Noetherian scheme

<h2 id="properties-and-noetherian-hypotheses">Properties and Noetherian hypotheses</h2>
<p>Having a (locally) Noetherian hypothesis for a statement about schemes generally makes a lot of problems more accessible because they sufficiently rigidify many of its properties.
</p><p>Any Noetherian scheme can only have finitely many <a href="/facts/Irreducible_component/MxWWaCM7">irreducible components</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>Every <a href="/facts/Morphism_of_schemes/55CKqHHo">morphism</a> from a Noetherian scheme 
  
    
      
        X
        →
        S
      
    
    {\displaystyle X\to S}
  
 is <a href="/facts/Quasi-compact_morphism/sUCP0wez">quasi-compact</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h3>Dévissage</h3><p>
One of the most important structure theorems about Noetherian rings and Noetherian schemes is the <a href="/facts/D%25C3%25A9vissage/XqHbEuav">dévissage theorem</a>. This makes it possible to decompose arguments about <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaves</a> into inductive arguments. Given a <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequence</a> of coherent sheaves</p><blockquote><p>
  
    
      
        0
        →
        
          
            
              E
            
          
          ′
        
        →
        
          
            E
          
        
        →
        
          
            
              E
            
          
          ″
        
        →
        0
        ,
      
    
    {\displaystyle 0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0,}
  
</p></blockquote><p>proving one of the sheaves has some property is equivalent to proving the other two have the property. In particular, given a fixed coherent sheaf 
  
    
      
        
          
            F
          
        
      
    
    {\displaystyle {\mathcal {F}}}
  
 and a sub-coherent sheaf 
  
    
      
        
          
            
              F
            
          
          ′
        
      
    
    {\displaystyle {\mathcal {F}}'}
  
, showing 
  
    
      
        
          
            F
          
        
      
    
    {\displaystyle {\mathcal {F}}}
  
 has some property can be reduced to looking at 
  
    
      
        
          
            
              F
            
          
          ′
        
      
    
    {\displaystyle {\mathcal {F}}'}
  
 and 
  
    
      
        
          
            F
          
        
        
          /
        
        
          
            
              F
            
          
          ′
        
      
    
    {\displaystyle {\mathcal {F}}/{\mathcal {F}}'}
  
. Since this process can only be non-trivially applied only a finite number of times, this makes many induction arguments possible.
</p><h3>Homological properties</h3>
<p>There are many nice <a href="/facts/Homology_(mathematics)/VFJmryEW">homological</a> properties of Noetherian schemes.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h4>Čech and sheaf cohomology</h4>
<p><a href="/facts/%25C4%258Cech_cohomology/HLWD7BnA">Čech cohomology</a> and <a href="/facts/Sheaf_cohomology/TrcpmD6k">sheaf cohomology</a> agree on an affine open cover. This makes it possible to compute the sheaf cohomology of 
  
    
      
        
          
            P
          
          
            S
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {P} _{S}^{n}}
  
 using Čech cohomology for the standard open cover.
</p>
<h4>Compatibility of colimits with cohomology</h4><p>
Given a <a href="/facts/Direct_system/HjyazYb1">direct system</a> 
  
    
      
        {
        
          
            
              F
            
          
          
            α
          
        
        ,
        
          ϕ
          
            α
            β
          
        
        
          }
          
            α
            ∈
            Λ
          
        
      
    
    {\displaystyle \{{\mathcal {F}}_{\alpha },\phi _{\alpha \beta }\}_{\alpha \in \Lambda }}
  
 of sheaves of abelian groups on a Noetherian scheme, there is a canonical isomorphism</p><blockquote><p>
  
    
      
        
          
            lim
            →
          
        
        ⁡
        
          H
          
            i
          
        
        (
        X
        ,
        
          
            
              F
            
          
          
            α
          
        
        )
        →
        
          H
          
            i
          
        
        (
        X
        ,
        
          
            lim
            →
          
        
        ⁡
        
          
            
              F
            
          
          
            α
          
        
        )
      
    
    {\displaystyle \varinjlim H^{i}(X,{\mathcal {F}}_{\alpha })\to H^{i}(X,\varinjlim {\mathcal {F}}_{\alpha })}
  
