Cohomology theories

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

cohomology with real coefficients computed using differential forms

cohomology theory based on the intersection properties of open covers of a topological space

Group comohology of Galois modules

cohomology theory associated to a group 𝐺 and a 𝐺‐module

right derived functors of the global sections functor Γ: AbSh → Ab

algebraic construction of the cohomology of (the underlying manifold of) a simply connected Lie group in terms of its Lie algebra

sheaf cohomology on the étale site

Mathematical term

formulation to quantize gauge field theories in physics

algebraically defined notion of differential form on an algebra over a ring or on a scheme

Weil cohomology theory for schemes over a base field, whose values are modules over the ring of Witt vectors over the base field, that replaces Zariski open sets by infinitesimal thickenings of Zariski open sets with divided power structures

invariant of algebraic varieties and of more general schemes