Koszul Complexes and Relative Homological Algebra of Functors Over Posets

被引：3

作者：

Chacholski, Wojciech ^{[1
]}

Guidolin, Andrea ^{[1
]}

Ren, Isaac ^{[1
]}

Scolamiero, Martina ^{[1
]}

Tombari, Francesca ^{[2
]}

机构：

[1] KTH, Dept Math, S-10044 Stockholm, Sweden

[2] Max Planck Inst Math Sci, D-04103 Leipzig, Germany

来源：

FOUNDATIONS OF COMPUTATIONAL MATHEMATICS | 2024年

关键词：

Relative homological algebra; Poset representations; Betti diagrams; Koszul complexes; Topological data analysis; Multi-parameter persistent homology;

D O I：

10.1007/s10208-024-09660-z

中图分类号：

TP301 [理论、方法];

学科分类号：

081202 ;

摘要：

Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

引用

页数：45

共 26 条

[1]

[Anonymous], 1956, Trans. Amer. Math. Soc.

[2]

[Anonymous], 2000, Relative Homological Algebra

[3] Approximation by interval-decomposables and interval resolutions of persistence modules * [J].