Betti splitting via componentwise linear ideals

Betti splitting via componentwise linear ideals

Davide Bolognini
Abstract.

A monomial ideal admits a Betti splitting if the Betti numbers of can be determined in terms of the Betti numbers of the ideals and Given a monomial ideal , we prove that is a Betti splitting of , provided and are componentwise linear, generalizing a result of Francisco, Hà and Van Tuyl. If has a linear resolution, the converse also holds. We apply this result recursively to the Alexander dual of vertex-decomposable, shellable and constructible simplicial complexes and to determine the graded Betti numbers of the defining ideal of three general fat points in the projective space. footnotetext: Key words: resolution of monomial ideals, componentwise linear ideals, Betti splittings, fat points.
AMS Mathematics Subject Classification 2010: Primary 13D02, 13A02; Secondary 05E40, 05E45

1. Introduction

Our aim is to pursue the spirit of Eliahou and Kervaire [3], Francisco, Hà and Van Tuyl [7] in order to find suitable decomposition of the Betti table of a monomial ideal, possibly available for recursive procedures.

Let be a field and let be a monomial ideal. Consider monomial ideals such that the set of minimal monomial generators of is the disjoint union of the sets of minimal monomial generators of and . We say that is a Betti splitting of if

where denotes the graded Betti numbers of a minimal -free graded resolution.

This approach was used for the first time by Eliahou and Kervaire [3], giving an explicit formula for the total Betti numbers of stable ideals. Fatabbi [5] developed the theory for graded Betti numbers. Many authors wrote papers applying the Eliahou-Kervaire technique to the resolution of special classes of monomial ideals (see e.g.[4], [5], [6], [8], [17], [19], [23], [24], [25]). Francisco, Hà and Van Tuyl proved in [7, Corollary 2.4] that if and have a linear resolution, then is a Betti splitting of . In Section 3 we generalize this result, proving that is a Betti splitting of , provided and are componentwise linear (Theorem 3.3). If has a linear resolution, the converse also holds (Proposition 3.1).

Componentwise linear ideals have been extensively studied (see e.g.[6], [8], [10],[11]). Stable ideals, ideals with linear quotients and ideals with linear resolution are examples of componentwise linear ideals.

In Section 4 we apply the theory of Betti splittings to the Alexander dual ideal of a simplicial complex . By the Stanley-Reisner correspondence, squarefree monomial ideals correspond to simplicial complexes (see e.g.[10]). This is an important bridge between Commutative Algebra and Combinatorics. In particular, by Hochster’s formula [15], the graded Betti numbers of reflect geometric and topological information on .

As a consequence of Theorem 3.3, we recover a result due to Moradi and Kosh-Ahang [17], proving that the Alexander dual of a vertex-decomposable simplicial complex admits a particular kind of splitting, the so-called -splitting (Corollary 4.3). In Corollary 4.5 we prove a Betti splitting result for shellable and constructible simplicial complexes, showing that in general they do not admit -splitting (Example 4.4).

A further application is an extension of a result proved by Valla in [25]. By using a recursive approach we can compute explicitly the graded Betti numbers of the defining ideal of three general fat points in the projective space (Corollary 5.2 and Corollary 5.3).

Acknowledgements. The results presented in this paper are part of my PhD thesis and I would like to thank my advisors, Maria Evelina Rossi, Leila De Floriani and Emanuela De Negri for their encouragements and their constant care about my work. The author is grateful to Hop Nguyen for pointing out [18, Lemma 2.8(ii)]. It gives substantial improvements to a previous version of Theorem 3.3. Thanks to Aldo Conca and Matteo Varbaro for helpful discussions and suggestions.

2. Preliminaries

Let be a field, the maximal homogeneous ideal of an homogeneous ideal. Denote by the graded Betti numbers of . In the following we omit the superscript . Given a monomial ideal , let be the minimal system of monomial generators of and be the lowest degree of a generator in

Definition 2.1.

(Francisco, Hà, Van Tuyl,[7]) Let , and be monomial ideals such that and is the disjoint union of and . Then is a Betti splitting of if

The previous definition can be given in terms of the vanishing of some modules maps, as stated in the following result.

