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 $f\colon X\to Y$.

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 could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, we define $f^{-1}{\mathcal {G}}$ to be the sheaf associated to the presheaf:

$U\mapsto \varinjlim _{V\supseteq f(U)}{\mathcal {G}}(V).$

(Here $U$ 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 $f\colon X\to Y$ 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 $f_{\ast }$, 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 $f_{\ast }$. 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 $i\colon Z\to Y$ denotes the inclusion of a closed subset, the stalk 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 ${\displaystyle 0}$ otherwise. A similar adjunction holds for the case of sheaves of modules, replacing ${\displaystyle i^{-1}}$ by $i^*$.

References

• Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 842190. See section II.4.