Monomial ideals with primary components given by powers of monomial prime ideals

We characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.


Introduction
In this paper we study monomial ideals which are intersections of powers of monomial prime ideals, and call them monomial ideals of intersection type. Any squarefree monomial ideal is of intersection type since it is actually an intersection of monomial prime ideals. Obviously, among the non-radical monomial ideals, the monomial ideals of intersection type are closest to squarefree monomial ideals. Hence one may expect that they are easier to understand than general monomial ideals. Indeed, various special types of such ideals have been studied in the literature, among them the so-called ideals of tetrahedral curves, see [21] and [11]. In [12] the authors discuss the problem when a monomial ideal of intersection type is componentwise linear. In that paper monomial ideals of intersection type are called intersections of Veronese type ideals.
In Section 1 of the present paper we deal with the problem to characterize those monomial ideals which are of intersection type. The answer is given in Theorem 1.1 where it is shown that a monomial ideal I ⊂ S = K[x 1 , . . . , x n ] is of intersection type if and only if for each associated prime ideal p of I the minimal degree of a generator of the monomial localization I(p) of I is bigger than the maximal degree of a non-zero socle element of S(p)/I(p). Moreover, it is shown that if I is of intersection type, its presentation as an intersection of powers of monomial prime ideals is uniquely determined. We call this presentation the canonical primary decomposition of I. One should notice that an ideal of intersection type, if it has embedded prime ideals, may have primary decompositions, different from the canonical one. The exponents of the powers of the monomial prime ideals appearing in the intersection are bounded above. Indeed, if I = p∈Ass(S/I) p dp , then d p ≤ reg(I(p)). We say that I is of strong intersection type, if d p = reg(I(p)) for all p ∈ Ass(S/I), and show that this is the case if and only if I(p) has a linear resolution for all p ∈ Ass(S/I). It is clear that any squarefree monomial ideal is of strong intersection type, while monomial ideals of intersection type with embedded prime ideals may or may not be of strong intersection type.
In Section 2 we consider classes of ideals which are of (strong) intersection type. It is shown in Proposition 2.1 that any polymatroidal ideal is of strong intersection type, and in Theorem 2.4 we show that the canonical primary decomposition of polymatroidal ideals is given in terms of the rank function of the underlying polymatroid. Unfortunately we do not have a uniform description of the associated prime ideals in terms of the rank function. However for polymatroidal ideals of Veronese type and for transversal polymatroidal ideals the canonical primary decomposition is made explicit, see Examples 2.5. On the other hand, a coherent primary decomposition of any polymatroidal ideal, which however may not always be irredundant, is given in Theorem 2.6. Among the ideals of Borel type, the principal Borel ideals are precisely those which are of strong intersection type, as shown in Proposition 2.8. In the remaining part of Section 2 we consider powers of edge ideals. In Theorem 2.12 we characterize, terms of their underlying graph, those edge ideals whose second power is of intersection type, and prove in Corollary 2.13 that if the square of an edge ideal is not of intersection type, then so are all its higher powers. It would be of interest to classify all graphs with the property that all its powers are of intersection type.
In the last section we list a few properties of ideals of intersection type. One nice property is that any ideal of intersection type is integrally closed, see Proposition 3.1. In Corollary 3.3 the supporting hyperplanes of the Newton polyhedron of an ideal of intersection type are given in terms of its canonical primary decomposition. Finally in Theorem 3.4 it is shown that if I k is of intersection type, then its ℓ th symbolic power I (ℓ) of I is contained in I k , where ℓ = reg(I k ). By using this fact and a result of Kodiyalam [19], and assuming that all powers of I are of intersection type, we show in the second part of Theorem 3.4 that I (rk) ⊂ I k for all k ≫ 0, where r = ρ(I) + 1 and where ρ(I) is the minimum of θ(J) over all graded reductions J of I. Here θ(J) denotes the maximal degree of a generator in a minimal set of generators of J.
