# Homological Algebra for Persistence Modules

## TL;DR

This paper develops the homological algebra framework for persistence modules, including tensor products, Hom functors, and derived functors, providing explicit computations and classification results in both sheaf and graded module contexts.

## Contribution

It introduces a comprehensive homological algebra theory for persistence modules, including classification, Kunneth and universal coefficient theorems, and enriched category structures.

## Key findings

- Explicit computations for interval modules
- Classification of injective, projective, and flat modules
- Establishment of Kunneth and universal coefficient theorems

## Abstract

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.