Every 1D persistence module is a restriction of some indecomposable 2D persistence module

A recent work by Lesnick and Wright proposed a visualisation of 2D persistence modules by using their restrictions onto lines, giving a family of 1D persistence modules. We give a constructive proof that any 1D persistence module with finite support can be found as a restriction of some indecomposable 2D persistence module with finite support. As consequences of our construction, we are able to exhibit indecomposable 2D persistence modules whose support has holes as well as an indecomposable 2D persistence module containing all 1D persistence modules with finite support as line restrictions. Finally, we also show that any finite-rectangle-decomposable nD persistence module can be found as a restriction of some indecomposable $$(n+1)$$(n+1)D persistence module.