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Understanding the structure of indecomposable $n$-dimensional persistencemodules is a difficult problem, yet is foundational for studyingmultipersistence. To this end, Buchet and Escolar showed that any finitelypresented rectangular $(n-1)$-dimensional persistence module with finitesupport is a hyperplane restriction of an $n$-dimensional persistence module.We extend this result to the following: If $M$ is any finitely presented$(n-1)$-dimensional persistence module with finite support, then there existsan indecomposable $n$-dimensional persistence module $M'$ such that $M$ is therestriction of $M'$ to a hyperplane. We also show that any finite zigzagpersistence module is the restriction of some indecomposable $3$-dimensionalpersistence module to a path.

Hyperplane Restrictions of Indecomposable $n$-Dimensional Persistence Modules