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| In [[mathematics]], especially in [[algebraic geometry]] and the theory of [[complex manifold]]s, '''coherent sheaves''' are a specific class of [[Sheaf (mathematics)|sheaves]] having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with reference to a [[Sheaf (mathematics)|sheaf of rings]] that codifies this geometrical information.
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| Coherent sheaves can be seen as a generalization of [[vector bundles]], or of [[locally free sheaf|locally free sheaves]] of finite rank. Unlike vector bundles, they form a "nice" [[category (mathematics)|category]] closed under usual operations such as taking [[kernel (category theory)|kernels]], [[cokernel]]s and finite [[direct sum]]s. The '''quasi-coherent sheaves''' are a generalization of coherent sheaves and include the locally free sheaves of infinite rank.
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| Many results and properties in algebraic geometry and [[complex analytic geometry]] are formulated in terms of coherent or quasi-coherent sheaves and their [[cohomology]].
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| == Definitions ==
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| A ''coherent sheaf'' on a [[ringed space]] <math>(X,\mathcal{O}_X)</math> is a [[sheaf (mathematics)|sheaf]] <math>\mathcal{F}</math> of <math>\mathcal{O}_X</math>-[[module (mathematics)|modules]] with the following two properties:
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| # <math>\mathcal{F}</math> is of ''finite type'' over <math>\mathcal{O}_X</math>, i.e., for any point <math>x\in X</math> there is an open neighbourhood <math>U\subset X</math> such that the restriction <math>\mathcal{F}|_U</math> of <math>\mathcal{F}</math> to <math>U</math> is generated by a finite number of sections (in other words, there is a surjective morphism <math>\mathcal{O}_X^n|_U \to \mathcal{F}|_U</math> for some <math>n\in\mathbb{N}</math>); and
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| # for any open set <math>U\subset X</math>, any <math>n\in\mathbb{N}</math> and any morphism <math>\varphi\colon \mathcal{O}_X^n|_U \to \mathcal{F}|_U</math> of <math>\mathcal{O}_X</math>-modules, the kernel of <math>\varphi</math> is of finite type.
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| The sheaf of rings <math>\mathcal{O}_X</math> is coherent if it is coherent considered as a sheaf of modules over itself. Important examples of coherent sheaves of rings include the sheaf of germs of [[holomorphic]] functions on a [[complex manifold]] and the structure sheaf of a [[Noetherian scheme]] from algebraic geometry.
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| A coherent sheaf is always a sheaf of ''finite presentation'', or in other words each point <math>x\in X</math> has an open neighbourhood <math>U</math> such that the restriction <math>\mathcal{F}|_U</math> of <math>\mathcal{F}</math> to <math>U</math> is isomorphic to the cokernel of a morphism <math>\mathcal{O}_X^n|_U \to \mathcal{O}_X^m|_U</math> for some integers <math>n</math> and <math>m</math>. If <math>\mathcal{O}_X</math> is coherent, then the converse is true and each sheaf of finite presentation over <math>\mathcal{O}_X</math> is coherent.
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| A sheaf <math>\mathcal{F}</math> of <math>\mathcal{O}_{X}</math>-modules is said to be '''quasi-coherent''' if it has a local presentation, i.e. if there exist an open cover by <math>U_i</math> of the topological space <math>X</math> and an exact sequence
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| :<math>\mathcal{O}^{(I_i)}|_{U_i} \to \mathcal{O}^{(J_i)}|_{U_i} \to \mathcal{F}|_{U_i} \to 0</math>
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| where the first two terms of the sequence are direct sums (possibly infinite) of copies of the structure sheaf.
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| ==Examples of coherent sheaves==
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| * On a Noetherian scheme, the structure sheaf is a coherent sheaf of rings. In the following examples, schemes are assumed to be Noetherian.
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| * The [[Oka coherence theorem]] states that the sheaf of holomorphic functions on a complex manifold is a coherent sheaf of rings.
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| * The sheaf of sections of a vector bundle (on a scheme, or a complex [[analytic space]]) is coherent.
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| * Ideal sheaves: If ''Z'' is a closed complex subspace of a complex analytic space ''X'', the sheaf ''I<sub>''Z''/''X''</sub>'' of all holomorphic functions vanishing on ''Z'' is coherent. Likewise, the ideal sheaf of regular functions vanishing on a closed subscheme is coherent.
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| * The structure sheaf ''O''<sub>''Z''</sub> of a closed subscheme ''Z'' of ''X'', or of a closed analytic subspace, is a coherent sheaf on X. The sheaf ''O''<sub>''Z''</sub> has fiber dimension (defined below) equal to zero at points in the open set ''X''−''Z'', and fiber dimension one at points in ''Z''.
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| ==Properties==
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| The category of coherent sheaves on <math>(X,\mathcal{O}_X)</math> is an [[abelian category]], a full subcategory of the (much more unwieldy) abelian category of all sheaves on <math>(X,\mathcal{O}_X)</math>.
