Generalized Newton Complementary Duals of Monomial Ideals
Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphisms between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.
Given a polynomial ring over a field and a graded ideal in , one would like to understand various algebraic properties of the ideal. For instance, the Castelnuovo-Mumford regularity, the projective dimension, and the Cohen-Macaulayness of , are of great importance. Finding the minimal free resolution of the ideal is the key to those properties. This has been an active area among commutative algebraists and algebraic geometers. Another central topic in algebraic geometry is the blow up algebras defined by the given ideal.
When is a monomial ideal, one can associate to its combinatorial objects such as (hyper)graphs, and use combinatorial methods to recover algebraic properties; see, for example, the surveys [H] and [MV]. However, describing the precise minimal free resolution of a squarefree monomial ideal is not easy; see, for instance, [AHH], [HV] and [Ho]. There are even fewer results on finding the minimal free resolution for non-squarefree monomial ideals. The first non-trivial class to consider is that of stable ideals, studied by Eliahou and Kervaire in [EK]. When is a monomial ideal generated in the same degree, its special fiber ring is the associated toric ring. Hence understanding the defining equations of the special fiber ring of the ideal is very important. Villarreal [V] found the explicit description of the defining equations of the special fiber ring of edge ideals, i.e., squarefree monomial ideal generated in degree two. There are some other work on this subject, but they are almost always centered around squarefree monomial ideals.
The motivation of this work comes from the paper [CN1] of Corso and Nagel, where they studied the specialization of generalized Ferrers graphs (see Definitions 2.9 and 2.11). They showed that every strongly stable ideal of degree two can be obtained via a specialization. The authors later explicitly described the minimal free resolutions of Ferrers ideals and the defining equations of the special fiber ring in [CN2]. For this purpose, they used cellular resolutions as introduced by Bayer and Sturmfels in [BS]. This construction provides a characteristic-free context. Note that the special fiber ring is a determinantal ring in this case.
Meanwhile, the Newton complementary duals of monomial ideals were first introduced by Costa and Simis in [CS]. There, the dual operation was applied to study the rational maps between the base ideals and the dual ideals. In this work, we will extend the dual operation to get the generalized Newton complementary duals of monomial ideals, as introduced in Definition 2.1. The generalized (Newton complementary) dual operation is indeed a dual operation, since the double dual will bring back the base ideal to itself (Remark 2.4). Properties of generalized Newton complementary duals were investigated in this work.
In Section 3, we establish an isomorphism of special fiber rings between the base ideal and the generalized dual ideal (Theorem 3.1), generalizing the corresponding result of Costa and Simis. In other words, we will prove that the base ideal defines a birational map if and only if its generalized dual ideal defines a birational map. We then focus on the generalized duals of monomial ideals that are related to classical Ferrers graphs. The ideals that we consider are duals of specializations of generalized Ferrers ideals. As a corollary of Theorem 3.1 and the work of Corso, Nagel, Petrović, and Yuen in [CNPY], we describe the special fiber rings of the generalized duals of specializations of generalized Ferrers ideals. In other words, we describe the toric rings associated to the generalized duals of such ideals (Corollary 3.3). Those toric rings are Koszul normal Cohen-Macaulay domains. In particular, we establish a new class of Koszul ideals.
In Section 4, we focus on the basic properties of duals of monomial ideals generated in the same degree. Whence, the generalized duals are also generated in the same degree. This class of monomial ideals has the nice property that it is closed under the ideal products. Moreover, when the base ideals are stable of degree two or strongly stable, the generalized duals have linear quotients (Theorem 4.2); in particular, their regularities coincide with the common degrees of the minimal monomial generators. We further show that Newton complementary duals of an ideal defined by a bipartite graph is the Alexander dual of the edge ideal of the complement of the base bipartite graph (Theorem 4.4). Notice that this property does not hold when the base ideal is not coming from a bipartite graph. This also shows that the Newton complementary dual is quite different from the Alexander duality functor in [M] (Example 4.5). Moreover, even for bipartite graphs, generalized Newton complement dual and generalized Alexander dual do not agree in general (Remark 4.6).
