We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky’s h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Čech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves.

Homology theories for algebraic varieties are often constructed using simplicial sets of algebraic cycles. For example, Bloch’s higher Chow groups and motivic cohomology, as defined by Suslin and Voevodsky [17], are given in this fashion. In this paper, we construct homology groups Hi(X,G), where G is an algebraic group and X is a variety, by considering cycles on the simplicial scheme BG×X, an...

Let A be a Banach algebra, not necessarily unital, and let B be a closed subalgebra of A. We establish a connection between the Banach cyclic cohomology group HC(A) of A and the Banach B-relative cyclic cohomology group HCnB(A) of A. We prove that, for a Banach algebra A with a bounded approximate identity and an amenable closed subalgebra B of A, up to topological isomorphism, HC(A) = HCnB(A) ...

Taking the `1-completion and the topological dual of the singular chain complex gives rise to `1-homology and bounded cohomology respectively. In contrast to `1-homology, major structural properties of bounded cohomology are well understood by the work of Gromov and Ivanov. Based on an observation by Matsumoto and Morita, we derive a mechanism linking isomorphisms on the level of homology of Ba...

We introduce a notion of ‘cover of level n’ for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram of n-covers biject with global homotopy classes when the target is an n-type. When the target is an Eilenberg–MacLane sheaf, this specializes to computing derived functor cohomology, up to degree ...

Let (MUn)n denote the Ω-spectra representing the cobordism cohomology theory. The coefficient ring MU∗ is the polynomial algebra over Z generated by elements xk of degree 2k, k ≥ 1. As we will consider spaces like map(CP,MUn) which already has infinite mod p cohomology group in each degree, we will use profinite completion and continuous cohomology. So we fix a prime number p and let Ŝ, respect...

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)ho...