1 Introduction

Systems of parameters play an important role in dimension theory. As a consequence of Krull’s generalized principal ideal theorem, it can be seen that in a Noetherian local ring \((R,{\mathfrak m})\) with \(\dim R=d\), there exist elements \(f_1,\ldots ,f_d\in {\mathfrak m}\) with \(\dim R/(f_1,\ldots ,f_d)=0\). Such a sequence of elements of R is called a system of parameters, or sop for short. A similar statement holds for standard graded K-algebras with K a field. In our applications, we mainly consider such algebras.

One of the central problems in Combinatorial Commutative Algebra is to show that a certain K-algebra attached to a combinatorial object is Cohen–Macaulay. Usually, the Cohen–Macaulay property has a nice combinatorial interpretation. In the case that the defining ideal of the algebra is a monomial ideal, Hochster’s formula [10] and its extension by Takayama [17] are powerful tools to investigate the homological properties of the algebra. In the case that the defining ideal is a binomial prime ideal, one may use the squarefree divisor complex [3] or one may use Gröbner basis theory to reduce the problem to the case of monomial ideals.

In this paper, we propose another approach which is based on the basic fact that R is Cohen–Macaulay if and only if one (equivalently all) of the sop’s of R form(s) a regular sequence. This approach confronts us with two problems. The first problem is to find a suitable sop, and the second is to decide whether the given sop forms a regular sequence. Regarding the first problem, Stanley [15, Proposition 4.3] finds an explicit special sop for the Stanley–Reisner ring of any balanced simplicial complex. This also includes the order complexes. In the cases considered here, we also use special sop’s.

In the first section of this paper, however, we first deal with the second problem. Based on results of Serre [13], see also [2, Theorem 4.6.10], one has a numerical condition for when a sop is a regular sequence. Indeed, let \(f_1,\ldots ,f_d\in {\mathfrak m}\) be a sop of R and let \({\overline{R}}=R/(f_1,\ldots ,f_d)\). Then, denoting by e(M) the multiplicity of an R-module M, one has \(e({\overline{R}})\ge e(R)\), and if \(e({\overline{R}})= e(R)\), then \(f_1,\ldots ,f_d\) is a regular sequence. (Equivalently, R is Cohen–Macaulay.) Moreover, if \((f_1,\ldots ,f_d)\) is a reduction ideal of \({\mathfrak m}\) and \(f_1,\ldots ,f_d\) is a regular sequence, then \(e({\overline{R}})= e(R)\), see Proposition 1.1. There is also a graded version of this criterion, see Proposition 1.2. In the case that R is a standard graded K-algebra and the sop \(f_1,\ldots ,f_d\) is homogeneous with \(\deg f_i =a_i\), then this sop is a regular sequence if and only if \(e(\overline{R})=a_1a_2\cdots a_d e(R)\). By a lack of good references, we provided the detailed proofs of these results.

In Proposition 1.3 we give in the graded case a measure for the difference \(e({\overline{R}})-e(R)\). As a consequence, we obtain in Corollary 1.4 the result that if the sop \(f_1,\ldots ,f_d\) is a superficial sequence, and \(R/(f_1,\ldots ,f_r)\) is Cohen–Macaulay for some \(r<d\), then R is Cohen–Macaulay.

In Sect. 2 we study a class of posets and their order complexes as well as König graphs by means of sop’s. We consider a poset P which as a set is the disjoint union of two sets \(C_1\) and \(C_2\), where \(C_1:x_1<x_2<\cdots <x_n\) and \(C_2:y_1<y_2<\cdots <y_n\) are maximal chains in P. For such a poset, the sequence \(x_1-y_1, \ldots , x_n-y_n\) is a sop of the Stanley–Reisner ring \(K[\Delta (P)]\), where \(\Delta (P)\) denotes the order complex of P. The covering relations \(x_i\lessdot y_j\) in P we call the diagonals of P. The Cohen–Macaulay property of \(K[\Delta (P)]\) can be expressed in terms of the diagonals of P. Indeed, in Theorem 2.1 it is shown \(K[\Delta (P)]\) is Cohen–Macaulay if and only if it is pure shellable, and that this is equivalent to the condition that the diagonals of P satisfy the following conditions: (i) if \(x_i\lessdot y_j\) or \(y_i\lessdot x_j\), then \(j=i+1\), and (ii) \(\{x_i,y_{i+1}\}\notin \Delta (P)\) implies that \(\{x_{i+1},y_i\}\in \Delta (P)\). In a similar fashion, it can be characterized when \(I_{\Delta (P)}\) has a linear resolution, see Proposition 2.2.

Note that \(I_{\Delta (P)}\) may be viewed as the edge ideal I(G) of a suitable bipartite graph G. So, the question arises for which graphs G can we find a sop \(f_1,\ldots , f_d\) of K[V(G)]/I(G), where each \(f_i\) is just a difference of two variables, like we have it for \(K[\Delta (P)]\). The advantage of such sop’s is that after reduction they preserve the monomial structure and just identify vertices. The surprising answer to the above question is that a graph G admits such a special sop if and only if G is a König graph. In fact, this is a corollary of a more general theorem. Let \(I\subset S\) be a monomial ideal in the polynomial ring \(S=K[x_1,\ldots ,x_n]\) over the field K in n variables. We denote by \({\text {m-grade}}(I)\) the maximal length of a regular sequence of monomials in I and call this number the monomial grade of I. One has \({\text {m-grade}}(I)\le {\text {grade}}(I)={\text {height}}(I)\). We call I a monomial ideal of König type if \(I\ne 0\) and \({\text {m-grade}}(I)={\text {height}}(I)\). The naming is justified by the fact that if \(I=I(G)\) for some graph G, then \({\text {height}}(I)=\tau (G)\) and \({\text {m-grade}}(I)=\nu (G)\), so that the edge ideal of a graph G is a monomial ideal of König type if and only if G is a König graph. Now our Theorem 2.3 says that a monomial ideal \(I\subset S=K[x_1,\ldots ,x_n]\) is a monomial ideal of König type if and only if S/I admits a sop \(f_1,\ldots ,f_d\), where each \(f_k\) is of the form \(x_i-x_j\) for suitable i and j.

Applied to graphs, this result reads as follows: let G be a graph without isolated vertices, \(S=K[V(G)]\) and for any edge \(e=\{x,y\}\in E(G)\), let \(f_e=x-y\) be an element in S. Then, G is a König graph if and only if there exists a subset \(\{e_1,\ldots ,e_d\}\) of edges of G such that \(f_{e_1},\ldots ,f_{e_d}\) is a sop for \(R=S/I(G)\). This sop has the nice property that \({\text {reg}}(R/(f_{e_1},\ldots ,f_{e_d})R)\le {\text {reg}}(R)\), as shown in Theorem 2.6.

For a graph G, we denote by \({\text {mi}}(G)\) the number of maximal independent sets of G. It is an important problem in graph theory to give upper bounds for \({\text {mi}}(G)\). For a König graph it was shown in [9, Corollary 3.4] that \(2^{\nu (G)}\) is an upper bound for \({\text {mi}}(G)\) where \(\nu (G)\) denotes the maximum size of matchings of G, and in [1, Theorem 1] it was proved that \({\text {mi}}(G)\le M(G)+1\), where M(G) is the number of induced matchings in G. By using our special sop for unmixed König graphs, we give a stronger bound for \({\text {mi}}(G)\) and at the same time provide a combinatorial criterion for the Cohen–Macaulay property for unmixed König graphs. A different combinatorial characterization of Cohen–Macaulay König graphs is known from [4, Proposition 28]. Our result (Theorem 2.7) is a follows: Let G be a König graph and \(\{e_1,\ldots ,e_m\}\) be a maximal matching of G with \(\nu (G)=m\), and let k be the number of induced matchings of G contained in \(\{e_1,\ldots ,e_m\}\). Then, \({\text {mi}}(G)\le k+1\) and equality holds if and only if G is a Cohen–Macaulay graph.