In Section 5, we find a cellular complex which supports the minimal free resolution for the generalized duals of strongly stable ideals generated in degree (Theorem 5.3). Consequently, we can easily recover the Betti numbers and projective dimensions of such ideals (Remark 5.4). The construction involved is inspired by the works of Corso and Nagel [CN1] and [CN2] as well as the work of Mermin [Mer]. To be more precise, we use the technique of iterated mapping cones for the proof. This provides a geometric description of the free resolution and hence is characteristic-free. One also notices that, in some sense, our results provide the dual version of the work of Dochtermann and Engström in [DE] or Nagel and Reiner in [NR], where they found cellular resolutions of the edge ideals of cointerval hypergraphs or squarefree strongly stable ideals generated in a fixed degree. It is worth pointing out that the generalized dual ideals are usually non-squarefree, whereas they focused on squarefree cases.
In the last section, inspired by the work of Nagel and Reiner in [NR], we consider a type of monomial ideals of degree 2 that generalizes the concept of stable ideal. When the base ideal is compatible in the sense of Definition 6.6, we can describe explicitly a cellular free resolution of the generalized dual. The construction uses the quasi-Borel move which is similar to the Borel move introduced in [FMS]. And the proof uses the properties of linear quotients and iterated mapping cones (Proposition 6.4 and Theorem 6.11). In the special case when the base ideal is stable, one has a very neat formula for the Betti numbers of the generalized dual ideal (Remark 6.13). This construction offers a quick and computationally efficient way to obtain the minimal free resolution from a graphical investigation.
Let be a polynomial ring over a field . We give a standard graded structure, where all variables have degree one. We write for the -vector space of homogeneous degree forms in so that . We use the notation to denote a rank-one free module with the generator in degree so that .
Analogously, the ring is endowed with a multigraded (-graded) structure. Whence, if , following [P, Section 26], we will usually say “multidegree ” instead of “-degree ”. Meanwhile, means the free -module with one generator in the multidegree .
Let be a finitely generated graded -module. We can compute the minimal graded free resolution of :
The minimal graded free resolution of is unique up to isomorphism. Hence, the numbers , called the graded Betti numbers of , are invariants of . Two coarser invariants measuring the complexity of this resolution are the projective dimension of , denoted by , and the Castelnuovo-Mumford regularity of , denoted by . They are defined as
Now, we introduce the generalized Newton complementary dual of a given monomial ideal in the ring . The Newton complementary dual was first introduced by Costa and Simis in [CS]. Here, with stands for the set .
Let be the minimal monomial generating set of . Let . If for each and , one has , we say is -determined. The generalized Newton complementary dual of determined by is the monomial ideal with . We simply call the -dual of . When the vector is clear from the context, we also call the generalized dual of .
We illustrate the concept of the generalized dual with a simple example.
Consider the ideal . Let . The generalized Newton complementary dual of determined by is with .
In Definition 2.1, when for each , the ideal is exactly the Newton complementary dual of defined in [CS]. We will write for the Newton complementary dual of .
Two easy observations:
double applications of the generalized dual bring back the base ideal: ;
when and are two -determined monomial ideals generated in degrees and respectively, one has where .
Next, we recall some of the definitions and theorems regarding the cellular resolution of a monomial ideal from [MS]. One of the main goals of this work is to establish a cellular complex that provides the minimal free resolution of the -dual of an ideal. This topic will be further investigated in later sections.
Definition 2.5 ([Ms, Section 4.1]).
A (polyhedral) cell complex is a finite collection of finite polytopes (in ) called the faces or cells of , satisfying the following conditions.
If is a polytope in and is a face of then .
If , then is a common face of and .
The maximal faces are called facets. The dimension of is determined by the maximal dimension of its facets. When are two faces, is called a facet of whenever is one dimensional less than . A cell complex is labeled if we can associate to each vertex a vector . The induced label of any face of is the exponent vector of .
Since each vector can be identified with the monomial , for simplicity, we will also say that the above face is labeled by the monomial .
Let be the set of faces of of dimension . Note that the empty set is the unique -dimensional face. A cell complex has an incidence function , where if is a facet of . Note that the sign is determined by whether the orientation of induces the orientation of where the orientation is determined by some ordering of the vertices.
Let be a cellular complex of dimension . The cellular free complex supported on is the complex of -graded -modules
where , and the differential map is defined on the basis element of in as
We may consider the componentwise comparison partial order on defined by whenever . If , we define a subcomplex , namely the subcomplex of faces whose labels are less than or equal to .
A common procedure to determine whether in (1) is a resolution is by applying the following criteria of Bayer and Sturmfels. This criteria is useful because it reduces the question of whether a cellular free complex is acyclic to a question of the geometry of the polyhedral cell complex.
Lemma 2.6 ([Bs]).
The complex is a cellular resolution if and only if for each the complex is acyclic over the base field .
