In their paper "The theory of multidimensional persistence", Carlsson and Zomorodian write "Our study of multigraded objects shows that no complete discrete invariant exists for multidimensional persistence. We still desire a discriminating invariant that captures persistent information, that is, homology classes with large persistence." In this talk I will discuss how tools from commutative algebra give computable invariants able to capture homology classes with large persistence. Specifically, multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.