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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}}
 
==Definition==
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]]:
 
:<math>U \mapsto \varinjlim_{V\supseteq f(U)}\mathcal{G}(V).</math>
(''U'' is an open subset of ''X'' and the [[colimit]] runs over all open subsets ''V'' of ''Y'' containing ''f(U)'').
 
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.
 
The restriction maps, as well as the functoriality of the inverse image follows from the universal property of [[direct limit]]s.
 
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
 
:<math>f^*\mathcal G := f^{-1}\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X</math>.
 
== Properties ==
 
* 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>.
* <math>f^{-1}</math> is an exact functor, as can be seen by the above calculation of the stalks.
* <math>f^*</math> is (in general) only right exact. If <math>f^*</math> is exact, ''f'' is called [[Flat morphism|flat]].
* <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:
:<math>\mathrm{Hom}_{\mathbf {Sh}(X)}(f^{-1} \mathcal G, \mathcal F ) = \mathrm{Hom}_{\mathbf {Sh}(Y)}(\mathcal G, f_*\mathcal F)</math>.
However, these morphisms are ''almost never'' isomorphisms.
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>.
 
==References==
* {{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.
 
{{DEFAULTSORT:Inverse Image Functor}}
[[Category:Algebraic geometry]]
[[Category:Sheaf theory]]

Latest revision as of 12:38, 12 November 2014

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