cohen Macaulay ring R=. n Section 4, we obtain a closed formula for the characteristic cycle of the local cohomology modules Hp p (H r(r)), where p is any homogeneous prime ideal of R, again in terms of the minimal primary decomposition of. The ideas we use to give this formula are a natural extension of those used in [1] to compute the characteristic cycle of Hm(H p r (R)). From this formula we are able to compute the Bass numbers p (p;h n i (R)) := dim k(p) Ext p R p (k(p);h n i R p (R p )) of the local cohomology modules H r (R) with respect to any prime ideal p R. n particular, we can give a description of the injective dimension of these modules as well we can describe their associated primes. This will be done in Section Preliminaries Let R = k[x 1 ;:::;x n ] be the polynomial ring in n independentvariables, where k is a eld of characteristic zero. n this section, we x the notation we will use throughout this work. A squarefree monomial in R is a product x := x 1 1 xn n, where {0; 1} n.an ideal R is said to be a squarefree monomial ideal if it may be generated by squarefree monomials. ts minimal primary decomposition is then given in terms of face ideals := (x i i 0), where {0; 1} n. Set 1 =(1;:::;1) {0; 1} n. The Alexander dual ideal of is the ideal =(x = x 1 1 xn n x 1 ): The minimal primary decomposition of can be easily described from the ideal. Namely, let {x 1 ;:::;x m } be a minimal system of generators of. Then, the minimal

3 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) primary decomposition of is of the form = 1 m, and we have =. The reader mightsee [16], among many others, for more information. Let = 1 m be the minimal primary decomposition of a squarefree monomial ideal R. n our work it will be very useful to use the following poset in order to encode the information given by the minimal primary decomposition. The poset : We consider = { 1 ; 2 ;:::; m }, where 1 = { i1 1 6 i 1 6 m}, 2 = { i1 + i2 1 6 i 1 i 2 6 m},. m = { m }. Namely, the sets j consistof the sums of j face ideals in the minimal primary decomposition of. Notice that the sums of face ideals in the poset are treated as dierent elements even if they describe the same ideal. The Taylor complex: Let {x 1 ;:::;x m } be a minimal system of generators of a squarefree monomial ideal R. Let F be the free R-module of rank m generated by e 1 ;:::;e m. By using the Taylor complex (see [17]) we have a resolution of the ideal : T ( ): d 0 F m d 1 m F1 0; where F j = j F is the jth exterior power of F and the dierentials d j are dened as d j (e i1 e ij )= ( 1) k LCM(x i 1 ;:::;x i j ) LCM(x i 1 ;:::; x i k ;:::;x i j ) 16k6j e i1 ê ik e ij : The Taylor complex T ( ) is a cellular free resolution supported on the simplex whose vertices are labelled by the generators of the ideal. More precisely, it constitutes a nonaugmented oriented chain complex up to a homological shift for. By using Alexander duality it is easy to see that the information given by the Taylor complex is also encoded by the poset. To this purpose, we only have to point out that LCM(x i 1 ;:::;x i j ) corresponds to the sum of face ideals i1 + + ij j. The ring of dierential operators: n the sequel, D will denote the ring of dierential operators corresponding to R. For details and unexplained terminology we refer to [5 7]. The ring D has a natural increasing ltration given by the order, such that the corresponding associated graded ring gr(d) is isomorphic to the polynomial ring R[ 1 ;:::; n ]. Let M be a nitely generated D-module equipped with a good ltration, i.e. an increasing sequence of nitely generated R-submodules, such that the associated graded module gr(m) is a nitely generated gr(d)-module. The characteristic ideal of M is the ideal in gr(d)=r[ 1 ;:::; n ] given by J (M) := rad(ann gr(d) (gr(m))). One may prove that J (M) is independent of the good ltration on M. The characteristic variety

4 4 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 of M is the closed algebraic set given by C(M) :=V(J(M)) Spec(gr(D)) = Spec(R[ 1 ;:::; n ]): The characteristic variety allows us to describe the support of a nitely generated D-module as R-module. Let : Spec(R[ 1 ;:::; n ]) Spec(R) be the map dened by (x; )=x. Then Supp R (M)=(C(M)). The characteristic cycle of M is dened as CC(M)= m i V i ; where the sum is taken over all the irreducible components V i = V (p i ) of the characteristic variety C(M), where p i Spec(gr(D)) and m i is the multiplicity of the module gr(m) pi. The varieties that appear in the characteristic cycle can be described in terms of conormal bundles (see [14] for details). f X X = Spec(R) is dened by the face ideal k[x 1 ;:::;x n ], {0; 1} n. The conormal bundle to X in X is the subvariety T X X Spec(R[ 1 ;:::; n ]) dened by the equations {x i =0 i =1} and { i =0 i =0}. 3. The characteristic cycle of H r (R) Let = 1 m be the minimal primary decomposition of a squarefree monomial ideal R. n [1] we computed the characteristic cycle of H r (R) by using the local cohomology modules supported on sums of face ideals included in the poset = { 1 ; 2 ;:::; m }. We briey recall the main steps of the process used in that computation. First we introduced the following: Denition 3.1. We say that i1 + + ij j and i1 + + ij + ij+1 j+1 are paired if i1 + + ij = i1 + + ij + ij+1. Then, a poset of nonpaired sums of face ideals in the minimal primary decomposition of was constructed by means of the following: Algorithm. Let m be the number of face ideals in the minimal primary decomposition = 1 m. For j from 1 to m 1, incrementing by 1. For all k 6 j COMPARE the ideals j+1 +( i1 + + ir ) and k + j+1 +( i1 + + ir ): For all k 6 j COMPARE the ideals k +( i1 + + ir ) and k + j+1 +( i1 + + ir ): More precisely, the poset of nonpaired sums of face ideals was obtained considering: nput: The set of all sums of face ideals obtained as a sum of face ideals in the minimal primary decomposition of.