In the last section, we introduce a sop for the Stanley–Reisner ring of any simplicial complex \(\Delta \). We call it the universal sop of \(K[\Delta ]\) because it is built in a uniform way for all simplicial complexes, and its construction does not depend on the base field K. The price we have to pay for this is that this is not a sop of linear forms; instead, it is defined as follows: \(p_i(\Delta )=\sum _{\begin{array}{c} F\in \Delta \\ |F|=i \end{array}}\prod _{j\in F}x_j\) for \(i=1,\ldots ,\dim \Delta +1\), see Theorem 3.1. By using Proposition 1.2, we obtain a Cohen–Macaulay criterion for \(K[\Delta ]\) in terms of this sop. This turns out to be a useful computational tool to check Cohen–Macaulayness, as we demonstrate at the example of a chessboard complex.

2 Criteria of Cohen–Macaulayness in terms of systems of parameters

In this section, we collect some results on sop’s which all are based on results of Serre (see [2, Theorem 4.6.10]) and which in terms of multiplicities allow to check whether a ring or a module is Cohen–Macaulay. One of the first efficient applications of these criteria was given by the first author of this paper in order to study the conormal module and the module of differentials of a K-algebra, see [7].

Proposition 1.1

Let R be a Noetherian local ring (or a standard graded K-algebra) with (graded) maximal ideal \({\mathfrak m}\), and let \(I\subset R\) be an ideal generated by a (homogeneous) sop of R. Then,

  1. (a)

    \(e(R/I)=\ell (R/I)\ge e(R)\).

  2. (b)

    If \(e(R/I)= e(R)\), then R is Cohen–Macaulay.

  3. (c)

    If I is a reduction ideal of \({\mathfrak m}\) and R is Cohen–Macaulay, then \(e(R/I)= e(R)\).

If (b) holds, then the sop which generates I is a regular sequence. In particular, \(r(R)=r(R/I)\), and so R is Gorenstein if and only if R/I is Gorenstein. (Here, we denote by r(M) the (Cohen–Macaulay) type of a Cohen–Macaulay module M.)

Proof

For the proof we recall a few facts: Let \(M\ne 0\) be a finitely generated R-module of dimension d and \(I\subseteq {\mathfrak m}\) be an ideal with \(\dim M/IM=0\). Then,

$$\begin{aligned} e(I,M)=\lim _{k\rightarrow \infty }(d!/k^d)\ell (M/I^{k+1}M) \end{aligned}$$

is called the multiplicity of M with respect to I. The multiplicity of M, denoted e(M), is the multiplicity of M with respect to \({\mathfrak m}\).

Obviously, if \(I\subseteq J\subseteq {\mathfrak m}\), then \( e(I,M)\ge e(J,M). \) In particular,

$$\begin{aligned} e(I,M)\ge e(M). \end{aligned}$$
(1)

On the other hand, if I is a reduction ideal of \({\mathfrak m}\) with respect to M, that is, if \(I{\mathfrak m}^k M={\mathfrak m}^{k+1}M\) for some k, then equality holds in (1), see [2, Lemma 4.5.5].

We also need the following result ([2, Corollary 4.6.11] or [7] where it first appeared: let I be generated by a sop and assume that M has positive rank. Then,

  1. (i)

    \(\ell (M/IM)\ge e(I,M){\text {rank}}M\).

  2. (ii)

    M is Cohen–Macaulay if and only if \(\ell (M/IM)= e(I,M){\text {rank}}M\).

Now we apply these results to the case that \(M=R\). We first notice that \(e(R/I)=\ell (R/I)\), since \(\dim R/I=0\). Next (1) and (i) imply

$$\begin{aligned} \ell (R/I)\ge e(I,R)\ge e(R). \end{aligned}$$
(2)

This proves (a). If \(\ell (R/I)= e(R)\), then (2) implies \(\ell (R/I)= e(I,R)\) and then (ii) yields (b). Finally, if I is a reduction ideal of \({\mathfrak m}\), then \(e(I,R)=e(R)\), and (2) together with (ii) implies (c).

If R is Cohen–Macaulay, then each sop is a regular sequence. \(\square \)

Now we turn to a graded version of Proposition 1.1.

Proposition 1.2

Let R be a standard graded K-algebra with graded maximal ideal \({\mathfrak m}\), and let I be generated by the homogeneous sop \(f_1,\ldots ,f_d\) with \(\deg f_i=a_i\) for \(i=1,\ldots ,d\). Then,

  1. (a)

    \(e(R/I)=\ell (R/I)\ge a_1a_2\cdots a_d e(R).\)

  2. (b)

    R is Cohen–Macaulay if and only if \(e(R/I)= a_1a_2\cdots a_d e(R).\)

If the equivalent conditions given in (b) hold, then \(f_1,\ldots ,f_d\) is a regular sequence. In particular, \(r(R)=r(R/I)\), and R is Gorenstein if and only if R/I is Gorenstein.

Proof

Let \(a=a_1a_2\cdots a_d\) and set \(b_i=a/a_i\) for \(i=1,\ldots ,d\). Then,

$$\begin{aligned} e(f_1^{b_1},\ldots ,f_d^{b_d},R)=b_1b_2\cdots b_de(f_1,\ldots ,f_d,R), \end{aligned}$$

see [16, Proposition 11.2.9]. Moreover, \(\deg f_i^{b_i}= a\) for \(i=1,\ldots ,d\). Therefore,

$$\begin{aligned} (f_1^{b_1},\ldots ,f_d^{b_d}) \subseteq {\mathfrak m}^a. \end{aligned}$$

Thus, since \(e({\mathfrak m}^a,R)=a^de(R)\) ( [16, Proposition 11.2.9]), we obtain that

$$\begin{aligned} b_1b_2\cdots b_de(f_1,\ldots ,f_d,R)=e(f_1^{b_1},\ldots ,f_d^{b_d},R)\ge e({\mathfrak m}^a,R)=a^de(R), \end{aligned}$$
(3)

which together with (i) in the proof of Proposition 1.1 imply the inequality in (a).

(b) Assuming that \(\ell (R/I)= a_1a_2\cdots a_d e(R)\), we obtain together with (3) that

$$\begin{aligned} a_1a_2\cdots a_d e(R)=\ell (R/I)\ge e(f_1,\ldots ,f_d,R)\ge a_1a_2\cdots a_d e(R), \end{aligned}$$

and hence \(\ell (R/I)= e(f_1,\ldots ,f_d,R)\). Thus, (ii) in the proof of Proposition 1.1 implies that R is Cohen–Macaulay. Conversely, suppose that R is Cohen–Macaulay. Then, \(f_1,\ldots ,f_d\) is a regular sequence. Let \({\text {Hilb}}_R(t)=Q_R(t)/(1-t)^d\) be the Hilbert series of R. Then,

$$\begin{aligned} Q_{R/I}(t)={\text {Hilb}}_{R/I}(t)={\text {Hilb}}_R(t)\prod _{i=1}^d(1-t^{a_i})=Q_R(t)(\prod _{i=1}^d(\sum _{j=0}^{a_{i-1}}t^j). \end{aligned}$$

It follows that

$$\begin{aligned} \ell (R/I)=Q_{R/I}(1)=Q_R(1)a_1a_2\cdots a_d=e(R)a_1a_2\cdots a_d. \end{aligned}$$

\(\square \)

The next result is a certain refinement of the statements given in Proposition 1.2. We first recall the following fact (see for example ( [2, Proposition A.4.]): Let \((R,{\mathfrak m})\) be a Noetherian local ring and \(f_1,\ldots ,f_m\) a sequence of elements in \({\mathfrak m}\). Then,

(\(\alpha \)):

\(\dim R\ge \dim R/(f_1,\ldots ,f_m)\ge \dim R-m\), and

(\(\beta \)):

\(\dim R/(f_1,\ldots ,f_m)= \dim R-m\) if and only \(f_1,\ldots , f_m\) can be completed to a sop of R.

A similar statement holds for graded K-algebras.

Proposition 1.3

With the assumptions and notation of Proposition 1.2, let

$$\begin{aligned} U_i={\text {Ker}}(R/(f_1,\ldots ,f_{i-1})\xrightarrow {f_i} R/(f_1,\ldots ,f_{i-1})).\end{aligned}$$

Then,

  1. (a)

    \(\dim U_i\le d-i\) for all i.

