Powers of the edge ideals and matchings in hypergraphs

In what follows $t,d\in\mathbb{N}$, $d\geq 2$, $R=K[x_{1},\dots,x_{n}]$ is the polynomial ring over the field $K$ and $S_{1},\dots,S_{m}$ are monomials in $R$. For our convenience, sometimes we use the same notation $S_{k}$ for both of the monomial $S_{k}$ and $\{x_{i}\ :\ x_{i}\mid S_{k}\}$. By identifying the vertex $x_{i}$ and the variable $x_{i}$, $\mathcal{H}$ stands for a simple hypergraph with $V(\mathcal{H})=\{x_{1},\dots,x_{n}\}$ and $\mathcal{E}(\mathcal{H})=\{S_{1},\dots,S_{m}\}$. Recall a hypergraph is $d$-uniform if all of its edges have the same cardinality $d$ and

is the edge ideal of $\mathcal{H}$. So there is a correspondence between $d$-uniform hypergraphs and square-free monomial ideals in degree $d$ (see [18]).

For a minimal graded free resolution

of $R/I$, $\beta_{i,j}(R/I)$ is called the $(i,j)$th graded Betti number of $R/I$. Also, recall that the regularity of $R/I$ is defined as

In the last few decades, studying the regularity of square-free monomial ideals because of their connections to other fields such as combinatorics, algebraic topology and computational algebra, has been extensively expanded. In particular, many researchers in combinatorial commutative algebra are interested in computing or bounding the regularity of the edge ideal of graphs and hypergraphs and their powers (see [1] and [9] for some surveys in this context).

In this paper, we firstly investigate free resolutions of powers of a square-free monomial ideal supported on a simplicial complex. Then we use the gained results in Section 2 for a simplicial complex introduced in [8] for powers of the edge ideal of a hypergraph. These yield to characterize or bound the graded Betti numbers and regularity of $R/I^{t}$ in Section 3. For instance in Corollary 3.5 a combinatorial characterization is gained for vanishing of $\beta_{i,2di}(R/I^{2})$ in terms of the number of matchings of size $i$ in a $d$-uniform hypergraph $\mathcal{H}$. The needed definitions of some kinds of matchings are presented in Definition 3.1. Moreover the main result of this note is the following result:

The above result is indeed a generalization of some parts of [10, Theorem 6.5], [12, Lemma 2.2 and Proposition 2.5], [16, Theorems 3.5(1), 3.7(1) and Corollary 3.6] for arbitrary powers of the edge ideal of a hypergraph. Also, it gives a lower bound better than [3, Theorem 3.7(1) and Corollary 3.8(1)] and [17, Corollary 3.9] for the regularity of powers of the edge ideals of uniform hypergraphs, since there are examples of $d$-uniform hypergraphs achieving our bound such that $(s.s.i.m)^{\prime}_{\mathcal{H}}$ is reasonably larger that $(d-1)(i.m)_{\mathcal{H}}$ as presented in Example 3.8.

In this section, for each $1\leq k\leq m$,

Hence $I$ is a square-free monomial ideal of $R$ generated in degree $d$. Now assume that

Hereafter we order the elements of $\mathcal{B}_{m,t}$ in such a way that if $\textbf{b}=(b_{1},\dots,b_{m})$ and $\textbf{b}^{\prime}=(b^{\prime}_{1},\dots,b^{\prime}_{m})$ are two elements of $\mathcal{B}_{m,t}$, then $\textbf{b}<\textbf{b}^{\prime}$ if and only if there exists an integer $1\leq k\leq m$ such that for all $1\leq\ell\leq k-1$ we have $b_{\ell}=b^{\prime}_{\ell}$ and $b_{k}>b^{\prime}_{k}$. If we set $S^{\textbf{b}}=S_{1}^{b_{1}}\dots S_{m}^{b_{m}}$ for all $\textbf{b}=(b_{1},\dots,b_{m})\in\mathcal{B}_{m,t}$, then $I^{t}=\langle S^{\textbf{b}}\ :\ \textbf{b}\in\mathcal{B}_{m,t}\rangle$. But note that $\{S^{\textbf{b}}\ :\ \textbf{b}\in\mathcal{B}_{m,t}\}$ is not necessarily a minimal set of generators of $I^{t}$. So, $I^{t}$ is minimally generated by $p_{m,t}$ elements, when $p_{m,t}\leq|\mathcal{B}_{m,t}|=\left(\begin{array}[]{c}t+m-1\\ m-1\end{array}\right)$. In this section we are going to study some simplicial resolutions of $R/I^{t}$. To this aim we first need some notation.