Proposition 2.2.

(Francisco, Hà and Van Tuyl, [7, Proposition 2.1]) Let , and be monomial ideals such that and is the disjoint union of and . Consider the short exact sequence

and the corresponding long exact sequence of modules:

Then the following are equivalent:

  • is a Betti splitting;

  • , for all .

If is generated in degree we say that has a -linear resolution if for every and . When the context is clear, we simply write that has a linear resolution. The Castelnuovo-Mumford regularity of is defined by An ideal generated in degree has a -linear resolution if and only if . Let and be monomial ideals, with and is the disjoint union of and . Francisco, Hà and Van Tuyl proved in [7, Corollary 2.4] that if and have a linear resolution, then is a Betti splitting. In Section 3 we generalize this result assuming and componentwise linear. We recall some definitions and results that will be useful later. These hold also in a more general setting, but from now on we assume to be the standard graded polynomial ring with coefficients in a field (see [12],[18], [21] and [22] for more details). Componentwise linear ideals have been introduced by Herzog and Hibi in [11]. Denote by the ideal generated by all the homogeneous polynomials of degree belonging to . In the monomial case, is simply the ideal generated by all monomials of degree belonging to .

Definition 2.3.

A homogeneous ideal is called componentwise linear if has a linear resolution, for every .

Notice that we cannot detect if an ideal is componentwise linear from its graded Betti numbers. In the following example we show an ideal that is not componentwise linear but with the same graded Betti numbers of a componentwise linear ideal.

Example 2.4.

([14, Example 5.5]) Let be the ideals and . Using CoCoA [2], one can see that and have the same graded Betti numbers. The ideal is stable hence componentwise linear, while has not a linear resolution, thus is not componentwise linear.

Linearity defect was introduced by Herzog and Iyengar in [12] and measures how far a resolution is from being linear.

Let be a finitely generated graded -module and its minimal graded free resolution over . Let be the chain complex obtained by replacing by zero each entry of degree greater than one in the matrices of the differentials of .

Definition 2.5.

Let be a finitely generated graded -module . The linearity defect of is defined by

where denotes the -th homology of the chain complex

We close this section by stating a special case of a useful characterization of componentwise linear modules due to Römer (see also [26, Proposition 4.9]).

Theorem 2.6.

(Römer,[21, Theorem 3.2.8]) For any homogeneous ideal , is componentwise linear if and only if .

3. Results

Let and be monomial ideals, with and is the disjoint union of and . Francisco, Hà and Van Tuyl proved in [7, Corollary 2.4] that if and have a linear resolution, then is a Betti splitting. Provided with a linear resolution, the converse also holds, as stated in the following result.

Proposition 3.1.

Let be a positive integer, be a monomial ideal with a -linear resolution, monomial ideals such that , and . Then the following facts are equivalent:

  1. is a Betti splitting of ;

  2. and have -linear resolutions.

If this is the case, then has a -linear resolution.

Proof.

Assume holds. Then for all . Let . For we have thus and have a -linear resolution.

Conversely one has that implies , by [7, Corollary 2.4].

By [7, Corollary 2.2] we have . Since , then , thus has a -linear resolution. ∎

Notice that there exist ideals with a linear resolution that do not admit any Betti splitting (see Example 5.4).

Now we extend [7, Corollary 2.4], assuming and componentwise linear ideals. In the proof we use the following key-lemma.

Lemma 3.2.

(Nguyen,[18, Lemma 2.8(ii)]) Let be an -linear map between finitely generated -modules. If for some , the map is zero, then the map

is zero for all and all .

Theorem 3.3.

Let and be monomial ideals such that and is the disjoint union of and . If and are componentwise linear, then is a Betti splitting of .

Proof. By Proposition 2.2, it suffices to prove that all the maps

are zero for all . Since and are componentwise linear, by Theorem 2.6 we have . Then . Since is the disjoint union of and , hence and . Then the map

is zero. By Lemma 3.2 with the result follows.

In view of Proposition 3.1, the assumptions on and in Theorem 3.3 cannot be weakened in general, without further assumptions.

