- Inverse image functor
In
mathematics , the inverse image functor is acontravariant construction of sheaves. Thedirect image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features.Definition
Suppose given a sheaf mathcal{G} on "Y" and that we want to transport mathcal{G} to "X" using a continuous map "f" : "X" → "Y". We will call the result the "inverse image" or "pullback" sheaf f^{-1}mathcal{G}. If we try to imitate the
direct image by setting f^{-1}mathcal{G}(U) = mathcal{G}(f(U)) 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 f^{-1}mathcal{G} to be thesheaf associated to the presheaf ::U mapsto varinjlim_{Vsupseteq f(U)}mathcal{G}(V).("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 "f"-1("F") is just the stalk of "F" 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 schemes in
algebraic geometry , one often works with sheaves of mathcal{O}_Y-modules, where mathcal{O}_Y is the structure sheaf of "Y". Then the functor "f"-1 is inappropriate, because (in general) it does not even give sheaves of mathcal{O}_X-modules. In order to remedy this, one defines in this situation for a sheaf of mathcal O_Y-modules mathcal G its inverse image by:f^*mathcal G := f^{-1}mathcal{G} otimes_{f^{-1}mathcal{O}_Y} mathcal{O}_X.
Properties
* While "f"-1 is more complicated to define than "f"∗, the stalks are easier to compute: given a point x in X, one has f^{-1}mathcal{G})_x cong mathcal{G}_{f(x)}.
* f^{-1} is an exact functor, as can be seen by the above calculation of the stalks.
* f^* is (in general) only right exact. If f^* is exact, "f" is called flat.
* f^{-1} is the left adjoint of thedirect image functor "f"∗. This implies that there are natural unit and counit morphisms mathcal{G} ightarrow f_*f^{-1}mathcal{G} and f^{-1}f_*mathcal{F} ightarrow mathcal{F}. However, these are "almost never" isomorphisms. For example, if i : Z ightarrow Y denotes the inclusion of a closed subset, the stalks of i_* i^{-1} mathcal G at a point y in Y is canonically isomorphic to mathcal G_y if y is in Z and 0 otherwise. A similar adjunction holds for the case of sheaves of modules, replacing f^{-1} by f^*.References
* | year=1986. See section II.4.
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