Category: Cohomology theories

List of cohomology theories
This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For othe
Monsky–Washnitzer cohomology
In algebraic geometry, Monsky–Washnitzer cohomology is a p-adic cohomology theory defined for non-singular affine varieties over fields of positive characteristic p introduced by Paul Monsky and Gerar
Gelfand–Fuks cohomology
In mathematics, Gelfand–Fuks cohomology, introduced in, is a cohomology theory for Lie algebras of smooth vector fields. It differs from the Lie algebra cohomology of Chevalley-Eilenberg in that its c
Chromatic homotopy theory
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work
Quantum cohomology
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two v
Topological modular forms
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these
Lie algebra cohomology
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 Lie groups and homogeneous spaces by relating co
P-adic cohomology
No description available.
André–Quillen cohomology
In commutative algebra, André–Quillen cohomology is a theory of cohomology for commutative rings which is closely related to the cotangent complex. The first three cohomology groups were introduced by
Alexander–Spanier cohomology
In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.
Complex-oriented cohomology theory
In algebraic topology, a complex-orientable cohomology theory is a multiplicative cohomology theory E such that the restriction map is surjective. An element of that restricts to the canonical generat
Sheaf cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructio
In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likew
Goncharov conjecture
In mathematics, the Goncharov conjecture is a conjecture introduced by Goncharov suggesting that the cohomology of certain motivic complexes coincides with pieces of K-groups. It extends a conjecture
Coherent sheaf cohomology
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questio
Group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous
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 fr
Elliptic cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
L² cohomology
In mathematics, 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 that one uses square-integrab
Nonabelian cohomology
In mathematics, a nonabelian cohomology is any cohomology with coefficients in a nonabelian group, a sheaf of nonabelian groups or even in a topological space. If homology is thought of as the abelian
Spencer cohomology
In mathematics, Spencer cohomology is cohomology of a manifold with coefficients in the sheaf of solutions of a linear partial differential operator. It was introduced by Donald C. Spencer in 1969.
Bredon cohomology
The Bredon cohomology, introduced by Glen E. Bredon, is a type of equivariant cohomology that is a contravariant functor from the category of G-complex with equivariant homotopy maps to the category o
Hodge–de Rham spectral sequence
In mathematics, the Hodge–de Rham spectral sequence (named in honor of W. V. D. Hodge and Georges de Rham) is an alternative term sometimes used to describe the Frölicher spectral sequence (named afte
Morava K-theory
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early
Pullback (cohomology)
In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism: from the cohomology r
De Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about
Bivariant theory
In mathematics, a bivariant theory was introduced by Fulton and MacPherson, in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring
BRST quantization
In theoretical physics, the BRST formalism, or BRST quantization (where the BRST refers to the last names of Carlo Becchi, , Raymond Stora and Igor Tyutin) denotes a relatively rigorous mathematical a
Rigid cohomology
In mathematics, rigid cohomology is a p-adic cohomology theory introduced by . It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology
Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a
Dolbeault cohomology
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a
Cohomology of a stack
In algebraic geometry, the cohomology of a stack is a generalization of étale cohomology. In a sense, it is a theory that is coarser than the Chow group of a stack. The cohomology of a quotient stack
Good cover (algebraic topology)
In mathematics, an open cover of a topological space is a family of open subsets such that is the union of all of the open sets. A good cover is an open cover in which all sets and all non-empty inter
Čech cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Ed
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as st
Deligne cohomology
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for
Crystalline cohomology
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 Witt vectors over k. It was introduced by Alexan
Motivic cohomology
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 Chow ring of algebraic cycles as a special case. S
Weil cohomology theory
In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of
Cartan pair
In the mathematical fields of Lie theory and algebraic topology, the notion of Cartan pair is a technical condition on the relationship between a reductive Lie algebra and a subalgebra reductive in .
Koszul cohomology
In mathematics, the Koszul cohomology groups are groups associated to a projective variety X with a line bundle L. They were introduced by Mark Green , and named after Jean-Louis Koszul as they are cl
Infinitesimal cohomology
In mathematics, infinitesimal cohomology is a cohomology theory for algebraic varieties introduced by Grothendieck. In characteristic 0 it is essentially the same as crystalline cohomology. In nonzero
Local cohomology
In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES writt
Atiyah conjecture
In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.
Intersection homology
In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherso
Factor system
In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary
Witt vector cohomology
In mathematics, Witt vector cohomology was an early p-adic cohomology theory for algebraic varieties introduced by Serre. Serre constructed it by defining a sheaf of truncated Witt rings Wn over a var
Brown–Peterson cohomology
In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced byEdgar H. Brown and Franklin P. Peterson, depending on a choice of prime p. It is described in detail by Dougla
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grot
Cohomology with compact support
In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.