1. When is a monomial ideal the intersection of powers of monomial prime ideals?
. , x n ] be the polynomial ring in n variables over the field K with graded maximal ideal m = (x 1 , . . . , x n ), and let I ⊂ S be a monomial ideal. In general, unless I is squarefree, the ideal I can not be written as intersection of powers of monomial prime ideals. In the following we give a necessary and sufficient condition for I to have such a presentation, and show that this presentation is unique once it is irredundant. To describe the main result of this section we introduce some notation. Let p be a monomial prime ideal, that is, p is a prime ideal generated by a subset of the variables. As in [18] we denote by I(p) the monomial localization of I, and by S(p) the polynomial ring over K in the variables which belong to p. Recall that I(p) ⊂ S(p) is the monomial ideal which is obtained from I as the image of the K-algebra homomorphism ϕ : and ϕ(x i ) = 1, otherwise. Monomial localizations are compatible with products and intersections. In other words, if I and J are monomial ideals, then (IJ)(p) = I(p)J(p) and (I ∩ J)(p) = I(p) ∩ J(p).
We denote by V * (I) the set of monomial prime ideals containing I. It is known that Ass(S/I) ⊂ V * (I) (see for example [14,Corollary 1.3.9]), and that p ∈ V * (I) belongs to Ass(S/I) if and only if depth(S(p)/I(p)) = 0. The latter is the case if and only if I(p) : m p = I(p) where m p denotes the graded maximal ideal of S(p). The K-vector space (I(p) : m p )/I(p) ⊂ S(p)/I(p) is called the socle of S(p)/I(p), and denoted by Soc(S(p)/I(p)). Thus p ∈ Ass(S/I) if and only if Soc(S(p)/I(p)) = 0. Since the colon ideal of monomial ideals is again a monomial ideal it follows that Soc(S(p)/I(p)) is generated by the elements u + I(p) where u ∈ S(p) is a monomial with u ∈ I(p) and m p u ⊂ I(p). Since Soc(S(p)/I(p)) is a finite dimensional graded K-vector space we may define the number max(Soc(S(p)/I(p))) = max{i : (Soc(S(p)/I(p))) i = 0} for any p ∈ Ass(S/I). We also set min(M) = min{i : M i = 0} for any finitely generated graded S(p)-module M. Finally we denote by I(p) the saturation of I(p) which is defined to be the monomial ideal k I(p) : m k p . In the sequel we will also use the following notation: for a graded ideal I and an integer k ≥ 0 we denote by I ≥k the ideal generated by all graded components I i of I with i ≥ k. Proof. (a) For the proof of (i) ⇒ (ii) let I = r i=1 p a i i . We may assume that this presentation is irredundant. Then Ass(S/I) = {p 1 , . . . , p r }. Fix some integer j.
It follows from this presentation that The proof of (ii) ⇒ (iii) follows immediately once that it is noticed that (ii) is equivalent to saying that I(p) = I(p) ≥ap for all p ∈ Ass(I). This identity implies that min(I(p)) ≥ a p , and that the maximal degree of a non-zero element u + I(p) ∈ Soc(S(p)/I(p)), u a monomial, is at most a p − 1 since u ∈ I(p) \ I(p).
(iii) ⇒ (i): For all p ∈ Ass(S/I) let d p = max(Soc(S(p)/I(p))) + 1. We claim that I = p∈Ass(S/I) p dp . In order to prove this, we first show that I ⊂ p∈Ass(S/I) p dp . Indeed, let p ∈ Ass(S/I) and suppose that I ⊂ p dp . Then I(p) = I(p) ≥dp . It follows that min(I(p)) < d p , a contradiction.