  2. (b)

    Set \(\dim U_i=-1\) if \(U_i=0\). Then,

    $$\begin{aligned} e(R/I)=a_1\cdots a_de(R)+\sum _{\begin{array}{c} i=1 \\ \dim U_i=d-i \end{array}}^da_{i+1}\cdots a_de(U_i). \end{aligned}$$

In particular, if \(\deg f_i=1\) for all i, then

$$\begin{aligned} e(R/I)=e(R)+\sum _{\begin{array}{c} i=1 \\ \dim U_i=d-i \end{array}}^de(U_i). \end{aligned}$$

Proof

  1. (a)

    \(U_i\) is a submodule of \(R/(f_1,\ldots ,f_{i-1})\) with \(f_iU_i=0\). Thus, \(U_i\) is a \(R/(f_1,\ldots ,f_{i})\)-module and hence \(\dim U_i\le \dim R/(f_1,\ldots ,f_{i}) =d-i\), where the equation follows from (\(\beta \)).

  2. (b)

    From the following exact sequence

    $$\begin{aligned} 0\rightarrow U_i\rightarrow (R/(f_1,\ldots ,f_{i-1}))(-a_i)\xrightarrow {f_i} R/(f_1,\ldots ,f_{i-1})\rightarrow R/(f_1,\ldots ,f_{i})\rightarrow 0, \end{aligned}$$

we deduce the equality

$$\begin{aligned} {\text {Hilb}}_{R/(f_1,\ldots ,f_{i})}(t)=(1-t^{a_i}){\text {Hilb}}_{R/(f_1,\ldots ,f_{i-1})}(t)+{\text {Hilb}}_{U_i}(t). \end{aligned}$$
(4)

We have \({\text {Hilb}}_{R/(f_1,\ldots ,f_{i})}(t)=Q_i(t)/(1-t)^{d-i}\) with \(Q_i(1)=e(R/(f_1,\ldots ,f_{i}))\), similarly, \({\text {Hilb}}_{R/(f_1,\ldots ,f_{i-1})}(t)=Q_{i-1}(t)/(1-t)^{d-(i-1)}\) with \(Q_{i-1}(1)=e(R/(f_1,\ldots ,f_{i-1}))\) and \({\text {Hilb}}_{U_i}(t) =P_i(t)/(1-t)^{\delta _i}\) with \(P_i(1)=e(U_i)\) and \(\delta _i\le d-i\).

Thus, (4) implies that

$$\begin{aligned} Q_i(t)/(1-t)^{d-i}=(1-t^{a_i})(Q_{i-1}(t)/(1-t)^{d-(i-1)})+ P_i(t)/(1-t)^{\delta _i}, \end{aligned}$$

from which we deduce that

$$\begin{aligned} Q_i(t)=Q_{i-1}(t)(\sum _{j=0}^{a_i-1}t^j)+(1-t)^{d-i-\delta _i}P_i(t). \end{aligned}$$

Substituting t by 1, we get

$$\begin{aligned} e(R/(f_1,\ldots ,f_i))= {\left\{ \begin{array}{ll} a_ie(R/(f_1,\ldots ,f_{i-1})), \;\; \; \;\; \;\;\;\ \text {if}~{ \dim U_i<d-i},\\ a_ie(R/(f_1,\ldots ,f_{i-1}))+e(U_i), \text { if}~{ \dim U_i=d-i}. \end{array}\right. } \end{aligned}$$

These formulas together with induction on i complete the proof. \(\square \)

Proposition 1.3 together with Proposition 1.2 has the following a surprising consequence

Corollary 1.4

With the assumptions and notation of Proposition 1.3, let \(r<d\) be an integer with the property that \(\dim U_i<d-i\) for \(i=1,\ldots ,r\) and that \(R/(f_1,\ldots ,f_r)\) is Cohen–Macaulay. Then, R is Cohen–Macaulay and \(U_i=0\) for all i. In particular, if \(f_1,\dots ,f_d\) is a superficial sequence and \(R/(f_1,\ldots ,f_i)\) is Cohen–Macaulay for some \(i<d\), then R is Cohen–Macaulay.

Proof

Proposition 1.3 implies that \(e(R/(f_1,\ldots ,f_r))=a_1\cdots a_re(R)\). Since by assumption \(R/(f_1,\ldots ,f_r)\) is Cohen–Macaulay, it follows that \(f_{r+1},\cdots , f_d\) is a regular \(R/(f_1,\ldots ,f_r)\)-sequence. Hence, \(e(R/(f_1,\ldots ,f_d))=a_{r+1}\cdots a_de(R/(f_1,\ldots ,f_r))\), and we deduce that \(e(R/(f_1,\ldots ,f_d))=a_1\cdots a_d e(R)\). Thus, the desired result follows from Proposition 1.2. \(\square \)

We close this section with a remark and a question. Let \(S=K[x_1,\ldots ,x_n]\) be the polynomial ring over the field K, \(R=S/I\) with \(I\subset (x_1,\ldots ,x_n)^2\) a graded ideal. Let \(f_1,\ldots ,f_d\) be linear forms of S which form a sop for R. Let \({\overline{S}}=S/(f_1,\ldots ,f_d)\). Then, \({\overline{S}}\) is isomorphic to a polynomial ring in \(n-d\) variables, and \(R/(f_1,\ldots ,f_d)={\bar{S}}/{\overline{I}}\), where \({\overline{I}}=I{\overline{S}}\).

Remark 1.5

With the notation introduced, we have \({\text {proj dim}}_{{\overline{S}}} {\overline{I}} \le {\text {proj dim}}_S I \) and \(\mu ({\overline{I}})\le \mu (I)\). Equality holds in both inequalities, if S/I is Cohen–Macaulay.

Proof

By the Auslander–Buchsbaum formula,

$$\begin{aligned} {\text {proj dim}}_S S/I=n-{\text {depth}}S/I\ge n-\dim S/I=n-d ={\text {proj dim}}_{{\overline{S}}}{\overline{S}}/{\overline{I}}. \end{aligned}$$

The last equation holds since \(\dim {\overline{S}}/{\overline{I}}=0\). This implies the first assertion. It is obvious that \(\mu (I)\ge \mu (I{\overline{S}})\). Finally, if S/I is Cohen–Macaulay, then I is generated by a regular sequence and the desired equalities hold. \(\square \)

Remark 1.5 implies in particular that if \(\mu ({\overline{I}})< \mu (I)\), then R cannot be Cohen–Macaulay. On the other hand, \(\mu ({\overline{I}})=\mu (I)\), does not necessarily imply that R is Cohen–Macaulay. For example for the cycle graph \(C_6:x_1,\ldots ,x_6\), the sequence \(x_1-x_2,x_3-x_4,x_5-x_6\) is a sop for \(R=K[x_1,\ldots ,x_6]/I(C_6)\) and \(\overline{I(C_6)}=(x_1^2,x_3^2,x_5^2,x_1x_3,x_3x_5,x_1x_5)\). Then, \(\mu (\overline{I(C_6)})=\mu (I(C_6))=6\), while R is not Cohen–Macaulay.

In view of these inequalities, one is tempted to ask whether under the assumptions of Remark 1.5 we have \({\text {reg}}({\overline{I}})\le {\text {reg}}(I)\). In the next section, we show that this inequality for the regularity indeed holds for the edge ideal of König graphs and suitable natural sop’s.

3 Special systems of parameters applied to order complexes and König graphs

In this section, we define monomial ideals of König type which include edge ideals of König graphs and give a characterization for these ideals in terms some sop’s for their quotient rings. Also we apply Proposition 1.1 to the Stanley–Reisner ring of two families of simplicial complexes, namely the order complex of a finite poset and the independence complex of a König graph and give combinatorial descriptions for the Cohen–Macaulay property of these rings.