Clearly the converse of Theorem 3.3 does not hold in general, as shown in the following example.

Example 3.4.

Let be the monomial ideal defined by

Define and . It is easy to check that is a Betti splitting of . In fact the graded Betti numbers of and are given by

Nevertheless is not componentwise linear, since is generated in one degree and has no linear resolution.

Several examples and Proposition 3.1 suggest the following question.

Question: Assume componentwise linear. Does the converse of Theorem 3.3 hold?

4. Betti splitting for simplicial complexes

In this section we present some applications of Theorem 3.3 to simplicial complexes. For more definitions about simplicial complexes, their properties and the Stanley-Reisner correspondence we refer to [10, Chapter 1] and [16, Chapter 3] and references over there.

Definition 4.1.

An abstract simplicial complex on vertices is a collection of subsets of , called faces, such that if , , then .

Denote a face by , with . A facet is a maximal face of with respect to the inclusion of sets. Denote by the collection of facets of . A simplicial complex is called pure if all the facets of have the same cardinality. The Alexander dual ideal of is defined by

, where and .

A decomposition , such that is the disjoint union of and induces a decomposition . We call a Betti splitting of if is a Betti splitting of . Note that all the Alexander dual ideals involved are computed with respect to the vertices of . In the following diagram (see e.g.[16]) we recall the hierarchy of some properties of (possibly non-pure) simplicial complexes.

(4.1)

It can be proved ([13], see also [10, Theorem 8.2.20]) that a simplicial complex is sequentially Cohen-Macaulay if and only if is componentwise linear.

All these three properties are defined recursively and for this reason we can apply Theorem 3.3 to .

In [7, Theorem 2.3] Francisco, Hà and Van Tuyl give conditions on and forcing to be a Betti splitting of . The splitting given in Theorem 3.3 is not a consequence of [7, Theorem 2.3], as shown in the following example.

Example 4.2.

Let be a field of characteristic zero and let be the Alexander dual ideal of the simplicial complex whose set of facets:

Let , with and .

Let be the Alexander dual ideal of , for . Both and are shellable (use Macaulay [9]). Then and are componentwise linear. By Theorem 3.3, is a Betti splitting of . Nevertheless the assumptions of [7, Theorem 2.3] are not satisfied since and both and are not zero.

In the next result we focus on a special splitting of a monomial ideal . Let be a variable of . Let be the ideal generated by all monomials of divided by a and let be the ideal generated by the remaining monomials of . If is a Betti splitting, we call a -splitting of . We recover the following known result.

Corollary 4.3.

([17, Theorem 2.8, Corollary 2.11]) If is a vertex decomposable simplicial complex then there exists such that admits -splitting.

Proof.

Since is vertex decomposable, there exists a vertex such that and are both vertex decomposable. By [17, Lemma 2.2] we have . A vertex decomposable simplicial complex is sequentially Cohen-Macaulay by Diagram 4.1. Then and are componentwise linear. The statement follows from Theorem 3.3. ∎

In Corollary 4.3, vertex decomposability cannot be replaced by shellability or constructibility, as it is shown in the next example.

Example 4.4.

Consider the simplicial complex whose set of facets is

The given order of the facets of is indeed a shelling, thus is shellable (use Macaulay [9]). By Diagram 4.1, is componentwise linear. Since is generated in degree , has actually a -linear resolution. Consider the splitting , for every . The resolution of is not linear, for every By Proposition 3.1, does not admit -splitting.

Let be a constructible simplicial complex. Although in general does not admit any -splitting, it admits a Betti splitting, as a consequence of Theorem 3.3.

Corollary 4.5.

Let be a constructible simplicial complex. Then admits Betti splitting.

Proof.

By definition of constructible complexes, there exist two constructible simplicial complexes and such that and is the disjoint union of and . Then and are componentwise linear ideals, by Diagram 4.1. By Theorem 3.3, is a Betti splitting of . ∎

In the pure case the previous result is [19, Corollary 3.4]. With the same notations in the proof of Corollary 4.5, for shellable complexes a more precise result holds: consists of a single facet, because in this case has linear quotients (see [1]).