Next assume that I is properly contained in J = p∈Ass(S/I) p dp . Then there exists p ∈ Min(J/I) ⊂ Ass(S/I) such that J(p)/I(p) = (J/I)(p) = 0. Let u ∈ J(p) \ I(p). Then deg u ≥ d p because u ∈ m p . Since p ∈ Min(J/I), it follows that J(p)/I(p) has finite length. Thus max(Soc(S(p)/I(p))) ≥ deg u ≥ d p , a contradiction. It follows that max(Soc(S(p i )/I(p i ))) = d i − 1, as desired.
We call a monomial ideal satisfying the equivalent conditions of Theorem 1.1 to be a monomial ideal of intersection type. The presentation of an ideal of intersection type as given in Theorem 1.1(b) is called the canonical primary decomposition of I. Of course, a monomial ideal of intersection type, if it has embedded prime ideals, may have also other primary decompositions than just the canonical one. We apply our Theorem 1.1 to show that the ideal J 2 is not of intersection type. For this, note that m ∈ Ass(S/J 2 ) with xyztuv + J 2 a non-zero socle element of S/J 2 of degree 6. Since J 2 is generated in degree 6, Theorem 1.1(a) implies that J 2 is not of intersection type.
If the monomial ideal I is of intersection type, then the powers of the prime ideals in the intersection are naturally bounded. More precisely we have Theorem 1.3. Let I be a monomial ideal of intersection type with presentation I = p∈Ass(S/I) p dp . Then the following statements hold: Since I(p)/I(p) is a finite length module, [8,Corollary 20.19] combined with Theorem 1.1 implies ≤ reg(S(p)/I(p)) + 1 = reg(I(p)).
(b) By Theorem 1.1 we have I(p) = I(p) ≥dp . Thus if we assume that I(p) has a linear resolution it follows that reg(I(p)) = min(I(p)) = d p .
A monomial ideal satisfying the equivalent conditions of Theorem 1.3 is said to be of strong intersection type. Obviously, any squarefree monomial ideal is of strong intersection type.

Example 1.4. The ideal
of intersection type, and m = (x, y, z, t) ∈ Ass(S/I). But I is not generated in a single degree, let alone has a linear resolution. Therefore, I is not of strong intersection type.

Classes of monomial ideals of (strong) intersection type
In this section we consider classes of ideals which are of intersection type or strong intersection type.
Polymatroidal ideals. We fix a field K, and let P be a discrete polymatroid on the ground set [n] of rank d, see [15] where these concepts are explained. The polymatroidal ideal associated with P is the monomial ideal I ⊂ K[x 1 , . . . , x n ] generated by all monomials x u where u ∈ B(P) ⊂ N n . Here B(P) denotes the set of bases of P, namely the set of all u ∈ P whose modulus |u| = i u i is maximal. All bases u ∈ B(P) have the same modulus, namely d, which is defined to be the rank of P. In particular, I is generated in the single degree d.

Proposition 2.1. Any polymatroidal ideal is of strong intersection type.
Proof. We recall that a polymatroidal ideal has linear quotients [6, Theorem 5.2] and thus has a linear resolution. Moreover, I(p) is polymatroidal for any monomial prime ideal p which contains I (see [18,Corollary 3.2]). In particular, I(p) has a linear resolution for all p ∈ Ass(S/I). Thus Theorem 1.3 yields the desired conclusion.
In [1] it is conjectured that a monomial ideal is polymatroidal, if all its monomial localizations have a linear resolution. The following example shows that it does not suffice to require that I is of strong intersection type to conclude that I is polymatroidal, that is, to require that I(p) has a linear resolution for all p ∈ Ass(S/I).

Example 2.2.
