Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian [1] which gave, together with G.Singh [4], an algorithm to compute their Gröbner bases. Although their algorithm has polynomial complexity when the chain modules are free, i.e. in the one-critical case, it might be exponential in general. We give a new presentation of multipersistence homology modules, which allows...

This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A localglobal cycle is defined to be a family of such local cycles that is, at the same time, a global cycle. The motivating problem is the numerical characterisation of the flag vectors of convex polytopes. Centra...

Let  be a local Cohen-Macaulay ring with infinite residue field,  an Cohen - Macaulay module and  an ideal of  Consider  and , respectively, the Rees Algebra and associated graded ring of , and denote by  the analytic spread of  Burch’s inequality says that  and equality holds if  is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of  as  In this paper we ...

We describe Koszul type complexes associated with a linear map from any module to a free module, and vice versa with a linear map from a free module to an arbitrary module, generalizing the classical Koszul complexes. Given a short complex of finite free modules, we assemble these complexes to what we call Koszul bicomplexes. They are used in order to investigate the homology of the Koszul comp...

We study the multi-dimensional persistence of Carlsson and Zomorodian [3] and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicia...

This work concerns finite free complexes with length homology over a commutative noetherian local ring $R$. The focus is on that have $\mathrm{dim}\, R$, which the smallest possible value, and in particular resolutions of modules projective dimension. Lower bounds are obtained Euler characteristic such short when $R$ strict complete intersection, also Dutta multiplicity, localization at its max...

We investigate the abelian category which is the target of intersection homology. Recall that, given a stratified space X, we get intersection homology groups IHnX depending on the choice of an n-perversity p. The n-perversities form a lattice, and we can think of IHnX as a functor from this lattice to abelian groups, or more generally R-modules. Such perverse R-modules form a closed symmetric ...

We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduc...