Homological algebra of modules over posets
Abstract
Homological algebra of modules over posets is developed, as closely parallel as possible to that of finitely generated modules over noetherian commutative rings, in the direction of finite presentations and resolutions. Centrally at issue is how to define finiteness to replace the noetherian hypothesis which fails. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both theoretical and computational purposes: it guarantees finite presentations and resolutions of various sorts, all related by a syzygy theorem, amenable to algorithmic manipulation. Tameness and its homological theory are new even in the finitely generated discrete setting of $\mathbb{N}^n$-gradings, where tame is materially weaker than noetherian. In the context of persistent homology of filtered topological spaces, especially with multiple real parameters, the algebraic theory of tameness yields topologically interpretable data structures in terms of birth and death of homology classes.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2020
- DOI:
- 10.48550/arXiv.2008.00063
- arXiv:
- arXiv:2008.00063
- Bibcode:
- 2020arXiv200800063M
- Keywords:
-
- Mathematics - Algebraic Topology;
- Mathematics - Commutative Algebra;
- Mathematics - Algebraic Geometry;
- Mathematics - Combinatorics;
- Mathematics - Representation Theory;
- 05E40;
- 13E99;
- 06B15;
- 13D02;
- 55N31;
- 06A07;
- 32B20;
- 14P10;
- 52B99;
- 13A02;
- 13P20;
- 68W30;
- 13P25;
- 62R40;
- 06A11;
- 06F20;
- 06F05;
- 68T09
- E-Print:
- 43 pages, 16 figures. Supersedes the homological portion of arXiv:1908.09750 (which, in turn, superseded the first few sections of arXiv:1709.08155). The material concerning primary decomposition in partially ordered groups and proofs of conjectures due to Kashiwara and Schapira are now in separate manuscripts