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Combinatorial resolutions of multigraded modules and multipersistent homology

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arxiv 1206.1819 v2 pith:6EAATF7O submitted 2012-06-08 math.AT math.AC

Wojciech Chacholski , Martina Scolamiero , Francesco Vaccarino This is my paper · ORCID
classification math.ATmath.AC
keywords homologymultigradedcellularmodulecombinatorialcomplexmodulesmultipersistence
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abstract

Let $ R=k[x_1...x_r]$ and $M$ a multigraded $R-$module. In this work we interpret $M$ as a multipersistent homology module and give a multigraded resolution of it. The construction involves cellular resolutions of monomial ideals and reflects the combinatorial structure of multipersistence homology modules. In the one critical case, a multifiltration is represented by a labelled cellular complex. A multipersistence homology module measures the defect of acyclicity of the associated multigraded cellular chain complex.

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  1. Modules over posets: commutative and homological algebra

    math.AC 2019-08 conditional novelty 8.0 of 10

    Tame modules over arbitrary posets are proved to have finite encodings, fringe presentations, indicator resolutions, and primary decompositions, leading to proofs of two sheaf-theoretic conjectures.

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