5 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) We apply the algorithm where COMPARE means remove both ideals in case they are paired. Output: The set P of all nonpaired sums of face ideals obtained as a sum of face ideals in the minimal primary decomposition of. We ordered P by the number of summands P = {P 1 ; P 2 ;:::;P m }, in such a way that no sum in P j is paired with a sum in P j+1. Observe that some of these sets can be empty. Finally we dened the sets of nonpaired sums of face ideals with a given height P j;r := { i1 + + ij P j ht( i1 + + ij )=r +(j 1)}: The formula given in the main result of [1] is expressed in terms of these sets of nonpaired sums of face ideals. Theorem 3.2 ( Alvarez Montaner [1]). Let R be an ideal generated by squarefree monomials and let = 1 m be its minimal primary decomposition. Then CC(H r (R)) = CC(H r i (R)) + CC(H r+1 i1 + i2 (R)) + i P 1;r i1 + i2 P 2;r Betti numbers and multiplicities CC(H r+(m 1) m (R)): m P m;r Our aim in this section is to explain how the computation of the characteristic cycle of the local cohomology modules H r (R) is related to the process of minimizing the free resolution of the ideal given by the Taylor complex. n general, the free resolution of given by the Taylor complex is not minimal. Precisely, itis notminimal when there existj and k such that LCM(x i 1 ;:::;x i j ) =1: LCM(x i 1 ;:::; x i k ;:::;x i j ) By the Alexander duality correspondence this is equivalent to the equality i1 + + ij = i1 + + Î ik + + ij of sums of face ideals in the minimal primary decomposition of, i.e. itis equivalentto say that i1 + + ij and i1 + + Î ik + + ij are paired ideals: We can interpret our algorithm for cancelling paired sums of face ideals as an algorithm to minimize the free resolution of the Alexander dual ideal given by the Taylor complex. More precisely, the minimal free resolution we obtain is labelled by the elements in the poset P, i.e. itis in the form d F : 0 F m d 1 m F1 0; where F j = R( ) j 1;( ) : { {0;1} n P j}

6 6 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 We can easily describe the Betti numbers of from the multiplicities of the characteristic cycle of the local cohomology modules supported on. However, we should notice that the description of the dierentials of this minimal free resolution is more involved than the description of the dierentials of the Taylor complex. Our main result in this section is the following: Proposition 3.3. Let R be Alexander dual ideal of a squarefree monomial ideal R. Then we have j; ( )=m n +j; (R=): A dierent approach to this result is also given in [3, Corollary 2.2]. Remark 3.4. Observe that the multiplicities m i; of the characteristic cycle of a xed local cohomology module H n i (R) describe the modules and the Betti numbers of the (n i)-linear strand of. Namely, itdescribes the subpieces F n i j := R( ) j 1;( ) of the resolution F. =n i+j 1 f R= is Cohen Macaulay then there is only one nonvanishing local cohomology module (see [1, Proposition 3.1]), so by the previous remark, we recover the following fundamental result of Eagon and Reiner [8]. Corollary 3.5. Let R be the Alexander dual ideal of a squarefree monomial ideal R. Then, R= is Cohen Macaulay if and only if has a linear free resolution. A generalization of that result expressed in terms of the projective dimension of R= and the Castelnuovo Mumford regularity of is given by Terai in [18]. We can also give a dierent approach by using the previous results. Corollary 3.6. Let R be the Alexander dual ideal of a squarefree monomial ideal R. Then we have pd(r=) = reg( ): Proof. By using Proposition 3.3 we have reg( ):=max{ j j; ( ) 0} = max{ j m n +j; (R=) 0}: Then, by Alvarez Montaner [1, Corollary 3.13] we getthe desired resultsince pd(r=)=cd(r; ) = max{ j m n +j; (R=) 0}; where the rst assertion comes from [11].