For a poset P, a nonempty subposet C of P which is totally ordered is called a chain in P. The order complex of P denoted by \(\Delta (P)\) is the simplicial complex whose faces are the chains in P. The length of a chain C in P is defined to be \(|C|-1\). The height of an element x in P is defined to be the maximal length of a chain descending from x. For elements x and y in a poset P, it is said that y covers x, denoted \(x\lessdot y\), if \(x<y\) and there exists no \(z\in P\) such that \(x<z<y\). Also for a monomial ideal I, the cardinality of any minimal generating set of monomials of I is denoted by \(\mu (I)\).

The order complex \(\Delta (P)\) considered in the next theorem is indeed the independence complex of a bipartite graph, and the characterization of Cohen–Macaulay bipartite graphs is known. Here, by using systems of parameters, we recover this result and formulate it in terms of the poset P and present an explicit shelling for \(\Delta (P)\).

Theorem 2.1

Let P be a poset which as a set is the disjoint union of two sets \(C_1\) and \(C_2\), where \(C_1:x_1<x_2<\cdots <x_n\) and \(C_2:y_1<y_2<\cdots <y_n\) are maximal chains in P and let \(\Delta =\Delta (P)\). Then, the following conditions are equivalent:

  1. (a)

    \(\Delta \) is Cohen–Macaulay.

  2. (b)

    \(\Delta \) is pure shellable.

  3. (c)

    \(\Delta \) satisfies the following conditions:

    1. (1)

      If \(x_i\lessdot y_j\) or \(y_i\lessdot x_j\), then \(j=i+1\), and

    2. (2)

      \(\{x_i,y_{i+1}\}\notin \Delta \) implies that \(\{x_{i+1},y_i\}\in \Delta \).

Proof

(b)\({}\Rightarrow {}\)(a): By [8, Theorem 8.2.6] the assertion holds.

(a) \({}\Rightarrow {}\)(c): Suppose that \(\Delta \) is Cohen–Macaulay. Then, \(\Delta \) is pure. Suppose \(x_i\lessdot y_j\) for some i and j. If \(j\le i\), then the chain \(x_1<\cdots<x_i<y_j<y_{j+1}<\cdots <y_n\) is a chain of cardinality at least \(n+1\), which is included in some maximal chain of P. But by purity of \(\Delta \) any maximal chain should have cardinality \(|C_1|=n\), which gives a contradiction. Thus, \(i<j\). Similarly, if \(y_i\lessdot x_j\) for some i and j, then \(i<j\). Now assume that \(x_i\lessdot y_j\) and by contradiction let \(j\ne i+1\). Since \(j>i\), one should have \(j>i+1\). Then, \(x_1<\cdots<x_i<y_j<y_{j+1}<\cdots <y_n\) is a maximal chain of cardinality at most \(n-1\) in P, which is again a contradiction to purity of \(\Delta \). So \(j=i+1\). The argument for the case \(y_i\lessdot x_j\) is similar.

To prove (2), first we show that the sequence \(x_1-y_1,x_2-y_2,\ldots ,x_n-y_n\) is a sop for the ring \(R=S/I_{\Delta }\), where \(S=K[x_1,\ldots ,x_n,y_1,\ldots ,y_n]\). Indeed by (1), \(x_iy_i\in I_{\Delta }\) for any \(1\le i\le n\). So we have \(x_i^2,y_i^2\in (I_{\Delta },x_1-y_1,x_2-y_2,\ldots ,x_n-y_n)\) for all i and then \(\dim R/(x_1-y_1,x_2-y_2,\ldots ,x_n-y_n)R=0\). Also \(\dim R=\dim \Delta +1=n\). Now, by contradiction suppose that for some i, \(\{x_i,y_{i+1}\}\notin \Delta \) and \(\{x_{i+1},y_i\}\notin \Delta \). This means that \(x_iy_{i+1}\) and \(x_{i+1}y_i\) belong to the set of minimal generators \({\mathcal {G}}(I_{\Delta })\) of \(I_{\Delta }\). One has \(R/(x_1-y_1,x_2-y_2,\ldots ,x_n-y_n)R\cong K[x_1,\ldots ,x_n]/I'\), where \(I'=(x_ix_j: x_i\) and \(y_j\) are non-comparable in P). Since R is Cohen–Macaulay, by Remark 1.5, one should have \(\mu (I_{\Delta })=\mu (I')\). But since \({\mathcal {G}}(I_{\Delta })=\{x_iy_j: x_i\) and \(y_j\) are non-comparable in\({ P}\}\), and \(x_iy_{i+1},x_{i+1}y_i\in {\mathcal {G}}(I_{\Delta })\) correspond to just one element in \({\mathcal {G}}(I')\) that is \(x_ix_{i+1}\), we have \(\mu (I')<\mu (I_{\Delta })\), a contradiction. Thus \(\{x_i,y_{i+1}\}\in \Delta \) or \(\{x_{i+1},y_i\}\in \Delta \).

(c)\({}\Rightarrow {}\)(b): Let P be a poset satisfying the assumptions of (c). Let \(F=\{z_1<z_2<\cdots <z_k\}\) be an arbitrary facet of \(\Delta \). First, note that \(z_1\lessdot z_2\lessdot \cdots \lessdot z_k\). Also assumption (1) of (c) implies that \(\{x_i,y_i\}\notin \Delta \) for any \(1\le i\le n\), and then \(|\{x_i,y_i\}\cap F|\le 1\). We claim that for each facet F of \(\Delta \), \(|\{x_i,y_i\}\cap F|=1\) for any \(1\le i\le n\). One can easily see that for \(i=1\) the claim holds true. Indeed if \(z_1\in C_1\), then \(z_1=x_1\), because otherwise \(F \subsetneq F\cup \{x_1\}\in \Delta \), a contradiction. Similarly, if \(z_1\in C_2\), then \(z_1=y_1\). So \(|\{x_1,y_1\}\cap F|=1\). Assume inductively that for any \(i=1,\ldots ,m-1\), \(|\{x_i,y_i\}\cap F|=1\). We show that \(|\{x_m,y_m\}\cap F|=1\). We have \(x_{m-1}\in F\) or \(y_{m-1}\in F\). Without loss of generality, suppose \(x_{m-1}\in F\). If \(x_m\in F\), we are done. So, assume that \(x_m\notin F\). Note that \(x_{m-1}\ne z_k\), since otherwise \(F \subsetneq F\cup \{x_m\}\in \Delta \), a contradiction. So, there exists \(z_t\in F\) such that \(x_{m-1}\lessdot z_t\). If \(z_t=x_j\) for some j, then \(j>m\) and hence \(x_{m-1}< x_m< z_t\), which contradicts to \(x_{m-1}\lessdot z_t\). Thus, \(z_t=y_j\) for some j and by (1), \(j=m\). Thus, \(z_t=y_m\in F\). So \(|\{x_m,y_m\}\cap F|=1\). Therefore, any facet F of \(\Delta \) has cardinality n such that for any \(1\le i\le n\), either \(x_i\in F\) or \(y_i\in F\). Thus, \(\Delta \) is pure.

Let \({\mathcal {F}}(\Delta )\) denotes the set of facets of \(\Delta \). To prove the shellability, consider the ordering on \({\mathcal {F}}(\Delta )\) as follows. For the facets \(F_i\) and \(F_j\) of \(\Delta \), we set \(F_i\prec F_j\) if there exists \(1\le t\le n\) such that \(y_t\in F_j\setminus F_i\) and for any \(d<t\), the elements of height d in \(F_i\) and \(F_j\) are the same. Now, let \(F_i\prec F_j\) and \(1\le t\le n\) be such that \(y_t\in F_j\setminus F_i\) and for any \(d<t\), the elements of height d in \(F_i\) and \(F_j\) are the same. If \(\{x_t,y_{t+1}\}\in \Delta \), then we set \(F_k=(F_j\setminus \{y_t\})\cup \{x_t\}\). Then, \(F_k\in {\mathcal {F}}(\Delta )\), \(F_k\prec F_j\) and \(F_j\setminus F_k=\{y_t\}\). So, we may assume that \(\{x_t,y_{t+1}\}\notin \Delta \). Thus, by assumption, \(\{x_{t+1},y_t\}\in \Delta \). Let r be the greatest integer with \(y_t,y_{t+1},\ldots ,y_r\in F_j\setminus F_i\) and \(\{x_{s+1},y_s\}\in \Delta \) for any \(t\le s\le r\). Note that r is well-defined, since \(y_t\in F_j\setminus F_i\) and \(\{x_{t+1},y_t\}\in \Delta \). Two cases may happen:

(1) Suppose that \(y_{r+1}\in F_j\setminus F_i\). Then, by the maximality of r, \(\{x_{r+2},y_{r+1}\}\notin \Delta \). So by assumption \(\{x_{r+1},y_{r+2}\}\in \Delta \). Then, \(F_k=(F_j\setminus \{y_{r+1}\})\cup \{x_{r+1}\}\in {\mathcal {F}}(\Delta )\), \(F_k\prec F_j\) and \(F_j\setminus F_k=\{y_{r+1}\}\).

