Sheaf of modules

<h2 id="examples">Examples</h2>
<ul><li>Given a ringed space (X, O), if F is an O-submodule of O, then it is called the sheaf of ideals or <a href="/facts/Ideal_sheaf/TfqpB5WW">ideal sheaf</a> of O, since for each open subset U of X, F(U) is an <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> of the ring O(U).</li>
<li>Let X be a <a href="/facts/Smooth_variety/Y76o7b9T">smooth variety</a> of dimension n. Then the <a href="/facts/Tangent_sheaf/gkcxvXVu">tangent sheaf</a> of X is the dual of the <a href="/facts/Cotangent_sheaf/gkcxvXVu">cotangent sheaf</a> 
 
 
 
 
 Ω
 
 X
 
 
 
 
 {\displaystyle \Omega _{X}}
 
 and the <a href="/facts/Canonical_sheaf/nJKN8mpG">canonical sheaf</a> 
 
 
 
 
 ω
 
 X
 
 
 
 
 {\displaystyle \omega _{X}}
 
 is the n-th exterior power (<a href="/facts/Determinant_line_bundle/CupAl1ft">determinant</a>) of 
 
 
 
 
 Ω
 
 X
 
 
 
 
 {\displaystyle \Omega _{X}}
 
.</li>
<li>A <a href="/facts/Sheaf_of_algebras/nPcFLdka">sheaf of algebras</a> is a sheaf of modules that is also a sheaf of rings.</li></ul>
<h2 id="operations">Operations</h2>
Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by

F
 
 ⊗
 
 O
 
 
 G
 
 
 {\displaystyle F\otimes _{O}G}
 
 or 
 
 
 
 F
 ⊗
 G
 
 
 {\displaystyle F\otimes G}
 
,
is the O-module that is the sheaf associated to the presheaf 
 
 
 
 U
 ↦
 F
 (
 U
 )
 
 ⊗
 
 O
 (
 U
 )
 
 
 G
 (
 U
 )
 .
 
 
 {\displaystyle U\mapsto F(U)\otimes _{O(U)}G(U).}
 
 (To see that sheafification cannot be avoided, compute the global sections of 
 
 
 
 O
 (
 1
 )
 ⊗
 O
 (
 −
 1
 )
 =
 O
 
 
 {\displaystyle O(1)\otimes O(-1)=O}
 
 where O(1) is <a href="/facts/Serre%2527s_twisting_sheaf/gcvPmjpU">Serre's twisting sheaf</a> on a projective space.)
Similarly, if F and G are O-modules, then

H
          
        
        o
        
          m
          
            O
          
        
        (
        F
        ,
        G
        )
      
    
    {\displaystyle {\mathcal {H}}om_{O}(F,G)}

denotes the O-module that is the sheaf 
 
 
 
 U
 ↦
 
 Hom
 
 O
 
 
 |
 
 
 U
 
 
 
 
 ⁡
 (
 F
 
 
 |
 
 
 U
 
 
 ,
 G
 
 
 |
 
 
 U
 
 
 )
 
 
 {\displaystyle U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> In particular, the O-module

H
          
        
        o
        
          m
          
            O
          
        
        (
        F
        ,
        O
        )
      
    
    {\displaystyle {\mathcal {H}}om_{O}(F,O)}

is called the dual module of F and is denoted by 
 
 
 
 
 
 
 F
 ˇ
 
 
 
 
 
 {\displaystyle {\check {F}}}
 
. Note: for any O-modules E, F, there is a canonical homomorphism

E
 ˇ
 
 
 
 ⊗
 F
 →
 
 
 H
 
 
 o
 
 m
 
 O
 
 
 (
 E
 ,
 F
 )
 
 
 {\displaystyle {\check {E}}\otimes F\to {\mathcal {H}}om_{O}(E,F)}
 
,
which is an isomorphism if E is a <a href="/facts/Locally_free_sheaf/h2NoUN7e">locally free sheaf</a> of finite rank. In particular, if L is locally free of rank one (such L is called an <a href="/facts/Invertible_sheaf/D8RxXPEd">invertible sheaf</a> or a <a href="/facts/Line_bundle/CupAl1ft">line bundle</a>),<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> then this reads:

L
              ˇ
            
          
        
        ⊗
        L
        ≃
        O
        ,
      
    
    {\displaystyle {\check {L}}\otimes L\simeq O,}

implying the isomorphism classes of invertible sheaves form a group. This group is called the <a href="/facts/Picard_group/lWH0CoWQ">Picard group</a> of X and is canonically identified with the first cohomology group 
 
 
 
 
 H
 
 1
 
 
 ⁡
 (
 X
 ,
 
 
 
 O
 
 
 
 ∗
 
 
 )
 
 
 {\displaystyle \operatorname {H} ^{1}(X,{\mathcal {O}}^{*})}
 
 (by the standard argument with <a href="/facts/%25C4%258Cech_cohomology/HLWD7BnA">Čech cohomology</a>).
If E is a locally free sheaf of finite rank, then there is an O-linear map 
 
 
 
 
 
 
 E
 ˇ
 
 
 
