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

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

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...

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

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

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

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...

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

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...

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

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...

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...

their theory to show that sheaf cohomology may be defined as certain derived functors in this context. Homological methods and sheaf theory had already been...

equations, and the GAGA principle says that sheaf cohomology of an algebraic variety is the same as the sheaf cohomology of the analytic variety defined by the...

define group cohomology is to use topological cohomology theories (such as simplicial cohomology, singular cohomology or sheaf cohomology). More precisely...

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}^{*})...

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

{g}}} is the universal enveloping algebra. For a topological space X, sheaf cohomology can be defined as H ∗ ( X , A ) = Ext ∗ ⁡ ( Z X , A ) . {\displaystyle...

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

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...

sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of 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...

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

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...

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...

ideal sheaf is denoted O ( D ) {\displaystyle {\mathcal {O}}(D)} or L(D). By the exact sequence above, there is an exact sequence of sheaf cohomology groups:...

points, and its investigation was a major reason for the development of sheaf cohomology. Suppose f is an analytic function defined on a non-empty open subset...

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...