The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear resolution, as do all of its powers, if and only if the colouring satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular resolution supported on a cubical complex. From this, we give the graded Betti numbers in terms of the face-vector of the underlying simplicial complex. Moreover, we explicitly describe the Boij-Söderberg decompositions of both the ideal and its quotient. We also give explicit formulæ for the codimension, Krull dimension, multiplicity, projective dimension, depth, and regularity. Further still, we describe the associated primes, and we show that they are persistent.
Key words and phrases:Monomial ideal, linear resolution, cellular resolution, Betti numbers, simplicial complex, vertex colouring, face ideal
2010 Mathematics Subject Classification:13F55, 05E45, 13D02, 05C15, 06A12
One method of generating ideals with specific properties, e.g., linear resolutions or persistent associated primes, is to construct an ideal from a combinatorial object which has structure that can be exploited to force the desired properties on the ideal. A classical approach to generating ideals has been the Stanley-Reisner correspondence that associates to a simplicial complex on vertices a squarefree monomial ideal in an -variate polynomial ring (see, e.g., the books of Herzog and Hibi , Miller and Sturmfels , and Stanley ). For example, Eagon and Reiner [14, Corollary 5] showed that if the Alexander dual of the simplicial complex is pure and shellable, both of which are combinatorial conditions, then the associated Stanley-Reisner ring has a linear resolution and the Betti numbers can be derived from the -vector of .
In an alternate use of squarefree ideals, Herzog and Hibi  studied the Hibi ideal of a poset . The Hibi ideal is in a polynomial ring with two variables for each element: one each to encode the presence and absence of the element from an order ideal. They showed that every power of a Hibi ideal has a linear resolution, and the Betti numbers and the primary decomposition of the Hibi ideal can be described in terms of structural properties of the poset.
More recently, Biermann and Van Tuyl  constructed a new simplicial complex from a given simplicial complex and proper vertex colouring of . This new complex is pure and vertex-decomposable (hence shellable), and its -vector is the -vector of the original complex. Hence using Eagon and Reiner’s aforementioned result, they showed that has a linear resolution with Betti numbers derived from the -vector of independently from the colouring . As it turns out (see Section 4.2), if the colouring is the collection of singletons of the vertices of , then the ideal is in a polynomial ring with two variables for each vertex: one each to encode the presence and absence of the vertex from a face of .
Given a simplicial complex and an ordered proper vertex colouring of , we follow the approach of using two variables to encode the presence and absence of an element from a given subset to define the uniform face ideal of with respect to , which we denote ; see Definition 4.2. While is generated in a single degree, as in the above cases, it is not generally a squarefree ideal, contrary to the above cases. However, is squarefree precisely when the colouring is the collection of singletons of the vertices of ; in this case, is one of the ideals studied by Biermann and Van Tuyl .
This manuscript is devoted to the study of the family of uniform face ideals. In particular, we show that the presence of several ideal properties are equivalent to the colouring being of a special type. We define a nested proper vertex colouring to be an ordered colouring so that the links of the vertices of a given colour are linearly ordered by containment (Definition 3.2); this is a simplicial analogue of the graph theoretic concept defined by the author . Francisco, Mermin, and Schweig  defined the -Borel property for a poset , which is a generalisation of the Borel property. We define a poset such that is -Borel precisely when is a nested -colouring (Theorem 5.2). Using this connection, we show that the product of two uniform face ideals coming from nested colourings is also a uniform face ideal coming from a nested colouring (Corollary 5.3).
One desired property that an ideal generated in a single degree may enjoy is having a linear resolution, that is, all minimal syzygies are linear. Biermann and Van Tuyl , Corso and Nagel  and , Herzog and Hibi , Nagel and Reiner , and Nagel and Sturgeon  each studied squarefree monomial ideals associated to some combinatorial structure (simplicial complexes, posets, or Ferrers hypergraphs) that have linear resolutions. Indeed, the ideals studied by Herzog and Hibi have the additional property that their powers always have linear resolutions. This need not always happen; indeed, Sturmfels  gave an example of a squarefree monomial ideal in six variables that has a linear resolution but whose second power does not. We show here that the uniform face ideal has a linear resolution precisely when is a nested colouring. Since products of uniform face ideals coming from nested colourings are also uniform face ideals coming from nested colourings, all powers of uniform face ideals coming from nested colourings also have linear resolutions (Theorem 6.8).
