One of the main approaches to topological data analysis is to study discrete invariants of (multiparameter) persistence modules. In this talk, we discuss recent research which utilizes techniques from (relative) homological algebra to interpret classical examples of invariants and define new invariants. The Hilbert function and (generalizations of) the rank invariant serve as our main examples. The talk will also serve as a light introduction to the fields of topological data analysis and relative homological algebra. Portions of the talk are based on joint works with Claire Amiot, Benjamin Blanchette, Thomas Brüstle, Steve Oudot, and Luis Scoccola.