mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated...

specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic...

In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of...

Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The...

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of...

Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the...

In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape...

visualized. More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the...

In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological...

In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a...

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients...

symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes...

similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically,...

for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. Especially in algebraic...

mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces. It was introduced by James W. Alexander (1935)...

and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex...

In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of...

mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into...

L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except...

coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological...

In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking...

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for...

specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product...

In mathematics, Deligne cohomology sometimes called Deligne-Beilinson cohomology is the hypercohomology of the Deligne complex of a complex manifold. It...

the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms. The homotopy invariance of cohomology states...

In mathematics, a nonabelian cohomology is any cohomology with coefficients in a nonabelian group, a sheaf of nonabelian groups or even in a topological...

In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. Historically...

In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have...

In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not...

developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002. He is also known for the proof...