7 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) Arithmetical properties n [1] we reduce the Cohen Macaulay property of the quotient ring R= to a question on the vanishing of the local cohomology modules H n i (R). Then, by using Theorem 3.2, we give new characterizations of this arithmetical property in terms of the face ideals in the minimal primary decomposition of the monomial ideal. The characterization of the Gorenstein property of R= is more involved and we pointoutthatapartfrom the results given in Section 3.1, we will also use the fact that the local cohomology modules H n i (R) supported on squarefree monomial ideals have a natural Z n -graded structure. Our main result in this section is the following: Proposition 3.7. Let R be an ideal generated by squarefree monomials and = 1 m be its minimal primary decomposition. Then, the following are equivalent: (i) R= is Gorenstein such that x i is a zero divisor in R= for all i. (ii) R= is Cohen Macaulay and m n ht ; =1 for all j, j =1;:::;m. Remark 3.8. The condition that x i is a zero divisor in R= for all i means that any variable x i divides a generator of. Proof. R= is Gorenstein if and only if it is isomorphic to a shifting of its canonical module Ext ht R (R=; R( 1)). Since both modules have a natural structure of squarefree module (see [21] for details), it will be enough to study the graded pieces of these modules in degrees {0; 1} n and the morphism of multiplication by the variables x i between these pieces. Namely, let 1 ;:::; n be the natural basis of Z n. Then, we only have to check when the following diagram is commutative for all i: x (R=) i (R=)+i = = Ext ht R x (R=; R( 1)) i Ext ht R (R=; R( 1)) +i First, we have to point out that R= is Cohen Macaulay if and only if the modules Ext j R (R=; R( 1)) vanish for all j ht. More precisely, by Corollary 3.5, R= is Cohen Macaulay if and only if j; ( )=0 for all j ht, where j; ( ) denotes the jth Betti number of the Alexander dual ideal. On the other side, by Mustata [13, Corollary 3.1] we have dim k Ext j R (R=; R( 1)) 1 = dim k Ext j R (R=; R) = j; ( ) so, in order to compute the dimension of the graded pieces of Ext ht R (R=; R( 1)) we have to describe the Betti numbers ht ; ( ). Now, we are ready to check when the vertical arrows in the diagram are isomorphisms. First, note that the dimensions of the pieces of R= are dim k (R=) = 1 for all 6 1 j ; j =1;:::;m:

8 8 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 Then, by using the results of Section 3.1, we will see that the following conditions are equivalent: (i) dim k Ext ht R (R=; R( 1)) = 1 for all 6 1 j, j =1;:::;m, (ii) m n ht ; = 1 for all j, j =1;:::;m. The proof follows from the fact that the Betti numbers that describe the dimension of the graded pieces are related to the multiplicities m j; of the characteristic cycle of the local cohomology modules H n i (R) by means of Proposition 3.3. Namely, we have ht ; ( )=m n ht ; : Once we have checked when the graded pieces of both modules coincide we have to study the morphism of multiplication by the variables x i between these pieces. The morphism of multiplication x i :(R=) (R=) +i between the graded pieces of the quotient ring R= is the identity if both pieces are dierent from zero and i =0. On the other side, a topological interpretation of the multiplication by the variables x i between the pieces of the canonical module Ext ht R (R=; R( 1)) has been given in [13]. Namely, let be the full simplicial complex whose vertices are labelled by the minimal system of generators {x 1 ;:::;x s } of. Let T := { 1 := {x 1 ;:::;x s }\{x i i =1} } be a simplicial subcomplex of (see also [4] for details). Then, the morphism of multiplication is determined by the relative simplicial cohomology morphism i : H r 2 (T +i ; k) H r 2 (T ; k); induced by the inclusion T T +i. Namely, we have the following commutative diagram: H r 2 (T +i ; k) i H r 2 (T ; k) = = Ext ht R (R=; R( 1)) x i Ext ht R (R=; R( 1)) +i n the case we are considering, it is easy to check that this morphism is the identity if both pieces are dierent from zero and i =0. Remark 3.9. The isomorphism between the Z n -graded pieces of the quotient ring R= and the canonical module Ext ht R (R=; R( 1)) states that the corresponding Z n -graded Hilbertseries coincide. Stanley [15] studied the Gorenstein property by using the coarser Z-graded Hilbert series. n particular, he gave the following example of a nongorenstein ring R= having the same Hilbert series as its canonical module: Example Let R = k[x 1 ;x 2 ;x 3 ;x 4 ]. Consider the ideal =(x 1 ;x 2 ) (x 1 ;x 4 ) (x 2 ;x 4 ) (x 3 ;x 4 ):

