Combinatorial resolutions of multigraded modules and multipersistent homology
arXiv:1206.1819
Abstract
Let and a multigraded module. In this work we interpret 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.
This paper has been overcome by Combinatorial presentation of multidimensional persistent homology. Wojciech Chacholski, Martina Scolamiero, Francesco Vaccarino. arXiv:1409.7936