In each of , , , and , the -graded—and hence total—Betti numbers of the studied squarefree monomial ideal can be derived from underlying properties of the associated combinatorial structure. We similarly describe the -graded Betti numbers of all uniform face ideals coming from nested colourings using the -vector of the simplicial complex (Theorem 7.11). Nagel and Sturgeon also gave the Boij-Söderberg decomposition (see, e.g., ) of the Betti tables of both the ideal they studied and its quotient. Using this, they classified the Betti tables possible for ideals with a linear resolution (see also ). In our case, we provide the explicit Boij-Söderberg decomposition—with combinatorial interpretations of the coefficients—for the Betti table of both (Proposition 8.3) and (Proposition 8.7).
Another useful property that an ideal may enjoy is having a minimal free resolution supported on a CW-complex, as described by Bayer and Sturmfels . Velasco  showed that not all monomial ideals have such a resolution. Despite this, many classes of monomial ideals do have cellular resolutions. Indeed, Nagel and Reiner  described a cellular resolution of several ideals associated to Ferrers hypergraphs; with a similar approach, Dochtermann and Engström  described a cellular resolution of edge ideals of cointerval hypergraphs. Recently, Engström and Norén  gave cellular resolutions for all powers of certain edge ideals. We give a cellular resolution for a uniform face ideal coming from a nested colouring that is supported on a cubical complex (Theorem 7.10).
Moreover, it is desirable for an ideal to have persistent associated primes, that is, for . Francisco, Hà, and Van Tuyl  described the associated primes of the cover ideals of graphs. Using this, they showed that the cover ideals of perfect graphs have persistent associated primes. Martínez-Bernal, Morey, and Villarreal  showed that all edge ideals of graphs (i.e., quadratically generated squarefree monomial ideals) have persistent primes. Further, Bhat, Biermann, and Van Tuyl  described a new family of squarefree monomial ideals, the partial -cover ideal of a graph, which also have persistent associated primes. In each case, the persistence is established via exploitation of the underlying combinatorial structure. We similarly exploit the structure of a nested colouring to describe the associated primes of a uniform face ideal coming from a nested colouring (Corollary 9.8), and thus show that they have persistent associated primes (Theorem 9.12).
The remainder of the manuscript is organised as follows. In Section 2, we recall the necessary algebraic and combinatorial definitions. In Section 3, we describe nested colourings of simplicial complexes, as well as the connection to the graph theoretic analogue. In Section 4, we define the uniform face ideal, and we consider the special case of the singleton colouring. In Section 5, we classify the presence and absence of various exchange properties of ideals. In Section 6, we study the first syzygies of a uniform face ideal. For the remainder of the manuscript, we consider only nested colourings. In Section 7, we give a minimal cellular resolution of a uniform face ideal, and we also give the -graded Betti numbers. In Section 8, we demonstrate the Boij-Söderberg decompositions of both a uniform face ideal and its quotient. In the final section, Section 9, we classify several algebraic properties of , including giving explicit formulæ for the codimension, Krull dimension, multiplicity, projective dimension, depth, and regularity. We also consider the Cohen-Macaulay and unmixed properties, and we describe the associated primes, which we show to be persistent.
In this section, we recall the necessary algebraic and combinatorial background that will be used throughout this manuscript.
2.1. Free resolutions & derivative algebraic invariants
Let be the -variate polynomial ring over a field , and let be a finitely generated graded -module, e.g., a homogeneous ideal of or its quotient . The -module given by the relations is the twist of . A minimal free resolution of is an exact sequence of free -modules of the form
where , , is the free -module
The numbers are the graded Betti numbers of , and is the total Betti number of . The -module has a -linear resolution if for all and .
The Poincaré polynomial of is , and the Hilbert series of is the formal power series . If we write , where , then is the -polynomial of and is the Krull dimension of . The codimension of is . Finally, the multiplicity (or degree) of is .
The regularity of is . The projective dimension of is , that is, is the length of the minimal free resolution of . The depth of is the length of the longest homogeneous -sequence, denoted . By the Auslander-Buchsbaum formula, , we may more easily compute the depth of . The -module is Cohen-Macaulay if .
2.2. Simplicial complexes
For a positive integer , we define to be the set . A simplicial complex on the vertex set is a collection of subsets of closed under inclusion. The elements of are faces, and the maximal faces are facets. The dimension of a face is , and the dimension of is the maximum dimension of its faces. The -vector (or face vector) of is the -tuple , where and is the number of faces of dimension in .
