Exceptional inverse image functor

<h2 id="definition">Definition</h2>

Let f: X → Y be a <a href="/facts/Continuous_map/sKbl02pB">continuous map</a> of <a href="/facts/Topological_spaces/v2ut7CbK">topological spaces</a> or a <a href="/facts/Morphism/Kdpuyz3S">morphism</a> of <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a>. Then the exceptional inverse image is a functor

Rf!: D(Y) → D(X)
where D(–) denotes the <a href="/facts/Derived_category/pBQHnFYo">derived category</a> of <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaves</a> of abelian groups or modules over a fixed ring.
It is defined to be the <a href="/facts/Adjoint_functor/bJ1P7AN2">right adjoint</a> of the <a href="/facts/Derived_functor/YdjtF5ds">total derived functor</a> Rf! of the <a href="/facts/Direct_image_with_compact_support/KWGudJ5q">direct image with compact support</a>. Its existence follows from certain properties of Rf! and general theorems about existence of adjoint functors, as does the unicity.
The notation Rf! is an abuse of notation insofar as there is in general no functor f! whose derived functor would be Rf!.

<h2 id="examples-and-properties">Examples and properties</h2>
<ul><li>If f: X → Y is an <a href="/facts/Immersion_(mathematics)/QMqIgkVY">immersion</a> of a <a href="/facts/Locally_closed/DVkS2W7J">locally closed</a> subspace, then it is possible to define</li></ul>
f!(F) := f∗ G,
where G is the subsheaf of F of which the sections on some open subset U of Y are the sections s ∈ F(U) whose <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> is contained in X. The functor f! is <a href="/facts/Left_exact_functor/4tsQzwBp">left exact</a>, and the above Rf!, whose existence is guaranteed by <a href="/facts/Abstract_nonsense/Ptjpdjpd">abstract nonsense</a>, is indeed the derived functor of this f!. Moreover f! is right adjoint to <a href="/facts/Direct_image_with_compact_support/KWGudJ5q">f!</a>, too.
<ul><li>Slightly more generally, a similar statement holds for any <a href="/facts/Quasi-finite_morphism/xkWkqD4R">quasi-finite morphism</a> such as an <a href="/facts/%25C3%2589tale_morphism/DyM6taVS">étale morphism</a>.</li>
<li>If f is an <a href="/facts/Open_immersion/eh7NLhmS">open immersion</a>, the exceptional inverse image equals the usual <a href="/facts/Inverse_image_functor/wDG4ARH0">inverse image</a>.</li></ul>
<h2 id="duality-of-the-exceptional-inverse-image-functor">Duality of the exceptional inverse image functor</h2>
Let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a smooth manifold of dimension 
 
 
 
 d
 
 
 {\displaystyle d}
 
 and let 
 
 
 
 f
 :
 X
 →
 ∗
 
 
 {\displaystyle f:X\rightarrow *}
 
 be the unique map which maps everything to one point. For a ring 
 
 
 
 Λ
 
 
 {\displaystyle \Lambda }
 
, one finds that 
 
 
 
 
 f
 
 !
 
 
 Λ
 =
 
 ω
 
 X
 ,
 Λ
 
 
 [
 d
 ]
 
 
 {\displaystyle f^{!}\Lambda =\omega _{X,\Lambda }[d]}
 
 is the shifted 
 
 
 
 Λ
 
 
 {\displaystyle \Lambda }
 
-<a href="/facts/Orientation_sheaf/cThhN158">orientation sheaf</a>.
On the other hand, let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a smooth 
 
 
 
 k
 
 
 {\displaystyle k}
 
-variety of dimension 
 
 
 
 d
 
 
 {\displaystyle d}
 
. If 
 
 
 
 f
 :
 X
 →
 Spec
 ⁡
 (
 k
 )
 
 
 {\displaystyle f:X\rightarrow \operatorname {Spec} (k)}
 
 denotes the structure morphism then 
 
 
 
 
 f
 
 !
 
 
 k
 ≅
 
 ω
 
 X
 
 
 [
 d
 ]
 
 
 {\displaystyle f^{!}k\cong \omega _{X}[d]}
 
 is the shifted <a href="/facts/Canonical_bundle/nJKN8mpG">canonical sheaf</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
Moreover, let 
 
 
 
 X
 
 
 {\displaystyle X}
 
 be a smooth 
 
 
 
 k
 
 
 {\displaystyle k}
 
-variety of dimension 
 
 
 
 d
 
 
 {\displaystyle d}
 
 and 
 
 
 
 ℓ
 
 
 {\displaystyle \ell }
 
 a prime invertible in 
 
 
 
 k
 
 
 {\displaystyle k}
 
. Then 
 
 
 
 
 f
 
 !
 
