Picard group

<h2 id="examples">Examples</h2>
<ul><li>The Picard group of the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum</a> of a <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domain</a> is its <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal class group</a>.</li>
<li>The invertible sheaves on <a href="/facts/Projective_space/7MSpA9cM">projective space</a> Pn(k) for k a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>, are the <a href="/facts/Twisting_sheaf/Vc2CmshX">twisting</a> <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaves</a> 
 
 
 
 
 
 O
 
 
 (
 m
 )
 ,
 
 
 
 {\displaystyle {\mathcal {O}}(m),\,}
 
 so the Picard group of Pn(k) is isomorphic to Z.</li>
<li>The Picard group of the affine line with two origins over k is isomorphic to Z.</li>
<li>The Picard group of the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-dimensional <a href="/facts/Complex_affine_space/f9JkK6N3">complex affine space</a>: 
 
 
 
 Pic
 ⁡
 (
 
 
 C
 
 
 n
 
 
 )
 =
 0
 
 
 {\displaystyle \operatorname {Pic} (\mathbb {C} ^{n})=0}
 
, indeed the <a href="/facts/Exponential_sequence/7apRHy9o">exponential sequence</a> yields the following long exact sequence in cohomology

⋯
 →
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 Z
 
 _
 
 
 )
 →
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 O
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 )
 →
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 O
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 ⋆
 
 
 )
 →
 
 H
 
 2
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 Z
 
 _
 
 
 )
 →
 ⋯
 
 
 {\displaystyle \dots \to H^{1}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\to H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}})\to H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}}^{\star })\to H^{2}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\to \cdots }
 
</li></ul>
and since 
 
 
 
 
 H
 
 k
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 Z
 
 _
 
 
 )
 ≃
 
 H
 
 
 
 
 s
 i
 n
 g
 
 
 
 
 
 k
 
 
 (
 
 
 C
 
 
 n
 
 
 ;
 
 Z
 
 )
 
 
 {\displaystyle H^{k}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq H_{\scriptscriptstyle {\rm {sing}}}^{k}(\mathbb {C} ^{n};\mathbb {Z} )}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> we have 
 
 
 
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 Z
 
 _
 
 
 )
 ≃
 
 H
 
 2
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 Z
 
 _
 
 
 )
 ≃
 0
 
 
 {\displaystyle H^{1}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq H^{2}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq 0}
 
 because 
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {C} ^{n}}
 
 is contractible, then 
 
 
 
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 O
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 )
 ≃
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 O
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 ⋆
 
 
 )
 
 
 {\displaystyle H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}})\simeq H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}}^{\star })}
 
 and we can apply the <a href="/facts/Dolbeault_cohomology/BLg7w4Dh">Dolbeault isomorphism</a> to calculate 
 
 
 
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 
 
 O
 
 
 
 
 
 C
 
 
 n
 
 
 
 
 )
 ≃
 
 H
 
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 ,
 
 Ω
 
 
 
 C
 
 
 n
 
 
 
 
 0
 
 
 )
 ≃
 
 H
 
 
 
 ∂
 ¯
 
 
 
 
 0
 ,
 1
 
 
 (
 
 
 C
 
 
 n
 
 
 )
 =
 0
 
 
 {\displaystyle H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}})\simeq H^{1}(\mathbb {C} ^{n},\Omega _{\mathbb {C} ^{n}}^{0})\simeq H_{\bar {\partial }}^{0,1}(\mathbb {C} ^{n})=0}
 