Let J be the monomial ideal of Example 1.2, and let p ∈ V * (J 3 ). We claim that J 3 (p) has a linear resolution if and only if p ∈ Ass(S/J 3 ). Calculations with Singular [13] show that Ass(S/J 3 ) consists of all the prime ideals of height 3, 5 and 6 which belong to V * (J 3 ), altogether these are 17 prime ideals. Moreover we have the following irredundant primary decomposition of J 3 9 . We notice that m ∈ Ass(S/J 3 ), and by [3, Corollary 3.3] we know that J 3 (= J 3 (m)) has a linear resolution. Next consider any prime ideal p ∈ V * (J 3 ) of height 5. Then J 3 (p) is just the third power of the edge ideal of the cycle of length 5. For example, if p = (x, y, z, t, u) then By Singular one can check that also in this case J 3 (p) has linear resolution. Finally let p ∈ V * (J 3 ) be of height 3. Then p is a minimal prime ideal of J 3 . Hence J 3 (p) = m 3 p , and thus has a linear resolution. We conclude that J 3 (p) has a linear resolution for any associated prime ideal p ∈ Ass(S/J 3 ), and hence J 3 is an ideal of strong intersection type.
On the other hand, if J 3 would be a polymatroidal ideal, then J 3 (p) would be polymatroidal for any monomial prime ideal p ∈ V * (J 3 ), see [18,Corollary 3.2]. Any height 4 monomial prime ideal p belongs to V * (J 3 ) and contains exactly two minimal prime ideals associated to J 3 . It follows that J 3 (p) is no longer generated in a single degree and thus can not be polymatroidal. To exemplify this, let p = (x, y, z, t). Then J 3 (p) = (t, x, y) 3 ∩ (z, x, y) 3 is minimally generated in degrees 3, 4, 5 and 6.
Our discussion showed that Ass(S/J 3 ) is a proper subset of V * (J 3 ), and that J 3 (p) has a linear resolution if and only p ∈ Ass(S/J 3 ).
Next we want to describe the canonical primary decomposition of a polymatroidal ideal. Let P be a discrete polymatroid on the ground set [n], and denote by 2 [n] the set of all subsets of [n]. For a vector u ∈ Z n and F ∈ 2 [n] we set u(F ) = i∈F u i .
The ground set rank function of P is the function ρ : 2 [n] → Z + defined by setting together with ρ(∅) = 0.

Lemma 2.3. Let P be a discrete polymatroid of rank d on the ground set [n]
and I the polymatroidal ideal associated with P. Furthermore, let F ⊂ [n] and p F be the monomial prime ideal generated by the variables Proof. As observed before in the proof of Proposition 2.1, the ideal I(p F ) is polymatroidal for all p F which contain I. If I ⊂ p F , then I(p F ) = S(p F ) which by definition we may also consider as a polymatroidal ideal. The ideal I(p F ) is generated in a single degree, say t F . By the definition of monomial localization we have the desired formula follows.
The canonical primary decomposition of a polymatroidal ideal is now given as follows: Theorem 2.4. Let P be a discrete polymatroid of rank d with rank function ρ, and let I be the polymatroidal ideal associated with P. Then is the canonical primary decomposition of I.
Proof. Let p F ∈ Ass(S/I). Then reg I(p F ) is equal to the common degree of the generators of I(p F ), since I(p F ) has a linear resolution. By Lemma 2.3, this common degree is d − ρ([n] \ F ). Since I is of strong intersection type, the desired conclusion follows from Theorem 1.3.
In general it is not so easy to identify the associated prime ideals of a polymatroidal ideal. The complete answer is known for ideals of Veronese type as well as for polymatroidal ideals associated with transversal polymatroids. We describe these cases in the following examples. d, a 1 , . . . , a n be positive integers. The ideal I d;a 1 ,...,an generated by all monomials x c 1 1 · · · x cn n of degree d with c i ≤ a i for i = 1, . . . , n is called of Veronese type. Ideals of Veronese type are polymatroidal.

Examples 2.5. (a) Let
Let I = I d;a 1 ,...,an be the ideal of Veronese type with the property that d ≥ a i for all i and a 1 , . . . , a n ≥ 1. The set of associated prime ideals is described in [24, Proposition 3.1] as follows Thus by applying Theorem 2.4 the canonical primary decomposition of I is given as since the rank function of the corresponding polymatroid P is given by ρ(F ) = min{d, i∈F a i } for all F ⊂ [n].