(2) Suppose that \(y_{r+1}\notin F_j\setminus F_i\). Then, either \(y_{r+1}\notin F_j\) or \(y_{r+1}\in F_i\cap F_j\). If \(r>t\), then by our assumption on r, \(\{x_r,y_{r-1}\}\in \Delta \). So in both cases, if we set \(F_k=(F_j\setminus \{y_r\})\cup \{x_r\}\in {\mathcal {F}}(\Delta )\), then \(F_k\) is the facet which fulfils the desired condition for shellability. Now let \(r=t\). Then, by assumption on t, \(F_i\cap \{x_k,y_k: 1\le k\le t-1\}=F_j\cap \{x_k,y_k: 1\le k\le t-1\}\). Then, by setting \(F_k=(F_j\setminus \{y_t\})\cup \{x_t\}\in {\mathcal {F}}(\Delta )\), we have the desired facet which fulfils the condition for shellability. \(\square \)

The next result shows that the linear resolution property for \(I_{\Delta (P)}\) can be again expressed in terms of conditions on chains of the form \(x_i<y_j\) in P. For a graph G by V(G) and E(G), we mean the vertex set and the edge set of G, respectively. Also for a subset \(S\subseteq V(G)\), the induced subgraph of G on the set S is denoted by \(G_S\).

Proposition 2.2

Let P be a poset with the assumptions of Theorem 2.1 and \(\Delta =\Delta (P)\). Then, \(I_{\Delta }\) has a linear resolution if and only if whenever \(\{x_i,y_j\},\{x_r,y_s\}\in \Delta \), then \(\{x_i,y_s\}\in \Delta \) or \(\{x_r,y_j\}\in \Delta \).

Proof

We have \(I_{\Delta }=I(G)\), where G is a bipartite graph with bipartition \(X=\{x_1,x_2,\ldots ,x_n\}\), \(Y=\{y_1,y_2,\ldots ,y_n\}\) and the edge set \(E(G)=\{\{x_i,y_j\}:\ \{x_i,y_j\}\notin \Delta \}\). Thus, by [6, Theorem 1], \(I_{\Delta }\) has a linear resolution if and only if \(G^c\) is a chordal graph. Any cycle C of length \(m\ge 5\) has a chord in \(G^c\), because C has at least 3 vertices from X or from Y, and \(G^c_X\) and \(G^c_Y\) are complete graphs. So, \(G^c\) is chordal if and only if any cycle of length 4 in \(G^c\) has a chord. Note that \(C:x_i,x_r,y_s,y_j\) is a cycle in \(G^c\) if and only if \(\{x_i,y_j\},\{x_r,y_s\}\in \Delta \) and it has a chord if and only if \(\{x_i,y_s\}\in \Delta \) or \(\{x_r,y_j\}\in \Delta \). Thus, \(G^c\) is a chordal graph if and only if whenever \(\{x_i,y_j\},\{x_r,y_s\}\in \Delta \), then \(\{x_i,y_s\}\in \Delta \) or \(\{x_r,y_j\}\in \Delta \). \(\square \)

For a graph G, let \(\tau (G)\) be the minimum cardinality of a vertex cover of G and \(\nu (G)\) denotes the maximum cardinality of a matching of G. One can see that for any graph G, \(\tau (G)\ge \nu (G)\). Recall that a graph G is called a König graph, when this inequality becomes an equality. Let \(I\subset S\) be a monomial ideal in the polynomial ring \(S=K[x_1,\ldots ,x_n]\) over the field K in n variables. We denote by \({\text {m-grade}}(I)\) the maximal length of a regular sequence of monomials in I and call this number the monomial grade of I. One has \({\text {m-grade}}(I)\le {\text {grade}}(I)={\text {height}}(I)\). We call I a monomial ideal of König type if \(I\ne 0\) and \({\text {m-grade}}(I)={\text {height}}(I)\). The naming is justified by the fact that if \(I=I(G)\) for some graph G, then \({\text {height}}(I)=\tau (G)\) and \({\text {m-grade}}(I)=\nu (G)\), so that the edge ideal of König graphs are the monomial ideals of König type among edge ideals.

The following theorem characterizes monomial ideals of König type in terms of existence of some forms of sop’s for their quotient rings.

Theorem 2.3

Let \(I\subset S=K[x_1,\ldots ,x_n]\) be a monomial ideal. Then, the following conditions are equivalent:

  1. (a)

    I is a monomial ideal of König type.

  2. (b)

    S/I admits a sop \(f_1,\ldots ,f_d\), where each \(f_k\) is of the form \(x_i-x_j\) for suitable i and j.

Proof

(a) \({}\Rightarrow {}\)(b): Let \(h={\text {height}}(I)\). Since I is a monomial ideal of König type, there exists a regular sequence of monomials \(u_1,\ldots ,u_h\) with \(u_i\in I\) for \(i=1,\ldots ,h\). We may assume that each \(u_i\) belongs to the unique minimal set of monomial generators \({\mathcal {G}}(I)\) of I. Indeed, if \(u_i=wv\) with \(v\in {\mathcal {G}}(I)\), then we may replace \(u_i\) by v in the above regular sequence.

For a monomial \(u\in S\), we set \({\text {supp}}(u)=\{i\, x_i|u\}\). We may assume that \(\bigcup _{u\in {\mathcal {G}}(I)}{\text {supp}}(u)=[n]\). Indeed, suppose this is not the case. Then, for simplicity we may assume that \(\bigcup _{u\in {\mathcal {G}}(I)}{\text {supp}}(u)=\{1,\ldots ,r\}\). Note that \(r\ge 1\), since \(I\ne 0\). Let \(R=K[x_1,\ldots ,x_r]/(u\,\; u\in {\mathcal {G}}(I))\). Then, \(S/I=R[x_{r+1},\ldots , x_n]\), and \(x_1-x_{r+1},\ldots , x_1-x_n\) is part of a sop of S/I and \((S/I)/(x_1-x_{r+1},\ldots , x_1-x_n)\cong R\). Thus, if R has the desired sop, then so does S/I.

We proceed by induction on \(d=\dim S/I\). If \(d=0\), then there is nothing to show. So, we assume that \(d>0\). First, suppose the case that \(u_1,\ldots ,u_h\) are all pure powers, say \(u_i=x_i^{a_i}\) for any \(1\le i\le h\). We show that \(x_1-x_{h+1},\ldots ,x_1-x_n\) is a sop for S/I. Indeed, this is a sequence of length \(n-h=d\) and if \(J=(u_1,\ldots ,u_h)\), then \((S/J)/(x_1-x_{h+1},\ldots ,x_1-x_n)\cong K[x_1,\ldots ,x_h]/J\). So, \(\dim (S/I)/(x_1-x_{h+1},\ldots ,x_1-x_n)\le \dim (S/J)/(x_1-x_{h+1},\ldots ,x_1-x_n)=0\).

