Differential modules and Deformations of Free Complexes
Maya Banks, Keller VandeBogert
TL;DR
This paper develops a theory for deforming free complexes into differential modules, classifies such modules up to quasi-isomorphism, and applies the results to study rigidity properties and rank conjectures.
Contribution
It introduces an iterative deformation approach parameterized by Ext groups, providing an algorithmic classification of differential modules with prescribed homology.
Findings
Classified free differential modules up to quasi-isomorphism.
Developed an algorithmic method for deforming free complexes.
Applied theory to rigidity properties and rank conjectures.
Abstract
We classify (up to quasi-isomorphism) the free differential modules whose homology is equal to a given module by developing a theory for deforming an arbitrary free complex into a differential module. We use an iterative approach to parameterize the deformations and obstructions in terms of certain Ext groups, giving an algorithmic realization of a result of Brown-Erman. We apply this theory to study certain rigidity properties of free resolutions and related rank conjectures.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models