Consider for example I = I 4;3,2,1 ⊂ S = K[x 1 , x 2 , x 3 ]. Then and the above formulas yield and the canonical primary decomposition In [18,Theorem 4.7] it is shown that the associated prime ideals of I corresponded to the trees of the graph G I in the following way: p F ∈ Ass(S/I) if and only if there exists a tree T such that p F = p T .
In the following we show that the irredundant primary decomposition given in terms of the rank function as described in Theorem 2.4 coincides with the one given in [18,Corollary 4.10]. There the irredundant primary decomposition of I is given as where a F is the number of vertices of a tree T in G I which is maximal with respect to the property that p F = p T , where p T = i∈V (T ) p F i . On the other hand, let ρ be the rank function of the transversal polymatroid attached to I. In [15,Section 9] it is shown that It follows that Now let T be a tree in G I which is maximal with respect to the property that p F = p T . Suppose we have shown that |{i : . Suppose this inequality is strict. Then there exists j ∈ V (T ) with F j ⊂ F . Since F = i∈V (T ) F i , we then notice that F j ∩ F i = ∅ for some i ∈ V (T ), contradicting the maximality of T .
A redundant primary decomposition of a polymatroidal ideal, which in some cases is even irredundant and which is given only in terms of the rank function of the corresponding polymatroid can be described as follows: Let P be a discrete polymatroid of rank d on the ground set [n] and ρ : 2 [n] → Z + its rank function. Then ρ satisfies the following conditions: We define the complementary rank function τ : . The function τ also satisfies the properties (i) and (ii), but instead of (iii) one has: A subset F ⊂ [n] will be called τ -closed, if τ (G) < τ (F ) for any proper subset G of F , and F will be called τ -separable if there exist non-empty subsets G and H of F with G ∩ H = ∅ and G ∪ H = F such that τ (G) + τ (H) = τ (F ). If F is not τ -separable, then it is called τ -inseparable. Corresponding concepts exist for ρ.
This is the canonical primary decomposition of I 4;3,2,1 presented already in Example 2.5(a). Thus in this case Theorem 2.6 yields an irredundant primary decomposition.
On the other hand if we consider the squarefree Veronese ideal I = I n−1;1,...,1 ⊂ K[x 1 , . . . , x n ] we will see that the intersection given in Theorem 2.6 is very far of being an irredundant primary decomposition. Indeed, the canonical primary decomposition of I is The rank function ρ of the associated discrete polymatroid of I can be easily computed: ρ(F ) = |F | for any F [n], and ρ([n]) = n − 1. Thus we obtain that τ (∅) = 0 and τ (F ) = |F | − 1 for any nonempty subset F ⊂ [n]. This implies that every set F ⊂ [n] with |F | ≥ 2 is τ -closed and thus the intersection given by Theorem 2.6 is the following Ideals of Borel type. We recall (see [ The principal Borel ideal generated by the monomial u is the smallest Borel ideal containing u, and is denoted u . Proposition 2.8. Let I be an ideal of Borel type. The following conditions are equivalent: (a) I is of strong intersection type; i is an irredundant primary decomposition of I. Since I is of Borel type it follows from the previous comments that we may assume that p 1 ⊂ p 2 ⊂ · · · ⊂ p r . Therefore we must have d 1 < d 2 < · · · < d r , otherwise the given primary decomposition would not be irredundant. We claim that I is the principal Borel ideal generated by the monomial , where for all i = 1, . . . , r, x n i is the variable of highest index appearing in the minimal system of generators of p i . Indeed, note first that u = p d 1 1 p d 2 −d 1 2 · · · p ds−d s−1 s . Thus the ideal u is a transversal polymatroidal ideal and applying either [18,Corollary 4.10] or [10,Theorem 4.3] we obtain that u = r i=1 p d i i is the irredundant primary decomposition of u . Therefore, I = u . In order to prove the implication (c) ⇒ (a) we use the fact that a principal Borel ideal is polymatroidal and by Proposition 2.1 we obtain the desired conclusion.