Now, assume at least one of the \(u_i\)’s is not a pure power. Without loss of generality, assume that \(u_1\) is not a pure power, say \(u_1=x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}\) with \(a_1, a_2>0\). Let \(f_1=x_1-x_2\). We claim that \(f_1\) is a part of a sop. In other words, \(f_1\) is not contained in any minimal prime ideal P of I with \({\text {height}}(P) ={\text {height}}(I)=h\). Indeed, let P be a minimal prime ideal containing \(f_1\). Since I is a monomial ideal, P is a monomial prime ideal. Therefore, if \(f_1\in P\), then \(x_1,x_2\in P\). For each i there exists \(j_i\) such that \(x_{j_i}\) divides \(u_i\) and \(x_{j_i}\in P\). Thus, \(Q=(x_1,x_2, x_{j_2},\cdots , x_{j_h}) \subset P\). Since \(u_1,\ldots ,u_h\) is a regular sequence, the supports of the \(u_i\) are pairwise disjoint. This implies that the variables generating Q are pairwise distinct. It follows that \({\text {height}}(P)\ge h+1\), and \(f_1\) is a part of a sop.

Identifying \(x_1\) with \(x_2\), we see that \((S/I)/(f_1)\cong K[x_2,\ldots ,x_n]/{\overline{I}}\), where \({\overline{I}}= (x_2^{a_1+a_2}\cdots x_n^{a_n}, u_2,\ldots ,u_h, \cdots )\). This shows that \({\text {m-grade}}({\overline{I}})\ge h\). Since \(f_1\) is a parameter element of S/I it follows that \({\text {height}}({\overline{I}})={\text {height}}(I)=h\) and since in general \({\text {m-grade}}({\overline{I}})\le {\text {height}}({\overline{I}})\), we must have \({\text {m-grade}}({\overline{I}})= {\text {height}}({\overline{I}})\). This means that \({\overline{I}}\) is a again monomial ideal of König type. Since \(\dim ((S/I)/(f_1))= d-1\), we may apply our induction hypothesis and find a sop \(f_2,\ldots , f_d\) of \((S/I)/(f_1)\) as required in (b). Then, \(f_1, f_2,\ldots ,f_d\) is the desired sop for S/I,

(b)\({}\Rightarrow {}\)(a): Let \(R=S/I\) and \({\overline{R}}=R/(f_1,\ldots ,f_d)R\). Since \(f_1,\ldots ,f_d\) is a sop for R, it is a linearly independent sequence in \(S_1=\bigoplus _{i=1}^n Kx_i\). So \(\dim _K(S_1/(f_1,\ldots ,f_d)S_1)=n-d=h\), where \(h={\text {height}}(I)\). Since reduction modulo \(f_1,\ldots ,f_d\) simply identifies variables, for simplicity we may assume that \(S_1/(f_1,\ldots ,f_d)S_1\cong \bigoplus _{i=1}^h Kx_i\) and then \({\overline{R}}\cong K[x_1,\ldots ,x_h]/{\overline{I}}\), where \({\overline{I}}\) is a monomial ideal. Since \(\dim ({\overline{R}})=0\) and since \({\overline{I}}\) is a monomial ideal, it follows that \({\mathcal {G}}({\overline{I}})\) contains a pure power \(x_i^{a_i}\) of each the variables \(x_1,\ldots ,x_h\). Let \(u_1,\ldots ,u_h\) be generators of I with the property that \(u_i\) specializes to \(x_i^{a_i}\) under the reduction modulo \(f_1,\ldots ,f_d\). Suppose \(u_i\) and \(u_j\) have a common factor for \(i\ne j\). Then, this is also the case for \(x_i^{a_i}\) and \(x_j^{a_j}\), a contradiction. Therefore, \(u_1,\ldots ,u_h\) is a regular sequence, and so \({\text {m-grade}}(I)={\text {height}}(I)\). \(\square \)

Applying Theorem 2.3 to the edge ideal of König graphs, we have the following algebraic characterization for a König graph G in terms of special sop’s for \(R=S/I(G)\).

Corollary 2.4

Let G be a graph without isolated vertices, \(S=K[V(G)]\) and for any edge \(e=\{x,y\}\in E(G)\), let \(f_e=x-y\) be an element in S. Then, G is a König graph if and only if there exists a subset \(\{e_1,\ldots ,e_d\}\) of edges of G such that \(f_{e_1},\ldots ,f_{e_d}\) is a sop for \(R=S/I(G)\).

Proof

The if part follows from Theorem 2.3. Now let G be a König graph and \(\{e_1,\ldots ,e_m\}\) be a maximal matching of G such that \(\tau (G)=m\). Suppose that \(e_i=\{x_i,y_i\}\) for any \(1\le i\le m\), and without loss of generality assume that \(C=\{x_1,\ldots ,x_m\}\) is a minimal vertex cover of G. Let \(V(G)\setminus \bigcup _{i=1}^m e_i=\{z_1,\ldots ,z_k\}\). Since C is a vertex cover of G and G has no isolated vertex, each \(z_i\) is adjacent to some \(x_{j_i}\in C\) for \(1\le i\le k\). We set \(e'_i=\{z_i,x_{j_i}\}\) for any \(1\le i\le k\). Then, \(f_{e_1},\ldots ,f_{e_m},f_{e'_1},\ldots ,f_{e'_k}\) is a sop for R. Note that \(m+k=n-m=d\), where \(n=|V(G)|\). Also \(R/(f_{e_1},\ldots ,f_{e_m},f_{e'_1},\ldots ,f_{e'_k})\cong K[x_1,\ldots ,x_m]/J\), where J is an ideal containing \((x_1^2,\ldots ,x_m^2)\). Hence, \(\dim K[x_1,\ldots ,x_m]/J=0\). \(\square \)

Let R be a graded ring and M a finitely generated graded R-module. It is known that for an M-superficial sequence \(f_1,\ldots ,f_m\) of linear forms \({\text {reg}}(M/(f_1,\ldots ,f_m)M)\le {\text {reg}}(M)\), see [5, Proposition 20.20]. In view of this fact, it is natural to ask the following

Question 2.5

Let R be a graded ring, M be a finitely generated graded R-module and \(f_1,\ldots ,f_m\) be a sop of linear forms for M. Does the inequality

$$\begin{aligned} {\text {reg}}(M/(f_1,\ldots ,f_m)M)\le {\text {reg}}(M) \end{aligned}$$

hold?

In the following theorem, we prove the expected inequality for the S-module \(R=S/I(G)\), when G is a König graph and the sop is of a natural special form. First, we recall some definitions. Two disjoint edges e and \(e'\) of a graph G form a gap, when there exists no edge in G with one endpoint in e and the other in \(e'\). Otherwise, we say that e and \(e'\) are adjacent. Moreover, a subset A of edges of G is called an induced matching if any two edges in A form a gap. By \({\text {a}}(G)\) we mean the maximum cardinality of an induced matching of G. The maximum cardinality of an independent set of G is denoted by \(\alpha (G)\). For a graph G, with the vertex set \(\{x_1,\ldots ,x_n\}\) the whiskered graph of G is a graph which is obtained by adding new vertices \(\{y_1,\ldots ,y_n\}\) and edges \(\{\{x_i,y_i\}:\ 1\le i\le n\}\) to G. This new graph is denoted by \(G\cup W(G)\), and the edges \(\{x_i,y_i\}\) are called whiskers.