The Alexander dual of a simplicial complex on is the simplicial complex on with faces , where . For any face of , the link of in is the simplicial complex
Let be a simplicial complex on . The Stanley-Reisner ideal of is the ideal of , where . It is well-known that the Stanley-Reisner ideals are precisely the squarefree monomial ideals. The quotient ring is the Stanley-Reisner ring of .
A simple graph is a vertex set , e.g., , together with a collection of edges , i.e., -subsets of . Two vertices and of are adjacent if is an edge of . The neighbourhood of in is the set of vertices adjacent to in . A subset of the vertices of is an independent set of if no two vertices in are adjacent. Similarly, is a clique of if every pair of vertices in are adjacent.
A partition of is a proper vertex -colouring of if the are independent sets of ; in this case, the are the colour classes of . The chromatic number of is the least integer such that a proper vertex -colouring of exists.
There is a natural graph associated to a simplicial complex on . The underlying graph (or -skeleton) of is the simple graph on vertex set with an edge between and if and only if . Moreover, there are a pair of natural simplicial complexes associated to a simple graph . The clique complex of is the simplicial complex of cliques of , and the independence complex of is the simplicial complex of independent sets of . Notice that the minimal non-faces of are the non-edges of , and further the minimal non-faces of are the edges of . Hence the independence complex of is the clique complex of the complement of . Moreover, the Stanley-Reisner ideal of a clique (independence) complex is generated by quadrics; these are precisely the flag complexes. We also note that for any graph .
A poset is a set endowed with a partial order that is antisymmetric, reflexive, and transitive. A relation in is a covering relation if implies either or . For any two elements and of , an element of is the meet of and if it is the unique element of such that and . Further, is a meet-semilattice if every pair of elements of has a meet.
A subposet of is an order ideal of if and in implies . The interval of and in is the subposet . For any , the -boolean poset (or -boolean lattice) is the poset of all subsets of ordered by inclusion. A meet-semilattice is meet-distributive if every interval of such that is the meet of the elements of covered by is isomorphic to the -boolean poset.
3. Colouring simplicial complexes
In this section, we define a colouring of a simplicial complex, and further define a special family of proper vertex colourings.
3.1. Proper vertex colourings
The following concepts are simplicial analogues of some common graph theoretic concepts. A set is an independent set of if no two members of are in a common face of . A partition of is a proper vertex -colouring of if the are independent sets of ; in this case, the are the colour classes of . The chromatic number of is the least integer such that a proper vertex -colouring of exists.
It is clear that the proper vertex colourings of are precisely the proper vertex colourings of .
Let be a simplicial complex on . If is a partition of , then is a proper vertex colouring of if and only if it is a proper vertex colouring of .
In particular, .
This follows immediately as the edges of are precisely the -faces of . ∎
3.2. Nested colourings
As defined and studied in , an independent set of a finite simple graph is nested if the vertices of can be linearly ordered so that implies ; such an order is a nesting order of . A proper vertex colouring of is nested if every colour class of is nested. The nested chromatic number is the least integer such that a nested -colouring of exists.
We define here the simplicial analogue of a nested colouring. The link of a vertex in a simplicial complex will play the role of the neighbourhood of a vertex in a graph.
Let be a simplicial complex, and let be vertices of . An independent set of is nested if the vertices of can be linearly ordered so that implies ; such an order is a nesting order of . A proper vertex -colouring of is nested if every colour class of is nested. The nested chromatic number is the least integer such that a nested -colouring of exists.
The nesting orders on the vertices of a nested independent set are the same, up to permutations of vertices that have precisely the same link. Such vertices are indistinguishable except for their label.
Let . There are proper vertex colourings of of which are nested. For example, is a nested colouring of ; thus . However, is a non-nested colouring of as . See Figure 3.1 for illustrations.
Exchanging a vertex of a face for a lesser (in the nesting order) vertex of the same colour generates another face of the simplicial complex. This gives an alternate, and perhaps more useful, condition on a partition of the vertices that is equivalent to being a nested colouring.
Let be a simplicial complex on . If is a partition of , then is a nested -colouring if and only if there is an ordering on the vertices of each class such that if is less than in that order, and is a face of containing , then is a face of .
This follows immediately since is a face of containing if and only if is a face of . ∎
Nested colourings of are nested colourings of ; the converse holds when is the flag.
Let be a simplicial complex on . If is a nested colouring of , then is a nested colouring of , and the converse holds when is flag.
In particular, , and equality holds when is flag.