 ⊗
 E
 ≃
 
 End
 
 O
 
 
 ⁡
 (
 E
 )
 →
 O
 
 
 {\displaystyle {\check {E}}\otimes E\simeq \operatorname {End} _{O}(E)\to O}
 
 given by the pairing; it is called the <a href="/facts/Trace_map/K3oaFIS4">trace map</a> of E.
For any O-module F, the <a href="/facts/Tensor_algebra/H7CKKryf">tensor algebra</a>, <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> and <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric algebra</a> of F are defined in the same way. For example, the k-th exterior power

⋀
          
            k
          
        
        F
      
    
    {\displaystyle \bigwedge ^{k}F}

is the sheaf associated to the presheaf 
 
 
 
 U
 ↦
 
 ⋀
 
 O
 (
 U
 )
 
 
 k
 
 
 F
 (
 U
 )
 
 
 {\textstyle U\mapsto \bigwedge _{O(U)}^{k}F(U)}
 
. If F is locally free of rank n, then 
 
 
 
 
 ⋀
 
 n
 
 
 F
 
 
 {\textstyle \bigwedge ^{n}F}
 
 is called the <a href="/facts/Determinant_line_bundle/CupAl1ft">determinant line bundle</a> (though technically <a href="/facts/Invertible_sheaf/D8RxXPEd">invertible sheaf</a>) of F, denoted by det(F). There is a natural perfect pairing:

⋀
          
            r
          
        
        F
        ⊗
        
          ⋀
          
            n
            −
            r
          
        
        F
        →
        det
        (
        F
        )
        .
      
    
    {\displaystyle \bigwedge ^{r}F\otimes \bigwedge ^{n-r}F\to \det(F).}

Let f: (X, O) →(X', O') be a morphism of ringed spaces. If F is an O-module, then the <a href="/facts/Direct_image_sheaf/z3mckxvp">direct image sheaf</a> 
 
 
 
 
 f
 
 ∗
 
 
 F
 
 
 {\displaystyle f_{*}F}
 
 is an O'-module through the natural map O' →f*O (such a natural map is part of the data of a morphism of ringed spaces.)
If G is an O'-module, then the module inverse image 
 
 
 
 
 f
 
 ∗
 
 
 G
 
 
 {\displaystyle f^{*}G}
 
 of G is the O-module given as the tensor product of modules:

f
          
            −
            1
          
        
        G
        
          ⊗
          
            
              f
              
                −
                1
              
            
            
              O
              ′
            
          
        
        O
      
    
    {\displaystyle f^{-1}G\otimes _{f^{-1}O'}O}

where 
 
 
 
 
 f
 
 −
 1
 
 
 G
 
 
 {\displaystyle f^{-1}G}
 
 is the <a href="/facts/Inverse_image_sheaf/wDG4ARH0">inverse image sheaf</a> of G and 
 
 
 
 
 f
 
 −
 1
 
 
 
 O
 ′
 
 →
 O
 
 
 {\displaystyle f^{-1}O'\to O}
 
 is obtained from 
 
 
 
 
 O
 ′
 
 →
 
 f
 
 ∗
 
 
 O
 
 
 {\displaystyle O'\to f_{*}O}
 
 by <a href="/facts/Adjoint_functor/bJ1P7AN2">adjuction</a>.
There is an adjoint relation between 
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f_{*}}
 
 and 
 
 
 
 
 f
 
 ∗
 
 
 
 
 {\displaystyle f^{*}}
 
: for any O-module F and O'-module G,

Hom
          
            O
          
        
        ⁡
        (
        
          f
          
            ∗
          
        
        G
        ,
        F
        )
        ≃
        
          Hom
          
            
              O
              ′
            
          
        
        ⁡
        (
        G
        ,
        
          f
          
            ∗
          
        
        F
        )
      
    
    {\displaystyle \operatorname {Hom} _{O}(f^{*}G,F)\simeq \operatorname {Hom} _{O'}(G,f_{*}F)}

as abelian group. There is also the <a href="/facts/Projection_formula/V458LRhM">projection formula</a>: for an O-module F and a locally free O'-module E of finite rank,

f
          
            ∗
          
        
        (
        F
        ⊗
        
          f
          
            ∗
          
        
        E
        )
        ≃
        
          f
          
            ∗
          
        
        F
        ⊗
        E
        .
      
    
    {\displaystyle f_{*}(F\otimes f^{*}E)\simeq f_{*}F\otimes E.}

<h2 id="properties">Properties</h2>

Let (X, O) be a ringed space. An O-module F is said to be generated by global sections if there is a surjection of O-modules:

⨁
          
            i
            ∈
            I
          
        
        O
        →
        F
        →
        0.
      
