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

cohomology Bounded cohomology BRST cohomology Čech cohomology Coherent sheaf cohomology Crystalline cohomology Cyclic cohomology Deligne cohomology Equivariant...

Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties...

to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts...

in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many...

singular cohomology of topological spaces, but in fact, any constant sheaf on an irreducible variety has trivial cohomology (all higher cohomology groups...

de Rham cohomology and the sheaf cohomology of R _ {\textstyle {\underline {\mathbb {R} }}} . (Note that this shows that de Rham cohomology may also...

has enough injectives, and consequently one can and does define the sheaf cohomology H i ⁡ ( X , − ) {\displaystyle \operatorname {H} ^{i}(X,-)} as the...

Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. A quasi-coherent sheaf on a ringed...

cohomology which are a special case of this: De Rham cohomology is the sheaf cohomology of the sheaf of locally constant R {\displaystyle \mathbb {R} }...

scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided...

subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in étale cohomology constructible sheaves are...

Here H0 means simply the sections over U, and the sheaf cohomology H1(2πiZ|U) is the singular cohomology of U. One can think of H1(2πiZ|U) as associating...

topology which interpolates between cohomology with coefficients in a fixed abelian group A, and general sheaf cohomology in which coefficients vary from...

of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate...

branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version...

coefficients in sheaf cohomology. Let X {\displaystyle X} be a topological space, and A {\displaystyle A} a set. The sections of the constant sheaf A _ {\displaystyle...

ordinary cohomology theories, it holds in extraordinary cohomology theories (such as topological K-theory and cobordism). From the point of view of sheaf cohomology...

Rham cohomology to the singular cohomology given by integration is an isomorphism. The Poincaré lemma implies that the de Rham cohomology is the sheaf 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...

complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group H 1 ( X , O X ∗ ) . {\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})...

complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero. The implications...

{exp} }{\longrightarrow }}{\mathcal {O}}_{X}^{\times }\to 0.} Taking sheaf cohomology of this exact sequence gives maps H 1 ( X , O X × ) → c 1 H 2 ( X ...

the limit, are called Stein manifolds and their nature was to make sheaf cohomology groups vanish, on the other hand, the Grauert–Riemenschneider vanishing...

n-groups constructed from a complex manifold X {\displaystyle X} and sheaf cohomology on X {\displaystyle X} . This is particularly applicable for complex...

nilpotent sheaf of ideals on T; for example, Spec(k)→ Spec(k[x]/(x2)). Grothendieck showed that for smooth schemes X over C, the cohomology of the sheaf OX on...

Noetherian schemes. Čech cohomology and sheaf cohomology agree on an affine open cover. This makes it possible to compute the sheaf cohomology of P S n {\displaystyle...

formal properties. A perverse sheaf is an object C of the bounded derived category of sheaves with constructible cohomology on a space X such that the set...

inflation-restriction exact sequence. Group cohomology is closely related to topological cohomology theories such as sheaf cohomology, by means of an isomorphism H...

the Dolbeault operator of E. This is therefore a resolution of the sheaf cohomology of O ( E ) {\displaystyle {\mathcal {O}}(E)} . In particular associated...