Multi-Parameter Persistence Modules are Generically Indecomposable

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.