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Coherent sheaf

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

Examples of vector bundles, Basic constructions of coherent sheaves & Definitions

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Overview

Definitions

Properties

Basic constructions of coherent sheaves

Functoriality

Local behavior of coherent sheaves

Examples of vector bundles

Serre construction and vector bundles

Chern classes and algebraic K-theory

Bundle homomorphism vs. sheaf homomorphism

The category of quasi-coherent sheaves

Coherent cohomology

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Coherent sheaf

Nodes115
Edges114
Triples104
Avg. degree1.98
Density0.017391
Components1

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Coherent sheaf

Top relations

related to References · 51
Coherent sheaf → Alexandre, Algebraic Geometry, Annals, Antieau, Benjamin, Berlin, Coherent, Coherent Analytic Sheaves, Commutative Algebra, Curves, David, Dieudonné, EMS PressGrauert, EMS PressOnishchik, EMS PressSerre, Encyclopedia, Faisceaux, Graduate Texts, Grothendieck, Grundlehren
related to Coherent cohomology · 10
Coherent sheaf → Although, Among, Broadly, Euler, Hodge, In, Riemann, Roch, Serre, The
related to Examples of vector bundles · 10
Coherent sheaf → Delta, For, If, Let, Omega, Sections, Then, TX, X/k, X/Y
related to The case of schemes · 6
Coherent sheaf → Gamma, Noetherian, On, Spec, The, When
has application · 5
Coherent sheaf → Chern, For, If, Noetherian, Since
related to Functoriality · 5
Coherent sheaf → An, For, If, Let, Noetherian
related to Local behavior of coherent sheaves · 5
Coherent sheaf → An, For, In, Nakayama's, Thus
see also · 3
Coherent sheaf → Picard, Reflexive, Rosenberg
is a · 2
Coherent sheaf → vector bundle if and only if its rank is locally constant, vector bundle just from its fibers

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Important terminology

displaystyle sheaf mathcal coherent sheaves scheme bundle vector space quasi-coherent mathbb bundles line example locally -module algebraic associated given every

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Coherent sheafis avector bundle just from its fibers0.90text
Coherent sheafis avector bundle if and only if its rank is locally constant0.90text
taking kernelsinstance ofand so they are closed under operations0.80text
imagesinstance ofand so they are closed under operations0.80text
and cokernelsinstance ofand so they are closed under operations0.80text
Serre dualityinstance ofduality theorems0.80text
relations between topologyinstance ofduality theorems0.80text
algebraic geometry such as Hodge theoryinstance ofduality theorems0.80text
and formulas for Euler characteristics of coherent sheaves such as the Riemanninstance ofduality theorems0.80text
Coherent sheafhas applicationSince0.60section
Coherent sheafhas applicationNoetherian0.60section
Coherent sheafhas applicationChern0.60section

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