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• Local_cohomology abstract "In algebraic geometry, local cohomology is an analog 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).In the geometric form of the theory, sections ΓY are considered of a sheaf F of abelian groups, on a topological space X, with support in a closed subset Y. The derived functors of ΓY form local cohomology groupsHYi(X,F)There is a long exact sequence of sheaf cohomology linking the ordinary sheaf cohomology of X and of the open set U = X \\Y, with the local cohomology groups.The initial applications were to analogues of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic variety, except for some 'loss' that can be controlled. These results applied to the algebraic fundamental group and to the Picard group.In commutative algebra for a commutative ring R and its spectrum Spec(R) as X, Y can be replaced by the closed subscheme defined by an ideal I of R. The sheaf F can be replaced by an R-module M, which gives a quasicoherent sheaf on Spec(R). In this setting the depth of a module can be characterised over local rings by the vanishing of local cohomology groups, and there is an analogue, the local duality theorem, of Serre duality, using Ext functors of R-modules and a dualising module.".
• Local_cohomology wikiPageExternalLink books?id=5HgmUQsbe5sC.
• Local_cohomology wikiPageExternalLink S0273-0979-99-00785-5.pdf.
• Local_cohomology wikiPageID "3212091".
• Local_cohomology wikiPageLength "4091".
• Local_cohomology wikiPageOutDegree "39".
• Local_cohomology wikiPageRevisionID "683186281".
• Local_cohomology wikiPageWikiLink Abelian_group.
• Local_cohomology wikiPageWikiLink Alexander_Grothendieck.
• Local_cohomology wikiPageWikiLink Algebraic_geometry.
• Local_cohomology wikiPageWikiLink Algebraic_variety.
• Local_cohomology wikiPageWikiLink American_Mathematical_Society.
• Local_cohomology wikiPageWikiLink Category:Cohomology_theories.
• Local_cohomology wikiPageWikiLink Category:Commutative_algebra.
• Local_cohomology wikiPageWikiLink Category:Duality_theories.
• Local_cohomology wikiPageWikiLink Category:Sheaf_theory.
• Local_cohomology wikiPageWikiLink Category:Topological_methods_of_algebraic_geometry.
• Local_cohomology wikiPageWikiLink Closed_set.
• Local_cohomology wikiPageWikiLink Coherent_sheaf.
• Local_cohomology wikiPageWikiLink Commutative_algebra.
• Local_cohomology wikiPageWikiLink Commutative_ring.
• Local_cohomology wikiPageWikiLink Depth_(ring_theory).
• Local_cohomology wikiPageWikiLink Derived_functor.
• Local_cohomology wikiPageWikiLink Dualizing_module.
• Local_cohomology wikiPageWikiLink Exact_sequence.
• Local_cohomology wikiPageWikiLink Ext_functor.
• Local_cohomology wikiPageWikiLink Glossary_of_algebraic_geometry.
• Local_cohomology wikiPageWikiLink Graduate_Studies_in_Mathematics.
• Local_cohomology wikiPageWikiLink Grothendieck_local_duality.
• Local_cohomology wikiPageWikiLink Hyperplane_section.
• Local_cohomology wikiPageWikiLink Lefschetz_hyperplane_theorem.
• Local_cohomology wikiPageWikiLink Local_ring.
• Local_cohomology wikiPageWikiLink Module_(mathematics).
• Local_cohomology wikiPageWikiLink Open_set.
• Local_cohomology wikiPageWikiLink Picard_group.
• Local_cohomology wikiPageWikiLink Relative_cohomology.
• Local_cohomology wikiPageWikiLink Serre_duality.
• Local_cohomology wikiPageWikiLink Sheaf_(mathematics).
• Local_cohomology wikiPageWikiLink Sheaf_cohomology.
• Local_cohomology wikiPageWikiLink Société_mathématique_de_France.
• Local_cohomology wikiPageWikiLink Spectrum_of_a_ring.
• Local_cohomology wikiPageWikiLink Springer_Science+Business_Media.
• Local_cohomology wikiPageWikiLink Support_(mathematics).
• Local_cohomology wikiPageWikiLink Séminaire_de_Géométrie_Algébrique_du_Bois_Marie.
• Local_cohomology wikiPageWikiLink Topological_space.
• Local_cohomology wikiPageWikiLink Étale_fundamental_group.
• Local_cohomology wikiPageWikiLinkText "Local cohomology".
• Local_cohomology wikiPageWikiLinkText "local cohomology".
• Local_cohomology wikiPageUsesTemplate Template:Citation.
• Local_cohomology wikiPageUsesTemplate Template:Cite_book.
• Local_cohomology wikiPageUsesTemplate Template:Harvtxt.
• Local_cohomology subject Category:Cohomology_theories.
• Local_cohomology subject Category:Commutative_algebra.
• Local_cohomology subject Category:Duality_theories.
• Local_cohomology subject Category:Sheaf_theory.
• Local_cohomology subject Category:Topological_methods_of_algebraic_geometry.
• Local_cohomology hypernym Analog.
• Local_cohomology type Drug.
• Local_cohomology type Method.
• Local_cohomology type Theory.
• Local_cohomology comment "In algebraic geometry, local cohomology is an analog 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).In the geometric form of the theory, sections ΓY are considered of a sheaf F of abelian groups, on a topological space X, with support in a closed subset Y.".
• Local_cohomology label "Local cohomology".
• Local_cohomology sameAs Q6664297.
• Local_cohomology sameAs m.08zmyn.
• Local_cohomology sameAs Q6664297.
• Local_cohomology wasDerivedFrom Local_cohomology?oldid=683186281.
• Local_cohomology isPrimaryTopicOf Local_cohomology.