    
    {\displaystyle \bigoplus _{i\in I}O\to F\to 0.}

Explicitly, this means that there are global sections si of F such that the images of si in each stalk Fx generates Fx as Ox-module.
An example of such a sheaf is that associated in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> to an R-module M, R being any <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>, on the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum of a ring</a> Spec(R).
Another example: according to <a href="/facts/Cartan%2527s_theorem_A/KgMmbQ37">Cartan's theorem A</a>, any <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaf</a> on a <a href="/facts/Stein_manifold/PWdqgPPp">Stein manifold</a> is spanned by global sections. (cf. Serre's theorem A below.) In the theory of <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a>, a related notion is <a href="/facts/Ample_line_bundle/XRQKvXGD">ample line bundle</a>. (For example, if L is an ample line bundle, some power of it is generated by global sections.)
An injective O-module is <a href="/facts/Flasque_sheaf/ykaN5Ivq">flasque</a> (i.e., all restrictions maps F(U) → F(V) are surjective).<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> Since a flasque sheaf is acyclic in the category of abelian sheaves, this implies that the i-th right derived functor of the global section functor 
 
 
 
 Γ
 (
 X
 ,
 −
 )
 
 
 {\displaystyle \Gamma (X,-)}
 
 in the category of O-modules coincides with the usual i-th sheaf cohomology in the category of abelian sheaves.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="sheaf-associated-to-a-module">Sheaf associated to a module</h2>
Let 
 
 
 
 M
 
 
 {\displaystyle M}
 
 be a module over a ring 
 
 
 
 A
 
 
 {\displaystyle A}
 
. Put 
 
 
 
 X
 =
 Spec
 ⁡
 (
 A
 )
 
 
 {\displaystyle X=\operatorname {Spec} (A)}
 
 and write 
 
 
 
 D
 (
 f
 )
 =
 {
 f
 ≠
 0
 }
 =
 Spec
 ⁡
 (
 A
 [
 
 f
 
 −
 1
 
 
 ]
 )
 
 
 {\displaystyle D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])}
 
. For each pair 
 
 
 
 D
 (
 f
 )
 ⊆
 D
 (
 g
 )
 
 
 {\displaystyle D(f)\subseteq D(g)}
 
, by the universal property of localization, there is a natural map

ρ
          
            g
            ,
            f
          
        
        :
        M
        [
        
          g
          
            −
            1
          
        
        ]
        →
        M
        [
        
          f
          
            −
            1
          
        
        ]
      
    
    {\displaystyle \rho _{g,f}:M[g^{-1}]\to M[f^{-1}]}

having the property that 
 
 
 
 
 ρ
 
 g
 ,
 f
 
 
 =
 
 ρ
 
 g
 ,
 h
 
 
 ∘
 
 ρ
 
 h
 ,
 f
 
 
 
 
 {\displaystyle \rho _{g,f}=\rho _{g,h}\circ \rho _{h,f}}
 
. Then

D
        (
        f
        )
        ↦
        M
        [
        
          f
          
            −
            1
          
        
        ]
      
    
    {\displaystyle D(f)\mapsto M[f^{-1}]}

is a contravariant functor from the category whose objects are the sets D(f) and morphisms the inclusions of sets to the <a href="/facts/Category_of_abelian_groups/jF3sXRFn">category of abelian groups</a>. One can show<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> it is in fact a <a href="/facts/B-sheaf/V3MnSb2f">B-sheaf</a> (i.e., it satisfies the gluing axiom) and thus defines the sheaf 
 
 
 
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {M}}}
 
 on X called the sheaf associated to M.
The most basic example is the structure sheaf on X; i.e., 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 =
 
 
 
 A
 ~
 
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}
 
. Moreover, 
 
 
 
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {M}}}
 
 has the structure of 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 =
 
 
 
 A
 ~
 
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}
 
-module and thus one gets the <a href="/facts/Exact_functor/4tsQzwBp">exact functor</a> 
 
 
 
 M
 ↦
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle M\mapsto {\widetilde {M}}}
 
 from ModA, the <a href="/facts/Category_of_modules/AnnF9JKf">category of modules</a> over A to the category of modules over 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
. It defines an <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalence</a> from ModA to the category of <a href="/facts/Quasi-coherent_sheaves/h2NoUN7e">quasi-coherent sheaves</a> on X, with the inverse 
 
 
 
 Γ
 (
 X
 ,
 −
 )
 
 
 {\displaystyle \Gamma (X,-)}
 
, the <a href="/facts/Global_section_functor/MM3hcRw4">global section functor</a>. When X is <a href="/facts/Noetherian_scheme/amyCoMDf">Noetherian</a>, the functor is an equivalence from the category of finitely generated A-modules to the category of coherent sheaves on X.
The construction has the following properties: for any A-modules M, N, and any morphism 
 
 
 
 φ
 :
 M
 →
 N
 
 
 {\displaystyle \varphi :M\to N}
 
,

<ul><li>
 
 
 
 M
 [
 
 f
 
 −
 1
 
 
 
 ]
 
 ∼
 
 
 =
 
 
 
 M
 ~
 
 
 
 
 
 |
 
 
 D
 (
 f
 )
 
 
 
 
 {\displaystyle M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}}
 
.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a></li>
<li>For any prime ideal p of A, 
 
 
 
 
 
 
 
 M
 ~
 
 
 
 
 p
 
 
 ≃
 
 M
 
 p
 
 
 
