局部化持续交换代数

中文摘要 AI 辅助

我们针对Stanley-Reisner环发展了一种局部化的交换代数持续理论，其基于在坐标素理想而非极大理想处支撑的局部上同调。该构造以Suwayyid和Wei的持续Stanley-Reisner理论（arXiv:2503.23482）及其针对图和超图的函子性发展（arXiv:2512.17619）为模型，在这些工作中，面环的不变量如分次Betti数、f-向量和h-向量会在滤过过程中被持续化。该框架构建于极小自由分解之上，因此是Tor-理论性的；我们则转而在内射侧开展工作，所得模记录了在单个顶点处局部化的信息，补充了由极大支撑局部上同调给出的全局图景。对于顶点素理想 $p_i = (x_j: j \neq i)$，我们证明了 $H^q_{p_i}(k[\Delta])$ 的一个 $\mathbb{Z}^n$ 分次正合分解，该分解对应于顶点i的删除和链的极大支撑局部上同调，前者位于 $x_i$ 次数0处，后者在所有正 $x_i$ 次数处重复出现；在分次维数层面，这恢复了Rahimi双分次公式的顶点素理想情形。结合Hochster公式，这给出了每个多重分次分量的闭合组合描述。基于该结构，我们引入了按顶点划分的持续局部上同调数，证明了持续链-Hochster公式，得到了所得反向箭头持续模的区间分解和瓶颈稳定性定理，保留了非逆变量的乘法作为持续模的一个态射（该态射无法由两个条形码单独确定），并将该理论推广到任意坐标素理想，其中非逆变量的乘法映射组装成一个交换的布尔持续模图。

英文摘要

We develop a localized persistent theory of commutative algebra for Stanley-Reisner rings, based on local cohomology supported at a coordinate prime rather than at the maximal ideal. The construction is modeled on the persistent Stanley-Reisner theory of Suwayyid and Wei (arXiv:2503.23482) and its functorial development for graphs and hypergraphs (arXiv:2512.17619), in which invariants of the face ring such as graded Betti numbers and f- and h-vectors are persisted across a filtration. That framework is built from the minimal free resolution and is thus Tor-theoretic; we work instead on the injective side, and the resulting modules record information localized at a single vertex, complementing the global picture given by maximal-support local cohomology. For a vertex prime $p_i = (x_j : j \neq i)$ we prove an exact $\mathbb{Z}^n$-graded decomposition of $H^q_{p_i}(k[Δ])$ into the maximal-support local cohomology of the deletion and of the link of the vertex $i$, the first in $x_i$-degree zero and the second repeated in every positive $x_i$-degree; at the level of graded dimensions this recovers the vertex-prime case of Rahimi's bigraded formula. With Hochster's formula this yields a closed combinatorial description of every multigraded piece. Building on this structure we introduce per-vertex persistent local cohomology numbers, prove a persistent links-Hochster formula, obtain interval decompositions of the resulting reversed-arrow persistence modules and a bottleneck stability theorem, retain multiplication by the uninverted variable as a morphism of persistence modules that the two barcodes alone do not determine, and extend the theory to an arbitrary coordinate prime, where the multiplication maps of the uninverted variables assemble into a commuting Boolean diagram of persistence modules.

发表机构

• Syracuse University（雪城大学）
• King Fahd University of Petroleum and Minerals（法赫德国王石油与矿产大学）
• University of Georgia（佐治亚大学）

机构由 AI 辅助整理，请以论文原文为准。