Breaker Chains 与 Zigzag 模中区间子模的分解

中文摘要 AI 辅助

有限维 zigzag 模可分解为区间模。对于普通持久性模，限制的每个完全支撑的区间子模都作为直和项分裂，而对于 zigzag 模则未必如此。我们研究这种分裂的失效。一个 breaker 对是局部配置，由两个重叠的区间直和项组成，在第二个区间的左边界有一个后向箭头，在第一个区间的右边界有一个前向箭头。单个 breaker 对给出一个局部非分裂的区间子模，但它未必控制一个给定的更大的限制。因此我们引入 breaker 链。我们的主要结果表明，限制 $V[p,q]$ 包含一个同构于 $I[p,q]$ 的非分裂子模，当且仅当其区间分解包含一个跨越 $[p,q]$ 的 breaker 链。等价地，满足 $\operatorname{Hom}(I[p,q],J)\ne0$ 的适当区间直和项 $J$ 的支撑覆盖 $[p,q]$。这种覆盖表述给出了非分裂嵌入的构造性描述、其分裂行为的三分法，以及一个从具有 $m$ 个区间的有向条形码出发的 $O(m\log m)$ 算法。

英文摘要

Finite-dimensional zigzag modules decompose into interval modules. For an ordinary persistence module, every fully supported interval submodule of a restriction splits as a direct summand, whereas this need not hold for a zigzag module. We study this failure of splitting. A breaker pair is a local configuration of two overlapping interval summands with a backward arrow at the left boundary of the second interval and a forward arrow at the right boundary of the first. A single breaker pair gives a local nonsplitting interval submodule, but it need not control a prescribed larger restriction. We therefore introduce breaker chains. Our main result shows that a restriction $V[p,q]$ contains a nonsplitting submodule isomorphic to $I[p,q]$ if and only if its interval decomposition contains a breaker chain spanning $[p,q]$. Equivalently, the proper interval summands $J$ for which $\operatorname{Hom}(I[p,q],J)\ne0$ have supports covering $[p,q]$. This cover formulation gives a constructive description of nonsplitting embeddings, a trichotomy for their splitting behavior, and an $O(m\log m)$ algorithm from an oriented barcode with $m$ intervals.

发表机构

• Indian Institute of Science Education and Research (IISER) Mohali（印度科学教育研究所（莫哈利））

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