Inverse image functor

Given a continuous map $f:X\rightarrow Y$, the inverse image functor $f^{-1}$ associates to any sheaf ${\mathcal {G}}$ on Y its inverse image $f^{-1}{\mathcal {G}}$, which is a sheaf on $X$.

It is defined by:

$f^{-1}{\mathcal {G}}(U)=\varinjlim _{V\supseteq f(U)}{\mathcal {G}}(V)$

To define the restriction maps, we use the universal property of direct limits.

It is possible to define the direct image and the inverse image of a morphism as well, and using this definition, f_* and f^-1 become functors.

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.

• $f^{-1}$ 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 $f^{-1}f_{*}{\mathcal {F}}\rightarrow {\mathcal {F}}$. However, these 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.