Combinatorial resolutions of multigraded modules and multipersistent homology

Wojciech Chacholski; Martina Scolamiero; Francesco Vaccarino

Combinatorial resolutions of multigraded modules and multipersistent homology

Algebraic Topology 2015-12-22 v2 Commutative Algebra

Authors: Wojciech Chacholski , Martina Scolamiero , Francesco Vaccarino

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.

Keywords

module theory persistent homology combinatorics

Cite

@article{arxiv.1206.1819,
  title  = {Combinatorial resolutions of multigraded modules and multipersistent homology},
  author = {Wojciech Chacholski and Martina Scolamiero and Francesco Vaccarino},
  journal= {arXiv preprint arXiv:1206.1819},
  year   = {2015}
}

Comments

This paper has been overcome by Combinatorial presentation of multidimensional persistent homology. Wojciech Chacholski, Martina Scolamiero, Francesco Vaccarino. arXiv:1409.7936

Combinatorial resolutions of multigraded modules and multipersistent homology

Abstract

Keywords

Cite

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