Remark 2.9. (a) Every monomial ideal of the form I = r i=1 p a i i with the property that p i ⊂ p j for all i = j is of strong intersection type. In particular all squarefree monomial ideals are of strong intersection type, as we already noticed before.
(b) Let p 1 ⊂ p 2 ⊂ · · · ⊂ p s be a chain of monomial prime ideals in S and (2) Let G be a graph with the property that I(G) k has a linear resolution for all k ≥ 2 and such that Ass(S/I k ) ⊂ Ass(S/I)∪{m} for all k. Then Theorem 1.3 implies that I(G) k is of strong intersection type for all k. The 5-cycle C 5 satisfies this condition. Indeed, it follows from [2, Theorem 6.12] that I(C 5 ) k has a linear resolution for all k ≥ 2, and from [5, Lemma 3.1] that Ass(S/I(C 5 ) k ) ⊂ Ass(S/I(C 5 )) ∪ {m} for all k.
(3) Let G = C 2k+1 be the odd cycle of length 2k + 1 with k ≥ 3. Since G c contains an induced 4-cycle, it follows that I(G) s has no linear resolution for any s ≥ 1, see [22,Proposition 1.8]. On the other hand it is known that Ass(S/I(G) s ) = Ass(S/I(G)) for s ≤ k and Ass(S/I(G) s ) = Ass(S/I(G)) ∪ {m} for s > k, see [5,Lemma 3.1]. It follows that I(G) s is of strong intersection type for s ≤ k, and not of strong intersection type for s > k. We do not know whether or not I(G) s is of intersection type for all s ≥ 1.
Let G be a graph. We call a 3-cycle C of G central if all vertices of G are neighbors of C.
A slight generalization of the [16, Theorem 2.1] is the following lemma, which we shall need in the proof of the next theorem. ∈ I 2 such that u · (x 1 , . . . , x n , y 1 , . . . , y k ) ⊂ I 2 . By [16, Corollary 1.2] we know that u is a squarefree monomial. We claim that u is not divisible by any y j . Assuming the claim proved, it follows that u / ∈ I(G) 2 and u · (x 1 , . . . , x n ) ⊂ I(G) 2 . Therefore depth (K[x 1 , . . . , x n ]/I(G) 2 ) = 0, and by [16, Theorem 2.1.] we obtain the desired conclusion.
In order to prove the claim we argue by contradiction. Without loss of generality we may assume that y 1 divides u. Since u / ∈ I 2 it follows that u can not be further divisible by any y i . Thus u = x i 1 · · · x is y 1 for some integers 1 ≤ i 1 < . . . < i s ≤ n with s ≥ 1 because y 1 does not belong to the socle of R/I 2 . Furthermore for every integers p = q we have {i p , i q } is not an edge of G, otherwise u ∈ y 1 I(G), a contradiction. This implies that ux i 1 is not divisible by any of the minimal generators of y 1 I(G) or I(G) 2 and therefore ux i 1 / ∈ I 2 , a contradiction. Hence our claim is proved and we are done. . Assume now that there exists a monomial prime ideal p ∈ Ass(S/I 2 ) which does not belong to Ass(S/I)∪{m}. If I(p) is only generated by variables, then I(p) = pS(p), because otherwise depth(S(p)/I(p) 2 ) > 0. It follows that p ∈ Ass(S/I), a contradiction. Therefore, I(p) contains also generators of degree 2, and we may apply Lemma 2.11, and conclude that max(Soc(S(p)/I(p) 2 )) ≥ 3. On the other hand, by the assumption on G, each variable x i divides some generator of I. Since at least one variable is mapped to 1 in S(p), it follows that I(p) contains also variables, and hence min(I(p) 2 ) = 2. By Theorem 1.1 this contradicts our assumption that I 2 is of intersection type.
