# 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 we are given a sheaf ${\displaystyle {\mathcal {G}}}$ on ${\displaystyle Y}$ and that we want to transport ${\displaystyle {\mathcal {G}}}$ to ${\displaystyle X}$ using a continuous map ${\displaystyle f\colon X\to Y}$.

We will call the result the inverse image or pullback sheaf ${\displaystyle f^{-1}{\mathcal {G}}}$. If we try to imitate the direct image by setting

${\displaystyle f^{-1}{\mathcal {G}}(U)={\mathcal {G}}(f(U))}$

for each open set ${\displaystyle U}$ of ${\displaystyle X}$, we immediately run into a problem: ${\displaystyle f(U)}$ 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 ${\displaystyle f^{-1}{\mathcal {G}}}$ to be the sheaf associated to the presheaf:

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

(Here ${\displaystyle U}$ is an open subset of ${\displaystyle X}$ and the colimit runs over all open subsets ${\displaystyle V}$ of ${\displaystyle Y}$ containing ${\displaystyle f(U)}$.)

For example, if ${\displaystyle f}$ is just the inclusion of a point ${\displaystyle y}$ of ${\displaystyle Y}$, then ${\displaystyle f^{-1}({\mathcal {F}})}$ is just the stalk of ${\displaystyle {\mathcal {F}}}$ 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 ${\displaystyle f\colon X\to Y}$ of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of ${\displaystyle {\mathcal {O}}_{Y}}$-modules, where ${\displaystyle {\mathcal {O}}_{Y}}$ is the structure sheaf of ${\displaystyle Y}$. Then the functor ${\displaystyle f^{-1}}$ is inappropriate, because in general it does not even give sheaves of ${\displaystyle {\mathcal {O}}_{X}}$-modules. In order to remedy this, one defines in this situation for a sheaf of ${\displaystyle {\mathcal {O}}_{Y}}$-modules ${\displaystyle {\mathcal {G}}}$ its inverse image by

${\displaystyle f^{*}{\mathcal {G}}:=f^{-1}{\mathcal {G}}\otimes _{f^{-1}{\mathcal {O}}_{Y}}{\mathcal {O}}_{X}}$.

## Properties

• While ${\displaystyle f^{-1}}$ is more complicated to define than ${\displaystyle f_{\ast }}$, the stalks are easier to compute: given a point ${\displaystyle x\in X}$, one has ${\displaystyle (f^{-1}{\mathcal {G}})_{x}\cong {\mathcal {G}}_{f(x)}}$.
• ${\displaystyle f^{-1}}$ is an exact functor, as can be seen by the above calculation of the stalks.
• ${\displaystyle f^{*}}$ is (in general) only right exact. If ${\displaystyle f^{*}}$ is exact, f is called flat.
• ${\displaystyle f^{-1}}$ is the left adjoint of the direct image functor ${\displaystyle f_{\ast }}$. This implies that there are natural unit and counit morphisms ${\displaystyle {\mathcal {G}}\rightarrow f_{*}f^{-1}{\mathcal {G}}}$ and ${\displaystyle f^{-1}f_{*}{\mathcal {F}}\rightarrow {\mathcal {F}}}$. These morphisms yield a natural adjunction correspondence:
${\displaystyle \mathrm {Hom} _{\mathbf {Sh} (X)}(f^{-1}{\mathcal {G}},{\mathcal {F}})=\mathrm {Hom} _{\mathbf {Sh} (Y)}({\mathcal {G}},f_{*}{\mathcal {F}})}$.

However, the morphisms ${\displaystyle {\mathcal {G}}\rightarrow f_{*}f^{-1}{\mathcal {G}}}$ and ${\displaystyle f^{-1}f_{*}{\mathcal {F}}\rightarrow {\mathcal {F}}}$ are almost never isomorphisms. For example, if ${\displaystyle i\colon Z\to Y}$ denotes the inclusion of a closed subset, the stalk of ${\displaystyle i_{*}i^{-1}{\mathcal {G}}}$ at a point ${\displaystyle y\in Y}$ is canonically isomorphic to ${\displaystyle {\mathcal {G}}_{y}}$ if ${\displaystyle y}$ is in ${\displaystyle Z}$ and ${\displaystyle 0}$ otherwise. A similar adjunction holds for the case of sheaves of modules, replacing ${\displaystyle i^{-1}}$ by ${\displaystyle i^{*}}$.

## 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.