Symbolic Powers and Matroids

We prove that all the symbolic powers of a Stanley-Reisner ideal are Cohen-Macaulay if and only if the associated simplicial complex is a matroid.


Introduction
Stanley-Reisner rings supply a bridge between combinatorics and commutative algebra, attaching to any simplicial complex ∆ on n vertices the Stanley-Reisner ideal I ∆ and the Stanley-Reisner ring K[∆] = S/I ∆ , where S is the polynomial ring on n variables over a field K. One of the most interesting part of this theory is finding relationships between combinatorial and topological properties of ∆ and ring-theoretic ones of K [∆]. For instance, it is a wide open problem to characterize graph-theoretically the graphs G for which K[∆(G)] is Cohen-Macaulay, where ∆(G) denotes the independence complex of G. In [TY,Theorem 3], Terai and Yoshida proved that S/I m ∆ is Cohen-Macaulay for any m ∈ N ≥1 if and only if I ∆ is a complete intersection. Because it is a general fact that all the powers of any homogeneous complete intersection ideal are Cohen-Macaulay, somehow the above result says that there are no Stanley-Reisner ideals with this property but the trivial ones. Since if S/I m ∆ is Cohen-Macaulay then I m ∆ is equal to the mth symbolic power I (m) ∆ of I ∆ , it is natural to ask: For which ∆ the ring S/I (m) ∆ is Cohen-Macaulay for any m ∈ N ≥1 ? The answer is amazing. In this paper we prove that S/I (m) ∆ is Cohen-Macaulay for any m ∈ N ≥1 if and only if ∆ is a matroid (Theorem 2.1). The above result is proved independently and with different methods by Minh and Trung in [MT,Theorem 3.5]. Matroid is a well-studied concept in combinatorics, and it was originally introduced as an abstraction of the notion of the set of bases of a vector space. The approach to prove the above result is not direct, passing through the study of some blowup algebras related to ∆. Among the consequences of Theorem 2.1 we remark Corollary 2.9: After localizing at the maximal irrelevant ideal, I ∆ is a set-theoretic complete intersection whenever ∆ is a matroid.

The result
In this section we prove the main theorem of the paper.

Definition of the basic objects
First of all we define the basic objects involved in the statement. For the part concerning commutative algebra and Stanley-Reisner rings, we refer to Bruns and Herzog [BH], Stanley [St] or Miller and Sturmfels [MS]. For what concerns the theory of matroids, some references are the book of Welsh [We] or that of Oxley [Ox].
Let K be a field, n a positive integer and S = K[x 1 , . . . , x n ] the polynomial ring on n variables over K. Also, m is the maximal irrelevant ideal of S. We denote the set {1, . . . , n} by [n]. By a simplicial complex ∆ on [n] we mean a collection of subsets of [n] such that for The faces of ∆ which are maximal under inclusion are called facets. We denote the set of the facets of ∆ by F (∆). For a simplicial complex ∆ we can consider a square-free monomial ideal, known as the Stanley-Reisner ideal of ∆, The K-algebra K[∆] = S/I ∆ is called the Stanley-Reisner ring of ∆, and it turns out that More precisely, with the convention of denoting by ℘ A = (x i : i ∈ A) the prime ideal of S generated by the variables associated to a given subset A ⊆ [n], we have Given any ideal I ⊆ S its mth symbolic power is I (m) = (I m S W ) ∩ S, where W is the complement in S of the union of the associated primes of I and S W denotes the localization of S at the multiplicative system W . If I is a square-free monomial ideal then I (m) is just the intersection of the (ordinary) powers of the minimal prime ideals of I. Thus The last concept which is needed to understand the main theorem of the paper is a matroid.
A simplicial complex ∆ on [n] is said to be a matroid if, for any two facets F and G of ∆ and any i ∈ F, there exists a j ∈ G such that (F \ {i}) ∪ { j} is a facet of ∆. It is well known that if ∆ is a matroid then K[∆] is Cohen-Macaulay. In particular all the facets of a matroid have the same dimension. An useful property of matroids is the following.

