AI 中文总结
本文研究循环闭幂补图边环的同调不变量，通过独立复形诱导子复形的同调刻画，得到分次Betti数、正则度等闭式公式。
AI 中文摘要
设 $G_{n,m}=\overline{\C_n^{[m]}}$，其中 $\C_n^{[m]}$ 表示 $n$-循环的闭 $m$ 次幂。我们在 $n\geq 3m+1$ 的范围内研究 $G_{n,m}$ 的边环的分次 Betti 数和同调不变量。这些图构成了一类自然族，用于研究其正则度可以与诱导匹配数显式比较的边环。特别地，对于 $n\geq4m+1$，图 $G_{n,m}$ 的诱导匹配数为 1，而其边环的正则度为 2。我们的方法基于对独立复形 $\Delta(G_{n,m})$ 的诱导子复形的同调的表征。我们引入一族由其在循环周围的连续间隙刻画的顶点子集族 $\mathcal{S}_V(k,m)$，并证明对于 $W\in\mathcal{S}_V(k,m)$，诱导子复形 $\Delta[W]$ 具有 $\mathbb{S}^1$ 的同伦型，而对于 $W\notin\mathcal{S}_V(k,m)$，其所有正维约化同调群均消失。将此表征与 Hochster 公式以及 $\mathcal{S}_V(k,m)$ 的显式枚举相结合，我们获得了第二链上的分次 Betti 数的闭式公式。我们进一步确定了 $G_{n,m}$ 的边环的极值 Betti 数、正则度和投射维数。最后，我们计算了独立复形的 $f$- 和 $h$-向量，并使用 Hilbert 级数确定线性链上的分次 Betti 数。$m=2$ 的情形恢复了文献~\cite{RatherSquare} 中获得的循环平方补图的相关结果。
英文摘要
Let $G_{n,m}=\overline{\C_n^{[m]}}$, where $\C_n^{[m]}$ denotes the closed $m$th power of the $n$-cycle. We study the graded Betti numbers and homological invariants of the edge ring of $G_{n,m}$ in the range $n\geq 3m+1$. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for $n\geq4m+1$, the graph $G_{n,m}$ has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex $Δ(G_{n,m})$. We introduce a family $\mathcal{S}_V(k,m)$ of vertex subsets characterized by their successive gaps around the cycle and show that, for $W\in\mathcal{S}_V(k,m)$, the induced subcomplex $Δ[W]$ has the homotopy type of $\mathbb{S}^1$, whereas for $W\notin\mathcal{S}_V(k,m)$ all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of $\mathcal{S}_V(k,m)$, we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of $G_{n,m}$. Finally, we compute the $f$- and $h$-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case $m=2$ recovers the corresponding results for complements of squares of cycles obtained in~\cite{RatherSquare}.
