A multifiltration is a functor indexed by N^r that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural N^r-graded R[x_1,...,x_r]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and R-modules. We prove in particular that the N^r-graded R[x_1,...,x_r]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals.

Combinatorial presentation of multidimensional persistent homology / Chacholski, W.; Scolamiero, Martina; Vaccarino, Francesco. - In: JOURNAL OF PURE AND APPLIED ALGEBRA. - ISSN 0022-4049. - STAMPA. - (2017). [10.1016/j.jpaa.2016.09.001]

### Combinatorial presentation of multidimensional persistent homology

#### Abstract

A multifiltration is a functor indexed by N^r that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural N^r-graded R[x_1,...,x_r]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and R-modules. We prove in particular that the N^r-graded R[x_1,...,x_r]-modules that can occur as R-spans of multifiltrations of sets are the direct sums of monomial ideals.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11583/2651578`