Statement and proof
What we are going to prove is the following theorem. Remark 2.3. If ∆ is the k-skeleton of the (n − 1)-simplex, −1 ≤ k ≤ n − 1, then ∆ is a matroid. So Theorem 2.1 implies that all the symbolic powers of I ∆ are Cohen-Macaulay.
In order to prove Theorem 2.1 it is useful to introduce another square-free monomial ideal associated to a simplicial complex ∆, namely the cover ideal of ∆ We have dim(S/J(∆)) = n − dim ∆ − 1. The name "cover ideal" comes from the following fact: Then it is easy to see that Let ∆ c be the simplicial complex on [n] whose facets are . Actually the matroid ∆ c is known as the dual of ∆.
In order to have a good combinatorial description of J(∆) (m) we need a concept that is more general than vertex cover: For a natural number k, a k-cover of ∆ is a nonzero function Of course vertex covers and 1-covers with values on {0, 1} are the same things. It is not difficult to see that A k-cover α of ∆ is said to be basic if for any nonzero function β : , if β is a k-cover of ∆ then β = α. Of course to the basic m-covers of ∆ corresponds a minimal system of generators of J(∆) (m) . Now let us consider the multiplicative filtration S ymb(∆) = {J(∆) (m) } m∈N ≥1 . We can form the Rees algebra of S with respect to the filtration S ymb(∆), In [HHT,Theorem 3.2], Herzog, Hibi and Trung proved that A(∆) is noetherian. In particular, the associated graded ring of S with respect to S ymb(∆) and the special fiberĀ are noetherian too. The algebra A(∆) is known as the vertex cover algebra of ∆, and its properties have been intensively studied in [HHT]. The name comes from the fact that, writing and denoting by ( The algebraĀ(∆), instead, is called the algebra of basic covers of ∆, and its properties have been studied by the author with Benedetti and Constantinescu in [BCV] and with Constantinescu in [CV] for a 1-dimensional simplicial complex ∆. Clearly, the grading defined above on A(∆) induces a grading onĀ(∆), and it turns out that a basis for (Ā(∆)) m , m ≥ 1, as a K-vector space is . This implies that (Ā(∆)) m is a finite K-vector space for any m ∈ N. So we can speak about the Hilbert function ofĀ(∆), denoted by HFĀ (∆) , and from what said above we have, for k ≥ 1, The key to prove Theorem 2.1 is to compute the dimension ofĀ(∆). So we need a combinatorial description of dim(Ā(∆)). Being in general non-standard graded, the algebrā A(∆) could not have a Hilbert polynomial. However by [HHT,Corollary 2 has a Hilbert polynomial, we get a useful criterion to compute the dimension ofĀ(∆). First remind that, for two functions f , g : N → R, the writing f (k) = O(g(k)) means that there exists a positive real number λ such that The following proposition justifies the introduction ofĀ(∆). is a Cohen-Macaulay (n + 1)-dimensional (see [BH,Theorem 4.5.6]) ring. Thus H i M (A(∆) + ) = 0 for any i ≤ n by the above long exact sequence. Now let us look at the other short exact se- where A(∆) + (1) means A(∆) + with the degrees shifted by 1, and the corresponding long exact sequence on local cohomology is a n-dimensional ring (see [BH,Theorem 4.5.6]) this implies, using once again the independence of the base in computing local cohomology, that G(∆) is Cohen-Macaulay.
We are almost ready to show Theorem 2.1. We need just a technical lemma which allows us to construct "many" basic covers. 1 , a 2 , . . . , a d , b 1 , b 2

Lemma 2.5. Let s ≥ −1 and d be integer numbers such that s ≤ d − 3. For any positive integer k consider the set
Then |A k | = Ω(k 2d−s−3 ).