By Proposition 3.4, if in , and is a face of , then is a face of . In particular, if in , and is an edge of , then is an edge of . By [8, Proposition 2.16], the latter is equivalent to being a nested colouring of .
There exist nested colourings of that are not nested colourings of , and the nested chromatic numbers need not be the same.
First, recall given in Example 3.3. This complex is not flag, as . However, the nested colourings of are precisely the nested colourings of .
Now, let . In this case, but . In particular, is a nested colouring of . See Figure 3.2 for illustrations.
Most of the results about nested colourings of graphs in  have analogues for nested colourings of simplicial complexes. For instance, Proposition 3.4 is the simplicial analogue of [8, Proposition 2.16]. Moreover, since the nested chromatic number of a simplicial complex can be computed as the Dilworth number of a poset, as in [8, Corollary 2.19], we see that computing the nested chromatic number of a simplicial complex can be done in polynomial time; see [8, Theorem 2.21] for the graph theoretic analogue.
4. The uniform face ideals
In this section, we introduce the objects of interest in this manuscript: the uniform face ideal of a simplicial complex with respect to a proper vertex colouring of the complex. We further study a specific nested colouring that has many nice properties.
4.1. The uniform face ideals
A colouring of a simplicial complex is ordered if each colour class is endowed with a linear order on the vertices in the colour class. In this case, we may use a vector to identify the faces of a simplicial complex.
Let be a simplicial complex, and let be an ordered proper vertex -colouring of . For a face of , the index vector of with respect to is the vector , where if and if .
The index vector of with respect to any colouring is the zero vector, and the index vectors with precisely one nonzero entry are the index vectors of vertices.
We are now ready to define the object of interest in this manuscript.
Let be a simplicial complex on , and let be an ordered proper vertex -colouring of . Let be the polynomial ring in variables over the field . For any face , the uniform monomial of with respect to is the monomial
in , where is the index vector of with respect to . Further, the uniform face ideal of with respect to is the -ideal
The uniform monomial of with respect to any colouring is . Clearly, is a lower bound for the number of necessary variables to construct a uniform face ideal of , and empty colour classes have no effect on the generators of but do increase the number of variables in the polynomial ring.
Let as given in Example 3.3. Since has a total of faces, the associated uniform face ideals will each have monomial generators.
Consider the nested colouring of given in Example 3.3. Since is a -colouring, we let , and further
Further, recall the non-nested colouring , as given in Example 3.3. Since is a -colouring, we consider . In this case, we have
Recall that a pair of nesting orders on the vertices of a nested colouring differ only by permuting vertices with the same link. This implies that all nesting orders produce the same uniform face ideal, and so we will henceforth refer to the nesting order on the nested colouring.
Let be a simplicial complex, and let be a nested colouring of . If and are copies of endowed with (possibly distinct) nesting orders on the colour classes, then .
Since the nesting order on each class is independent of the nesting orders of the other classes, we may assume the nesting orders on and only differ for . Moreover, since all nesting orders are the same up to the permutation of vertices with the same link, we may assume that the nesting order on differs only for two vertices and such that and no other vertex of is between and in both orders.
Suppose and are the and vertices, respectively, in in . Hence and are the and vertices, respectively, in in . Since if and only if , for any face of , we have that is an index vector of a face in under if and only if is an index vector of a face in under . Similarly, this holds if we replace with . As the uniform face ideals are constructed using the index vectors, and the set of index vectors for with respect to and are the same, the uniform face ideals are also the same. ∎
Moreover, the product of uniform face ideals is again a uniform face ideal. However, the resultant simplicial complex depends on the colouring, ordering, and labeling of the factors.
Let and be simplicial complexes. If and are proper vertex -colourings of and , respectively, then for some simplicial complex and proper vertex -colouring of .
Let , where for .
Notice that for any simplicial complex and proper vertex -colouring , by the construction of , the sum of the exponents of and in is , for every and every minimal generator of . Hence the sum of the exponents of and in is , for every . Thus , where is some subset of .
Let be the set of subsets of such that , for some and . Clearly then, if is a simplicial complex, then is a proper vertex -colouring of and .
Since and , we have . Let be any member of that is not empty, and suppose , where and . Let be any vertex in , and let be the index so that . Since and , . Notice that is with the exponent on changed to and the exponent on reduces to zero. That is, . Thus , and is closed under inclusion. Therefore, is a simplicial complex. ∎
Let and be simplicial complexes on and vertices, respectively, with nested colourings and , respectively. We then have , where and . Note that is a nested colouring of ; in Corollary 5.3, we show that this always occurs.
