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, these morphisms 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 ${\displaystyle i^{*}}$.