Proof. Let us set
For a fixed a 1 ∈ X k , set The vectors (a 2 , . . . , a d ) ∈ Y k (a 1 ) are so many as the integer partitions of k − a 1 with at most d − 1 parts. Because a 1 ∈ X k these are at least so many as the partitions ⌊k/(d + 3)⌋ with at most d − 1 parts. These, in general, are less than all the monomials of degree ⌊k/(d + 3)⌋ in , since a permutation of the variables gives the same partitions but may give different monomials. Anyway, since this is the only reason, the number of the possible (a 2 , . . . , a d ) is at least So there exists a positive real number λ 2 , independent on a 1 , such that |Y k (a 1 )| ≥ λ 2 · k d−2 . Let a = (a 1 , a 2 , . . . , a d ) be a vector such that a 1 ∈ X k and (a 2 , . . . , a d ) ∈ Y k (a 1 ). Then set It is easy to notice that the vectors (b 1 , . . . , b d−s−1 ) ∈ Z k (a) are so many as all the monomials of degree a 1 + . . .
. So we get So the elements of Z k (a) are at least so many as the monomials of degree ⌊k/(d + 2)⌋ in d − s − 1 variables. Therefore there is a positive real number λ 3 , not depending on a, such that |Z k (a)| ≥ λ 3 · k d−s−2 . Finally, we have that Now we are ready to prove Theorem 2.1.
Proof. By the duality on the matroids it is enough to prove that S/J(∆) (m) is Cohen-Macaulay for any m ∈ N ≥1 if and only if ∆ is a matroid. Suppose that ∆ is (d − 1)dimensional. If-part. Let us consider a basic k-cover α of ∆. Let F be a facet of ∆ such that ∑ j∈F α( j) = k (F exists because α is basic). Set We claim that for any i ∈ [n] we have α(i) ∈ A F . In fact, if i 0 ∈ [n] does not belong to F, then, because α is basic, there exists a facet G of ∆ such that i 0 ∈ G and ∑ i∈G α(i) = k. By the exchange property there exists a vertex Hence α(i 0 ) = α( j 0 ) ∈ A F . The number of ways to give values on vertices of F such that the sum of the values on the whole F is k are k + d − 1 d − 1 . This implies that, for k ≥ 1, Only if-part. Suppose contrary that ∆ is not a matroid. Then there exist two facets F and G of ∆ and a vertex i ∈ F such that (F \ {i}) ∪ { j} is not a facet of ∆ for any j ∈ G. Let s be the greatest integer such that there exists an s-dimensional subface F ′ of F \ {i} such that there is a (d − s − 2)-dimensional subface of G whose union with F ′ is a facet of ∆. Notice that s ≤ d − 3 and s might be −1. Let F 0 ⊆ F \ {i} be an sdimensional face and G 0 ⊆ G a (d −s−2)-dimensional face satisfying the above conditions. Let (a 1 , . . . , a d , b 1 We claim that α ′ is a k-cover, not necessarily basic. By the definition of α ′ we have to check that for any facet If i ∈ H, then we have Reducing the values of α ′ where possible we can make it in a basic k-cover α. However we cannot reduce the values at the vertices of F ∪ G 0 because the equalities Thus the basic k-covers of F (∆) are at least so many as the cardinality of A k . So by Lemma 2.5 there exists a positive real number λ such that for k ≫ 0 we have which contradicts the hypothesis that S/J(∆) (m) is Cohen-Macaulay for any m ∈ N ≥1 .
We end the paper stating two corollaries of Theorem 2.1. First we recall that the multiplicity of a standard graded K-algebra R, denoted by e(R), is the leading coefficient of the Hilbert polynomial times (dim(R) − 1)!. Geometrically, let ProjR ⊆ P N , i.e. R = K[X 0 , . . . , X N ]/J for a homogeneous ideal J. The multiplicity e(R) counts the number of distinct points of ProjR ∩ H, where H is a generic linear subspace of P N of dimension N − dim(ProjR).