However, we will take a different approach by applying iterated mapping cones. Let us recall some of the basic constructions in [P].
Definition 2.7 ([P, Section 27]).
Let be a map of complexes of finitely generated -modules. The map is also called a comparison map. The mapping cone of is the complex with the differential , defined as follows:
for each .
Remark 2.8 ([P, Construction 27.3 and the discussion before it]).
Suppose that and above are free resolutions of finitely generated modules and respectively, while is an injective homomorphism of modules. Then, there is a lifting of to , which will also be denoted by . The mapping cone of provides a free resolution of the quotient module . We are interested in the case when monomial ideals and , while and . Notice that there is a short exact sequence here:
In later sections, we will work on duals relative to Ferrers ideals, stable ideals and strongly stables. We recall some definitions first. For a monomial , we write for the set and for the subset . We also write for . A monomial ideal is called stable if for each monomial , for each , one has . The ideal is called strongly stable if for each monomial , for each dividing and , one has . For both types, it is easy to see that when the ideal is generated in the same degree, it suffices to check the monomials in . Throughout this paper, we will assume that all the monomials in have degree . The definition below is defined in [CN1] and we will use a similar concept on the duals later.
Definition 2.9 ([Cn1, Definition 3.1]).
Let be a polynomial ring over a field and be a monomial ideal in . Let be a map that sends to where and are (possibly) additional variables. By abuse of notation, we use the same symbol to denote the substitution homomorphism , given by and . We call a specialization map and the monomial ideal the specialization of .
Here is an example of the specialization of an ideal.
Let . Consider the monomial ideal generated by
in . The specialization of is the ideal
Since the specialization map sends both and to the same element, has minimal generators while has .
Motivated by this example, we consider the following class of monomial ideals of degree .
Definition 2.11 ([Cn1, Definition 3.4]).
Let be a partition with , and . Let be a vector with
Since , in particular, . The ideal
is called a generalized Ferrers ideal.
When is the zero vector, we get back the original Ferrers ideal . On the other hand, when for , the generalized Ferrers ideal and its specialization have the same number of minimal generators, since no colliding phenomenon as in Example 2.10 will ever happen. The common number of minimal generators is simply . We will investigate similar patterns in Section 3 and 6. Notice that the most interesting case is when for each . In this situation, the specialization of the generalized Ferrers ideal is a strongly stable ideal, as observed in [CN1].
Let and be the Ferrers ideal for , that is,
Let for . The generalized Ferrers ideal is
Then the specialization map yields the ideal
Note that is a strongly stable ideal in .
3. Toric rings associated to dual ideals
Given , a monomial ideal generated in the same degree, we can define a toric ring, , which is isomorphic to the special fiber ring of . Recall that the special fiber ring of is the subring where is a new variable. Geometrically, the special fiber ring is the homogeneous coordinate ring of the image of a map . There is a natural surjective map . Consequently, we have a short exact sequence
where is generated by all forms such that . Note that is graded. In this section, we work on finding the defining equations of where is the -dual of a monomial ideal generated in the same degree.
The following theorem shows that the special fiber rings of the generalized Newton complementary duals and the given monomial ideals are isomorphic. This is a straightforward generalization of Costa and Simis [CS, Lemma 1.7].
Let be a polynomial ring in variables over a field . Let be an -determined monomial ideal such that is generated in the same degree. Then the special fiber ring of and that of are isomorphic:
We give a proof for completeness.
For the given vector , we write instead of to simplify the notation. Suppose . Let be the degree piece of the kernel ideal above. If is a non-decreasing sequences of integers in , we write and . By [Taylor], is generated by polynomials of the form
Since is again a monomial ideal, there is a surjective map given by . Let be the kernel of . Likewise, its degree piece is generated by polynomials of the form
where and . One notice immediately that for ,
To show that the two special fiber rings are isomorphic, we define the natural map
since . Thus, and in turn . This also induces a map
which is well-defined.
Since , we can define similar maps and as above. By the same argument, we have that . Since and are obviously inverse maps, we have . Thus . Similarly, . Thus we have
The work of Corso, Nagel, Petrovic̀, and Yuen [CNPY] considered the special fiber ring of the specialization of the generalized Ferrers ideal with and , such that for each . For this purpose, consider the polynomial ring
Let . Thus, we can think of as an matrix with the variable as the entry, when ; otherwise, the entry is . The symmetrized matrix is the matrix obtained by reflecting along the main diagonal. Notice that and is upper-triangular.
Theorem 3.2 ([Cnpy, Theorem 4.2 and Proposition 4.1]).
