Combinatorial resolutions of multigraded modules and multipersistent homology

2012/06/08 by Chacholski, Wojciech, Scolamiero, Martina, Vaccarino, Francesco
#55Nxx (Primary) 13D02 (Secondary) #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1206.1819

Abstract

Let R=k[x₁...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.

Related

• Combinatorial presentation of multidimensional persistent homology
• On the support of Betti tables of multiparameter persistent homology modules
• A presentation of general multipersistence modules computable in polynomial time?
• A refinement of multi-dimensional persistence
• Representations of matroids and free resolutions for multigraded modules