[Paper Review] Modules over posets: commutative and homological algebra

Ezra Miller|arXiv (Cornell University)|Aug 26, 2019

Topological and Geometric Data Analysis36 references4 citations

TL;DR

This paper develops a finiteness condition—tameness—for modules over posets, enabling finite primary decompositions, resolutions, and syzygy theorems analogous to those in noetherian commutative algebra. It proves two conjectures of Kashiwara and Schapira on constructible sheaves with microsupport in a cone by interpreting tameness via derived categories of subanalytically constructible sheaves, yielding computable, topologically interpretable data structures for persistent homology over arbitrary posets.

ABSTRACT

The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multiple real parameters. The algebraic theory yields computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology indexed by arbitrary posets. The exposition focuses on the nature and ramifications of a suitable finiteness condition to replace the noetherian hypothesis. 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 the theoretical and computational purposes: it guarantees finite primary decompositions, as well as various finite presentations and resolutions all related by a syzygy theorem, and the data structures thus produced are computable in addition to being interpretable. The tameness condition and its resulting theory are new even in the finitely generated discrete setting, where being tame is materially weaker than being noetherian.

Motivation & Objective

• To develop a finiteness condition replacing the noetherian hypothesis for modules over posets, enabling finite algebraic structures in persistent homology.
• To establish a theory of primary decomposition, presentations, and resolutions for poset modules under a new tameness condition.
• To prove two conjectures of Kashiwera and Schapira concerning sheaves with microsupport in a given cone using derived category methods.
• To provide computable, topologically interpretable data structures for persistent homology indexed by arbitrary posets, especially in multi-parameter settings.
• To unify topological, algebraic, combinatorial, and homological manifestations of finiteness in poset modules through the concept of tameness.

Proposed method

• Introduces the tameness condition as a finiteness hypothesis for modules over posets, characterized by equivalent topological, algebraic, combinatorial, and homological properties.
• Develops fringe presentations using upsets and downsets to encode modules finitely when tameness holds.
• Constructs indicator resolutions—specifically upset and downset resolutions—using subanalytic or PL structures in derived categories.
• Applies the theory to constructible sheaves by reducing compactly supported cases to finite subdivisions via Theorem 8.22.
• Uses the syzygy theorem for poset modules to relate finite presentations, resolutions, and primary decompositions in the tame setting.
• Establishes that tameness implies finite encoding of modules via constant subdivisions and uptight posets, enabling computation.

Experimental results

Research questions

• RQ1 Can a finiteness condition replace the noetherian hypothesis in the algebra of modules over posets to enable finite presentations and resolutions?
• RQ2 How can persistent homology over arbitrary posets be represented by computable, interpretable data structures based on birth and death of homology classes?
• RQ3 What is the relationship between tameness and the existence of finite primary decompositions in poset modules?
• RQ4 Can the derived category framework be used to prove conjectures about constructible sheaves with microsupport in a cone?
• RQ5 How does tameness unify topological, algebraic, combinatorial, and homological finiteness in poset modules?

Key findings

• The tameness condition guarantees finite primary decompositions, fringe presentations, and resolutions for modules over posets, even in the absence of noetherianity.
• The syzygy theorem holds for tame modules over posets, linking finite presentations, resolutions, and primary decompositions in a coherent algebraic framework.
• The theory proves Conjecture 3.20 of Kashiwara and Schapira, showing that compactly supported PL sheaves with microsupport in the negative polar cone are finite direct sums of constant sheaves on bounded polyhedra.
• The theory proves Conjecture 3.17 of Kashiwara and Schapira, establishing that subanalytically constructible sheaves with microsupport in a negative polar cone admit a subordinate conic stratification.
• The derived category approach yields a subanalytic indicator resolution for constructible sheaves, leading to a finite partition into locally closed subanalytic strata with constant homology.
• Tameness is strictly weaker than noetherianity in the discrete setting, making the theory applicable to a broader class of modules, including those arising in multi-parameter persistent homology.