Licci binomial edge ideals☆
Introduction
Binomial edge ideals associated to simple graphs have been intensively studied in the last decade. Their algebraic and homological properties are intimately related to the combinatorics of the underlying graph. A lot of effort has been dedicated to study the Cohen-Macaulay property of these ideals. As in the case of classical edge ideals, an exhaustive classification of graphs whose binomial edge ideals are Cohen-Macaulay seems to be a hopeless task. There are successful attempts to characterize graphs with specific properties which have Cohen-Macaulay binomial edge ideals. For example, the Cohen-Macaulay property of binomial edge ideals is known for block graphs which include the trees [3] and for bipartite graphs [1]. We refer also to the papers [12], [20], [21], [22] for other classes of Cohen-Macaulay binomial edge ideals.
Let G be a simple graph (that is, undirected, with no loops, and no multiple edges) on the vertex set and the polynomial ring in 2n variables. The binomial edge ideal of G is generated by all the binomials of the form where is an edge of G. In other words, is generated by the 2-minors of the generic matrix which correspond to the edges of G.
In this paper, we study binomial edge ideals which are in the linkage class of a complete intersection. We call such ideals licci, in brief. Besides the Cohen-Macaulay property, they satisfy some extra conditions which make possible a full characterization of graphs whose binomial edge ideals are licci. Linkage theory has a rich history in commutative algebra and algebraic geometry. Peskine and Szpiro [19] in 1974 reduced general linkage to questions on ideals over commutative algebras and after then, a lot of work has been done to develop this theory in commutative algebra and algebraic geometry. If are proper ideals in a local regular ring R, they are called directly linked and we write if there exists a regular sequence in such that and . One says that I and J belong to the same linkage class if there exists a sequence of direct links If J is a complete intersection ideal, then I is said to be licci. The ideals in the same linkage class share several properties. For example, if I and J are linked, then I is Cohen-Macaulay if and only if J is Cohen-Macaulay. In particular, it follows that a licci ideal is Cohen-Macaulay.
The following natural question arises. May we give a full characterization of the graphs G with the property that the associated binomial edge ideal is licci?
In this paper, we give a complete answer to this question. In [9] a necessary condition for a Cohen-Macaulay homogeneous ideal in a polynomial ring to be licci is given. In the case of binomial edge ideals, this condition implies that if (here is the maximal graded ideal of the ring S) is licci, then . This condition turns to be also sufficient for Cohen-Macaulay binomial edge ideals as we are going to show in this paper.
The regularity of binomial edge ideals have been intensively studied in the last years. In [15] it was proved that the regularity of is upper bounded by and it was conjectured that this upper bound is attained if and only if G is a path graph. This conjecture was later proved in [13]. Inspired by the paper [13], we prove a new upper bound for which is stronger than and it plays an essential role in the characterization of the graphs G whose binomial edge ideal is licci.
The structure of the paper is as follows. In Section 1, we recall the basic results on licci and binomial edge ideals needed in the next sections. In Section 2, we prove that if G is a connected graph, then , where is the clique complex of G (Theorem 2.1). We believe that this new general upper bound for the regularity of binomial edge ideals will inspire new results on their resolution. In brief, in Theorem 2.1, we prove that for every clique of the connected graph G, we have . The proof is based on a double induction. First we make induction on and, secondly, on a combinatorial invariant of G.
The characterization of graphs whose binomial edge ideal is licci is given in Section 3. In Theorem 3.5 we show that, for a connected graph G on n vertices, the following statements are equivalent:
- (i)
is licci.
- (ii)
is Cohen-Macaulay and .
- (iii)
G is a path graph or it is a triangle with possibly some paths attached to some of its vertices.
A straightforward consequence of Theorem 3.5 is Corollary 3.7 which says that for a connected bipartite graph G, the ideal is licci if and only if G is a path graph. The case when G is a disconnected graph is treated in Proposition 3.8.
In the last section of the paper, we show that for chordal graphs, in the equivalent statements of Theorem 3.5, we may replace the Cohen-Macaulay property with the unmixedness of the ideal (Theorem 4.2). For the proof we use a theorem of Dirac which characterizes the chordal graphs in terms of their clique complex.
Section snippets
Preliminaries
We recall some notions and fundamental results needed in the later sections.
A new upper bound for the regularity of binomial edge ideals
In this section, we give a new general upper bound for the regularity of .
Theorem 2.1 Let G be a connected graph on . Then .
When G is not connected, we derive the following upper bound for the regularity of .
Corollary 2.2 Let G be a graph on n vertices with the connected components . Then
Let us make some short remarks before proving the above theorem. This new bound will be an essential tool in proving Theorem 3.5. Moreover, it is a substantial
Licci binomial edge ideals
As in the previous section, let G be a simple graph on the vertex set and the polynomial ring over a field K. Let be the maximal graded ideal of S and set .
We recall the notion of decomposable graphs from [8].
Definition 3.1 A connected graph G is called decomposable if there exists two subgraphs and of G such that with where v is a simplicial vertex in and . In this case we say that G is decomposable in the vertex v. Otherwise, the graph G is
Licci binomial edge ideals of chordal graphs
In this section we show that if we restrict to chordal graphs, we may relax the condition (ii) in Theorem 3.5, namely, we may ask that is only unmixed instead of being Cohen-Macaulay. Before proving the main theorem of this section, we need a preparatory result. We first recall that for a graph G, denotes the number of maximal cliques of G, that is, the number of facets of the clique complex .
Lemma 4.1 Let G be a connected chordal graph with n vertices. Then if and only if the
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