  Motivated by recent progress in topological data analysis, we establish a
Matlis duality between injective hulls and flat covers of persistence modules.
This extends to a duality between minimal flat and minimal injective
resolutions. We utilize the theory of flat cotorsion modules and flat covers
developed by Enochs and Xu. By means of this theory we can work with
persistence modules which are not tame or even pointwise finite-dimensional.