9 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) The multiplicities of the characteristic cycle of the local cohomology module H 2 (R) allow us to compute the Z n -graded Hilbertseries of the canonical module by means of [3, Proposition 2.1]; [22, Proposition 2.8]. By using Theorem 3.2 we have CC(H 2 (R)) = T X (1; 1; 0; 0) X + T X (1; 0; 0; 1) X + T X (0; 1; 0; 1) X + T X (0; 0; 1; 1) X +2T X (1; 1; 0; 1) X + T X (1; 0; 1; 1) X + T X (0; 1; 1; 1) X + T X (1; 1; 1; 1) X: Notice that the corresponding multiplicities do not satisfy m 2; =1 for all bigger than the subindices corresponding to the face ideals in the minimal primary decomposition of so the Z n -graded Hilbert series of the quotient ring and the canonical module are dierent. Nevertheless, even though the quotient ring R= is not Gorenstein, the Z-graded Hilbert series of the quotient ring and the canonical module coincide (see [15]). Recall that the type of a Cohen Macaulay quotient ring R= of dimension d is the Bass number r(r=) := d (m; R=): By duality (see [9, Corollary 21.16]), it is the nonzero total Betti number of highest homological degree of the ring R=. We will give a description of the type by using the results that appear in Section 3.1. For this purpose let P (resp. P ) be the poset of nonpaired sums of face ideals obtained from the poset (resp. ) associated to the ideal (resp. ). n the sequel, CC(H n i (R)) = m i; (R= )TX X will be the characteristic cycle of a local cohomology module H n i (R), where is the Alexander dual ideal of a squarefree monomial ideal R. Proposition Let R be a squarefree monomial ideal of height ht = h, such that the quotient ring R= is Cohen Macaulay. Then, the type of R= is r(r=)= m n +h 1; (R= ): { p Ph } Proof. The projective dimension of R= is h =ht so we have r(r=)= h (R=). This total Betti number is obtained as a sum of the graded Betti numbers h; (R=) labelled by the set Ph so we have r(r=) := d (m; R=)= h; (R=)= h 1; (): } { p Ph } { p P h By Proposition 3.3 we get the desired result. A Cohen Macaulay ring R= of type 1 is Gorenstein. By using the previous description, we can give another criterion for the Gorenstein property in terms of the Alexander dual ideal this time:

10 10 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 Proposition Let R be a squarefree monomial ideal of height ht = h such that the quotient ring R= is Cohen Macaulay. Then, the following are equivalent: (i) R= is Gorenstein. (ii) { p P h } m n +h 1;(R= )=1. 4. The characteristic cycle of Hp(H p n i (R)) Let R be a squarefree monomial ideal. n [1], we computed the characteristic cycle of Hm(H p r(r)), where m =(x 1;:::;x n ) is the homogeneous maximal ideal, in order to describe the Lyubeznik numbers p;i (R=) dened as the Bass numbers p (m;h n i (R)) := dim k Ext p n i R (k; H (R)). Our aim in this section is to compute the characteristic cycle of Hp p (H r (R)), where p is any face ideal. The techniques we will use throughout this chapter are a natural continuation of those used in [1]. Namely, we will apply the long exact sequence of local cohomology to the sequences we have obtained in the process of computing CC(H n i (R)). Roughly speaking, we will divide the local cohomology module Hp p (H r(r)) into smaller pieces, the modules H p p (H r i1 + + ij (R)) that are labelled by the poset P occurring in Theorem 3.2. To shed some light to the situation, we present the case with two face ideals in the minimal primary decomposition of. First, we have to point out that in the case where there is only one nonvanishing local cohomology module, the spectral sequence E p;r 2 = Hp p (H r (R)) H p+r p (R) + degenerates at the E 2 -term so we have Hp p (H r p+r (R)) = Hp (R). + The case m=2: Let = 1 2 be the minimal primary decomposition of a squarefree monomial ideal. Denote h i := ht i i=1; 2 and h 12 := ht( ) and suppose h 1 6 h 2. Then we getthe following cases: (1) f h 1 h 2 h 12 1 then (H h1 (R))) = CC(H p+h1 p + 1 (R)); (R))) = CC(H p+h2 p + 2 (R)) and (H h12 1 (R))) = CC(H p+h12 p (R)): (2) f h 1 = h 2 h 12 1 then (H h1 (R))) = CC(H p+h1 p + 1 (R)) CC(H p+h2 p + 2 (R)) and (H h12 1 (R))) = CC(H p+h12 p (R)):

11 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) (3) f h 1 h 2 = h 12 1 then (H h1 (R))) = CC(H p+h1 p + 1 (R)) and from the exact sequence Hp p 2 (R)) Hp p (R)) Hp p (H h (R)) ; we have to determine the characteristic cycle of Hp p (R)). Let Z p 1 be the kernel of the morphism Hp p 2 (R)) Hp p (R)): Since CC(Z p 1 ) 1 (H h (R))); CC(Z p 1 ) CC(H p p 2 (R))); we have to study the equality between the initial pieces Hp p 1 (H h (R)) and Hp p 2 (R)) in order to compute these modules. Notice that this is equivalent to study the equality between the sums of face ideals p and p + 2.We have to consider the following cases: (i) p + 2 p : Letht(p + 2 )=p + h 2. n this case ht(p )=p + h 12 so we getthe shortexactsequence 0 Hp p 2 (R)) Hp p (R)) Hp p (H h (R)) 0: By the additivity of the characteristic cycle: (R)) = CC(H p+h2 p + 2 (R)) + CC(H p+h12 p (R)): (ii) p + 2 =p : Letht(p + 2 )=p+h 2. We have an exactsequence 0 H p 1 p H p p (R)) 0: (R)) Hp p 1 (H h (R)) Hp p 2 (R)) We will prove that the local cohomology modules Hp p (R)) vanish for all (H h (R)) 0. p, sothat Z p 1 = H p 1 p Let x i1 1 be an independent variable such that we have the equality = 2 +(x i1 ). Since p + 2 =p we have x i1 p.soletp =(x i1 ;x i2 ;:::;x is ) be the face ideal. Firstwe will compute H p (x i1 ) h2 (H (R)) by using the spectral sequence E p;r 2 = H p (x i1 ) (H r (R)) H p+r +(x i1 ) (R): The spectral sequence degenerates at the E 2 -term because we have H p h2 (x i1 )(H (R))=0 p 2. (H h1 (R))=0 p 1, due to the fact that H h1 H p (x i1 ) As a consequence we have H p h2 (x i1 )(H (R)) = H p+h2 p+h2 +(x i1 )(R)=H 1 (R)=0 p because +(x i1 )= 1 and ht 1 h 2. (R) = H h1 1 (R) and x i1 1.