Theorem 2.6

Let G be a graph, \(S=K[V(G)]\) and \(\{e_1,\ldots ,e_d\}\subseteq E(G)\) such that \(f_{e_1},\ldots ,f_{e_d}\) is a sop for \(R=S/I(G)\). Then,

$$\begin{aligned} {\text {reg}}(R/(f_{e_1},\ldots ,f_{e_d})R)\le {\text {reg}}(R). \end{aligned}$$

Proof

Set \({\overline{R}}=R/(f_{e_1},\ldots ,f_{e_d})R\). For each \(1\le i\le d\), we choose a vertex in \(e_i\) which we denote by \(x_i\). Let \(h=\tau (G)\). As shown in the proof of Theorem 2.3(b)\({}\Rightarrow {}\)(a), we may assume that \({\overline{R}}\cong K[x_1,\ldots ,x_h]/L\), where L contains \(x_1^2,\ldots ,x_h^2\) and there exists a regular sequence \(x_1y_1,\ldots ,x_hy_h\in I(G)\). So, if we set \(e_i=\{x_i,y_i\}\), this implies that \(\{e_1,\ldots ,e_h\}\) is a matching of G. Note that by Corollary 2.4, G is a König graph. So, the matching number of G is equal to h, from which we conclude that \(\{e_1,\ldots ,e_h\}\) is a maximal matching of G. The isomorphism \({\overline{R}}\cong K[x_1,\ldots ,x_h]/L\) implies that each edge of G has an endpoint in \(C=\{x_1,x_2,\ldots ,x_h\}\). So C is a vertex cover of G. Indeed, it is a minimal one, because \(h=\tau (G)\). We have \(L=(x_1^2,\ldots x_h^2)+I(H)\), where H is a graph on \(\{x_1,\ldots ,x_h\}\) such that

$$\begin{aligned} \{\{x_i,x_j\}:\ i<j\le h,\ e_i~ \text {and}~ e_j~ \text {are adjacent in} ~{G}\}\subseteq E(H). \end{aligned}$$
(5)

Since polarization does not change the regularity, one has \({\text {reg}}( K[x_1,\ldots ,x_h]/L)={\text {reg}}(T/I(H'))\), where \(T=K[x_1,\ldots ,x_h,x'_1,\ldots ,x'_h]\) for new variables \(x'_1,\ldots ,x'_h\), \(H'=H\cup W(H)\) and \(\{x_i,x'_i\}\) is a whisker of \(H'\) for any \(1\le i\le h\). Since any whiskered graph is very well covered, by [12, Theorem 1.3], \({\text {reg}}(T/I(H'))={\text {a}}(H')\). One can easily see that \({\text {a}}(H')\) is precisely the maximum size of independent sets of vertices in H. Moreover, by (5), if \(\{x_{s_1},\ldots ,x_{s_t}\}\) is an independent set of H, then \(\{e_{s_1},\ldots ,e_{s_t}\}\subseteq \{e_1,\ldots ,e_d\}\) is an induced matching of G. Therefore, \({\text {a}}(H')\le {\text {a}}(G)\). Thus, \({\text {reg}}({\overline{R}})={\text {reg}}( K[x_1,\ldots ,x_h]/L)={\text {reg}}(T/I(H'))={\text {a}}(H')\le {\text {a}}(G)\le {\text {reg}}(R)\). The last inequality holds by [11, Lemma 2.2]. \(\square \)

In general, the inequality of Theorem 2.6 may be strict. Let \(G=C_{n}\) denote the cycle graph with n vertices. Suppose \(E(G)=\{e_i=\{x_i,x_{i+1}\}: 1\le i\le n-1\}\cup \{e_{n}=\{x_1,x_{n}\}\}\) and \(n\equiv 2 {\text {mod}}4\). Let \(n=4m+2\) for some \(m\ge 1\). Then, \(\{f_{e_{2i-1}}: \ 1\le i\le 2m+1\}\) is a sop for \(R=S/I(G)\). With the notation used in the proof of Theorem 2.6, \(H'=C_{2m+1}\cup W(C_{2m+1})\) and \({\text {reg}}({\overline{R}})={\text {a}}(H')=\alpha (H)=m\). Since \(\{e_{3i+1}: 0\le i\le m\}\) is an induced matching of G of cardinality \(m+1\), \({\text {reg}}(R)\ge a(G)\ge m+1>m={\text {reg}}({\overline{R}})\).

For a graph G, let \({\text {mi}}(G)\) denote the number of maximal independent sets of G. After Erdös and Moser considered the problem of determining the largest value of \({\text {mi}}(G)\) in terms of the number of vertices of G, investigating this number and upper bounds for it has been studied for various classes of graphs. In [9, Corollary 3.4] it was shown that for a König graph G, \(2^{\nu (G)}\) is an upper bound for \({\text {mi}}(G)\). Also in [1, Theorem 1] it was proved that \({\text {mi}}(G)\le M(G)+1\), where M(G) is the number of induced matchings in G. In the following for an unmixed König graph G, we use a sop of the form \(f_{e_1},\ldots ,f_{e_d}\) for S/I(G) and improve the upper bound for \({\text {mi}}(G)\). Moreover, we give a combinatorial description of the Cohen–Macaulay property for unmixed König graphs. A different combinatorial characterization of Cohen–Macaulay König graphs is presented in [4, Proposition 28]. Recall that a graph G is called unmixed if all the minimal vertex covers (maximal independent sets) of G are of the same cardinality.

Theorem 2.7

Let G be an unmixed König graph, \(\{e_1,\ldots ,e_m\}\) be a maximal matching of G with \(\tau (G)=\nu (G)=m\) and k be the number of induced matchings of G contained in \(\{e_1,\ldots ,e_m\}\). Then

(a):

\({\text {mi}}(G)\le k+1\), and

(b):

G is a Cohen–Macaulay graph if and only if \({\text {mi}}(G)=k+1\).

Proof

Let \(S=K[V(G)]\), \(R=S/I(G)\) and \(\dim (R)=d\). By the proof of Theorem 2.4, there exist \(e_{m+1},\ldots ,e_d\in E(G)\) such that \(f_{e_1},\ldots ,f_{e_d}\) is a sop for R. We may assume that \(\{x_1,x_2,\ldots ,x_m\}\) is a minimal vertex cover of G, where \(x_i\in e_i\) for all \(1\le i\le m\). Then, \(R/(f_{e_1},\ldots ,f_{e_d})R\cong K[x_1,x_2,\ldots ,x_m]/L\), where

$$\begin{aligned} L=(x_1^2,x_2^2,\ldots ,x_m^2,x_ix_j:\ i<j\le m,\ ~e_i ~\text {and} ~e_j~ \text {are adjacent in}~ {G}). \end{aligned}$$

So, \(\ell (R/(f_{e_1},\ldots ,f_{e_d})R)=\ell (K[x_1,x_2,\ldots ,x_m]/L)\). Since \(x_1^2,x_2^2,\ldots ,x_m^2\in L\), any basis element of \(K[x_1,x_2,\ldots ,x_m]/L\) other than 1, is an squarefree monomial of the form \(x_{i_1}\cdots x_{i_r}\), where \(\{e_{i_1},\ldots ,e_{i_r}\}\subseteq \{e_1,\ldots ,e_m\}\) is an induced matching of G. So \(\ell (K[x_1,x_2,\ldots ,x_m]/L)=k+1\). Note that R is the Stanley–Reisner ring of the independence complex \(\Delta _G\) of G. Since G is unmixed, e(R) is the number of facets of \(\Delta _G\) which is equal to \({\text {mi}}(G)\). Now, by using Proposition 1.2,

$$\begin{aligned} {\text {mi}}(G)=e(R)\le \ell (R/(f_{e_1},\ldots ,f_{e_d})R)=\ell (K[x_1,x_2,\ldots ,x_m]/L)=k+1 \end{aligned}$$

and equality holds if and only if G is Cohen–Macaulay. \(\square \)

Theorem 2.7 shows that in a Cohen–Macaulay König graph G, no matter which maximal matching M of G we choose, if \(|M|=\nu (G)\), then the number of induced matchings of G contained in M is equal to \({\text {mi}}(G)\).

4 A universal system of parameters for Stanley–Reisner rings

Let K be a field and \(\Delta \) an arbitrary simplicial complex on [n] of dimension \(d-1\), and let \(S=K[x_1,\ldots , x_n]\) be the polynomial ring in n variables. We show that there exists a universal standard sop for \(K[\Delta ]=S/I_{\Delta }\). Using this sop, we present a criterion for the Cohen–Macaulayness of \(K[\Delta ]\).

For \(i=1,\dots ,d \), we set

$$\begin{aligned} p_i(\Delta )=\sum _{\begin{array}{c} F\in \Delta \\ |F|=i \end{array}}{{\mathbf {x}}}_F, \end{aligned}$$

where \({{\mathbf {x}}}_F=\prod _{i\in F}x_i\).

Theorem 3.1

The residue classes of the elements \(p_1(\Delta ),\ldots ,p_d(\Delta )\) in \(K[\Delta ]\) form a sop of \(K[\Delta ]\).

