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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 → → Y {\displaystyle 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.

• Definition

• Properties

• References

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

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

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

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

U ↦ ↦ lim → → V ⊇ ⊇ f ( U ) ⁡ ⁡ G ( V ) . {\displaystyle U\mapsto \varinjlim _{V\supseteq f(U)}{\mathcal {G}}(V).}

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

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

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

• While f − − 1 {\displaystyle f^{-1}} is more complicated to define than f ∗ ∗ {\displaystyle f_{\ast }} , the stalks are easier to compute: given a point x ∈ ∈ X {\displaystyle x\in X} , one has ( f − − 1 G ) x ≅ ≅ G f ( x ) {\displaystyle (f^{-1}{\mathcal {G}})_{x}\cong {\mathcal {G}}_{f(x)}} .

• f − − 1 {\displaystyle f^{-1}} is an exact functor, as can be seen by the above calculation of the stalks.

• f ∗ ∗ {\displaystyle f^{*}} is (in general) only right exact. If f ∗ ∗ {\displaystyle f^{*}} is exact, f is called flat.

• f − − 1 {\displaystyle f^{-1}} is the left adjoint of the direct image functor f ∗ ∗ {\displaystyle f_{\ast }} . This implies that there are natural unit and counit morphisms G → → f ∗ ∗ f − − 1 G {\displaystyle {\mathcal {G}}\rightarrow f_{*}f^{-1}{\mathcal {G}}} and f − − 1 f ∗ ∗ F → → F {\displaystyle f^{-1}f_{*}{\mathcal {F}}\rightarrow {\mathcal {F}}} . These morphisms yield a natural adjunction correspondence:

H o m S h ( X ) ( f − − 1 G , F ) = H o m S h ( Y ) ( G , f ∗ ∗ F ) {\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 G → → f ∗ ∗ f − − 1 G {\displaystyle {\mathcal {G}}\rightarrow f_{*}f^{-1}{\mathcal {G}}} and f − − 1 f ∗ ∗ F → → F {\displaystyle f^{-1}f_{*}{\mathcal {F}}\rightarrow {\mathcal {F}}} are almost never isomorphisms. For example, if i : : Z → → Y {\displaystyle i\colon Z\to Y} denotes the inclusion of a closed subset, the stalk of i ∗ ∗ i − − 1 G {\displaystyle i_{*}i^{-1}{\mathcal {G}}} at a point y ∈ ∈ Y {\displaystyle y\in Y} is canonically isomorphic to G y {\displaystyle {\mathcal {G}}_{y}} if y {\displaystyle y} is in Z {\displaystyle Z} and 0 {\displaystyle 0} otherwise. A similar adjunction holds for the case of sheaves of modules, replacing i − − 1 {\displaystyle i^{-1}} by i ∗ ∗ {\displaystyle i^{*}} .

• citerefiversen1986Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190. See section II.4.

• citerefhartshorne1977Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157