Tameness of Local Cohomology of Monomial Ideals with Respect to Monomial Prime Ideals

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame. Introduction Let R be a graded ring. Recall that a graded R-module N is tame, if there exists an integer j0 such that Nj = 0 for all j ≤ j0, or else Nj 6= 0 for all j ≤ j0. Brodmann and Hellus [4] raised the question whether for a finitely generated, positively graded algebra R with R0 Noetherian, the local cohomology modules H i R+ (M) for a finitely generated graded R-moduleM are all tame. Here R+ = ⊕ i>0Ri is graded irrelevant ideal of R. See [1] for a survey on this problem. In this paper we only consider rings defined by monomial relations. We first consider the squarefree case, since from a combinatorial point of view this is the more interesting case and also since in this case the formula which we obtain are more simple. So let K be a field and ∆ a simplicial complex on V = {v1, . . . , vn}. In Section 1, as a generalization of Hochster’s formula [3], we compute (Theorem 1.3) the Hilbert series of local cohomology of the Stanley-Reisner ring K[∆] with respect to a monomial prime ideal. With the choice of the monomial prime ideal, the Stanley-Reisner ring K[∆], and hence also all the local cohomology of it, can be given a natural bigraded structure. In Proposition 1.7 we give a formula for the Kdimension of the bigraded components of the local cohomology modules. Using this formula we deduce that the local cohomology of K[∆] with respect to a monomial prime ideal is always tame. In [6] Takayama generalized Hochster’s formula to any graded monomial ideal which is not necessarily squarefree. In Section 2, as a generalization of Takayama’s result, we compute the Hilbert series of local cohomology of monomial ideals with respect to monomial prime ideals and observe that again all these modules are tame. The result proved here is surprising because in a recent paper, Cutkosky and Herzog [5] gave an example which shows that in general not all local cohomology modules are tame. 1. Local cohomology of Stanley-Reisner rings with respect to monomial prime ideals Let K be a field and let S = K[Y1, . . . , Yr] be a polynomial ring with the standard grading. For a squarefree monomial ideal I ⊂ S we set R = S/I. We denote by 1 yi the residue classes of indeterminates Yi in R for i = 1, . . . , n. Thus we have R = K[y1, . . . , yr]. We may view R as the Stanley-Reisner ring of some simplicial complex ∆ with vertices {w1, . . . , wr}. Let P be any monomial prime ideal ofR. We may assume that P = (y1, . . . , yn) for some integer n ≤ r. After this choice of P we view R as a bigraded K-algebra. We rename some of the variables, and set xi = yn+i for i = 1, . . . , m wherem = r−n, and assign the following bidegrees: deg xi = (1, 0) for i = 1, . . . , m and deg yj = (0, 1) for j = 1, . . . , n. We decompose the vertex set of the corresponding simplicial complex ∆ accordingly, so that ∆ has vertices {v1, . . . , vm, w1, . . . , wn} where vertices V = {v1, . . . , vm} and W = {w1, . . . , wn} correspond to the variables of x1, . . . , xm and y1, . . . , yn, respectively. By [1, Theorem 5.1.19] we have H i P (R) ∼= H i(C.) for all i ≥ 0, where C. is the Čech complex C. : 0 → C → C → · · · → C → 0 with C = ⊕ 1≤j1<··· 0} and the support of b is the set supp b = {wj : w1 ≤ wj ≤ wn, bj 6= 0}. Note that supp b = Gb ∪Hb. We set Na = {vi : v1 ≤ vi ≤ vm, ai 6= 0} = supp a for a ∈ Z and denote by Z+ and Z n − the sets of {a ∈ Z : ai ≥ 0 for i = 1, . . . , m} and {b ∈ Z : bi ≤ 0 for i = 1, . . . , n}, respectively. With the notation introduced one has Lemma 1.1. The following statements hold: (a) dimK(Ry)(a,b) ≤ 1, for all a ∈ Z and b ∈ Z. (b) (Ry)(a,b) ∼= K, if and only if F ⊃ Gb, F ∪Hb ∪Na ∈ ∆ and a ∈ Z+ . Proof. As explained before, we may view the standard graded polynomial ring S as a standard bigraded polynomial ring and then R = K[x1, . . . , xm, y1, . . . , yn] with m + n = r is also naturally bigraded. Thus part (a) follows from [3, Lemma 5.3.6 (a)]. For the proof (b) we set c = (a, b). By [3, Lemma 5.3.6 (b)] we have F ⊃ Gc and F ∪Hc ∈ ∆. Thus (1) implies that a ∈ Z+ and hence Gc = Gb. We also note that Hc = Hb ∪Na. 2 As a consequence of Lemma 1.1 for a ∈ Z+ , b ∈ Z and i ∈ Z we observe that (C)(a,b) has the following Kbasis: {bF : F ⊃ Gb, F ∪Hb ∪Na ∈ ∆, |F | = i}. Therefore, since C. is Z × Z-bigraded complex one obtains for each (a, b) ∈ Z × Z a complex (C)(a,b) : 0 → (C)(a,b) → (C)(a,b) → · · · → (C)(a,b) → 0, of finite dimensional K-vector spaces (C)(a,b) = ⊕ F⊃Gb F∪Hb∪Na∈∆ |F |=i KbF . The differential ∂ : (C)(a,b) −→ (C)(a,b) is given by ∂(bF ) = ∑ (−1)∂(F,F ′bF ′ where the sum is taken over all F ′ such that F ′ ⊃ F , F ∪Hb∪Na ∈ ∆ and |F ′| = i+1, and where ∂(F, F ) = s for F ′ = [w0, . . . , wi] and F = [w0, . . . , ŵs, . . . , wi]. Then we describe the (a, b)th component of the local cohomology in terms of this subcomplex: H i P (K[∆])(a,b) = H (C)(a,b) = H i(C. (a,b)). (2) Let ∆ be a simplicial complex with vertex set V and C̃(∆) the augumented oriented chain complex of ∆, see [3, Section 5.3] for details. For an abelian group G, the ith reduced simplicial cohomology of ∆ with values in G is defined to be H̃ (∆;G) = H (HomZ(C̃(∆), G)) for all i. (3) Given F ⊆ V , we recall the following definitions : The star of F is the set st∆F = {G ∈ ∆ : F ∪G ∈ ∆}, and the link of F is the set lk∆ F = {G : F ∪G ∈ ∆, F ∩G = ∅}. We write st and lk instead of st∆ and lk∆ (for short). We see that stF is a subcomplex of ∆, lkF a subcomplex of stF , and that stF = lkF = ∅ if F / ∈ ∆. For W ⊆ V , we denote by ∆W the simplicial complex restricted to W . i.e. the simplicial complex consisting of all faces F ∈ ∆ whose vertices belong to W . Now in order to compute H i(C. (a,b)), we prove the following Lemma 1.2. For all a ∈ Z+ and b ∈ Z there exists an isomorphism of complexes (C)(a,b) −→ HomZ(C̃(lkstHb Gb ∪Na)W [−j − 1];K), j = |Gb| Proof. The assignment F 7→ F ′ = F −Gb establishes a bijection between the set β = {F ∈ ∆W : F ⊃ Gb, F ∪Hb ∪Na ∈ ∆, |F | = i}. and the set β ′ = {F ′ ∈ ∆W : F ′ ∈ (lkstHb Gb ∪ Na)W , |F ′| = i − j}. Here F ′ ∈ (lkstHb Gb ∪Na)W , since F ′ ∩ (Gb ∪ Na) = ∅ and F ′ ∪ (Gb ∪ Na) ∈ stHb. Therefore we see that α : (C)(a,b) −→ HomZ(C̃(lkstHb Gb ∪Na)i−j−1;K), bF 7→ φF−Gb is an isomorphism of vector spaces. Here φF ′ is defined by φF ′(F ) = { 1 if F = F , 0 otherwise. 3 As a generalization of Hochster’s formula [3, Theorem 5.3.8] we prove the following Theorem 1.3. Let I ⊂ S = K[X1, . . . , Xm, Y1, . . . , Yn] be a squarefree monomial ideal with the natural Z × Z-bigrading. Then the bigraded Hilbert series of the local cohomology modules of R = S/I = K[∆] with respect to the Z ×Zn-bigrading is given by HHi P (K[∆])(s, t) = ∑