We introduce a bounded version of Bredon cohomology for groups relative to family subgroups. Our theory generalizes and differs from Mineyev--Yaman's pairs. obtain cohomological characterizations amenability hyperbolicity, analogous the results Johnson Mineyev cohomology.

Let H be a Hopf algebra over a field k, and A an Hcomodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev (i.e. unipotent) completion of its fundamental group π1(X,x0) and the cohomology groups H (X,Lρ) with coefficients in rank one local systems. In this paper, we develop a tool that unifies these two approaches. This tool is the Malcev completion of π1(X,x0) relative to a hom...

The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of Kähler differentials. Here we generalize it for simplicial, graded and anticommutative objects in “good pairs of categories”. We apply this generalization to complex spaces and noetherian schemes and deduce two decomposition theorems for their (relati...

This paper uses relative symplectic cohomology, recently studied by Varolgunes, to understand rigidity phenomena for compact subsets of manifolds. As an application, we consider a crossings divisor in Calabi–Yau manifold [Formula: see text] whose complement is Liouville manifold. We show that, carefully chosen structure, the skeleton as subset exhibits strong properties akin superheavy Entov–Po...

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider Leibniz (rRB algebras) as object our study. We construct an L∞-algebra that characterizes rRB its Maurer–Cartan elements. Then define representations algebra and introduce cohomology with coefficients a representation. As applications ...