Persistence modules stratify their underlying parameter space, a quality that make persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multi-parameter persistence modules. Namely, we show the $K$-theory of grid multi-parameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multi-parameter notions of zig-zag persistence. We compare our calculations for the specific group $K_0$ with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

翻译：持续模对其底层参数空间进行分层，这一特性使其可通过分层空间的不变量进行研究。本文将一个此前仅适用于单参数持续模的结果推广至网格多参数持续模。具体而言，我们证明了网格多参数持续模的K理论在分层上具有可加性。这一结论同时适用于标准单调多参数持续以及多参数锯齿形持续概念。我们针对特定群K₀的计算结果与Botnan、Oppermann和Oudot的最新研究进行了比较，通过一个显式投影映射揭示了计算群之间的差异，并对二者结果的区别进行了说明。