Results on the regularity of square-free monomial ideals
Let \(k\) be a field and \(R=k[x_1,\ldots,x_n]\) a graded polynomial ring over \(k\) with variables having degree \(1\) (where \(n\geq 1\)). The paper studies squarefree monomial ideals in \(R\), which are in one-to-one correspondence with simple hypergraphs over the vertex set \(\{x_1,\ldots,x_n\}\). For a simple hypergraph \(\mathcal{H}\) with the edge set \(\mathcal E\), denote \(I(\mathcal H)=(x^e: e\in \mathcal E)\) the edge ideal of \(\mathcal H\). One wants to bound/compute the regularity \(\text{reg}~ I(\mathcal H)\) in terms of the combinatorics of \(\mathcal H\).NEWLINENEWLINEThe first main result gives an upper bound for \(\text{reg}~ I(\mathcal H)\) in terms of the 2-collages of \(\mathcal H\). A set of edges of \(\mathcal H\) is a 2-collage if for every edge \(E\) there exists an edge \(F\) in the 2-collage such that \(E\setminus F\) has at most 1 element. Then we have the following:NEWLINENEWLINETheorem 1.2. For any 2-collage \(E_1,\ldots, E_c\) of \(\mathcal H\), there is an inequality NEWLINE\[CARRIAGE_RETURNNEWLINE\text{reg}~ I(\mathcal H)\leq \sum_{i=1}^{c}(|E_i|-1).CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINENEWLINEIn the case when \(\mathcal H\) is a simple graph, i.e., every edge of \(\mathcal H\) has 2 elements, Theorem 1.2 implies a result proven by \textit{H. T. Hà} and \textit{A. Van Tuyl} [J. Algebr. Comb. 27, No. 2, 215--245 (2008; Zbl 1147.05051)]. The proof is based on Kalai-Meshulam's theorem on regularity of sum of squarefree monomial ideals [\textit{G. Kalai} and \textit{R. Meshulam}, J. Comb. Theory, Ser. A 113, No. 7, 1586--1592 (2006; Zbl 1105.13026)].NEWLINENEWLINEThe second main result gives an instance where regularity can be computed recursively:NEWLINENEWLINE Theorem 1.5. The equality \(\text{reg}~ I=\max\{\text{reg}~ (I+(x)),\text{reg}~ (I:x)+1\}\) holds for \(I\) being the edge ideal of a vertex-decomposable hypergraph, and \(x\) be the initial vertex in its shedding order.NEWLINENEWLINELet \(\mathcal H=G\), a simple graph.NEWLINENEWLINE The third main result (Theorem 1.6) bounds \(\text{reg}~ I(G)\) in terms of the maximal number of star graphs in \(G\) satisfying certain separatedness condition. This gives another improvement of Ha-Van Tuyl's result mentioned above.
scientific article; zbMATH DE number 6335978
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