Inverse image functor

In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map $f:X\to Y$, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features. Template:Sidebar

Definition

Suppose we are given a sheaf $𝒢$ on $Y$ and that we want to transport $𝒢$ to $X$ using a continuous map $f:X\to Y$.

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

$f^{-1}𝒢(U)=𝒢(f(U))$

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 $f^{-1}𝒢$ to be the sheaf associated to the presheaf:

$U\mapsto \underset{V\supseteq f(U)}{\underrightarrow{\lim }}𝒢(V).$

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

$f^{*}𝒢:=f^{-1}𝒢{\otimes }_{f^{-1}{𝒪}_Y}{𝒪}_X$.

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 $(f^{-1}𝒢{)}_x\cong {𝒢}_{f(x)}$.
• $f^{-1}$ 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 $𝒢\to f_{*}{f}^{-1}𝒢$ and $f^{-1}{f}_{*}ℱ\to ℱ$. These morphisms yield a natural adjunction correspondence:

${\mathrm{H}\mathrm{o}\mathrm{m}}_{𝐒𝐡(X)}(f^{-1}𝒢,ℱ)={\mathrm{H}\mathrm{o}\mathrm{m}}_{𝐒𝐡(Y)}(𝒢,f_{*}ℱ)$.

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

References

• Template:Citation. See section II.4.
• Template:Hartshorne AG