Complex Matching Distance and Stability for Minimal Projective Resolutions, with Applications to Persistence

Abstract

We develop a stability theory for minimal projective resolutions of $\mathbf{P}$-modules, where $\mathbf{P}$ is a finite metric poset. We use the G\"ulen--McCleary distance on $\mathbf{P}$-modules together with a new complex matching distance on bounded complexes of finitely generated projective $\mathbf{P}$-modules. The latter yields an extended metric on homotopy classes of such complexes and restricts to minimal projective resolutions. Our main theorem shows that this induced distance on minimal projective resolutions is bounded above by the G\"ulen--McCleary distance. As an application, we pass to the interval poset and kernel construction, interpreting persistence diagrams as the homotopy class of minimal projective resolutions of kernel modules. This gives a corresponding stability theorem, which in the one-parameter case recovers classical bottleneck stability and in the multiparameter case addresses existing notions of signed diagrams.