 
 
 
 Q
 
 
 ℓ
 
 
 ≅
 
 
 Q
 
 
 ℓ
 
 
 (
 d
 )
 [
 2
 d
 ]
 
 
 {\displaystyle f^{!}\mathbb {Q} _{\ell }\cong \mathbb {Q} _{\ell }(d)[2d]}
 
 where 
 
 
 
 (
 d
 )
 
 
 {\displaystyle (d)}
 
 denotes the <a href="/facts/Tate_twist/r6Q4kdje">Tate twist</a>.
Recalling the definition of the compactly supported cohomology as <a href="/facts/Direct_image_with_compact_support/KWGudJ5q">lower-shriek pushforward</a> and noting that below the last 
 
 
 
 
 
 Q
 
 
 ℓ
 
 
 
 
 {\displaystyle \mathbb {Q} _{\ell }}
 
 means the constant sheaf on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and the rest mean that on 
 
 
 
 ∗
 
 
 {\displaystyle *}
 
, 
 
 
 
 f
 :
 X
 →
 ∗
 
 
 {\displaystyle f:X\to *}
 
, and

H
          
          
            c
          
          
            n
          
        
        (
        X
        
          )
          
            ∗
          
        
        ≅
        Hom
        ⁡
        
          (
          
            
              f
              
                !
              
            
            
              f
              
                ∗
              
            
            
              
                Q
              
              
                ℓ
              
            
            [
            n
            ]
            ,
            
              
                Q
              
              
                ℓ
              
            
          
          )
        
        ≅
        Hom
        ⁡
        
          (
          
            
              
                Q
              
              
                ℓ
              
            
            ,
            
              f
              
                ∗
              
            
            
              f
              
                !
              
            
            
              
                Q
              
              
                ℓ
              
            
            [
            −
            n
            ]
          
          )
        
        ,
      
    
    {\displaystyle \mathrm {H} _{c}^{n}(X)^{*}\cong \operatorname {Hom} \left(f_{!}f^{*}\mathbb {Q} _{\ell }[n],\mathbb {Q} _{\ell }\right)\cong \operatorname {Hom} \left(\mathbb {Q} _{\ell },f_{*}f^{!}\mathbb {Q} _{\ell }[-n]\right),}

the above computation furnishes the 
 
 
 
 ℓ
 
 
 {\displaystyle \ell }
 
-adic <a href="/facts/Poincar%25C3%25A9_duality/lwNjUCIa">Poincaré duality</a>

H
          
          
            c
          
          
            n
          
        
        
          
            (
            
              X
              ;
              
                
                  Q
                
                
                  ℓ
                
              
            
            )
          
          
            ∗
          
        
        ≅
        
          
            H
          
          
            2
            d
            −
            n
          
        
        (
        X
        ;
        
          Q
        
        (
        d
        )
        )
      
    
    {\displaystyle \mathrm {H} _{c}^{n}\left(X;\mathbb {Q} _{\ell }\right)^{*}\cong \mathrm {H} ^{2d-n}(X;\mathbb {Q} (d))}

from the repeated application of the adjunction condition.

<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> treats the topological setting</li>
<li><a href="/facts/Michael_Artin/Tfk0f6rK">Artin, Michael</a> (1972). <a href="/facts/Alexandre_Grothendieck/jLQGPY7M">Alexandre Grothendieck</a>; <a href="/facts/Jean-Louis_Verdier/nR3cH3Lj">Jean-Louis Verdier</a> (eds.). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3. Lecture notes in mathematics (in French). Vol. 305. Berlin; New York: <a href="/facts/Springer_Science%252BBusiness_Media/nAesf6nT">Springer-Verlag</a>. pp. vi+640. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0070714">10.1007/BFb0070714</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-06118-2. treats the case of étale sheaves on schemes. See Exposé XVIII, section 3.</li>
<li>Gallauer, Martin, <a href="https://guests.mpim-bonn.mpg.de/gallauer/docs/m6ff.pdf">An Introduction to Six Functor Formalisms</a> (PDF), pp.10-11 gives the duality statements.</li></ul>

Exceptional inverse image functor open-in-new

Exceptional inverse image functor