</p></blockquote><p>meaning the <a href="/facts/Functor/l2u3rIBE">functors</a></p><blockquote><p>
  
    
      
        
          H
          
            i
          
        
        (
        X
        ,
        −
        )
        :
        
          Ab
        
        (
        X
        )
        →
        
          Ab
        
      
    
    {\displaystyle H^{i}(X,-):{\text{Ab}}(X)\to {\text{Ab}}}
  
</p></blockquote><p>preserve direct limits and <a href="/facts/Coproduct/0bJ8SCdu">coproducts</a>.
</p><h4>Derived direct image</h4>
<p>Given a <a href="/facts/Locally_of_finite_type/eh7NLhmS">locally finite type</a> morphism 
  
    
      
        f
        :
        X
        →
        S
      
    
    {\displaystyle f:X\to S}
  
 to a Noetherian scheme 
  
    
      
        S
      
    
    {\displaystyle S}
  
 and a complex of sheaves 
  
    
      
        
          
            
              E
            
          
          
            ∙
          
        
        ∈
        
          D
          
            C
            o
            h
          
          
            b
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {E}}^{\bullet }\in D_{Coh}^{b}(X)}
  
 with bounded <a href="/facts/Coherent_cohomology/BKAxIzgW">coherent cohomology</a> such that the sheaves 
  
    
      
        
          H
          
            i
          
        
        (
        
          
            
              E
            
          
          
            ∙
          
        
        )
      
    
    {\displaystyle H^{i}({\mathcal {E}}^{\bullet })}
  
 have proper support over 
  
    
      
        S
      
    
    {\displaystyle S}
  
, then the derived pushforward 
  
    
      
        
          R
        
        
          f
          
            ∗
          
        
        (
        
          
            
              E
            
          
          
            ∙
          
        
        )
      
    
    {\displaystyle \mathbf {R} f_{*}({\mathcal {E}}^{\bullet })}
  
 has bounded coherent cohomology over 
  
    
      
        S
      
    
    {\displaystyle S}
  
, meaning it is an object in 
  
    
      
        
          D
          
            C
            o
            h
          
          
            b
          
        
        (
        S
        )
      
    
    {\displaystyle D_{Coh}^{b}(S)}
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="examples">Examples</h2>
<p>Most schemes of interest are Noetherian schemes.
</p>
<h3>Locally of finite type over a Noetherian base</h3>
<p>Another class of examples of Noetherian schemes<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> are families of schemes 
  
    
      
        X
        →
        S
      
    
    {\displaystyle X\to S}
  
 where the base 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is Noetherian and 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is of <a href="/facts/Morphism_of_finite_type/djrynAmJ">finite type</a> over 
  
    
      
        S
      
    
    {\displaystyle S}
  
. This includes many examples, such as the <a href="/facts/Connected_component_(topology)/5CUTxfoA">connected components</a> of a <a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert scheme</a>, i.e. with a fixed Hilbert polynomial. This is important because it implies many <a href="/facts/Moduli_space/9sqrdpwR">moduli spaces</a> encountered in the wild are Noetherian, such as the <a href="/facts/Moduli_of_algebraic_curves/u4mAMEfJ">Moduli of algebraic curves</a> and Moduli of stable vector bundles. Also, this property can be used to show many schemes considered in algebraic geometry are in fact Noetherian.
</p>
<h3>Quasi-projective varieties</h3>
<p>In particular, quasi-projective varieties are Noetherian schemes. This class includes <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a>, <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a>, <a href="/facts/Abelian_varieties/dGSDVruG">abelian varieties</a>, <a href="/facts/Calabi%25E2%2580%2593Yau_manifold/LUyEX4kT">Calabi-Yau schemes</a>, <a href="/facts/Shimura_varieties/varFgmR6">Shimura varieties</a>, <a href="/facts/K3_surface/hqbR3jpu">K3 surfaces</a>, and <a href="/facts/Cubic_surface/jCvavESm">cubic surfaces</a>. Basically all of the objects from classical algebraic geometry fit into this class of examples.
</p>
<h3>Infinitesimal deformations of Noetherian schemes</h3>
<p>In particular, infinitesimal deformations of Noetherian schemes are again Noetherian. For example, given a curve 
  