 by the <a href="/facts/Dolbeault_cohomology/BLg7w4Dh">Dolbeault–Grothendieck lemma</a>.
<h2 id="picard-scheme">Picard scheme</h2>
The construction of a scheme structure on (<a href="/facts/Representable_functor/dILwN4oj">representable functor</a> version of) the Picard group, the Picard scheme, is an important step in algebraic geometry, in particular in the <a href="/facts/Duality_theory_of_abelian_varieties/xRSRm5KU">duality theory of abelian varieties</a>. It was constructed by Grothendieck (1962), and also described by Mumford (1966) and Kleiman (2005).
In the cases of most importance to classical algebraic geometry, for a <a href="/facts/Algebraic_curve/LnWsPmDP">non-singular</a> <a href="/facts/Complete_variety/4rhVhqAU">complete variety</a> V over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> zero, the <a href="/facts/Connected_space/5CUTxfoA">connected component</a> of the identity in the Picard scheme is an <a href="/facts/Abelian_variety/dGSDVruG">abelian variety</a> called the Picard variety and denoted Pic0(V). The dual of the Picard variety is the <a href="/facts/Albanese_variety/IC9tc5l0">Albanese variety</a>, and in the particular case where V is a curve, the Picard variety is naturally isomorphic to the <a href="/facts/Jacobian_variety/HFedCxIn">Jacobian variety</a> of V. For fields of positive characteristic however, <a href="/facts/Jun-Ichi_Igusa/5P56tJ0b">Igusa</a> constructed an example of a smooth projective surface S with Pic0(S) non-reduced, and hence not an <a href="/facts/Abelian_variety/dGSDVruG">abelian variety</a>.
The quotient Pic(V)/Pic0(V) is a <a href="/facts/Finitely-generated_abelian_group/dwtl7mp4">finitely-generated abelian group</a> denoted NS(V), the <a href="/facts/N%25C3%25A9ron%25E2%2580%2593Severi_group/MKKDDgAW">Néron–Severi group</a> of V. In other words, the Picard group fits into an <a href="/facts/Exact_sequence/leUJnwJb">exact sequence</a>

1
        →
        
          
            P
            i
            c
          
          
            0
          
        
        (
        V
        )
        →
        
          P
          i
          c
        
        (
        V
        )
        →
        
          N
          S
        
        (
        V
        )
        →
        1.
        
      
    
    {\displaystyle 1\to \mathrm {Pic} ^{0}(V)\to \mathrm {Pic} (V)\to \mathrm {NS} (V)\to 1.\,}

The fact that the rank of NS(V) is finite is <a href="/facts/Francesco_Severi/LRqMFxVM">Francesco Severi</a>'s theorem of the base; the rank is the Picard number of V, often denoted ρ(V). Geometrically NS(V) describes the <a href="/facts/Algebraic_equivalence/Bgj5eIU0">algebraic equivalence</a> classes of <a href="/facts/Divisor_(algebraic_geometry)/V0QDypeo">divisors</a> on V; that is, using a stronger, non-linear equivalence relation in place of <a href="/facts/Linear_equivalence_of_divisors/V0QDypeo">linear equivalence of divisors</a>, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to <a href="/facts/Numerical_equivalence/Bgj5eIU0">numerical equivalence</a>, an essentially topological classification by <a href="/facts/Intersection_number/ChzWmr4d">intersection numbers</a>.

<h2 id="relative-picard-scheme">Relative Picard scheme</h2>
Let f: X →S be a <a href="/facts/Morphism_of_schemes/55CKqHHo">morphism of schemes</a>. The relative Picard functor (or relative Picard scheme if it is a scheme) is given by:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> for any S-scheme T,

Pic
          
            X
            
              /
            
            S
          
        
        ⁡
        (
        T
        )
        =
        Pic
        ⁡
        (
        
          X
          
            T
          
        
        )
        
          /
        
        
          f
          
            T
          
          
            ∗
          
        
        (
        Pic
        ⁡
        (
        T
        )
        )
      
    
    {\displaystyle \operatorname {Pic} _{X/S}(T)=\operatorname {Pic} (X_{T})/f_{T}^{*}(\operatorname {Pic} (T))}

where 
 
 
 
 
 f
 
 T
 
 
 :
 
 X
 
 T
 
 
 →
 T
 
 
 {\displaystyle f_{T}:X_{T}\to T}
 
 is the base change of f and fT * is the pullback.
We say an L in 
 
 
 
 
 Pic
 
 X
 
 /
 
 S
 
 
 ⁡
 (
 T
 )
 
 
 {\displaystyle \operatorname {Pic} _{X/S}(T)}
 
 has degree r if for any geometric point s → T the pullback 
 
 
 
 
 s
 
 ∗
 
 
 L
 
 
 {\displaystyle s^{*}L}
 
 of L along s has degree r as an invertible sheaf over the fiber Xs (when the degree is defined for the Picard group of Xs.)