12 12 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 Now by using Brodmann s long exactsequence H p (x i1 ;x i2 ) h2 (H we get H p (x i1 ;x i2 ) p we get Hp p h2 (H (R)) H p (x i1 ) h2 (H (R)) (H p h2 (x i1 )(H (R))) xi2 ; (R))=0 p, and iterating this procedure on the generators of (R))=0 p. (4) f h 1 = h 2 = h 12 1, on use of the Mayer Vietoris sequence and as H h (R)= H h12 1 (R)=H h12 2 (R) = 0, the module H h1 (R) is the only nonvanishing local cohomology module. So, we have (H h1 (R))) = CC(H p+h1 p (R)): + This last characteristic cycle can be computed by using the results of Section 3. Namely: (i) f the primary decomposition p + =(p + 1 ) (p + 2 ) is minimal then (H h1 (R))) = CC(H p+h1 p + 1 (R)) + CC(H p+h2 p + 2 (R))) + CC(H p+h12 p (R))): n particular, the corresponding modules Z p vanish for all p. (ii) f the primary decomposition p + =(p + 1 ) (p + 2 ) is notminimal then we have p + = p + 1 or p + = p + 2 so or (H h1 (R))) = CC(H p+h1 p + 1 (R)); (H h1 (R))) = CC(H p+h2 p + 2 (R))); n particular Z p = H p+h12 p (R)) 0 for p =ht(p ) h 12. Remark 4.1. From the previous example, we can observe that the computation of the characteristic cycle of Hp p (H r(r)) is more involved than in the case where p = m is the homogeneous maximal ideal considered in [1]. The diculty lies in the fact that when we add the face ideal p to dierent sums of face ideals in the poset P we can obtain dierent face ideals. This contrasts with the case considered in [1] where we always obtain the homogeneous maximal ideal m when we add m to any sum in P. The general case: Let = 1 m be the minimal primary decomposition of a squarefree monomial ideal R and p R a face ideal. To compute (H r(r))) we consider the sums of face ideals in the poset P. For any face ideal R we dene the subsets P ;j; := { i1 + + ij P j p + i1 + + ij = }; and we introduce the following:

13 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) Denition 4.2. We say that i1 + + ij P ;j; and i1 + + ij + ij+1 P ;j+1; are almostpaired if ht( i1 + + ij )+1=ht i1 + + ij + ij+1. To getthe formula in Theorem 4.3 below, we will consider the poset of nonpaired and nonalmost paired sums of face ideals in the minimal primary decomposition of. This set is obtained by means of the algorithm introduced in Section 3, butnow we consider. nput: The sets P ; of all nonpaired sums of face ideals obtained as a sum of face ideals in the minimal primary decomposition of which give the face ideal when added to the ideal p. We apply the algorithm where COMPARE means remove both ideals in case they are almostpaired. Output: The sets Q ; of all nonpaired and nonalmostpaired sums of face ideals obtained as a sum of face ideals in the minimal primary decomposition of which give the face ideal when added to the ideal p. We order Q ; = {Q ;1; ; Q ;2; ;:::;Q ;m; } by the number of summands, in such a way that no sum in Q ;j; is almostpaired with a sum in Q ;j+1;. Observe that some of the sets Q ;j+1; can be empty. Finally, we dene the sets of nonpaired and nonalmost paired sums of face ideals with a given height Q ;j;r; := { i1 + + ij Q ;j; ht( i1 + + ij )=r +(j 1)}: The formula we will give in Theorem 4.3 is stated in terms of these sets of nonpaired and nonalmostpaired sums of face ideals. Theorem 4.3. Let = 1 m be the minimal primary decomposition of a squarefree monomial ideal R and p R a face ideal. Then (H r (R))) = ;p;n r; CC(H (R)); {0;1} n where ;p;n r; = ]Q ;j;r; such that = p +(r +(j 1)). The following remark will be very useful for the proof of the theorem. Remark 4.4. From the formula it is easy to see the following: f (H r+(j 1) i1 + + ij (R))) (H r (R))), then (H r+(j 1) i1 + + ij (R))) CC(Hp q (H r (R))); q p: Proof. We are going to use similar ideas as in the proof of [1, Theorem 4.4]. We proceed by induction on m, the number of ideals in the minimal primary decomposition. The case m = 1 is trivial. So, let m 1. We start with the Mayer Vietoris sequence HU+V r (R) HU r (R) HV r (R) HU V r (R) HU+V r+1 (R) ;

