Theorem on formal functions

<h2 id="the-construction-of-the-canonical-map">The construction of the canonical map</h2>
<p>Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map.
</p><p>Let 
  
    
      
        
          i
          ′
        
        :
        
          
            
              X
              ^
            
          
        
        →
        X
        ,
        i
        :
        
          
            
              S
              ^
            
          
        
        →
        S
      
    
    {\displaystyle i':{\widehat {X}}\to X,i:{\widehat {S}}\to S}
  
 be the canonical maps. Then we have the <a href="/facts/Base_change_map/8g3hymRF">base change map</a> of 
  
    
      
        
          
            
              O
            
          
          
            
              
                S
                ^
              
            
          
        
      
    
    {\displaystyle {\mathcal {O}}_{\widehat {S}}}
  
-modules
</p>

i
          
            ∗
          
        
        
          R
          
            q
          
        
        
          f
          
            ∗
          
        
        
          
            F
          
        
        →
        
          R
          
            p
          
        
        
          
            
              
                f
                ^
              
            
          
          
            ∗
          
        
        (
        
          i
          
            ′
            
              ∗
            
          
        
        
          
            F
          
        
        )
      
    
    {\displaystyle i^{*}R^{q}f_{*}{\mathcal {F}}\to R^{p}{\widehat {f}}_{*}(i'^{*}{\mathcal {F}})}
  
.
<p>where 
  
    
      
        
          
            
              f
              ^
            
          
        
        :
        
          
            
              X
              ^
            
          
        
        →
        
          
            
              S
              ^
            
          
        
      
    
    {\displaystyle {\widehat {f}}:{\widehat {X}}\to {\widehat {S}}}
  
 is induced by 
  
    
      
        f
        :
        X
        →
        S
      
    
    {\displaystyle f:X\to S}
  
. Since 
  
    
      
        
          
            F
          
        
      
    
    {\displaystyle {\mathcal {F}}}
  
 is coherent, we can identify 
  
    
      
        
          i
          
            ′
            
              ∗
            
          
        
        
          
            F
          
        
      
    
    {\displaystyle i'^{*}{\mathcal {F}}}
  
 with 
  
    
      
        
          
            
              
                F
              
              ^
            
          
        
      
    
    {\displaystyle {\widehat {\mathcal {F}}}}
  
. Since 
  
    
      
        
          R
          
            q
          
        
        
          f
          
            ∗
          
        
        
          
            F
          
        
      
    
    {\displaystyle R^{q}f_{*}{\mathcal {F}}}
  
 is also coherent (as <i>f</i> is proper), doing the same identification, the above reads:
</p>

(
        
          R
          
            q
          
        
        
          f
          
            ∗
          
        
        
          
            F
          
        
        
          )
          
            ∧
          
        
        →
        
          R
          
            p
          
        
        
          
            
              
                f
                ^
              
            
          
          
            ∗
          
        
        
          
            
              
                F
              
              ^
            
          
        
      
    
    {\displaystyle (R^{q}f_{*}{\mathcal {F}})^{\wedge }\to R^{p}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}}
  
.
<p>Using 
  
    
      
        f
        :
        
          X
          
            n
          
        
        →
        
          S
          
            n
          
        
      
    
    {\displaystyle f:X_{n}\to S_{n}}
  
 where 
  
    
      
        
          X
          
            n
          
        
        =
        (
        
          X
          
            0
          
        
        ,
        
          
            
              O
            
          
          
            X
          
        
        
          /
        
        
          
            
              J
            
          
          
            n
            +
            1
          
        
        )
      
    
    {\displaystyle X_{n}=(X_{0},{\mathcal {O}}_{X}/{\mathcal {J}}^{n+1})}
  
 and 
  
    
      
        
          S
          
            n
          
        
        =
        (
        
          S
          
            0
          
        
        ,
        
          
            
              O
            
          
          
            S
          
        
        
          /
        
        
          
            
              I
            
          
          
            n
            +
            1
          
        
        )
      
    
    {\displaystyle S_{n}=(S_{0},{\mathcal {O}}_{S}/{\mathcal {I}}^{n+1})}
  
, one also obtains (after passing to limit):
</p>

R
          
            q
          
        
        
          
            
              
                f
                ^
              
            
          
          
            ∗
          
        
        
          
            
              
                F
              
              ^
            
          
        
        →
        
          
            lim
            ←
          
        
        ⁡
        
          R
          
            p
          
        
        
          f
          
            ∗
          
        
        
          
            
              F
            
          
          
            n
          
        
      
    
    {\displaystyle R^{q}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}\to \varprojlim R^{p}f_{*}{\mathcal {F}}_{n}}

<p>where 
  
    
      
        
          
            
              F
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {F}}_{n}}
  
 are as before. One can verify that the composition of the two maps is the same map in the lede. (cf. EGA III-1, section 4)
</p>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, Alexandre</a>; <a href="/facts/Jean_Dieudonn%25C3%25A9/gWagzMdu">Dieudonné, Jean</a> (1961). <a href="http://www.numdam.org/item/PMIHES_1961__11__5_0">"Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie"</a>. <i><a href="/facts/Publications_Math%25C3%25A9matiques_de_l%2527IH%25C3%2589S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a></i>. 11. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684274">10.1007/bf02684274</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217085">0217085</a>.</li>
<li><a href="/facts/Robin_Hartshorne/mCBLyIW6">Hartshorne, Robin</a> (1977), <i><a href="/facts/Algebraic_Geometry_(book)/YcAhHWBj">Algebraic Geometry</a></i>, <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>
<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Luc_Illusie/Dltgzg7b">Illusie, Luc</a>. <a href="http://staff.ustc.edu.cn/~yiouyang/Illusie.pdf">"Topics in Algebraic Geometry"</a> (PDF).</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Grothendieck & Dieudonné 1961, 4.1.5 - Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085. <a href="http://www.numdam.org/item/PMIHES_1961__11__5_0" target="_blank">http://www.numdam.org/item/PMIHES_1961__11__5_0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Grothendieck & Dieudonné 1961, 4.2.1 - Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085. <a href="http://www.numdam.org/item/PMIHES_1961__11__5_0" target="_blank">http://www.numdam.org/item/PMIHES_1961__11__5_0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Hartshorne 1977, Ch. III. Corollary 11.2 - Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0463157</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>The same argument as in the preceding corollary <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Hartshorne 1977, Ch. III. Corollary 11.3 - Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0463157</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Theorem on formal functions open-in-new

Theorem on formal functions