Inverse image functor

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In mathematics, the inverse image functor is a 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.

Definition[edit]

Suppose given a sheaf \mathcal{G} on Y and that we want to transport \mathcal{G} to X using a continuous map f : XY. 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 the sheaf associated to the presheaf:

U \mapsto \varinjlim_{V\supseteq 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}(\mathcal{F}) is just the stalk of \mathcal{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 \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[edit]

  • 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 the direct image functor f. This implies that there are natural unit and counit morphisms \mathcal{G} \rightarrow f_*f^{-1}\mathcal{G} and f^{-1}f_*\mathcal{F} \rightarrow \mathcal{F}. These morphisms yield a natural adjunction correspondence:
\mathrm{Hom}_{\mathbf {Sh}(X)}(f^{-1} \mathcal G, \mathcal F ) = \mathrm{Hom}_{\mathbf {Sh}(Y)}(\mathcal G, f_*\mathcal F).

However, these morphisms are almost never isomorphisms. For example, if i : Z \rightarrow 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[edit]