Inverse image functor

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

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 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

• 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

• Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190. See section II.4.