Inverse image functor
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In mathematics, the inverse image functor is a contravariant 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.
Image functors for sheaves |
direct image f∗
|
[edit] Definition
Suppose given a sheaf on Y and that we want to transport to X using a continuous map f : X → Y. We will call the result the inverse image or pullback sheaf . If we try to imitate the direct image by setting 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 to be the sheaf associated to the presheaf :
(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 limits.
When dealing with morphisms f : X → Y of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of -modules, where is the structure sheaf of Y. Then the functor f-1 is inappropriate, because (in general) it does not even give sheaves of -modules. In order to remedy this, one defines in this situation for a sheaf of -modules its inverse image by
- .
[edit] Properties
- While f-1 is more complicated to define than f∗, the stalks are easier to compute: given a point , one has .
- 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 the direct image functor f∗. This implies that there are natural unit and counit morphisms and . However, these are almost never isomorphisms. For example, if denotes the inclusion of a closed subset, the stalks of at a point is canonically isomorphic to if y is in Z and 0 otherwise. A similar adjunction holds for the case of sheaves of modules, replacing f - 1 by f * .
[edit] References
- Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, MR842190, ISBN 978-3-540-16389-3. See section II.4.