We prove that for any monoid M , the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZM -modules.

Given a double complex X there are spectral sequences with the E2 terms being either HI (HII(X)) or HII(HI(X)). But if HI(X) = HII(X) = 0 both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So in this situation the spectral sequences may not give ...

The goal of this paper is to provide homological descriptions of monomial ideals. The key concepts in these descriptions are the minimal free resolution, the Koszul homology and the multigraded Betti numbers. These three objects are strongly related, being Tor modules a simple way to describe this relation. We introduce a new tool, Mayer-Vietoris trees which provides a good way to compute the h...

The Koszul homology of modules of the polynomial ring R is a central object in commutative algebra. It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we consider the case of modules of the form R/I where I is a monomial ideal. So far, some good algorithms have been given in the literature and implemented in different...

We extend the results of Adcock, Carlsson, and Carlsson ([ACC13]) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson, Singh, and Zomorodian in [CSZ10]. The use of topology to study point cloud data has been well established ([Car09], [Car14]). Given a finite metric space (e.g., a finite set in R n), one first constructs a filter...

We generalise Khovanov’s chain complex built from a “cube” of modules and homomorphisms, to a more general setting. We define the notion of a coloured poset and construct a homology functor for these objects, showing that for coloured Boolean lattices the resulting homology agrees with the homology of Khovanov’s complex.

We compute the homology of complexes finite Verma modules over annihilation superalgebra $\mathcal A(K'_{4})$, associated with conformal $K'_{4}$, obtained in \cite{K4}. use computation order to provide an explicit realization all irreducible quotients A(K'_{4})$.

A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.

We prove rigidity type results on the vanishing of stable Ext and Tor for modules of finite complete intersection dimension, results which generalize and improve upon known results. We also introduce a notion of prerigidity, which generalizes phenomena for modules of finite complete intersection dimension and complexity one. Using this concept, we prove results on length and vanishing of homolo...

This is a further investigation of our approach to group actions in homological algebra the settings homology $$\Gamma $$ -simplicial groups, particularly -equivariant and cohomology -groups. new direction called -homological algebra. The abstract kernel non-abelian extensions its relationship with obstruction theory, second are extended case -extensions We compute rational (co)homology groups ...