  The persistence barcode is a well-established complete discrete invariant for
finitely generated persistence modules [5] [1]. Its definition, however, does
not extend to multi- dimensional persistence modules. In this paper, we
introduce a new discrete invariant: the exterior critical series. This
invariant is complete in the one-dimensional case and can be defined for
multi-dimensional persistence modules, like the rank invariant [2]. However,
the exterior critical series can detect some features that are not captured by
the rank invariant.
