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 Künneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded module 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.

在单参数和多参数设置中，我们开发了持久性模块的同源代数的某些方面，这些参数被视为滑轮或渐变模块。两种理论是不同的。我们考虑了分级模块和捆张量乘积和Hom双函子及其派生的函子Tor和Ext，并给出了区间模块的显式计算。我们给出了内射，射影和平面间隔模块的分类。我们陈述了在捆和分级模块设置中持久性模块链复合物的同源性和同调性的Künneth定理和通用系数定理，并展示了这些定理如何应用于过滤后的细胞复合物产生的持久性模块。我们还为持久性模块给出了一个Gabriel-Popescu定理。最后，我们研究了丰富于持久性模块的类别。我们表明，渐变模块的观点产生了一个封闭的对称单项类别，该类别丰富了自身。