4.2. The singleton colouring
Let be a simplicial complex on . The proper vertex -colouring of with singleton colour classes, that is, , is the singleton colouring of . Clearly, the singleton colouring is a nested colouring. In this case, the index vector of any face of with respect to can be seen as the bit-vector encoding the presence of the vertices in .
The uniform face ideal is squarefree precisely when the colour classes of have cardinality at most one, i.e., is together with empty colour classes. Thus is the Stanley-Reisner ideal of a simplicial complex, which turns out to be related to a simplicial complex previously studied.
Given a proper vertex colouring of a simplicial complex , Biermann and Van Tuyl  defined a new simplicial complex with many nice properties. We recall their construction here.
[2, Construction 3] Let be a simplicial complex on , and let be a proper vertex -colouring of . Define to be the simplicial complex on with faces , where and is any subset of such that for all we have .
The preceding construction was implicitly introduced by Björner, Frankl, and Stanley [4, Section 5]. It was more recently introduced independently by Frohmader [22, Construction 7.1]. Biermann and Van Tuyl [2, Remark 4] have noted connections to related constructions.
Under an appropriate relabeling, is the Stanley-Reisner ideal of .
Let be a simplicial complex on . If is the Stanley-Reisner ideal of , where and are associated to and , respectively, for , then .
By construction, the minimal non-faces of are precisely the complements of the facets of . Further, by construction of , the facets are of the form , where and . The complement of is . Thus we have if and only if is in , and so the monomial associated to in is
This is precisely the uniform monomial of with respect to . ∎
In [2, Theorem 13], is shown to have a linear resolution with Betti numbers easily described by the -vector of , as we will see in Theorem 7.11. Moreover, a special case of Construction 4.7 was explored by the author with U. Nagel  in the case of flag complexes. As will be seen in Corollary 9.10, the squarefree uniform face ideals that are unmixed are precisely the ones coming from flag complexes.
Olteanu  defined the monomial ideal of independent sets of a graph which is the Stanley-Reisner ideal of . Thus the monomial ideal of independent sets of a graph is the uniform face ideal . Olteanu [36, Corollary 2.3] showed that this ideal always has a linear resolution and further gave explicit formulæ for the regularity, Betti numbers, projective dimension, and Krull dimension. Moreover, the presence of the Cohen-Macaulay property is classified therein. We note that these results all correlate with the results given in Theorems 6.8 and 7.11 and in Section 9.1.
5. Exchange properties of ideals
In this section, we classify precisely when the uniform face ideal of a simplicial complex with respect to a colouring has one of a variety of exchange properties of ideals. Through this we see that nested colourings force a nice structure on the uniform face ideal.
5.1. Stable and strongly stable
Let , where is a field. For a monomial , the maximum index of is the largest index such that divides . A monomial ideal is stable if is in for every monomial and each . Further still, a monomial ideal is strongly stable if is in for every monomial and such that divides . Clearly, if is strongly stable, then is stable. Furthermore, it suffices to only consider the monomials that minimally generate .
Eliahou and Kervaire  proved stable ideals have minimal free resolutions that are very easy to describe. Hulett  proved that the -graded Betti numbers can be easily derived from the maximum indices of the minimal generators. In particular, if a stable ideal is generated by monomials in one degree, then it has a linear resolution.
A uniform face ideal is stable (and strongly stable) precisely when is a -colouring. Note that -colourings are always nested.
Let be a simplicial complex on , and let be a -colouring of without trivial colour classes. The following statements are equivalent:
is strongly stable, and
is a -colouring.
Clearly, condition (ii) implies condition (i). Further, if is a -colouring, then , which is strongly stable. Thus condition (iii) implies condition (ii).
Suppose condition (i) holds, i.e., is stable. Without loss of generality, assume . Since is in , regardless of and , then . By the structure of , this implies that , i.e., condition (iii) holds. ∎
Let , where is a field. A monomial ideal is Borel if it is fixed under the action of the Borel group. If the characteristic of is zero, then a monomial ideal is strongly stable if and only if it is Borel (see, e.g., [25, Proposition 4.2.4]). Hence the Borel uniform face ideals are precisely those which come from -colourings.
Recently, Francisco, Mermin, and Schweig  generalised the Borel property using posets. Let be a poset on . A monomial ideal in is -Borel if is in for every monomial and each in