(b)⇒(c): Suppose that there exists a 3-cycle C of G which is not central. Let D be the set of vertices of G which are not neighbors of C, and let p be the monomial prime ideal generated by the variables x i with i ∈ D. Then I(p) = I(H) + Q where Q is generated by a set X of variables, and where I(H) is an edge ideal in a set of variables disjoint from X with H a connected graph containing the 3-cycle C which is central in H. Lemma 2.11 implies that depth(S(p)/I(p) 2 ) = 0. It follows that p ∈ Ass(S/I 2 ). The prime ideal p is not a minimal prime ideal of I since it contains monomial generators of I of degree 2, and it is not maximal since D = ∅. This is a contradiction. is generated by elements of the form u+I 2 where u is a squarefree monomial. Let u + I 2 be such a non-zero socle element, and let H be the induced subgraph of G whose vertex set is the support of u. In the proof of [16, Theorem 2.1, (a) ⇒ (b)] it is shown that H is either a 3-cycle or a line of length at most 2. It follows that deg u ≤ 3. Hence we have shown that max(Soc(S/I 2 )) ≤ 3. Since I 2 is generated in degree 4, Theorem 1.1 yields the desired conclusion. Corollary 2.13. Let I be an edge ideal such that I 2 is not of intersection type. Then for any k ≥ 2, the ideal I k is not of intersection type.
Proof. Let I = I(G). Since I 2 is not of intersection type, Theorem 2.12 implies that there exists a 3-cycle C contained in G which is not central. As in the proof of Theorem 2.12(b) ⇒ (c) it follows that there exists a monomial prime ideal p = m such that I(p) = I(H) + Q where H is a connected graph containing the 3-cycle C which now is central in H, and where Q is generated in a disjoint set of variables, say I(H) ⊂ K[x 1 , . . . , x k ] and Q = (x k+1 , . . . , x ℓ ). Furthermore we may assume that V (C) = {1, 2, 3}. It follows from Lemma 2.11 that u + I(p) 2 with u = x 1 x 2 x 3 is a non-zero socle element of S(p)/I(p) 2 . This implies that ( Since (x 1 x 2 ) k−2 u is not divisible by any x i ∈ Q, it follows that s = 0, so that (x 1 x 2 ) k−2 u ∈ I(H) k . This is a contradiction, since deg(x 1 x 2 ) k−2 u = 2k − 1, while the generators of I(H) k are of degree 2k.
We conclude that (x 1 x 2 ) k−2 u + I(p) k is a non-zero socle element of S(p)/I(p) k , and hence max(Soc(S(p)/I(p) k ) ≥ 2k − 1. On the other hand min(I(p) k ) = k. Thus Theorem 1.1 implies that I k is not of intersection type. Examples 2.14. (1) A simple example to which Corollary 2.13 applies is the following: let I = (xy, xz, yz, zt, tu) be the edge ideal of the graph G consisting of the 3-cycle {x, y, z} and the edges {z, t} and {t, u}. The 3-cycle {x, y, z} is not central. Therefore by Theorem 2.12, I 2 is not of intersection type, and thus by Corollary 2.13, I k is not of intersection type for all k ≥ 2.
(2) Let I = I(G) = (xy, yz, xz, yt, zt, tu) be the edge ideal of the graph G. Note that Ass(S/I 2 ) = Ass(S/I) ∪ {(x, y, z, t), m} and that the cycle {x, y, z} is not central. Since G c is chordal, [14,Theorem 10.2.6] implies that I k has a linear resolution for all k ≥ 1. On the other hand, it follows from Theorem 2.12 and Corollary 2.13 that I k is not of intersection type for all k ≥ 2.
