边添加下非匹配复形的同调及其在Stanley-Reisner理想中的应用
Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals
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中文总结 AI 辅助
该研究针对二分图的$t$-非匹配复形，证明边添加诱导同调单射，结合模型得到Leray数等结果，还确定了完全二分图对应理想的相关同调性质并解决了一个猜想。
中文摘要 AI 辅助
对于二分图$G$和整数$t\geq2$，令$\nM_t(G)$为其$t$-非匹配复形。我们证明，在保持二分性的同时添加一条边，会在次数$2t-3$的约化同调上诱导出一个单射。结合循环图非匹配复形的循环多面体模型，这表明当$G$包含长度至少为$2t$的环时，$\nM_t(G)$的Leray数为$2t-2$。在相同假设下，Hochster公式给出了Stanley-Reisner理想$I_{\nM_t(G)}$的正则性为$2t-1$，同时给出了其最高正则性分支上Betti数的显式下界及投射维数的下界。若$G$包含一个$2t$-环，我们还确定了$I_{\nM_t(G)}$在同调度$|E(G)|-2t+1$处的最大位移。对于$2\leq t\leq r\leq s$的$G=K_{r,s}$，我们确定了$I_{\nM_t(K_{r,s})}$的深度、投射维数、所有最大位移及唯一的极端Betti数，从而解决了关于棋盘复形面理想的一个猜想。
英文摘要
For a bipartite graph $G$ and an integer $t\geq2$, let $\NM_t(G)$ be its $t$-non-matching complex. We prove that adding an edge while preserving bipartiteness induces an injection on reduced homology in degree $2t-3$. Combined with the cyclic-polytope model for non-matching complexes of cycles, this shows that $\NM_t(G)$ has Leray number $2t-2$ whenever $G$ contains a cycle of length at least $2t$. Under the same hypothesis, Hochster's formula yields regularity $2t-1$ for the Stanley-Reisner ideal $I_{\NM_t(G)}$, together with explicit lower bounds for the Betti numbers on its top regularity strand and for its projective dimension. If $G$ contains a $2t$-cycle, we also determine the maximal shifts of $I_{\NM_t(G)}$ through homological degree $|E(G)|-2t+1$. For $G=K_{r,s}$ with $2\leq t\leq r\leq s$, we determine the depth, projective dimension, all maximal shifts, and the unique extremal Betti number of $I_{\NM_t(K_{r,s})}$, thereby settling a conjecture on facet ideals of chessboard complexes.
发表机构
- School of Mathematics and Systems Science, Shenyang Normal University（沈阳师范大学数学与系统科学学院）
- School of Mathematical Sciences, Soochow University（苏州大学数学科学学院）
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