A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations -- often referred to as multi-parameter persistence modules. One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode. This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability. We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes. We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules. We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset. This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.

翻译：现代拓扑数据分析的重要组成部分关注偏序集表示（通常称为多参数持久性模块）的代数不变量的设计与研究。其中一种不变量是最小秩分解，它通过单个有序矩形可分解模块对（解释为带符号条形码）对持久性模块的所有结构态射的秩进行编码。该带符号条形码将持久性条形码的概念从单参数持久性推广到任意参数数量，由此引发其瓶颈稳定性问题。本文证明：最小秩分解在带符号条形码的带符号瓶颈匹配自然概念下不具有稳定性。为解决这个问题，我们将重点转向秩精确分解——一种由模块相对于所谓秩精确结构的最小投射分解诱导的相关带符号条形码，并证明其在带符号匹配下具有瓶颈稳定性。作为证明的一部分，我们获得了两个具有独立意义的中介结果：计算了有限可表示多参数持久性模块范畴上秩精确结构的全局维数，并证明了钩形可分解模块的瓶颈稳定性结果。我们还给出了秩精确分解大小的上界（该上界与通常最小投射分解大小呈多项式关系），证明了由带符号匹配概念诱导的相异度函数的普适性结果，并在双参数情形下计算了与索引偏序集上集相关的不同精确结构的全局维数。这些结果融合了拓扑数据分析与偏序集表示理论的概念，我们认为对这两个领域都具有重要意义。