**Abstract:** A fundamental tool in topological data analysis is persistent homology, which allows detection and analysis of underlying structure in large datasets. Persistent homology (PH) assigns a module over a principal ideal domain to a filtered simplicial complex. While the theory of persistent homology for filtrations associated to a single parameter is well-understood, the situation for multifiltrations is more delicate; Carlsson-Zomorodian introduced multidimensional persistent homology (MPH) for multifiltered complexes via multigraded modules over a polynomial ring. We use tools of commutative and homological algebra to analyze MPH, proving that the MPH modules are supported on coordinate subspace arrangements, and that restricting an MPH module to the diagonal subspace $V(x_i-x_j | i \ne j)$ yields a PH module whose rank is equal to the rank of the original MPH module. This gives one answer to a question asked by Carlsson-Zomorodian. This is joint work with Nina Otter, Heather Harrington, Ulrike Tillman (Oxford).