 
 {\displaystyle {\widetilde {M}}_{p}\simeq M_{p}}
 
 as Op = Ap-module.</li>
<li>
 
 
 
 (
 M
 
 ⊗
 
 A
 
 
 N
 
 )
 
 ∼
 
 
 ≃
 
 
 
 M
 ~
 
 
 
 
 ⊗
 
 
 
 A
 ~
 
 
 
 
 
 
 
 N
 ~
 
 
 
 
 
 {\displaystyle (M\otimes _{A}N)^{\sim }\simeq {\widetilde {M}}\otimes _{\widetilde {A}}{\widetilde {N}}}
 
.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a></li>
<li>If M is <a href="/facts/Finitely_presented_module/6Njcpgzr">finitely presented</a>, 
 
 
 
 
 Hom
 
 A
 
 
 ⁡
 (
 M
 ,
 N
 
 )
 
 ∼
 
 
 ≃
 
 
 H
 
 
 o
 
 m
 
 
 
 A
 ~
 
 
 
 
 (
 
 
 
 M
 ~
 
 
 
 ,
 
 
 
 N
 ~
 
 
 
 )
 
 
 {\displaystyle \operatorname {Hom} _{A}(M,N)^{\sim }\simeq {\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}})}
 
.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li>
<li>
 
 
 
 
 Hom
 
 A
 
 
 ⁡
 (
 M
 ,
 N
 )
 ≃
 Γ
 (
 X
 ,
 
 
 H
 
 
 o
 
 m
 
 
 
 A
 ~
 
 
 
 
 (
 
 
 
 M
 ~
 
 
 
 ,
 
 
 
 N
 ~
 
 
 
 )
 )
 
 
 {\displaystyle \operatorname {Hom} _{A}(M,N)\simeq \Gamma (X,{\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}}))}
 
, since the equivalence between ModA and the category of quasi-coherent sheaves on X.</li>
<li>
 
 
 
 (
 
 
 lim
 →
 
 
 ⁡
 
 M
 
 i
 
 
 
 )
 
 ∼
 
 
 ≃
 
 
 lim
 →
 
 
 ⁡
 
 
 
 
 M
 
 i
 
 
 ~
 
 
 
 
 
 {\displaystyle (\varinjlim M_{i})^{\sim }\simeq \varinjlim {\widetilde {M_{i}}}}
 
;<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> in particular, taking a direct sum and ~ commute.</li>
<li>A sequence of A-modules is exact if and only if the induced sequence by 
 
 
 
 ∼
 
 
 {\displaystyle \sim }
 
 is exact. In particular, 
 
 
 
 (
 ker
 ⁡
 (
 φ
 )
 
 )
 
 ∼
 
 
 =
 ker
 ⁡
 (
 
 
 
 φ
 ~
 
 
 
 )
 ,
 (
 coker
 ⁡
 (
 φ
 )
 
 )
 
 ∼
 
 
 =
 coker
 ⁡
 (
 
 
 
 φ
 ~
 
 
 
 )
 ,
 (
 im
 ⁡
 (
 φ
 )
 
 )
 
 ∼
 
 
 =
 im
 ⁡
 (
 
 
 
 φ
 ~
 
 
 
 )
 
 
 {\displaystyle (\ker(\varphi ))^{\sim }=\ker({\widetilde {\varphi }}),(\operatorname {coker} (\varphi ))^{\sim }=\operatorname {coker} ({\widetilde {\varphi }}),(\operatorname {im} (\varphi ))^{\sim }=\operatorname {im} ({\widetilde {\varphi }})}
 
.</li></ul>
<h2 id="sheaf-associated-to-a-graded-module">Sheaf associated to a graded module</h2>
There is a graded analog of the construction and equivalence in the preceding section. Let R be a graded ring generated by degree-one elements as R0-algebra (R0 means the degree-zero piece) and M a graded R-module. Let X be the <a href="/facts/Proj_construction/gcvPmjpU">Proj</a> of R (so X is a <a href="/facts/Projective_scheme/aB2XdB7B">projective scheme</a> if R is <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a>). Then there is an O-module 
 
 
 
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {M}}}
 
 such that for any homogeneous element f of positive degree of R, there is a natural isomorphism

M
              ~
            
          
        
        
          
            |
          
          
            {
            f
            ≠
            0
            }
          
        
        ≃
        (
        M
        [
        
          f
          
            −
            1
          
        
        
          ]
          
            0
          
        
        
          )
          
            ∼
          
        
      
    
    {\displaystyle {\widetilde {M}}|_{\{f\neq 0\}}\simeq (M[f^{-1}]_{0})^{\sim }}

as sheaves of modules on the affine scheme 
 
 
 
 {
 f
 ≠
 0
 }
 =
 Spec
 ⁡
 (
 R
 [
 
 f
 
 −
 1
 
 
 
 ]
 
 0
 
 
 )
 
 
 {\displaystyle \{f\neq 0\}=\operatorname {Spec} (R[f^{-1}]_{0})}
 
;<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> in fact, this defines 
 
 
 
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {M}}}
 
 by gluing.
Example: Let R(1) be the graded R-module given by R(1)n = Rn+1. Then 
 
 
 
