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 Hartshorne (1967), and in 1961-2 at IHES written up as SGA2 - Grothendieck (1968), republished as Grothendieck (2005). Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The rational function , for example, is defined only on the complement of on the affine line over a field , and cannot be extended to a function on the entire space. The local cohomology module (where is the coordinate ring of ) detects this in the nonvanishing of a cohomology class . In a similar manner, is defined away from the and axes in the affine plane, but cannot be extended to either the complement of the -axis or the complement of the -axis alone (nor can it be expressed as a sum of such functions); this obstruction corresponds precisely to a nonzero class in the local cohomology module .[1]
Outside of algebraic geometry, local cohomology has found applications in commutative algebra,[2][3][4] combinatorics,[5][6][7] and certain kinds of partial differential equations.[8]
In algebraic geometry, localcohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in...
mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated...
often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Many earlier results...
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...
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for...
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...
lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The ∂ ∂ ¯ {\displaystyle \partial...
(this notion is closely related to the study of localcohomology). See also tight closure. Localcohomology can sometimes be used to obtain information on...
descriptions as a fallback List of things named after Alexander Grothendieck Localcohomology – Concept in algebraic geometry Nakai conjecture Moduli scheme – a...
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...
R-module. Then, for any finitely generated R-module M and integer i, the localcohomology group H m i ( M ) {\displaystyle H_{m}^{i}(M)} is dual to Ext R n −...
In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent...
a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with...
Galois cohomology, local Tate duality (or simply local duality) is a duality for Galois modules for the absolute Galois group of a non-archimedean local field...
Note that we can drop local contractibility as part of the hypothesis if we use Čech cohomology, which is designed to deal with local pathologies. This is...
geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize...
of a projective variety X {\displaystyle X} using Localcohomology. Another computation for local homology can be computed on a point p {\displaystyle...
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, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a...
local fields, ramification, group cohomology, and local class field theory. The book's end goal is to present local class field theory from the cohomological...
support concept. This was addressed in SGA2 in terms of localcohomology, and Grothendieck local duality; and subsequently. The Greenlees–May duality, first...
modules of generalized fractions. This topic later found applications in localcohomology, in the monomial conjecture, and other branches of commutative algebra...
Intersection cohomology was used to prove the Kazhdan–Lusztig conjectures and the Riemann–Hilbert correspondence. It is closely related to L2 cohomology. The...
Global Information
![{\displaystyle H_{(x)}^{1}(K[x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49cdb9e7bacb31ccc1058e4020da25a19934c89d)
![{\displaystyle K[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9e6c2ac2830d6a9abe078b47450777c41d69a9)
![{\displaystyle [1/x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6527ae3b375f61634d96aecaa5cb3eac03e3d3b8)
![{\displaystyle [1/xy]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bea4c6c4789f27aafebc1d652d773bd6c956b13)
![{\displaystyle H_{(x,y)}^{2}(K[x,y])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81db08fa5488b8f98b074e56745c5658e398faf1)