Proof

We first consider the case that \(\Delta \) is the n-simplex \(\Gamma _n\). In that case \(K[\Delta ]=S\), and we have to show that \(\dim S/(p_1(\Delta ),\ldots ,p_n(\Delta ))=0\). In order to simplify notation, we write \(p_i\) for \(p_i(\Delta )\) and all i.

Note that

$$\begin{aligned} p_i=\sum _{|F|=i}{{\mathbf {x}}}_F. \end{aligned}$$
(6)

Let < denote the reverse lexicographical order, and let \(J=(p_1,\ldots ,p_n)\). We claim that \(x_i^i\in {\text {in}}_<(J)\) for \(i=1,\ldots ,n\). Then, this shows that indeed \(\dim S/J=0\).

For \(i=1,\ldots ,n\), let

$$\begin{aligned} g_i=\sum _{j=1}^i(-1)^{j+1}x_i^{i-j}p_j. \end{aligned}$$

We will show that \(x_i^i={\text {in}}_<(g_i)\) for \(i=1,\ldots ,n\).

Observe first that \(g_i\) is homogeneous of degree i and that \(x_i^i\in {\text {supp}}(g_i)\). In order to complete the proof of the claim, we have to show that if u is a monomial of degree i with \(u>x_i^i\), then \(u\not \in {\text {supp}}(g_i)\).

Let v be a monomial in the support of \(g_i\), \(v=x_1^{a_1}\cdots x_n^{a_n}\). Then, \(a_j\le 1\) for all \(j\ne i\). Hence, if \(x_j^a\) divides u for \(j\ne i\) and \(a>1\), then \(u\not \in {\text {supp}}(g_i)\). Therefore, we may assume that \(u=x_Gx_i^k\) with \(0\le k\le i\), \(i\not \in G\) and \(|G|=i-k\). Moreover, since \(u>x_i^i\) in the reverse lexicographic order, it follows that \(G\subset [i-1]\). Hence, \(k>0\), because u is of degree i. Then, \(u\in {\text {supp}}(x_i^kp_{i-k})\) with coefficients 1, because \(x_G\in {\text {supp}}(p_{i-k})\), and we also have that \(u\in {\text {supp}}(x_{i}^{k-1}p_{i-k+1})\) with coefficient 1, because \(x_Gx_i=x_{G\cup \{i\}}\) belongs to \({\text {supp}}(p_{i-k+1})\).

Since u does not belong to the support of any other summand \(x_i^{i-j}p_j\) of \(g_i\), and since the summands in which u appears in the support have different signs, we conclude that \(u\not \in {\text {supp}}(g_i)\), as desired.

In order to deal with the general case, we observe that

$$\begin{aligned} S/(I_\Delta , p_1(\Delta ),\ldots , p_d(\Delta ))= S/(I_\Delta , p_1(\Gamma _n),\ldots , p_n(\Gamma _n)). \end{aligned}$$

In particular, it follows that \(\dim S/(I_\Delta , p_1(\Delta ),\ldots , p_d(\Delta ))=0\). This yields the desired conclusion. \(\square \)

Remark 3.2

Let \(p_i\in S\) be defined as in (6), and let \(J=(p_1,\ldots ,p_n)\). Then, for the reverse lexicographical order < we have

$$\begin{aligned} {\text {in}}_<(J)=(x_1,x_2^2,\ldots ,x_i^i,\ldots ,x_n^n). \end{aligned}$$

Indeed, Theorem 3.1 implies that \(p_1, \ldots , p_n\) is a regular sequence, and since \(x_1, x_2^2,\ldots ,x_n^n\) is also a regular sequence and since \(\deg x_i^i=\deg p_i\) for \(i=1,\ldots ,n\), we see that

$$\begin{aligned} \ell (S/J)=\ell (S/(x_1,x_2^2,\ldots ,x_n^n)\ge \ell (S/{\text {in}}_<(J))=\ell (S/J). \end{aligned}$$

Hence, \(\ell (S/(x_1,x_2^2,\ldots ,x_n^n)= \ell (S/{\text {in}}_<(J))\), which yields the desired conclusion.

Corollary 3.3

Let e be the number of facets F of \(\Delta \) with \(|F|=d\). Then, \(K[\Delta ]\) is Cohen–Macaulay if and only if

$$\begin{aligned} \ell (S/(I_\Delta , p_1(\Delta ),\ldots ,p_d(\Delta )))=d!e. \end{aligned}$$

Proof

We note that \(e=e(R)\) is the multiplicity of \(K[\Delta ]\). Thus, the assertion follows from Theorem 3.1 and Proposition 1.2. \(\square \)

Let \(\Delta \) be a simplicial complex of dimension \(d-1\). While d!e only depends on the simplicial complex \(\Delta \), in general the number \(\ell (S/(I_\Delta , p_1(\Delta ),\ldots ,p_d(\Delta )))\) also depends on the characteristic of the field K. This is not surprising since this is also the case for the Cohen–Macaulay property of \(K[\Delta ]\).

Corollary 3.3 can also be used as a computational tool to determine the depth of a Stanley–Reisner ring. We demonstrate this with the following example: consider the chessboard \({{\mathcal {P}}}_n\) of size \(n\times n\). The set of non-attacking rooks on \({{\mathcal {P}}}_n\) is a simplicial complex which we denote \(\Delta ({{\mathcal {P}}}_n)\). Fix a field K. For \(n>1\), the Stanley–Reisner ring \(K[\Delta ({{\mathcal {P}}}_n)]\) is not Cohen–Macaulay. Indeed, if \(n=2\), then \(\Delta ({{\mathcal {P}}}_n)\) is not connected and hence not Cohen–Macaulay and if \(n>2\), then for any face F with \(|F|=n-2\), \({\text {link}}_{\Delta ({{\mathcal {P}}}_n)}(F)=\Delta ({{\mathcal {P}}}_2)\). Hence, \(\Delta ({{\mathcal {P}}}_n)\) can not be Cohen–Macaulay. But what is the depth of \(K[\Delta ({{\mathcal {P}}}_n)]\)? For \(n=2,3\) the depth can be computed by using the depth command implemented in CoCoA. But already for \(n=4\), \({\text {depth}}K[\Delta ({{\mathcal {P}}}_4)]\) cannot be computed by CoCoA. Instead, we use the fact, first shown by Smith [14, Theorem 4.8], that for any simplicial complex \(\Delta \) one has

$$\begin{aligned} {\text {depth}}K[\Delta ] =\max \{i\, K[\Delta ^{(i)}] \text { is Cohen--Macaulay}\}, \end{aligned}$$

where \( \Delta ^{(i)}=\langle F\in \Delta \, |F|=i\rangle \) is the ith skeleton of \(\Delta \).

In order to obtain the Stanley–Reisner ideal of the \((n-1)\)-skeleton of \({{\mathcal {P}}}_n\), we have to add to \(I_{\Delta ({{\mathcal {P}}}_n)}\) the monomials corresponding to the facets of \({{\mathcal {P}}}_n\). Then, we use Corollary 3.3 to check the Cohen–Macaulayness of the \((n-1)\)-skeleton. For example, when \(n=4\), the calculation for \(I_{\Delta ({{\mathcal {P}}}_4)^{(3)}}\) gives

$$\begin{aligned} \ell (S/(I_{\Delta ({{\mathcal {P}}}_4)^{(3)}}, p_1(\Delta ({{\mathcal {P}}}_4)^{(3)}),p_2(\Delta ({{\mathcal {P}}}_4)^{(3)}), p_3(\Delta ({{\mathcal {P}}}_4)^{(3)})))= 6e(K[\Delta ({{\mathcal {P}}}_4)^{(3)}]). \end{aligned}$$

The length on the left-hand side can be computed by means of the multiplicity command of CoCoA. The output comes almost immediately. Hence, \(K[\Delta ({{\mathcal {P}}}_4)^{(3)}]\) is Cohen–Macaulay and so \({\text {depth}}K[\Delta ({{\mathcal {P}}}_4)]=3\).