Inverse image functor

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

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

• 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 ${\displaystyle i^{*}}$ .

• Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190. See section II.4.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157