The purpose of this paper is to calculate the cyclic homology of rings of integers of global fields. We accomplish this by explicitly computing the homology of the simple complex associated to Tsygan’s double complex. To accomplish this, we first compute the cyclic homology of cyclic algebras, i.e., algebras of the form A = R[t]/(P (t)), where P is a monic polynomial with coefficients in R. Mor...

Compression aims to reduce the size of an input, while maintaining its relevant properties. For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead large inputs, and tasks this area tend be expensive. We propose two methods for chain complexes free 2-parameter persistence modules. The first method extends multi-ch...

We consider homological edge percolation on a sequence (Gt)t of finite graphs covered by an infinite (quasi)transitive graph H and weakly convergent to H. In particular, we use the covering maps classify 1-cycles Gt as homologically trivial or non-trivial define several thresholds associated with rank thus defined first homology group open subgraphs generated Bernoulli (edge) process. identify ...

Abstract Simplicial complexes are generalizations of classical graphs. Their homology groups widely used to characterize the structure and topology data in e.g. chemistry, neuroscience, transportation networks. In this work we assume given a simplicial complex that can act on its underlying graph, formed by set 1-simplices, investigate stability with respect perturbations edges such graph. Prec...

We explore the codimension one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex ...

Let R be a commutative ring, (f) an ideal of R, and E = K(f ;R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T , the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.

In this work, we compute the topological coHochschild homology (coTHH) of interesting coalgebras such as Steenrod algebra spectrum. For this, start by extending Hess–Shipley definition coTHH to ?$\infty$-categories, following Nikolaus–Scholze approach Hochschild (THH). Furthermore, prove that what call quasi-proper can be obtained from THH via Spanier–Whitehead duality which provides further in...

Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a consequence, we deduce that for any quasi-alternating link, the total rank of its knot Floer homology is equal to the determinant of the link.

In this paper we show that the relation between KajiuraStasheff’s OCHA and A. Voronov’s swiss-cheese operad is analogous to the relation between SH Lie algebras and the little discs operad. More precisely, we show that the OCHA operad is quasi-isomorphic to the operad generated by the topdimensional homology classes of the swiss-cheese operad.