 O
 (
 1
 )
 =
 
 
 
 
 R
 (
 1
 )
 
 ~
 
 
 
 
 
 {\displaystyle O(1)={\widetilde {R(1)}}}
 
 is called <a href="/facts/Serre%2527s_twisting_sheaf/gcvPmjpU">Serre's twisting sheaf</a>, which is the dual of the <a href="/facts/Tautological_line_bundle/Vc2CmshX">tautological line bundle</a> if R is finitely generated in degree-one.
If F is an O-module on X, then, writing 
 
 
 
 F
 (
 n
 )
 =
 F
 ⊗
 O
 (
 n
 )
 
 
 {\displaystyle F(n)=F\otimes O(n)}
 
, there is a canonical homomorphism:

(
            
              
                ⨁
                
                  n
                  ≥
                  0
                
              
              Γ
              (
              X
              ,
              F
              (
              n
              )
              )
            
            )
          
          
            ∼
          
        
        →
        F
        ,
      
    
    {\displaystyle \left(\bigoplus _{n\geq 0}\Gamma (X,F(n))\right)^{\sim }\to F,}

which is an isomorphism if and only if F is quasi-coherent.

<h2 id="computing-sheaf-cohomology">Computing sheaf cohomology</h2>

Main article: <a href="/facts/Sheaf_cohomology/TrcpmD6k">sheaf cohomology</a>
Sheaf cohomology has a reputation for being difficult to calculate. Because of this, the next general fact is fundamental for any practical computation:

Theorem—Let X be a topological space, F an abelian sheaf on it and 
 
 
 
 
 
 U
 
 
 
 
 {\displaystyle {\mathfrak {U}}}
 
 an open cover of X such that 
 
 
 
 
 H
 
 i
 
 
 ⁡
 (
 
 U
 
 
 i
 
 0
 
 
 
 
 ∩
 ⋯
 ∩
 
 U
 
 
 i
 
 p
 
 
 
 
 ,
 F
 )
 =
 0
 
 
 {\displaystyle \operatorname {H} ^{i}(U_{i_{0}}\cap \cdots \cap U_{i_{p}},F)=0}
 
 for any i, p and 
 
 
 
 
 U
 
 
 i
 
 j
 
 
 
 
 
 
 {\displaystyle U_{i_{j}}}
 
's in 
 
 
 
 
 
 U
 
 
 
 
 {\displaystyle {\mathfrak {U}}}
 
. Then for any i,

H
          
            i
          
        
        ⁡
        (
        X
        ,
        F
        )
        =
        
          H
          
            i
          
        
        ⁡
        (
        
          C
          
            ∙
          
        
        (
        
          
            U
          
        
        ,
        F
        )
        )
      
    
    {\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {H} ^{i}(C^{\bullet }({\mathfrak {U}},F))}

where the right-hand side is the i-th <a href="/facts/%25C4%258Cech_cohomology/HLWD7BnA">Čech cohomology</a>.

<a href="/facts/Serre%2527s_vanishing_theorem/BKAxIzgW">Serre's vanishing theorem</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> states that if X is a projective variety and F a coherent sheaf on it, then, for sufficiently large n, the <a href="/facts/Serre_twist/Vc2CmshX">Serre twist</a> F(n) is generated by finitely many global sections. Moreover,

<ol>
<li> For each i, Hi(X, F) is finitely generated over R0, and</li>
<li> There is an integer n0, depending on F, such that

H
 
 i
 
 
 ⁡
 (
 X
 ,
 F
 (
 n
 )
 )
 =
 0
 ,
 
 i
 ≥
 1
 ,
 n
 ≥
 
 n
 
 0
 
 
 .
 
 
 {\displaystyle \operatorname {H} ^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}.}
 
</li>
</ol><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>
<h2 id="sheaf-extension">Sheaf extension</h2>
Let (X, O) be a ringed space, and let F, H be sheaves of O-modules on X. An extension of H by F is a <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequence</a> of O-modules

0
        →
        F
        →
        G
        →
        H
        →
        0.
      
    
    {\displaystyle 0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.}

As with group extensions, if we fix F and H, then all equivalence classes of extensions of H by F form an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> (cf. <a href="/facts/Baer_sum/ktggDk7I">Baer sum</a>), which is isomorphic to the <a href="/facts/Ext_functor/ktggDk7I">Ext group</a> 
 
 
 
 
 Ext
 
 O
 
 
 1
 
 
 ⁡
 (
 H
 ,
 F
 )
 
 
 {\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}
 
, where the identity element in 
 
 
 
 
 Ext
 
 O
 
 
 1
 
 
 ⁡
 (
 H
 ,
 F
 )
 
 
 {\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}
 
 corresponds to the trivial extension.
In the case where H is O, we have: for any i ≥ 0,

H
          
            i
          
        
        ⁡
        (
        X
        ,
        F
        )
        =
        
          Ext
          
            O
          
          
            i
          
        
        ⁡
        (
        O
        ,
        F
        )
        ,
      
    
    {\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {Ext} _{O}^{i}(O,F),}

since both the sides are the right derived functors of the same functor 
 
 
 
 Γ
 (
 X
 ,
 −
 )
 =
 
 Hom
 
 O
 
 
 ⁡
 (
 O
 ,
 −
 )
 .
 
