Essentially finite vector bundle

<h2 id="finite-vector-bundles">Finite vector bundles</h2>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a scheme and 
  
    
      
        V
      
    
    {\displaystyle V}
  
 a vector bundle on 
  
    
      
        X
      
    
    {\displaystyle X}
  
. For 
  
    
      
        f
        =
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        x
        +
        …
        +
        
          a
          
            n
          
        
        
          x
          
            n
          
        
        ∈
        
          
            Z
          
          
            ≥
            0
          
        
        [
        x
        ]
      
    
    {\displaystyle f=a_{0}+a_{1}x+\ldots +a_{n}x^{n}\in \mathbb {Z} _{\geq 0}[x]}
  
 an integral polynomial with nonnegative coefficients define
</p>

f
        (
        V
        )
        :=
        
          
            
              O
            
          
          
            X
          
          
            ⊕
            
              a
              
                0
              
            
          
        
        ⊕
        
          V
          
            ⊕
            
              a
              
                1
              
            
          
        
        ⊕
        
          
            (
            
              V
              
                ⊗
                2
              
            
            )
          
          
            ⊕
            
              a
              
                2
              
            
          
        
        ⊕
        …
        ⊕
        
          
            (
            
              V
              
                ⊗
                n
              
            
            )
          
          
            ⊕
            
              a
              
                n
              
            
          
        
      
    
    {\displaystyle f(V):={\mathcal {O}}_{X}^{\oplus a_{0}}\oplus V^{\oplus a_{1}}\oplus \left(V^{\otimes 2}\right)^{\oplus a_{2}}\oplus \ldots \oplus \left(V^{\otimes n}\right)^{\oplus a_{n}}}

<p>Then 
  
    
      
        V
      
    
    {\displaystyle V}
  
 is called finite if there are two distinct polynomials 
  
    
      
        f
        ,
        g
        ∈
        
          
            Z
          
          
            ≥
            0
          
        
        [
        x
        ]
      
    
    {\displaystyle f,g\in \mathbb {Z} _{\geq 0}[x]}
  
 for which 
  
    
      
        f
        (
        V
        )
      
    
    {\displaystyle f(V)}
  
 is isomorphic to 
  
    
      
        g
        (
        V
        )
      
    
    {\displaystyle g(V)}
  
. 
</p>
<h2 id="definition">Definition</h2>
<p>The following two definitions coincide whenever 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a reduced, connected and proper scheme over a perfect field.
</p>
<h3>According to Borne and <a href="/facts/Angelo_Vistoli/ouyu0Dc0">Vistoli</a></h3>
<p>A vector bundle is essentially finite if it is the kernel of a morphism 
  
    
      
        u
        :
        
          F
          
            1
          
        
        →
        
          F
          
            2
          
        
      
    
    {\displaystyle u:F_{1}\to F_{2}}
  
 where 
  
    
      
        
          F
          
            1
          
        
        ,
        
          F
          
            2
          
        
      
    
    {\displaystyle F_{1},F_{2}}
  
 are finite vector bundles. <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h3>The original definition of <a href="/facts/Madhav_V._Nori/KXBk0ylG">Nori</a></h3>
<p>A vector bundle is essentially finite if it is a <a href="/facts/Subquotient/tdBGumlX">subquotient</a> of a finite vector bundle in the category of <a href="/facts/Nori-semistable/b9B1gu28">Nori-semistable</a> vector bundles.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="properties">Properties</h2>
<ul><li>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a reduced and connected <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> over a perfect <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
  
    
      
        k
      
    
    {\displaystyle k}
  
 endowed with a section 
  
    
      
        x
        ∈
        X
        (
        k
        )
      
    
    {\displaystyle x\in X(k)}
  
.  Then a vector bundle 
  
    
      
        V
      
    
    {\displaystyle V}
  
 over 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is essentially finite if and only if there exists a <a href="/facts/Finite_morphism/djrynAmJ">finite</a> 
  
    
      
        k
      
    
    {\displaystyle k}
  
-<a href="/facts/Group_scheme/bJW14FE2">group scheme</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
 and a 
  
    
      
        G
      
    
    {\displaystyle G}
  
-<a href="/facts/Torsor_(algebraic_geometry)/nMKyR4eF">torsor</a>  
  
    
      
        p
        :
        P
        →
        X
      
    
    {\displaystyle p:P\to X}
  
 such that 
  
    
      
        V
      
    
    {\displaystyle V}
  
 becomes trivial over 
  
    
      
        P
      
    
    {\displaystyle P}
  
 (i.e. 
  
    
      
        
          p
          
            ∗
          
        
        (
        V
        )
        ≅
        
          O
          
            P
          
          
            ⊕
            r
          
        
      
    
    {\displaystyle p^{*}(V)\cong O_{P}^{\oplus r}}
  
, where 
  
    
      
        r
        =
        r
        k
        (
        V
        )
      
    
    {\displaystyle r=rk(V)}
  
).</li></ul>
<ul><li>When 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a reduced, connected and proper scheme over a perfect field with a point 
  
    
      
        x
        ∈
        X
        (
        k
        )
      
    
    {\displaystyle x\in X(k)}
  
 then the category 
  
    
      
        E
        F
        (
        X
        )
      
    
    {\displaystyle EF(X)}
  
 of essentially finite vector bundles provided with the usual tensor product 
  
    
      
        
          ⊗
          
            
              
                
                  O
                
              
              
                X
              
            
          
        
      
    
    {\displaystyle \otimes _{{\mathcal {O}}_{X}}}
  
, the trivial object 
  
    
      
        
          
            
              O
            
          
          
            X
          
        
      
    
    {\displaystyle {\mathcal {O}}_{X}}
  
 and the fiber functor 
  
    
      
        
          x
          
            ∗
          
        
      
    
    {\displaystyle x^{*}}
  
 is a <a href="/facts/Tannakian_formalism/j0apJbpX">Tannakian category</a>.</li></ul>
<ul><li>The 
  
    
      
        k
      
    
    {\displaystyle k}
  
-affine <a href="/facts/Group_scheme/bJW14FE2">group scheme</a> 
  
    
      
        
          π
          
            1
          
        
        (
        X
        ,
        x
        )
      
    
    {\displaystyle \pi _{1}(X,x)}
  
 naturally associated to the Tannakian category 
  
    
      
        E
        F
        (
        X
        )
      
    
    {\displaystyle EF(X)}
  
 is called the <a href="/facts/Fundamental_group_scheme/Ylnbp5U3">fundamental group scheme</a>.</li></ul>
<h2 id="notes">Notes</h2>

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

<ol>
<li id="fn:1"><p>Nori, Madhav V. (1976). "On the Representations of the Fundamental Group". Compositio Mathematica. 33 (1): 29–42. MR 0417179. <a href="https://eudml.org/doc/89294" target="_blank">https://eudml.org/doc/89294</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Szamuely, T. (2009). Galois Groups and Fundamental Groups. Vol. 117. Cambridge Studies in Advanced Mathematics. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>N. Borne, A. Vistoli The Nori fundamental gerbe of a fibered category, J. Algebr. Geom. 24, No. 2, 311-353 (2015) <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Nori, Madhav V. (1976). "On the Representations of the Fundamental Group". Compositio Mathematica. 33 (1): 29–42. MR 0417179. <a href="https://eudml.org/doc/89294" target="_blank">https://eudml.org/doc/89294</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Essentially finite vector bundle open-in-new

Essentially finite vector bundle