    
      
        C
        
          /
        
        
          Spec
        
        (
        
          
            F
          
          
            q
          
        
        )
      
    
    {\displaystyle C/{\text{Spec}}(\mathbb {F} _{q})}
  
, any <a href="/facts/Deformation_theory/tVLS3kkG">deformation</a> 
  
    
      
        
          
            C
          
        
        
          /
        
        
          Spec
        
        (
        
          
            F
          
          
            q
          
        
        [
        ε
        ]
        
          /
        
        (
        
          ε
          
            n
          
        
        )
        )
      
    
    {\displaystyle {\mathcal {C}}/{\text{Spec}}(\mathbb {F} _{q}[\varepsilon ]/(\varepsilon ^{n}))}
  
 is also a Noetherian scheme. A tower of such deformations can be used to construct formal Noetherian schemes.
</p>
<h3>Non-examples</h3>
<h4>Schemes over Adelic bases</h4>
<p>One of the natural rings which are non-Noetherian are the <a href="/facts/Ring_of_adeles/2LsbgeBE">ring of adeles</a> 
  
    
      
        
          
            A
          
          
            K
          
        
      
    
    {\displaystyle \mathbb {A} _{K}}
  
 for an <a href="/facts/Algebraic_number_field/e0K0TXSh">algebraic number field</a> 
  
    
      
        K
      
    
    {\displaystyle K}
  
. In order to deal with such rings, a topology is considered, giving <a href="/facts/Topological_ring/qxQqzhCy">topological rings</a>. There is a notion of algebraic geometry over such rings developed by <a href="/facts/Andr%25C3%25A9_Weil/Q4Dd0Gr4">André Weil</a> and <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h4>Rings of integers over infinite extensions</h4>
<p>Given an infinite <a href="/facts/Galois_field_extension/52fzNQbv">Galois field extension</a> 
  
    
      
        K
        
          /
        
        L
      
    
    {\displaystyle K/L}
  
, such as 
  
    
      
        
          Q
        
        (
        
          ζ
          
            ∞
          
        
        )
        
          /
        
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} (\zeta _{\infty })/\mathbb {Q} }
  
 (by adjoining all roots of unity), the <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> 
  
    
      
        
          
            
              O
            
          
          
            K
          
        
      
    
    {\displaystyle {\mathcal {O}}_{K}}
  
 is a non-Noetherian ring which is dimension 
  
    
      
        1
      
    
    {\displaystyle 1}
  
. This breaks the intuition that finite dimensional schemes are necessarily Noetherian. Also, this example provides motivation for why studying schemes over a non-Noetherian base; that is, schemes 
  
    
      
        
          Sch
        
        
          /
        
        
          Spec
        
        (
        
          
            
              O
            
          
          
            E
          
        
        )
      
    
    {\displaystyle {\text{Sch}}/{\text{Spec}}({\mathcal {O}}_{E})}
  
, can be an interesting and fruitful subject.
</p><p>
One special case<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>pg 93 of such an extension is taking the maximal unramified extension 
  
    
      
        
          K
          
            u
            r
          
        
        
          /
        
        K
      
    
    {\displaystyle K^{ur}/K}
  
 and considering the ring of integers 
  
    
      
        
          
            
              O
            
          
          
            
              K
              
                u
                r
              
            
          
        
      
    
    {\displaystyle {\mathcal {O}}_{K^{ur}}}
  
. The induced morphism</p><blockquote><p>
  
    
      
        
          Spec
        
        (
        
          
            
              O
            
          
          
            
              K
              
                u
                r
              
            
          
        
        )
        →
        
          Spec
        
        (
        
          
            
              O
            
          
          
            K
          
        
        )
      
    
    {\displaystyle {\text{Spec}}({\mathcal {O}}_{K^{ur}})\to {\text{Spec}}({\mathcal {O}}_{K})}
  
</p></blockquote><p>forms the <a href="/facts/Universal_covering/bDRGasl7">universal covering</a> of 
  
    
      
        
          Spec
        
        (
        
          
            
              O
            
          
          
            K
          
        
        )
      