 
 {\displaystyle \Gamma (X,-)=\operatorname {Hom} _{O}(O,-).}

Note: Some authors, notably Hartshorne, drop the subscript O.
Assume X is a projective scheme over a Noetherian ring. Let F, G be coherent sheaves on X and i an integer. Then there exists n0 such that

Ext
 
 O
 
 
 i
 
 
 ⁡
 (
 F
 ,
 G
 (
 n
 )
 )
 =
 Γ
 (
 X
 ,
 
 
 
 E
 x
 t
 
 
 
 O
 
 
 i
 
 
 (
 F
 ,
 G
 (
 n
 )
 )
 )
 ,
 
 n
 ≥
 
 n
 
 0
 
 
 
 
 {\displaystyle \operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {Ext}}_{O}^{i}(F,G(n))),\,n\geq n_{0}}
 
,
where 
 
 
 
 
 
 
 E
 x
 t
 
 
 
 O
 
 
 
 
 {\displaystyle {\mathcal {Ext}}_{O}}
 
 denotes the derived functors of 
 
 
 
 
 
 
 H
 o
 m
 
 
 
 O
 
 
 
 
 {\displaystyle {\mathcal {Hom}}_{O}}
 
.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

See also: <a href="/facts/Local-to-global_Ext_spectral_sequence/Fx98yI4t">local-to-global Ext spectral sequence</a>
<h3>Locally free resolutions</h3>

 
 
 
 
 
 E
 x
 t
 
 
 (
 
 
 F
 
 
 ,
 
 
 G
 
 
 )
 
 
 {\displaystyle {\mathcal {Ext}}({\mathcal {F}},{\mathcal {G}})}
 
 can be readily computed for any coherent sheaf 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 using a locally free <a href="/facts/Resolution_(algebra)/xEFm83U7">resolution</a>:<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> given a complex

⋯
        →
        
          
            
              L
            
          
          
            2
          
        
        →
        
          
            
              L
            
          
          
            1
          
        
        →
        
          
            
              L
            
          
          
            0
          
        
        →
        
          
            F
          
        
        →
        0
      
    
    {\displaystyle \cdots \to {\mathcal {L}}_{2}\to {\mathcal {L}}_{1}\to {\mathcal {L}}_{0}\to {\mathcal {F}}\to 0}

then

R
            H
            o
            m
          
        
        (
        
          
            F
          
        
        ,
        
          
            G
          
        
        )
        =
        
          
            H
            o
            m
          
        
        (
        
          
            
              L
            
          
          
            ∙
          
        
        ,
        
          
            G
          
        
        )
      
    
    {\displaystyle {\mathcal {RHom}}({\mathcal {F}},{\mathcal {G}})={\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}})}

hence

E
              x
              t
            
          
          
            k
          
        
        (
        
          
            F
          
        
        ,
        
          
            G
          
        
        )
        =
        
          h
          
            k
          
        
        (
        
          
            H
            o
            m
          
        
        (
        
          
            
              L
            
          
          
            ∙
          
        
        ,
        
          
            G
          
        
        )
        )
      
    
    {\displaystyle {\mathcal {Ext}}^{k}({\mathcal {F}},{\mathcal {G}})=h^{k}({\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}}))}

<h3>Examples</h3>
<h4>Hypersurface</h4>
Consider a smooth <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 of degree 
 
 
 
 d
 
 
 {\displaystyle d}
 
. Then, we can compute a resolution

O
          
        
        (
        −
        d
        )
        →
        
          
            O
          
        
      
    
    {\displaystyle {\mathcal {O}}(-d)\to {\mathcal {O}}}

and find that

E
              x
              t
            
          
          
            i
          
        
        (
        
          
            
              O
            
          
          
            X
          
        
        ,
        
          
            F
          
        
        )
        =
        
          h
          
            i
          
        
        (
        
          
            H
            o
            m
          
        
        (
        
          
            O
          
        
        (
        −
        d
        )
        →
        
          
            O
          
        
        ,
        
          
            F
          
        
        )
        )
      
    
    {\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})=h^{i}({\mathcal {Hom}}({\mathcal {O}}(-d)\to {\mathcal {O}},{\mathcal {F}}))}

<h4>Union of smooth complete intersections</h4>
Consider the scheme

X
        =
        
          Proj
        
        
          (
          
            
              
                
                  C
                
                [
                
                  x
                  
                    0
                  
                
                ,
                …
                ,
                
                  x
                  
                    n
                  
                
                ]
              
              
                (
                f
                )
                (
                
                  g
                  
                    1
                  
                
                ,
                
                  g
                  
                    2
                  
                
                ,
                
                  g
                  
                    3
                  
                
                )
              
            
          
          )
        
        ⊆
        
          
            P
          
          
            n
          
        
      
