Details

Autor(en) / Beteiligte

• Bubenik, Peter
• Milićević, Nikola

Titel

Homological Algebra for Persistence Modules

Ist Teil von

• Foundations of computational mathematics, 2021-10, Vol.21 (5), p.1233-1278

Ort / Verlag

New York: Springer US

Erscheinungsjahr

2021

Link zum Volltext

Quelle

SpringerLink (Online service)

Beschreibungen/Notizen

• 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.