Suppose that $\Delta$ is a simplicial complex whose vertices are labelled by the minimal monomial generators of $I$ and whose faces are labelled by the least common multiple of the vertices of that face. Then we say that $\Delta$ supports a free resolution $F_{\bullet}$ of $R/I$ or $F_{\bullet}$ is supported on $\Delta$ if the labelling of the vertices of $\Delta$ satisfies certain properties and the simplicial chain complex of $\Delta$ can be homogenized using the monomial labels on the faces. A well-known simplicial complex which supports a free resolution of $R/I$ is Taylor complex of $I$ which is a simplex on $m$ vertices labelled by the minimal monomial generators of $I$ and is denoted by $\mathrm{Taylor}(I)$.

In view of [4], if $\Delta$ supports a free resolution $F_{\bullet}$ of $I^{t}$, then $\Delta$ is a subcomplex of $\mathrm{Taylor}(I^{t})$ and $F_{\bullet}$ can be structured as follows.

where $F_{0}=R$ and $F_{i}$ is the free $R$-module whose free generators are $e_{\tau}$s for all faces $\tau$ of $\Delta$ of dimension $i-1$. If the vertices of $\tau$ is labelled by the monomials $S^{\textbf{b}_{\ell_{1}}},\dots,S^{\textbf{b}_{\ell_{i}}}$, then we sometimes denote $e_{\tau}$ by $e_{\ell_{1},\dots,\ell_{i}}$, where $1\leq\ell_{1}<\dots<\ell_{i}\leq p_{m,t}$. Also for all $1\leq i\leq\mathrm{dim}\Delta+1$ and $e_{\ell_{1},\dots,\ell_{i}}\in F_{i}$,

where

Note that for each $1\leq j\leq i$, if $\textbf{b}_{\ell_{j}}=(b_{1,\ell_{j}},\dots,b_{m,\ell_{j}})$, then we have

where $AB=(c_{r,s})_{n\times p_{m,t}}$. Now assume that $\mathrm{Max}(AB[\ell_{1},\dots,\ell_{i}])=(c_{1},\dots,c_{n})$ and

Then we have $\mu_{k}=\prod_{1\leq i\leq n}x_{i}^{c_{i}-d_{i,k}}$. By considering the degree of $e_{\ell_{1},\dots,\ell_{i}}$ as

$F_{\bullet}$ is a graded free resolution of $R/I^{t}$ which is not necessarily minimal as mentioned before, but we may use it for computing the graded Betti numbers $\beta_{i,j}(R/I^{t})$ as follows.

After tensoring $F_{\bullet}$ with $R/\langle x_{1},\dots,x_{n}\rangle$, for each $e_{\tau}$ where $\tau=\{S^{\textbf{b}_{\ell_{1}}},\dots,S^{\textbf{b}_{\ell_{i}}}\}\in\Delta$, we have

where for each $0\leq i\leq\mathrm{dim}\Delta+1$ and each member $u\in F_{i}$, $\overline{u}$ is the natural image of $u$ in $\overline{F_{i}}=F_{i}\otimes_{R}R/\langle x_{1},\dots,x_{n}\rangle$ and $\overline{\partial}_{i}=\partial_{i}\otimes_{R}\mathrm{id}_{R/\langle x_{1},\dots,x_{n}\rangle}$.

Afterwards suppose that $\Delta$ is a simplicial complex supporting a free resolution of $R/I^{t}$.

The above remarks illustrate that one can evaluate the graded Betti numbers of the edge ideal of a hypergraph by interaction of its edges as you see for instance in [10, Theorem 6.5], [12, Lemma 2.2] and [16, Section 3]. Note that all of these results just use Taylor resolution of $I^{t}$ (when $t=1$) which has easier structure as you may see above.

One can naturally generalize [16, Lemma 3.2] and obtain the following result, which is needed in the next section.

To start, we recall the following definitions.

The following observations may be helpful in the sequel.

Now we need some definitions from [7] and [8]. In [8] a simplicial complex $\mathbb{L}_{m}^{t}$ is defined and it is shown that it supports a free resolution of $I^{t}$. In [7], the case $t=2$ is investigated individually. So, for our next results of this section, we refer the reader to Definitions 3.1 and 3.4 in [7] and Definitions 4.2 and 5.1 and Proposition 4.3 in [8]. Hereinafter we apply the notions in Section 2 for the simplicial complex $\mathbb{L}_{m}^{t}$.

One can use Proposition 3.4 for vanishing of the special graded Betti numbers of the second power of the edge ideal as follows.