    
    {\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} [x_{0},\ldots ,x_{n}]}{(f)(g_{1},g_{2},g_{3})}}\right)\subseteq \mathbb {P} ^{n}}

where 
 
 
 
 (
 f
 ,
 
 g
 
 1
 
 
 ,
 
 g
 
 2
 
 
 ,
 
 g
 
 3
 
 
 )
 
 
 {\displaystyle (f,g_{1},g_{2},g_{3})}
 
 is a smooth <a href="/facts/Complete_intersection/UoFKxRqC">complete intersection</a> and 
 
 
 
 deg
 ⁡
 (
 f
 )
 =
 d
 
 
 {\displaystyle \deg(f)=d}
 
, 
 
 
 
 deg
 ⁡
 (
 
 g
 
 i
 
 
 )
 =
 
 e
 
 i
 
 
 
 
 {\displaystyle \deg(g_{i})=e_{i}}
 
. We have a complex

O
          
        
        (
        −
        d
        −
        
          e
          
            1
          
        
        −
        
          e
          
            2
          
        
        −
        
          e
          
            3
          
        
        )
        
          
            →
            
              
                [
                
                  
                    
                      
                        g
                        
                          3
                        
                      
                    
                  
                  
                    
                      −
                      
                        g
                        
                          2
                        
                      
                    
                  
                  
                    
                      −
                      
                        g
                        
                          1
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
                
                  
                    O
                  
                
                (
                −
                d
                −
                
                  e
                  
                    1
                  
                
                −
                
                  e
                  
                    2
                  
                
                )
              
            
            
              
                ⊕
              
            
            
              
                
                  
                    O
                  
                
                (
                −
                d
                −
                
                  e
                  
                    1
                  
                
                −
                
                  e
                  
                    3
                  
                
                )
              
            
            
              
                ⊕
              
            
            
              
                
                  
                    O
                  
                
                (
                −
                d
                −
                
                  e
                  
                    2
                  
                
                −
                
                  e
                  
                    3
                  
                
                )
              
            
          
        
        
          
            →
            
              
                [
                
                  
                    
                      
                        g
                        
                          2
                        
                      
                    
                    
                      
                        g
                        
                          3
                        
                      
                    
                    
                      0
                    
                  
                  
                    
                      −
                      
                        g
                        
                          1
                        
                      
                    
                    
                      0
                    
                    
                      −
                      
                        g
                        
                          3
                        
                      
                    
                  
                  
                    
                      0
                    
                    
                      −
                      
                        g
                        
                          1
                        
                      
                    
                    
                      
                        g
                        
                          2
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              
                
                  
                    O
                  
                
                (
                −
                d
                −
                
                  e
                  
                    1
                  
                
                )
              
            
            
              
                ⊕
              
            
            
              
                
                  
                    O
                  
                
                (
                −
                d
                −
                
                  e
                  
                    2
                  
                
                )
              
            
            
              
                ⊕
              
            
            
              
                
                  
                    O
                  
                
                (
                −
                d
                −
                
                  e
                  
                    3
                  
                
                )
              
            
          
        
        
          
            →
            
              
                [
                
                  
                    
                      f
                      
                        g
                        
                          1
                        
                      
                    
                    
                      f
                      
                        g
                        
                          2
                        
                      
                    
                    
                      f
                      
                        g
                        
                          3
                        
                      
                    
                  
                
                ]
              
            
          
        
        
          
            O
          
        
      
    
    {\displaystyle {\mathcal {O}}(-d-e_{1}-e_{2}-e_{3}){\xrightarrow {\begin{bmatrix}g_{3}\\-g_{2}\\-g_{1}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1}-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{1}-e_{3})\\\oplus \\{\mathcal {O}}(-d-e_{2}-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}g_{2}&g_{3}&0\\-g_{1}&0&-g_{3}\\0&-g_{1}&g_{2}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1})\\\oplus \\{\mathcal {O}}(-d-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}fg_{1}&fg_{2}&fg_{3}\end{bmatrix}}}{\mathcal {O}}}

resolving 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 ,
 
 
 {\displaystyle {\mathcal {O}}_{X},}
 
 which we can use to compute 
 
 
 
 
 
 
 E
 x
 t
 
 
 
 i
 
 
 (
 
 
 
 O
 
 
 
 X
 
 
 ,
 
 
 F
 
 
 )
 
 
 {\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/D-module/l1y6evYe">D-module</a> (in place of O, one can also consider D, the sheaf of differential operators.)</li>
<li><a href="/facts/Fractional_ideal/I5rvJ6nt">fractional ideal</a></li>
<li><a href="/facts/Holomorphic_vector_bundle/FeRrK9Bg">holomorphic vector bundle</a></li>
<li><a href="/facts/Generic_freeness/sd0em6cC">generic freeness</a></li></ul>
<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> (1960). <a href="http://www.numdam.org/item/PMIHES_1960__4__5_0">"Éléments de géométrie algébrique: I. Le langage des schémas"</a>. <a href="/facts/Publications_Math%25C3%25A9matiques_de_l%2527IH%25C3%2589S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a>. 4. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02684778">10.1007/bf02684778</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217083">0217083</a>.</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>
<li>Costa, Laura; Miró-Roig, Rosa María; Pons-Llopis, Joan (2021). <a href="https://books.google.com/books?id=v9IuEAAAQBAJ&pg=PT22">Ulrich Bundles</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2F9783110647686">10.1515/9783110647686</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783110647686.</li>
<li>"Links with sheaf cohomology". Local Cohomology. Cambridge Studies in Advanced Mathematics. Cambridge University Press. 2012. pp. 438–479. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9781139044059.023">10.1017/CBO9781139044059.023</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780521513630.</li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1955), <a href="https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf">"Faisceaux algébriques cohérents (§.66 Faisceaux algébriques cohérents sur les variétés projectives.)"</a> (PDF), Annals of Mathematics, 61 (2): 197–278, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1969915">10.2307/1969915</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1969915">1969915</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0068874">0068874</a></li></ul>

