论文标题
多参数持久性模块通常不可分解
Multi-Parameter Persistence Modules are Generically Indecomposable
论文作者
论文摘要
代数持久性研究持久性模块（通常，使用$ n \ geq 1 $的poset $ \ mathbf {r}^n $的线性表示形式）以及交织在一起的持久性模块之间的代数关系。持续模块之间的$ \ varepsilon $ - 间接的概念是同构概念的概括（当$ \ varepsilon = 0 $）的概念（恢复异构态度），可以用来量化任何两个持久模块与同位词的持续程度。此类研究的一个象征性例子是代数稳定定理，它增强了一个参数持续模块的krull-schmidt属性（$ \ mathbf {r} $的表示形式）通过概括同构的相互交流：如果其中一对单位持续的持久式模块是$ \ vareastial $ \ varemial $ \ varemial in ininine $ - 在两个模块的不可分解的求和之间匹配，以使匹配的不可分配为$ \ varepsilon $ - 互化和无与伦比的indecosables是$ \ varepsilon $ - 与零模块的互化。我们的第一个主要结果意味着，代数稳定性理论的明显扩展为多参数持续模块的情况（$ \ \ \ \ m i \ mathbf {r}^n $，具有$ n \ geq 2 $）的景观非常失败：任何有限呈现的多参数式式模块都可以在近似仲裁中近似仲裁。我们的第二个主要结果指出，足够接近不可分解的分解的模块是不可分解的和几乎微不足道的模块的直接总和。我们从这两个结果中得出了关于代数与多参数持续模块的拓扑特性之间相互作用的几个后果。这些结果提供了强大的动力，以不依赖不可分解的模块直接分解模块的方式来接近多参数持久性。
Algebraic persistence studies persistence modules (typically, linear representations of the poset $\mathbf{R}^n$ with $n \geq 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of $\varepsilon$-interleaving between persistence modules is a generalization of the notion of isomorphism (recovering isomorphism when $\varepsilon = 0$), which can be used to quantify how far any two persistence modules are from being isomorphic. An emblematic example of this kind of study is the algebraic stability theorem, which strengthens the Krull--Schmidt property of one-parameter persistence modules (representations of $\mathbf{R}$) by generalizing isomorphism to interleaving: If a pair of one-parameter persistence modules is $\varepsilon$-interleaved, then there exists a partial matching between the indecomposable summands of the two modules such that matched indecomposables are $\varepsilon$-interleaved and unmatched indecomposables are $\varepsilon$-interleaved with the zero module. Our first main result implies that the obvious extension of the algebraic stability theorem to the case of multi-parameter persistence modules (representations of $\mathbf{R}^n$ with $n \geq 2$) fails spectacularly: Any finitely presentable multi-parameter persistence module can be approximated arbitrarily well by an indecomposable module. Our second main result states that modules that are sufficiently close to an indecomposable decompose as a direct sum of an indecomposable and a nearly trivial module. We derive from these two results several consequences about the interplay between the algebraic and the topological properties of multi-parameter persistence modules. These results provide strong motivation for approaching multi-parameter persistence in a way that does not rely on directly decomposing modules by indecomposables.