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Sheaf_cohomology/TrcpmD6k">Sheaf cohomology</a></li>
<li><a href="/facts/Chow_variety/BKcqvb0o">Chow variety</a></li>
<li><a href="/facts/Cartier_divisor/V0QDypeo">Cartier divisor</a></li>
<li><a href="/facts/Holomorphic_line_bundle/FeRrK9Bg">Holomorphic line bundle</a></li>
<li><a href="/facts/Ideal_class_group/LLn5Pwj1">Ideal class group</a></li>
<li><a href="/facts/Arakelov_class_group/MlmGW0Gm">Arakelov class group</a></li>
<li><a href="/facts/Group-stack/ekIgMyw1">Group-stack</a></li>
<li>Picard category</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, A.</a> (1962), <a href="http://www.numdam.org/item?id=SB_1961-1962__7__143_0">V. Les schémas de Picard. Théorèmes d'existence</a>, Séminaire Bourbaki, t. 14: année 1961/62, exposés 223-240, no. 7, Talk no. 232, pp. 143–161</li>
<li><a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, A.</a> (1962), <a href="http://www.numdam.org/item?id=SB_1961-1962__7__221_0">VI. Les schémas de Picard. Propriétés générales</a>, Séminaire Bourbaki, t. 14: année 1961/62, exposés 223-240, no. 7, Talk no. 236, pp. 221–243</li>
<li><a href="/facts/Robin_Hartshorne/mCBLyIW6">Hartshorne, Robin</a> (1977), <a href="/facts/Algebraic_Geometry_(book)/YcAhHWBj">Algebraic Geometry</a>, 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/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/13348052">13348052</a></li>
<li><a href="/facts/Jun-Ichi_Igusa/5P56tJ0b">Igusa, Jun-Ichi</a> (1955), "On some problems in abstract algebraic geometry", Proc. Natl. Acad. Sci. U.S.A., 41 (11): 964–967, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1955PNAS...41..964I">1955PNAS...41..964I</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1073%2Fpnas.41.11.964">10.1073/pnas.41.11.964</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC534315">534315</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/16589782">16589782</a></li>
<li><a href="/facts/Steven_Kleiman/SLsWKylB">Kleiman, Steven L.</a> (2005), "The Picard scheme", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, pp. 235–321, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0504020">math/0504020</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005math......4020K">2005math......4020K</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2223410">2223410</a></li>
<li><a href="/facts/David_Mumford/WJoJ8K2u">Mumford, David</a> (1966), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, vol. 59, <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-07993-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0209285">0209285</a>, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/171541070">171541070</a></li>
<li><a href="/facts/David_Mumford/WJoJ8K2u">Mumford, David</a> (1970), Abelian varieties, Oxford: <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-560528-0, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/138290">138290</a></li></ul>

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

<ol>
<li id="fn:1">Sheaf cohomology#Sheaf cohomology with constant coefficients <a href="/wiki/Sheaf_cohomology#Sheaf_cohomology_with_constant_coefficients" target="_blank">/wiki/Sheaf_cohomology#Sheaf_cohomology_with_constant_coefficients</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kleiman 2005, Definition 9.2.2. - Kleiman, Steven L. (2005), "The Picard scheme", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: American Mathematical Society, pp. 235–321, arXiv:math/0504020, Bibcode:2005math......4020K, MR 2223410 <a href="https://arxiv.org/abs/math/0504020" target="_blank">https://arxiv.org/abs/math/0504020</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Picard group open-in-new

Picard group