Licci binomial edge ideals (Q778714)

Let \(I\) and \(J\) be ideals in a regular local ring. Write \(I\sim J\) if there exists a regular sequence \(z_1,\dots,z_g\) contained in \(I\cap J\) such that \(J= (z_1,\dots,z_g):I\) and \(I = (z_1,\dots,z_g):J\). The ideal \(I\) is said \textit{licci} if there exists a sequence \(I=I_0 \sim I_1 \sim \cdots \sim I_m = J\) with \(J\) an ideal generated by a regular sequence. Let \(G\) be a simple graph with vertex set \(\{1,\dots,n\}\) and let \(S=K[x_1,\dots,x_n,y_1,\dots,y_n]\) be a polynomial ring in \(2n\) variables with coefficients in a field. The binomial edge ideal of \(G\), denoted by \(J_G\), is the ideal generated by the \(2\times 2\) minors of \[ \left [\begin{array}{cccc}x_1 & x_2 & \cdots & x_n \\y_1 & y_2 & \cdots & y_n\end{array}\right ] \] with column indices \(i\) and \(j\), for every edge \(\{i,j\}\) of \(G\). The main result of this article (Theorem 3.5) is a characterization of the case \((J_G)_\mathfrak{m}\) licci, where \(\mathfrak{m}\) denotes the irrelevant ideal of \(S\). The authors prove that, if \(G\) is a connected graph, the following are equivalent: \((i)\) \((J_G)_\mathfrak{m}\) is licci; \((ii)\) \(J_G\) is Cohen-Macaulay and \(\operatorname{reg} S/J_G\) is equal to either \(n-1\) or \(n-2\); \((iii)\) \(G\) is either a path or a triangle with, possibly, paths attached to a subset of its vertices. The main part of the proof of this result regards \((ii)\implies (iii)\), in the case of \(\operatorname{reg} S/J_G = n-2\). The argument relies on a set of preliminary results on the behavior of the regularity with respect to the notion of decomposability of the graph. When \(\operatorname{reg} S/J_G = n-1\) in \((ii)\), it follows by a result of \textit{D. Kiani} and \textit{S. Saeedi Madani} [J. Comb. Theory, Ser. A 139, 80--86 (2016; Zbl 1328.05087)] that \(G\) is a path. In the proof that \((i)\implies (ii)\) the authors use Corollary 5.13 of \textit{C. Huneke} and \textit{B. Ulrich} [Ann. Math. (2) 126, 277--334 (1987; Zbl 0638.13003)]. This result states that for a Cohen-Macaulay homogeneous ideal, \(I\subset S\), in a polynomial ring, if \(I_\mathfrak{m}\) is licci then \[ \operatorname{reg} S/I \geq (\operatorname{height} I-1)(\operatorname{indeg} I-1). \] Finally, in \((iii)\implies (i)\) the authors use Theorem 2.6 [\textit{M. R. Johnson}, Trans. Am. Math. Soc. 350, No. 5, 1913--1930 (1998; Zbl 0897.13014)]. As part of the preliminary results used in the proof of Theorem 3.5, the article contains a new general upper bound for the regularity of \(S/J_G\), using the clique complex of the graph, \(\Delta(G)\). More precisely, the authors prove that \[ \operatorname{reg} S/J_G \leq n-\dim \Delta(G). \] The article also contains a brief introduction of the basic theory of binomial edge ideals, a characterization of disconnected graphs with licci binomial edge ideal (deduced from the main result), and also a characterization of connected chordal graphs with licci binomial ideal, in which the assumption on the Cohen-Macaulay property in \((ii)\) is replaced by the weaker assumption of unmixedness.