    
    {\displaystyle {\text{Spec}}({\mathcal {O}}_{K})}
  
.
</p><h4>Polynomial ring with infinitely many generators</h4>
<p>Another example of a non-Noetherian finite-dimensional scheme (in fact zero-dimensional) is given by the following quotient of a polynomial ring with infinitely many generators.
</p>
<blockquote><p>
  
    
      
        
          
            
              
                Q
              
              [
              
                x
                
                  1
                
              
              ,
              
                x
                
                  2
                
              
              ,
              
                x
                
                  3
                
              
              ,
              …
              ]
            
            
              (
              
                x
                
                  1
                
              
              ,
              
                x
                
                  2
                
                
                  2
                
              
              ,
              
                x
                
                  3
                
                
                  3
                
              
              ,
              …
              )
            
          
        
      
    
    {\displaystyle {\frac {\mathbb {Q} [x_{1},x_{2},x_{3},\ldots ]}{(x_{1},x_{2}^{2},x_{3}^{3},\ldots )}}}
  
</p></blockquote>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Excellent_ring/q4PGMtPF">Excellent ring</a> - slightly more rigid than Noetherian rings, but with better properties</li>
<li><a href="/facts/Chevalley%2527s_theorem_on_constructible_sets/ygTLFldz">Chevalley's theorem on constructible sets</a></li>
<li><a href="/facts/Zariski%2527s_main_theorem/p4QM6TN2">Zariski's main theorem</a></li>
<li><a href="/facts/Dualizing_complex/moQWLPdQ">Dualizing complex</a></li>
<li><a href="/facts/Nagata%2527s_compactification_theorem/UnMbe2ma">Nagata's compactification theorem</a></li></ul>

<ul><li><a href="/facts/Robin_Hartshorne/mCBLyIW6">Hartshorne, Robin</a> (1977). <i>Algebraic Geometry</i>. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <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>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0367.14001">0367.14001</a>.</li>
<li><a href="/facts/G%25C3%25BCnter_Harder/kMnTML70">Harder, Günter</a>. <a href="https://web.archive.org/web/20200724032357/http://www.math.uni-bonn.de/people/harder/Manuscripts/buch/Volume-III.Feb-26-2020.pdf">"Cohomology of Arithmetic Groups"</a> (PDF). Archived from <a href="http://www.math.uni-bonn.de/people/harder/Manuscripts/buch/Volume-III.Feb-26-2020.pdf">the original</a> (PDF) on 2020-07-24.</li>
<li>Danilov, V.I. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Noetherian_scheme&oldid=34135">"Noetherian scheme"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>

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

<ol>
<li id="fn:1"><p>"Lemma 28.5.7 (0BA8)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24. <a href="https://stacks.math.columbia.edu/tag/0BA8" target="_blank">https://stacks.math.columbia.edu/tag/0BA8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Lemma 28.5.8 (01P0)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24. <a href="https://stacks.math.columbia.edu/tag/01P0" target="_blank">https://stacks.math.columbia.edu/tag/01P0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Cohomology of Sheaves" (PDF). <a href="http://therisingsea.org/notes/Section3.2-CohomologyOfSheaves.pdf" target="_blank">http://therisingsea.org/notes/Section3.2-CohomologyOfSheaves.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"Lemma 36.10.3 (08E2)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24. <a href="https://stacks.math.columbia.edu/tag/08E2" target="_blank">https://stacks.math.columbia.edu/tag/08E2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Lemma 29.15.6 (01T6)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24. <a href="https://stacks.math.columbia.edu/tag/01T6" target="_blank">https://stacks.math.columbia.edu/tag/01T6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Conrad, Brian. "Weil and Grothendieck Approaches to Adelic Points" (PDF). Archived (PDF) from the original on 21 July 2018. <a href="http://math.stanford.edu/~conrad/papers/adelictop.pdf" target="_blank">http://math.stanford.edu/~conrad/papers/adelictop.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Neukirch, Jürgen (1999). "1.13". Algebraic Number Theory. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-03983-0. OCLC 851391469. <a href="978-3-662-03983-0" target="_blank">978-3-662-03983-0</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Noetherian scheme open-in-new

Noetherian scheme