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| (Analogously, the category of [[coherent module]]s over any ring ''R'' is a full abelian subcategory of the category of all ''R''-modules.)
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| If ''R'' denotes the ring of regular functions <math>\Gamma(X,\mathcal{O}_X)</math>, then every ''R''-module gives rise to a quasi-coherent sheaf of <math>\mathcal{O}_X</math>-modules in a natural fashion, yielding a functor from ''R''-modules to quasi-coherent sheaves. In general, not every quasi-coherent sheaf arises from an ''R''-module in this fashion. However, for an [[affine scheme]] ''X'' with [[coordinate ring]] ''R'', this construction gives an [[equivalence of categories]] between ''R''-modules and quasi-coherent sheaves on ''X''. In case the ring ''R'' is [[Noetherian ring|Noetherian]], coherent sheaves correspond exactly to finitely generated modules.
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| Some results in [[commutative algebra]] are naturally interpreted using coherent sheaves. For example, [[Nakayama's lemma]] says that if ''F'' is a coherent sheaf, then the fiber ''F''<sub>''x''</sub>⊗<sub>''O''<sub>''X'',''x''</sub></sub>''k''(''x'') of ''F'' at a point ''x'' (a vector space over the residue field ''k''(''x'')) is zero if and only if the sheaf ''F'' is zero on some open neighborhood of ''x''. A related fact is that the dimension of the fibers of a coherent sheaf is [[Semi-continuity|upper-semicontinuous]].<ref>R. Hartshorne. ''Algebraic Geometry.'' Springer-Verlag (1977). Example III.12.7.2.</ref> Thus a coherent sheaf has constant rank on an open set (where it is a vector bundle), while the rank can jump up on a lower-dimensional closed subset.
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| Given an (affine or projective) [[algebraic variety]] ''X'' (or more generally: a [[quasi-compact]] [[Glossary_of_scheme_theory#Separated_and_proper_morphisms|quasi-separated]] [[scheme (mathematics)|scheme]]), the category of quasi-coherent sheaves on ''X'' is a very well-behaved abelian category, a [[Grothendieck category]]. It follows that the category of quasi-coherent sheaves (unlike the category of coherent sheaves) has [[enough injectives]], which makes it a convenient setting for sheaf cohomology. The scheme ''X'' is determined up to isomorphism by the abelian category of quasi-coherent sheaves on ''X''.
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| ==Coherent cohomology==
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| The [[sheaf cohomology]] theory of coherent sheaves is called '''''coherent cohomology'''''. It is one of the major and most fruitful applications of sheaves, and its results connect quickly with classical theories.
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| Using a theorem of Schwartz on [[compact operator]]s in [[Fréchet space]]s, Cartan and Serre proved that [[compact manifold|compact]] complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension.
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| This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kähler manifolds. It plays a major role in the proof of the [[GAGA]] equivalence. An algebraic (and much easier) version of this theorem was proved by [[Jean-Pierre Serre|Serre]]. Relative versions of this result for a [[proper morphism]] were proved by [[Grothendieck]] in the algebraic case and by [[Hans Grauert|Grauert]] and [[Reinhold Remmert|Remmert]] in the analytic case. For example Grothendieck's result concerns the [[functor]] R''f''<sub>*</sub> or push-forward, in sheaf cohomology. (It is the [[right derived functor]] of the [[direct image of a sheaf]].) For a proper morphism in the sense of [[scheme theory]], this functor sends coherent sheaves to coherent sheaves. The result of [[Jean-Pierre Serre|Serre]] is the case of a morphism to a point.
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| The duality theory in scheme theory that extends [[Serre duality]] is called [[coherent duality]] (or ''Grothendieck duality''). Under some mild conditions of finiteness, the sheaf of [[Kähler differential]]s on an algebraic variety is a coherent sheaf Ω<sup>1</sup>. When the variety is smooth, Ω<sup>1</sup> is a vector bundle, the [[cotangent bundle]] of ''X''. For a smooth projective variety ''X'' of dimension ''n'', Serre duality says that the top [[exterior power]] Ω<sup>n</sup> = Λ<sup>n</sup>Ω<sup>1</sup> acts as the ''dualizing object'' for coherent sheaf cohomology.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *Section 0.5.3 of {{EGA|book=I}}
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| *[[Robin Hartshorne]], ''Algebraic Geometry'', Springer-Verlag, 1977, ISBN 0-387-90244-9
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| *{{eom|id=c/c022980|title=Coherent algebraic sheaf|first=V. I. |last= Danilov}}
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| *{{eom|id=c/c022990|title=Coherent analytic sheaf|first=A.L.|last= Onishchik}}
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| *{{Springer|title=Coherent sheaf|id=C/c023020|first=A.L.|last= Onishchik}}
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| ==External links==
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| * [http://stacks.math.columbia.edu/download/modules.pdf Sheaves of Modules], from the Stacks Project
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| [[Category:Topological methods of algebraic geometry]]
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| [[Category:Complex manifolds]]
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| [[Category:Sheaf theory]]
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