14 14 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 where U = 1 m 1 ; U V = = 1 m ; V = m ; U + V =( 1 m 1 )+ m : First, we split the Mayer Vietoris sequence into short exact sequences of kernels and cokernels: 0 B r H r U (R) H r V (R) C r 0; 0 C r H r U V (R) A r+1 0; 0 A r+1 HU+V r+1 (R) B r+1 0: Applying the long exact sequence of local cohomology we get Hp p (C r ) Hp p (HU V r (R)) Hp p (A r+1 ) Hp p+1 (C r ) : Assume we have proved the formula for ideals with less than m terms in the minimal primary decomposition. This allows to compute the characteristic cycles of Hp p (C r ) and Hp p (A r+1 ). To describe (C r )) we will denote by P(C), the set of face ideals that give the initial pieces which allow us to compute the characteristic cycle of C r. Applying the algorithm of cancellation to P(C) we obtain the poset Q(C) :={Q 1 (C);:::;Q m 1 (C)}. Ordering the ideals by heights and checking the sums with the ideal p, we obtain the sets Q ;j;r; (C) :={ i1 + + ij Q ;j; (C) ht( i1 + + ij )=r +(j 1)}: By induction we have (C r )) = ;p;n r; (C) CC(H (R)); {0;1} n where ;p;n r; (C)=]Q ;j;r; (C) such that = p +(r +(j 1)). To describe CC(H p p (A r+1 )), we will denote by P(A) the set of face ideals that give the initial pieces which allow us to compute the characteristic cycle of A r+1. Applying the algorithm of cancellation to P(A) we obtain the poset Q(A) :={Q 1 (A);:::; Q m 1 (A)}. Ordering the ideals by heights and checking the sums with the ideal p,we obtain the sets Q ;j;r+1; (A):={ i1 + + ij + m Q ;j; (A) ht( i1 + + ij + m )=r +1+(j 1)}: By induction we have (A r+1 )) = ;p;n (r+1); (A) CC(H (R)); {0;1} n where ;p;n (r+1); (A)=]Q ;j;r+1; (A) such that = p +(r +1+(j 1)).

15 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) Now we splitthe long exactsequence into shortexactsequences of kernels and cokernels: 0 Z p 1 H p p (C r ) X p 0; 0 X p H p p (H r U V (R)) Y p 0; 0 Y p H p p (A r+1 ) Z p 0: So, by additivity we have: CC(H p p (H r U V (R))) = CC(X p ) + CC(Y p ) = (CC(H p p (C r )) CC(Z p 1 )) + (CC(H p p (A r+1 )) CC(Z p )): To get the desired formula, it remains to cancel the possible almost pairs formed by ideals i1 + + ij Q ;j;r; (C) coming from the computation of (C r )) and ideals i1 + + ij + m Q ;j;r+1; (A) coming from the computation of (A r+1 )). t also remains to cancel the possible almost pair formed by the ideal m Q ;1;r; (C) coming from the computation of the cycles (C r )) and some i + m Q ;1;r+1; (A), coming from the computation of the cycles (A r+1 )). So, we only have to prove that the initial pieces coming from these almost paired ideals describe the cycles CC(Z p ) p. This will be done by means of the following: Claim. CC(Z p 1 )= CC(H (R)); where the sum is taken over the cycles that come from almost pairs of the form i1 + + ij Q ;j;r; (C) and i1 + + ij + m Q ;j;r+1; (A), with = p + i1 + + ij = p + i1 + + ij + m, and = p +(r +(j 1)). m Q 1;r; (C) and i1 + m Q 1;r+1; (A), with = p + m = p + i1 + m, and = p + r. The inclusion is obvious because CC(Z p 1 ) belongs to 1 (A r+1 )) and (C r )). To prove the opposite inclusion, let TX X = CC(H (R)) be a cycle which comes from an almostpair and suppose thatthis cycle is notcontained in CC(Z p 1 ). f T X X = CC(H p p (H r+(j 1) i1 + + ij (R))) = 1 (C r )) and 1 (A r+1 )) we set (H r+1+(j 1) U = i1 + + ij ; U V =( i1 + + ij ) m ; V = m ; U + V = i1 + + ij + m : i1 + + ij + m (R))) is contained in