3. Some properties of monomial ideals of (strong) intersection type We cannot expect that ideals of (strong) intersection type have, compared with arbitrary monomial ideals, much better properties, since for example any squarefree monomial ideal is of strong intersection type. However we have Proof. Let I = p∈Ass(S/I) p dp , and let u = x a 1 1 x a 2 2 · · · x an n be a monomial with u t ∈ I t for some t > 0. Then u t ∈ p tdp for all p ∈ Ass(S/I). Thus for p ∈ Ass(S/I), we see that i,x i ∈p ta i ≥ td p . Therefore, i,x i ∈p a i ≥ d p which implies that u ∈ p dp . It follows that u ∈ I, and this proves that I is integrally closed, see [ Not all integrally closed ideals are of intersection type. The simplest such example is I = (x, y 2 ).
For a monomial ideal I ⊂ K[x 1 , . . . , x n ] we denote by conv(I) the convex hull in R n of the set {a : x a ∈ I}. The set conv(I) is called the Newton polyhedron of I. It is well-known that I is integrally closed if and only if I is generated by the monomials x a with a ∈ conv(I).
For a monomial ideal I of intersection type, the supporting hyperplanes of conv(I) can be easily described. For a subset F ⊂ [n] we let, as before, p F be the monomial prime ideal generated by the variables x i with i ∈ F . Associated with p a F we consider the hyperplane H F,a = {ξ ∈ R n : i∈F ξ i = a}.
It is clear that x c ∈ p a F if and only if c belongs to the half space H + F,a = {ξ ∈ R n : i∈F ξ i ≥ a}.
Thus the following corollary follows directly from the definition of ideals of intersection type. N.V. Trung, in a discussion with the first author of this paper, sketched an argument which shows that for any squarefree monomial ideal I ⊂ S one has that I (rk) ⊂ I k for all k ≥ 1. Here r denotes the highest degree of a generator in the (unique) minimal set of monomial generators of I, and I (t) denotes the t th symbolic power of I which for a monomial ideal we define as follows: Let Q 1 , . . . , Q s be the uniquely determined primary components corresponding to the minimal prime ideals of I. Then I (t) = s i=1 Q t i . (It is also common to define the t th symbolic power of I to be S ∩ p∈Ass(S/I) I t S p . This symbolic power is contained in the one defined here.) In general we expect that I (rk) ⊂ I k for all k and all monomial ideals. On the other hand, it follows from a result of Ein, Lazarsfeld and Smith [7,Theorem A], that for all k, J (ck) ⊂ J k for any graded ideal J ⊂ S of codimension c. Of course there are cases where r > c and vice versa. In the squarefree case however both numbers are bounded by the dimension of S.
Here we prove a related result for monomial ideals of intersection type. Recall that an ideal J ⊂ I is said to be a reduction of I, if I k = JI k−1 for some k ≥ 2. (a) if I k is of intersection type, then I (reg(I k )) ⊂ I k ; (b) if I k is of intersection type for all k ≫ 0, then I (rk) ⊂ I k for all k ≫ 0, where r = ρ(I) + 1. In particular, r ≤ max(I) + 1. Moreover, if I k has a linear resolution for all k ≫ 0, then r can be chosen to be max(I).
Proof. (a) Set s = reg(I k ), and let q ∈ Ass(S/I k ). There exists p ∈ Min(S/I) with p ⊂ q. It follows that p s ⊂ q s ⊂ q dq , because d q ≤ reg(I k (q)) ≤ s, see Theorem 1. p dp = I k .
(b) By Kodiyalam [19] we know that reg(I k ) = θ(I)k + c for k ≫ 0. If I k has a linear resolution for all k ≫ 0, then c = 0, and the assertion follows from (a). Otherwise c > 0, and for all k ≥ c one has that (θ(I) + 1)k ≥ reg(I k ). Again (a) yields the desired conclusion.