Let be a specialization of generalized Ferrers ideal. The special fiber ring of is a determinantal ring arising from the minors of a symmetric matrix. More precisely, there is a graded isomorphism
by using notations above. Furthermore, the ring is a Koszul normal Cohen-Macaulay domain of Krull dimension .
By the above theorem and Theorem 3.1, we can describe the special fiber rings of generalized Newton complementary duals of specialization of generalized Ferrers ideals.
Let be an -determined specialization of generalized Ferrers ideal, and let be the -dual of . The special fiber ring of is a determinantal ring arising from the minors of a symmetric matrix. More precisely, there is a graded isomorphism
by using notations above. Furthermore, is a Koszul normal Cohen-Macaulay domain of Krull dimension .
4. Properties of the generalized Newton complementary dual
In this section, we provide additional nice properties of the generalized Newton complementary dual. Within this section, we will always consider the co-lexicographic total order on the monomials in of degree : we will say if there is a such that , while for . For our monomial ideal generated in degree , we will always assume that . The following observation is easy to verify.
Let be a (strongly) stable ideal generated in the same degree as above. Then the sub-ideal is also (strongly) stable.
Let be a polynomial ring in variables over a field . Let be an -determined monomial ideal such that is generated in the same degree . If is stable with , or is strongly stable, then the -dual has linear quotients. In particular, has a linear resolution.
Without loss of generality, we assume that . Write as above. By induction, it suffices to show that the colon ideal
is linear. Since , we may write with . Write
Notice that is stable. Thus . This means that . We want to show that . This, in particular, implies that is linear.
Notice that for each with and , it can be translated into . Thus . In turn, .
Now, take a minimal monomial generator of . By definition, for some integer and some monomial . This is equivalent to saying that . By the minimality of , one has . Thus, divides .
Suppose that is stable with . If is not linear, this reduces to . But we already have . This contradicts the minimality of .
Suppose that is strongly stable with . Whence, . As divides , for any , we have . Thus, by the minimality of , we have , unless for some . In the latter case, as and , we will have , which is a contradiction.
Therefore, we have shown that .
The “in particular” part follows from [HH, Proposition 8.2.1]. ∎
The following example shows that for ideals generated by elements of degree greater than , the strongly stable condition is necessary.
Let be the ideal generated by
in . It is not difficult to verify that is stable, but not strongly stable. The Newton complementary dual of does not have linear quotients. Indeed, computation by Macaulay2 [M2] suggests that does not have a linear resolution.
Next, we examine the -duals of edge ideals associated to bipartite graphs. More precisely, we focus on the Newton complementary dual when is such an edge ideal; whence, . For simplicity, we say Newton-dual instead of Newton complementary dual. We begin by recalling several definitions and results about edge ideals and graphs, and then consider a connection among squarefree monomial ideals, Alexander Duals and Newton-duals.
Let be the polynomial ring on variables. Suppose that is a finite simple graph (that is, a graph that does not have loops or multiple edges) with vertices labeled by . The edge ideal of , denoted by , is the ideal of generated by the squarefree monomials such that is an edge of . This gives a one-to-one correspondence between finite simple graphs and squarefree monomial ideals generated in degree .
The complement of a graph , is the graph with identical vertex set such that its edge set contains the edge if and only if is not an edge of .
For a subset , let . Note that any squarefree monomial in can be written in this way. Let be the prime ideal . For any squarefree monomial ideal , the Alexander dual of is
In Theorem 4.4, we will study the relation between the Alexander dual and the Newton-dual for bipartite graphs. Let be a bipartite graph with respect to vertex partition . Let be the edge ideal. To remove isolated vertices, let
and we similarly define . Let be the complement graph of with respect to the vertex set . We may think of it as the essential complement of . Meanwhile, we sometimes write for the edge ideal of .
Let be a finite bipartite graph corresponding to the vertex partition , and the associated edge ideal in the polynomial ring . Then
Without loss of generality, we may assume that , and . By the definitions of and the Alexander dual, it suffices to show that a primary decomposition for is given by
We proceed by induction on , the minimal number of generators of .
The base case when is trivial. For the general case when , we remove one edge from . Without loss of generality, we may assume the edge is . After this removal, we obtain the subgraph and its associated edge ideal . Now, or , and or . Consequently, we have the following four cases.
If and , while . Consequently,
By the induction hypothesis, we have
Write . This monomial belongs to every component in the above decomposition. Thus, we have
If and , while . Consequently,
By induction hypothesis, we have