Inverse image functor

<h2 id="definition">Definition</h2>
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 <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous map</a> 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
. 
We will call the result the inverse image or <a href="/facts/Pullback_(category_theory)/bC0QoX5U">pullback</a> <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> 
 
 
 
 
 f
 
 −
 1
 
 
 
 
 G
 
 
 
 
 {\displaystyle f^{-1}{\mathcal {G}}}
 
. If we try to imitate the <a href="/facts/Direct_image/z3mckxvp">direct image</a> 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 <a href="/facts/Presheaf/MM3hcRw4">presheaf</a> and not a sheaf. Consequently, we define 
 
 
 
 
 f
 
 −
 1
 
 
 
 
 G
 
 
 
 
 {\displaystyle f^{-1}{\mathcal {G}}}
 
 to be the <a href="/facts/Sheaf_associated_to_the_presheaf/V3MnSb2f">sheaf associated to the presheaf</a>:

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 <a href="/facts/Colimit/JoLugLs3">colimit</a> 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 <a href="/facts/Stalk_(sheaf)/qAD2etdW">stalk</a> of 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 at this point.
The restriction maps, as well as the <a href="/facts/Functoriality/l2u3rIBE">functoriality</a> of the inverse image follows from the <a href="/facts/Universal_property/5XJBYMNz">universal property</a> of <a href="/facts/Direct_limit/V3JuHnYK">direct limits</a>.
When dealing with <a href="/facts/Morphisms/Kdpuyz3S">morphisms</a> 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 of <a href="/facts/Locally_ringed_space/JyeaTMtX">locally ringed spaces</a>, for example <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a> in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, one often works with <a href="/facts/Sheaf_of_modules/grSMBXho">sheaves of 
 
 
 
 
 
 
 O
 
 
 
 Y
 
 
 
 
 {\displaystyle {\mathcal {O}}_{Y}}
 
-modules</a>, 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}}
 
.
<h2 id="properties">Properties</h2>
<ul><li>While 
 
 
 
 
 f
 
 −
 1
 
 
 
 
 {\displaystyle f^{-1}}
 
 is more complicated to define than 
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f_{\ast }}
 
, the <a href="/facts/Stalk_(sheaf)/qAD2etdW">stalks</a> 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)}}
 
.</li>
<li>
 
 
 
 
 f
 
 −
 1
 
 
 
 
 {\displaystyle f^{-1}}
 
 is an <a href="/facts/Exact_functor/4tsQzwBp">exact functor</a>, as can be seen by the above calculation of the stalks.</li>
<li>
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f^{*}}
 
 is (in general) only right exact. If 
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f^{*}}
 
 is exact, f is called <a href="/facts/Flat_morphism/CLM7e8d6">flat</a>.</li>
<li>
 
 
 
 
 f
 
 −
 1
 
 
 
 
 {\displaystyle f^{-1}}
 
 is the <a href="/facts/Adjoint_functor/bJ1P7AN2">left adjoint</a> of the <a href="/facts/Direct_image_functor/z3mckxvp">direct image functor</a> 
 
 
 
 
 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:</li></ul>

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^{*}}
 
.

<ul><li>Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-16389-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0842190">0842190</a>. See section II.4.</li>
<li><a href="/facts/Robin_Hartshorne/mCBLyIW6">Hartshorne, Robin</a> (1977), <a href="/facts/Algebraic_Geometry_(book)/YcAhHWBj">Algebraic Geometry</a>, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90244-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></li></ul>

Inverse image functor open-in-new

Inverse image functor