Homological Algebra of Modules over Posets

A theory of finite presentations and resolutions is developed for modules over an arbitrary poset Q to parallel the theory for finitely generated modules over polynomial rings. Since the Noetherian hypothesis fails, an alternative finiteness condition is introduced: the notion of a tame Q-module, which captures the hypothesis that, as a family of vector spaces indexed by Q, the module should be constant on finitely many regions, each marked by a single vector space of finite dimension. Four characterizations of tameness are provided: the initial topological definition in terms of constant regions; a combinatorial version by poset encoding; an algebraic version by fringe presentation that combines poset analogues of free presentation and injective copresentation; and a homological version culminating in a syzygy theorem, in analogy with the Hilbert syzygy theorem for modules over polynomial rings. Data structures for tame modules are introduced using monomial matrices to serve both theoretical and computational purposes. In the context of persistent homology of filtered topological spaces, especially with multiple real parameters, monomial matrices for tame modules yield topologically interpretable data structures in terms of birth and death of homology classes.

Duke Scholars

Author Ezra Miller Mathematics