16 16 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 f TX X = (H r m (R))) = 1 (H r+1 i1 + m (R))) is contained in (C r )) and 1 (A r+1 )) we set U = i ; U V = i m ; V = m ; U + V = i + m : Consider the Mayer Vietoris sequence HU r +V (R) H U r (R) H V r (R) H U r r+1 V (R) HU +V (R) : Notice that in the corresponding short exact sequences we have 0 Z p 1 H p p (C r) X p 0; 0 X p H p p (H r U V (R)) Y p 0; 0 Y p H p p (A r+1) Z p 0 T X X CC(H p p (C r)); TX X 1 (A r+1)); T X X CC(Z p 1): The last statement is due to the fact that, if the cycle TX X has notbeen cancelled during the computation of Hp p (H r(r)), i.e. itdoes notbelong to CC(Z p) then, it can not be cancelled in the computation of Hp p (HU r V (R)), i.e. itdoes notbelong to any CC(Z p) of the corresponding shortexactsequences. Finally, by induction and using Remark 4.4 we get a contradiction as T X X CC(H p 1 p (H r U V (R))) and T X X CC(H p p (H r U V (R))): Applying Theorem 4.3 to the case of the homogeneous maximal ideal m R we obtain the result of [1, Theorem 4.4], as for any sum of face ideals i1 + + ij P j;r we have m = m + i1 + + ij so that P j;r = P m; j;r; m j, r. 5. Bass numbers of local cohomology modules Let (H n i (R))) = ;p;i; TX X; be the characteristic cycle of the local cohomology module Hp p (H n i (R)). t provides much information on the modules Hp p (H n i (R)) as well on the ring R= because the multiplicities ;p;i; are invariants of R= (see [2]). Nevertheless, the aim of this section is to extract information on the modules H n i (R). Namely, we want to compute the Bass numbers of these latter modules.

17 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) First, we recall that the multiplicities ;p;i; (R=) of the characteristic cycle of Hp p (H n i (R)) are described in terms of the sets Q ;j;n i; of nonpaired and nonalmostpaired sums of j face ideals obtained by means of the algorithm. Namely we have: Proposition 5.1. Let R be a squarefree monomial ideal. Let Q be the poset of sums of face ideals obtained from the poset P by means of the algorithm of cancelling almost paired ideals. Then: ;p;i; (R=)=]Q ; +1 p (n i);n i; : This description gives an eective method to compute the Bass numbers of the local cohomology modules H n i (R) with respect to a face ideal p by means of [2, Proposition 2.1]. For completeness, we recall the precise statement. Proposition 5.2 ( Alvarez Montaner [2]). Let R be an ideal generated by squarefree monomials and let p R be a face ideal. f (H n i (R))) = ;p;i; TX X is the characteristic cycle of the local cohomology module Hp p (H n i (R)), then p (p ;H n i (R)) = ;p;i; : n order to compute the Bass numbers p (p;h n i (R)) of the local cohomology modules H n i (R) with respect to any prime ideal p R, one may use the following remark: Remark 5.3. For any prime ideal p R, let p denote the largest face ideal contained in p. Let d be the Krull dimension of R p =p R p. Then, by Goto and Watanabe [10, Theorem 1.2.3], we have p (p;h n i (R)) = p d (p ;H n i (R)); p: 5.1. njective dimension of local cohomology modules The rst goal of this section is to give a vanishing criterion for the Bass numbers p (p ;H n i (R)) of the local cohomology modules H n i (R). We remark that this criterion will be described in terms of the face ideals in the minimal primary decomposition of the monomial ideal. Proposition 5.4. Let R be an ideal generated by squarefree monomials and let p R be a face ideal. The following are equivalent: (i) p (p ;H n i (R)) 0.

18 18 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 (ii) TX X (H n i (R))). (iii) There exists i1 + + ij Q ;j; such that ht( i1 + + ij )=n i+(j 1)= p. Proof. By Proposition 5.2 the Bass number p (p ;H n i (R)) does notvanish if and only if the corresponding face ideal p R belongs to the support of the module Hp p (H n i (R)). Then, we conclude by Proposition 5.1. As a consequence, we can compute the graded injective dimension of the local cohomology modules H n i (R). Corollary 5.5. Let R be an ideal generated by squarefree monomials. The graded injective dimension of H n i (R) is id H n i We also have id H n i (R) = max {p ;p;i; 0}: ;p (R) = max { ht( i1 + + ij ) i1 + + ij Q ;j; }: ;j Remark 5.6. The injective dimension id H n i (R) can be computed from the graded injective dimension id H n i (R) by means of Remark Associated prime ideals of local cohomology modules Bass numbers allow us to describe the associated primes of a R-module M. Namely, we have Ass R (M) :={p Spec R 0 (p;m) 0}: Recall that the characteristic cycle of the local cohomology modules H n i (R) describe the support of these modules (see Section 2). Now, as a consequence of Proposition 5.4, we will distinguish the associated prime ideals among all prime ideals in the supportof H n i (R). Proposition 5.7. Let R be an ideal generated by squarefree monomials and let p R be a face ideal. The following are equivalent: (i) p Ass R H n i (R). (ii) TX X CC(Hp 0 (H n i (R))). (iii) There exists i1 + + ij Q ;j; such that ht( i1 + + ij )=n i+(j 1)=. The minimal primes in the support of any R-module are associated primes, i.e. the corresponding 0th Bass number does not vanish. As a consequence of Proposition 5.2, we can compute explicitly all the Bass numbers of local cohomology modules with respect to their minimal primes.