Corollary 2.6. A simplicial complex ∆ is a (d − 1)-dimensional matroid if and only if
Moreove, if ∆ is a matroid then Proof. The first fact follows putting together Theorem 2.1 and Proposition 2.4. For the second fact, we have to recall that, during the proof of Theorem 2.1, we showed that for a (d − 1)-dimensional matroid ∆ we have the inequality It is well known that if ∆ is a pure simplicial complex then |F (∆)| = e(K[∆]) (for instance see [BH,Corollary 5.1.9]), so we get the conclusion.
Example 2.7. If ∆ is not a matroid the inequality of Corollary 2.6 may not be true. For instance, take ∆ = C 10 the decagon (thus it is a 1-dimensional simplicial complex). Since C 10 is a bipartite graphĀ(C 10 ) is a standard graded K-algebra by [HHT,Theorem 5.1]. In particular it admits a Hilbert polynomial, and for k ≫ 0 we have In [CV] it is proved that for any bipartite graph G the algebraĀ(G) is a homogeneous algebra with straightening law on a poset described in terms of the minimal vertex covers of G. So the multiplicity ofĀ(G) can be easily read off from the above poset. In our case it is easy to check that e(Ā(C 10 )) = 20, whereas e(K[C 10 ]) = 10.
Let us introduce the last result of the paper. An ideal I of a ring R is a set-theoretic complete intersection if there exist f 1 , . . . , f h ∈ R, where h = ht(I), such that ( f 1 , . . . , f h ) = √ I. The importance of this notion comes from algebraic geometry, since if I is a set-theoretic complete intersection then the variety V (I) ⊆ Spec(R) can be defined set-theoretically "cutting" the "right" number of hypersurfaces of Spec(R). A necessary, in general not sufficient, condition for I to be a set-theoretic complete intersection is that the cohomological dimension of it, cd(R, I) = max{i : H i I (R) = 0}, is h. By a result of Lyubeznik in [Ly] it turns out that cd(S, I ∆ ) = n − depth(K[∆]), so if I ∆ is a set-theoretic complete intersection K[∆] will be Cohen-Macaulay.

Remark 2.8. In general if K[∆]
is Cohen-Maculay then I ∆ might not be a set-theoretic complete intersection. For instance, if ∆ is the triangulation of the real projective plane with 6 vertices described in [BH,p. 236], then K[∆] is Cohen-Macaulay whenever char(K) = 2. However, for any characteristic of K, I ∆ need at least (actually exactly) 4 polynomials of K[x 1 , . . . , x 6 ] to be defined up to radical (see the paper of Yan [Ya,p. 317,Example 2]), but ht(I ∆ ) = 3. Corollary 2.9. Let K be an infinite field. For any matroid ∆, the ideal I ∆ S m is a set-theoretic complete intersection in S m .
Proof. By the duality on matroids it is enough to prove that J(∆)S m is a set-theoretic complete intersection. For h ≫ 0 it follows by [HHT,Corollary 2.2] that the hth Veronese of A(∆),Ā (∆) (h) = m≥0Ā (∆) hm , is standard graded. ThereforeĀ(∆) (h) is the ordinary fiber cone of J(∆) (h) . MoreoverĀ(∆) is finite as aĀ(∆) (h) -module. So the dimensions ofĀ(∆) and ofĀ(∆) (h) are the same. Therefore, using Theorem 2.1 and Proposition 2.4, we get ht(J(∆)S m ) = ht(J(∆)) = dimĀ(∆) (h) = ℓ(J(∆) (h) ) = ℓ((J(∆)S m ) (h) ), where ℓ(·) is the analytic spread of an ideal, i.e. the Krull dimension of its ordinary fiber cone. From a result by Northcott and Rees in [NR,p. 151], since K is infinite, it follows that the analytic spread of (J(∆)S m ) (h) is the cardinality of a set of minimal generators of a minimal reduction of (J(∆)S m ) (h) . Clearly the radical of such a reduction is the same as the radical of (J(∆)S m ) (h) , i.e. J(∆)S m , so we get the statement.
Remark 2.10. Notice that a reduction of IS m , where I is a homogeneous ideal of S, might not provide a reduction of I. So localizing at the maximal irrelevant ideal is a crucial assumption of Corollary 2.9. It would be interesting to know whether I ∆ is a set-theoretic complete intersection in S whenever ∆ is a matroid.