Notice that Corollary 4.5 does not hold for (sequentially) Cohen-Macaulay simplicial complexes, as shown in the following example. We present an ideal that does not admit Betti splitting at all, in characteristic different from two. To our knowledge this is the first example in literature of an ideal that does not admit any Betti splitting (see [1, Example 5.31] for a characteristic-free example).

Example 4.6.

Let be a field of . Let be the following -vertex triangulation with facets of the real projective plane, due to Reisner [20].

Figure 1. A triangulation of real projective plane.

The graded Betti numbers of are given by

Francisco, Hà and Van Tuyl in [7] pointed out that does not admit -splitting (in this case the Stanley-Reisner ideal and coincide). Actually does not admit any Betti splitting. Using CoCoA [2] and Macaulay [9], we checked all the possible pairs of ideals and such that is the disjoint union of and . In each case, at least one between and has no linear resolution. Then, by Proposition 3.1, is not a Betti splitting.

5. The resolution of the ideal of three general fat points in

Let be the -dimensional scheme consisting of three general fat points in , with and . After a change of coordinates, we may assume that , and . Then the defining ideal of is

In the case of two fat points we denote by , with . If we denote by .

Francisco proved in [6] that the defining ideal of the zero-dimensional schemes of general fat points in is componentwise linear. In general the ideals are not stable (even if ). Valla [25, Corollary 3.5] computed the graded Betti numbers of the defining ideal of two general fat points in , , by using a Betti splitting argument. We prove a splitting result for and, as a consequence, we give a recursive procedure to compute the graded Betti numbers of in the case .

Theorem 5.1.

Let , . Let be the -dimensional scheme defined by three general fat points in , with . Assume . Then admits -splitting.

Proof.

Let and . We show that

Let Assume Then Since and , it follows

Let then . Since and then . For the other inclusion, note that and .

We prove now that . Let Then or . Assume If , there would be such that Since and , this is a contradiction. The proof for is the same, then the first inclusion is clear.

For the reverse inclusion, we first prove that Let be a monomial. Assume first that there is a variable with and . Then

Since and hence

Otherwise

Let . By definition of , there is such that If , then there are and a monomial such that . Since and then Hence and where This is a contradiction, because .

Let . If , then there is such that . Since , then . Hence a contradiction.

Clearly one has . By [6, Theorem 4.6], and are componentwise linear. In view of Theorem 3.3, we conclude that is a Betti splitting of . ∎

Notice that, in general, the splitting of Theorem 5.1 is not a particular case of [7, Theorem 2.3] (see for instance the case , , ).

In the next corollary we compute explicitly the graded Betti numbers of in the case by a recursive procedure.

Corollary 5.2.

Let , . Let be the -dimensional scheme consisting of three general fat points in , with and . Assume . Then

Proof.

Since , the assumptions of Theorem 5.1 are fullfilled, then admits -splitting. Let and be as in the proof of Theorem 5.1. Clearly . Note that and for each . We remark that, after a relabeling of the variables, is the ideal of two general fat points in , i.e. . Then

Since , for and , we can apply the same argument recursively to , for , to get

(1)

The result follows by [25, Corollary 3.5] and by the relation . ∎

Theorem 5.1 allows us to apply a recursive procedure for computing the Betti numbers of also in the case and This formula has the limit that the Betti numbers of , , could be involved. These ideals are studied in [5]. An explicit formula for the graded Betti numbers of is given only for [6, Proposition 3.2] and [6, Proposition 3.3].

Corollary 5.3.

Let , . Let be the -dimensional scheme consisting of three general fat points in , with , and . Let be the defining ideal of . Set ,

for

and

for .

Then

Proof.

Note that and . By Theorem 5.1, admits -splitting. The main difference with Corollary 5.2 is that for but . Then, following the proof of Corollary 5.2 and using Equation (1) we get

We focus our attention only on the first term of the equation. Now one has . If the claim follows. If , then and the assumptions of Theorem 5.1 are satisfied. Since we consider the new order of the ideals of the intersection given by multiplicities, then admits -splitting. One has for