19 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) Proposition 5.8. Let R be an ideal generated by squarefree monomials and p R be a face ideal. f p is a minimal prime in the support of the local cohomology module H n i (R) then 0 (p ;H n i (R)) 0; and p (p ;H n i (R))=0 for all p 0: Moreover, if CC(H n i (R)) = m i; TX X is the corresponding characteristic cycle, then: 0 (p ;H n i (R)) = m i; : The minimal primes of the local cohomology modules are easy to describe by the characteristic cycle. We have to point out that not all the associated primes are minimal as it can be seen in the following example where we prove the existence of embedded prime ideals. Example 5.9. Let R = k[x 1 ;x 2 ;x 3 ;x 4 ;x 5 ]. Consider the ideal =(x 1 ;x 2 ;x 5 ) (x 3 ;x 4 ;x 5 ) (x 1 ;x 2 ;x 3 ;x 4 ). By using Theorem 3.2 we compute the characteristic cycle of the corresponding local cohomology modules. Namely, we get CC(H 2 (R)) = TX (1; 1; 0; 0; 1) X + TX (0; 0; 1; 1; 1) X; CC(H 3 (R)) = TX (1; 1; 1; 1; 0) X +2TX (1; 1; 1; 1; 1) X: Collecting the multiplicities by the dimension of the corresponding varieties we obtain the following triangular matrix (see [1] for details) (R=)= 1 0 : 2 f we compute the Bass numbers with respect to the maximal ideal, i.e. the Lyubeznik numbers, we obtain the triangular matrix (see [20] for details) (R=)= 0 0 : 2 Then For the module H 2 id H 2 (R) we have 2 (R) = dim H (R)=2. (R)). Min R (H 2 (R)) = Ass R(H 2

20 20 J. A. Montaner / Journal of Pure and Applied Algebra 192 (2004) 1 20 For the module H 3 0= id H 3 (R) we have 3 (R) dim H (R)=1. (R)). Min R (H 3 (R)) = Ass R (H 3 More precisely, H 3 (R) = E(R=(x 1 ;x 2 ;x 3 ;x 4 )) E(R=(x 1 ;x 2 ;x 3 ;x 4 ;x 5 )) so, the maximal ideal m =(x 1 ;x 2 ;x 3 ;x 4 ;x 5 ) is an embedded associated prime ideal of this local cohomology module. References [1] J. Alvarez Montaner, Characteristic cycles of local cohomology modules of monomial ideals, J. Pure Appl. Algebra 150 (2000) [2] J. Alvarez Montaner, Some numerical invariants of local rings, Proc. Amer. Math. Soc. 132 (4) (2004) [3] J. Alvarez Montaner, R. Garca Lopez, S. Zarzuela, Local cohomology, arrangements of subspaces and monomial ideals, Adv. Math. 174 (2003) [4] J. Alvarez Montaner, S. Zarzuela, Linearization of local cohomology modules, in: L. Avramov, M. Chardin, M. Morales, C. Polini (Eds.), Commutative Algebra: nteractions with Algebraic Geometry, Contemporary Mathematics, Vol. 331, American Mathematical Society, Providence, R, 2003, pp [5] J.E. Bjork, Rings of Dierential Operators, North-Holland Mathematics Library, Amsterdam, [6] A. Borel, Algebraic D-modules, Perspectives in Mathematics, Academic Press, New York, [7] S.C. Coutinho, A Primer of Algebraic D-Modules, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, [8] J.A. Eagon, V. Reiner, Resolutions of Stanley Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (3) (1998) [9] D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, in: Graduate Text in Mathematics, Vol. 150, Springer, Berlin, [10] S. Goto, K. Watanabe, On graded rings, (Z n - graded rings), Tokyo J. Math. 1 (2) (1978) [11] G. Lyubeznik, On the local cohomology modules HA i (R) for ideals A generated by monomials in an R-sequence, in: S. Greco, R. Strano (Eds.), Complete ntersections, Lecture Notes in Mathematics, Vol. 1092, Springer, Berlin, 1984, pp [12] G. Lyubeznik, Finiteness properties of local cohomology modules, nvent. Math. 113 (1993) [13] M. Mustata, Local cohomology atmonomial ideals, J. Symbolic Comput. 29 (2000) [14] F. Pham, Singularites des systemes dierentiels de Gauss Manin, in: Progress in Mathematics, Vol. 2, Birkhauser, Basel, [15] R.P. Stanley, Hilbert functions of graded algebras, Adv. Math. 28 (1978) [16] R.P. Stanley, Combinatorics and Commutative Algebra, 2nd Edition, in: Progress in Mathematics, Vol. 41, Birkhauser, Basel, [17] D. Taylor, deals genreated by monomials in an R-sequence, Ph.D. Thesis, University of Chicago, [18] N. Terai, Generalization of Eagon Reiner theorem and h-vectors of graded rings, preprint, [19] U. Walther, Algorithmic computation of local cohomology modules and the cohomological dimension of algebraic varieties, J. Pure Appl. Algebra 139 (1999) [20] U. Walther, On the Lyubeznik numbers of a local ring, Proc. Amer. Math. Soc. 129 (6) (2001) [21] K. Yanagawa, Alexander duality for Stanley Reisner rings and squarefree N-graded modules, J. Algebra 225 (2000) [22] K. Yanagawa, Bass numbers of local cohomology modules with supports in monomial ideals, Math. Proc. Cambridge Philos. Soc. 131 (2001)