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

<ol>
<li id="fn:1">Vakil, Math 216: Foundations of algebraic geometry, 2.5. <a href="http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf" target="_blank">http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hartshorne, Ch. III, Proposition 2.2. - Hartshorne, Robin. Algebraic Geometry. pp. 233–235. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">This cohomology functor coincides with the right derived functor of the global section functor in the category of abelian sheaves; cf. Hartshorne, Ch. III, Proposition 2.6. - Hartshorne, Robin. Algebraic Geometry. pp. 233–235. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">There is a canonical homomorphism:

H
          
        
        o
        
          m
          
            O
          
        
        (
        F
        ,
        O
        
          )
          
            x
          
        
        →
        
          Hom
          
            
              O
              
                x
              
            
          
        
        ⁡
        (
        
          F
          
            x
          
        
        ,
        
          O
          
            x
          
        
        )
        ,
      
    
    {\displaystyle {\mathcal {H}}om_{O}(F,O)_{x}\to \operatorname {Hom} _{O_{x}}(F_{x},O_{x}),}

which is an isomorphism if F is of finite presentation (EGA, Ch. 0, 5.2.6.) <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">For coherent sheaves, having a tensor inverse is the same as being locally free of rank one; in fact, there is the following fact: if 
 
 
 
 F
 ⊗
 G
 ≃
 O
 
 
 {\displaystyle F\otimes G\simeq O}
 
 and if F is coherent, then F, G are locally free of rank one. (cf. EGA, Ch 0, 5.4.3.) <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Hartshorne, Ch III, Lemma 2.4. - Hartshorne, Robin. Algebraic Geometry. pp. 233–235. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">see also: https://math.stackexchange.com/q/447234 <a href="https://math.stackexchange.com/q/447234" target="_blank">https://math.stackexchange.com/q/447234</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Hartshorne, Ch. II, Proposition 5.1. - Hartshorne, Robin. Algebraic Geometry. pp. 233–235. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">EGA I, Ch. I, Proposition 1.3.6. harvnb error: no target: CITEREFEGA_I (help) <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">EGA I, Ch. I, Corollaire 1.3.12. harvnb error: no target: CITEREFEGA_I (help) <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">EGA I, Ch. I, Corollaire 1.3.12. harvnb error: no target: CITEREFEGA_I (help) <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">EGA I, Ch. I, Corollaire 1.3.9. harvnb error: no target: CITEREFEGA_I (help) <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Hartshorne, Ch. II, Proposition 5.11. - Hartshorne, Robin. Algebraic Geometry. pp. 233–235. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">"Section 30.2 (01X8): Čech cohomology of quasi-coherent sheaves—The Stacks project". stacks.math.columbia.edu. Retrieved 2023-12-07. <a href="https://stacks.math.columbia.edu/tag/01X8" target="_blank">https://stacks.math.columbia.edu/tag/01X8</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Costa, Miró-Roig & Pons-Llopis 2021, Theorem 1.3.1 - Costa, Laura; Miró-Roig, Rosa María; Pons-Llopis, Joan (2021). Ulrich Bundles. doi:10.1515/9783110647686. ISBN 9783110647686. <a href="https://books.google.com/books?id=v9IuEAAAQBAJ&pg=PT22" target="_blank">https://books.google.com/books?id=v9IuEAAAQBAJ&pg=PT22</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">"Links with sheaf cohomology". Local Cohomology. Cambridge Studies in Advanced Mathematics. Cambridge University Press. 2012. pp. 438–479. doi:10.1017/CBO9781139044059.023. ISBN 9780521513630. <a href="9780521513630" target="_blank">9780521513630</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Serre 1955, §.66 Faisceaux algébriques cohérents sur les variétés projectives. - Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents (§.66 Faisceaux algébriques cohérents sur les variétés projectives.)" (PDF), Annals of Mathematics, 61 (2): 197–278, doi:10.2307/1969915, JSTOR 1969915, MR 0068874 <a href="https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf" target="_blank">https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Hartshorne, Ch. III, Proposition 6.9. - Hartshorne, Robin. Algebraic Geometry. pp. 233–235. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Hartshorne, Robin. Algebraic Geometry. pp. 233–235. <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
</ol>

Sheaf of modules open-in-new

Sheaf of modules