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| In [[mathematics]], the '''inverse image functor''' is a [[Covariance and contravariance of functors|covariant]] construction of sheaves. The [[direct image functor]] is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.
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| {{Images of sheaves}}
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| ==Definition==
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| Suppose given a sheaf <math>\mathcal{G}</math> on ''Y'' and that we want to transport <math>\mathcal{G}</math> to ''X'' using a [[Continuous function (topology)|continuous map]] ''f'' : ''X'' → ''Y''. We will call the result the ''inverse image'' or ''pullback'' [[sheaf (mathematics)|sheaf]] <math>f^{-1}\mathcal{G}</math>. If we try to imitate the [[direct image]] by setting <math>f^{-1}\mathcal{G}(U) = \mathcal{G}(f(U))</math> for each open set ''U'' of ''X'', we immediately run into a problem: ''f''(''U'') is not necessarily open. The best we can do is to approximate it by open sets, and even then we will get a presheaf, not a sheaf. Consequently we define <math>f^{-1}\mathcal{G}</math> to be the [[sheaf associated to the presheaf]]:
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| :<math>U \mapsto \varinjlim_{V\supseteq f(U)}\mathcal{G}(V).</math>
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| (''U'' is an open subset of ''X'' and the [[colimit]] runs over all open subsets ''V'' of ''Y'' containing ''f(U)'').
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| For example, if ''f'' is just the inclusion of a point ''y'' of ''Y'', then <math>f^{-1}(\mathcal{F})</math> is just the [[Sheaf (mathematics)|stalk]] of <math>\mathcal{F}</math> at this point.
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| The restriction maps, as well as the functoriality of the inverse image follows from the universal property of [[direct limit]]s.
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| When dealing with morphisms ''f : X → Y'' of locally ringed spaces, for example [[scheme (mathematics)|schemes]] in [[algebraic geometry]], one often works with sheaves of <math>\mathcal{O}_Y</math>-modules, where <math>\mathcal{O}_Y</math> is the structure sheaf of ''Y''. Then the functor ''f''<sup>−1</sup> is inappropriate, because (in general) it does not even give sheaves of <math>\mathcal{O}_X</math>-modules. In order to remedy this, one defines in this situation for a sheaf of <math>\mathcal O_Y</math>-modules <math>\mathcal G</math> its inverse image by
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| :<math>f^*\mathcal G := f^{-1}\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X</math>.
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| == Properties ==
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| * While <math>f^{-1}</math> is more complicated to define than ''f''<sub>∗</sub>, the [[stalk (sheaf)|stalks]] are easier to compute: given a point <math>x \in X</math>, one has <math>(f^{-1}\mathcal{G})_x \cong \mathcal{G}_{f(x)}</math>.
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| * <math>f^{-1}</math> is an exact functor, as can be seen by the above calculation of the stalks.
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| * <math>f^*</math> is (in general) only right exact. If <math>f^*</math> is exact, ''f'' is called [[Flat morphism|flat]].
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| * <math>f^{-1}</math> is the [[adjoint functor|left adjoint]] of the [[direct image functor]] ''f''<sub>∗</sub>. This implies that there are natural unit and counit morphisms <math>\mathcal{G} \rightarrow f_*f^{-1}\mathcal{G}</math> and <math>f^{-1}f_*\mathcal{F} \rightarrow \mathcal{F}</math>. These morphisms yield a natural adjunction correspondence:
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| :<math>\mathrm{Hom}_{\mathbf {Sh}(X)}(f^{-1} \mathcal G, \mathcal F ) = \mathrm{Hom}_{\mathbf {Sh}(Y)}(\mathcal G, f_*\mathcal F)</math>.
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| However, these morphisms are ''almost never'' isomorphisms.
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| For example, if <math>i : Z \rightarrow Y</math> denotes the inclusion of a closed subset, the stalks of <math>i_* i^{-1} \mathcal G</math> at a point <math>y \in Y</math> is canonically isomorphic to <math>\mathcal G_y</math> if <math>y</math> is in <math>Z</math> and <math>0</math> otherwise. A similar adjunction holds for the case of sheaves of modules, replacing <math>f^{-1}</math> by <math>f^*</math>.
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| ==References==
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| * {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | id={{MathSciNet | id = 842190}} | year=1986}}. See section II.4.
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| {{DEFAULTSORT:Inverse Image Functor}}
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| [[Category:Algebraic geometry]]
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| [[Category:Sheaf theory]]
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The author is known as Araceli Gulledge. Delaware is our beginning location. Camping is something that I've carried out for many years. Interviewing is how he supports his family members but his promotion never arrives.
Also visit my blog post ... car warranty (website link)