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A group of classes of invertible sheaves (or line bundles). More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726601.png" /> be a ringed space. A [[Sheaf|sheaf]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726602.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726603.png" /> is called invertible if it is locally isomorphic to the structure sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726604.png" />. The set of classes of isomorphic invertible sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726605.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726606.png" />. The tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726607.png" /> defines an operation on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726608.png" />, making it an Abelian group called the Picard group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p0726609.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266010.png" /> is naturally isomorphic to the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266012.png" /> is the sheaf of invertible elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266013.png" />.
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A group of classes of invertible sheaves (or line bundles). More precisely, let $ (X,\mathcal{O}_{X}) $ be a ringed space. A [[Sheaf|sheaf]] $ \mathcal{L} $ of $ \mathcal{O}_{X} $-modules is called '''invertible''' if and only if it is locally isomorphic to the structure sheaf $ \mathcal{O}_{X} $. The set of classes of isomorphic invertible sheaves on $ X $ is denoted by $ \operatorname{Pic}(X) $. The tensor product $ \mathcal{L} \otimes_{\mathcal{O}_{X}} \mathcal{L}' $ defines an operation on the set $ \operatorname{Pic}(X) $, making it an Abelian group called the '''Picard group''' of $ X $. The group $ \operatorname{Pic}(X) $ is naturally isomorphic to the cohomology group $ {H^{1}}(X,\mathcal{O}_{X}^{*}) $, where $ \mathcal{O}_{X}^{*} $ is the sheaf of invertible elements in $ \mathcal{O}_{X} $.
  
For a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266014.png" />, the Picard group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266015.png" /> is the group of classes of invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266016.png" />-modules; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266017.png" />. For a Krull ring, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266018.png" /> is closely related to the [[Divisor class group|divisor class group]] for this ring.
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For a commutative ring $ A $, the Picard group $ \operatorname{Pic}(A) $ is the group of classes of invertible $ A $-modules; $ \operatorname{Pic}(A) \cong \operatorname{Pic}(\operatorname{Spec}(A)) $. For a Krull ring, the group $ \operatorname{Pic}(A) $ is closely related to the [[Divisor class group|divisor class group]] for this ring.
  
The Picard group of a complete normal [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266019.png" /> has a natural algebraic structure (see [[Picard scheme|Picard scheme]]). The reduced connected component of the zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266020.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266021.png" /> and is called the Picard variety for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266022.png" />; it is an [[Algebraic group|algebraic group]] (an Abelian variety if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266023.png" /> is a complete non-singular variety). The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266024.png" /> is called the Néron–Severi group and it has a finite number of generators; its rank is called the Picard number. In the complex case, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266025.png" /> is a smooth projective variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266026.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266027.png" /> is isomorphic to the quotient group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266028.png" /> of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266029.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266030.png" /> by the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072660/p07266031.png" />.
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The Picard group of a complete normal [[Algebraic variety|algebraic variety]] $ X $ has a natural algebraic structure (see [[Picard scheme|Picard scheme]]). The reduced connected component of the zero of $ \operatorname{Pic}(X) $ is denoted by $ {\operatorname{Pic}^{0}}(X) $ and is called the '''Picard variety''' for $ X $; it is an [[Algebraic group|algebraic group]] (an Abelian variety if $ X $ is a complete non-singular variety). The quotient group $ \operatorname{Pic}(X) / {\operatorname{Pic}^{0}}(X) $ is called the '''Néron–Severi group''', and it has a finite number of generators; its rank is called the '''Picard number'''. In the complex case, where $ X $ is a smooth projective variety over $ \mathbb{C} $, the group $ {\operatorname{Pic}^{0}}(X) $ is isomorphic to the quotient group of the space $ {H^{0}}(X,\Omega_{X}) $ of holomorphic $ 1 $-forms on $ X $ by the lattice $ {H^{1}}(X,\mathbb{Z}) $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR></table>
 
  
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<TR><TD valign="top">[1]</TD><TD valign="top">
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D. Mumford, “Lectures on curves on an algebraic surface”, Princeton Univ. Press (1966). {{MR|0209285}} {{ZBL|0187.42701}}</TD></TR>
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====References====
  
====Comments====
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<table>
 
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<TR><TD valign="top">[a1]</TD><TD valign="top">
 
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R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
====References====
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</table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 

Latest revision as of 07:58, 14 December 2016

A group of classes of invertible sheaves (or line bundles). More precisely, let $ (X,\mathcal{O}_{X}) $ be a ringed space. A sheaf $ \mathcal{L} $ of $ \mathcal{O}_{X} $-modules is called invertible if and only if it is locally isomorphic to the structure sheaf $ \mathcal{O}_{X} $. The set of classes of isomorphic invertible sheaves on $ X $ is denoted by $ \operatorname{Pic}(X) $. The tensor product $ \mathcal{L} \otimes_{\mathcal{O}_{X}} \mathcal{L}' $ defines an operation on the set $ \operatorname{Pic}(X) $, making it an Abelian group called the Picard group of $ X $. The group $ \operatorname{Pic}(X) $ is naturally isomorphic to the cohomology group $ {H^{1}}(X,\mathcal{O}_{X}^{*}) $, where $ \mathcal{O}_{X}^{*} $ is the sheaf of invertible elements in $ \mathcal{O}_{X} $.

For a commutative ring $ A $, the Picard group $ \operatorname{Pic}(A) $ is the group of classes of invertible $ A $-modules; $ \operatorname{Pic}(A) \cong \operatorname{Pic}(\operatorname{Spec}(A)) $. For a Krull ring, the group $ \operatorname{Pic}(A) $ is closely related to the divisor class group for this ring.

The Picard group of a complete normal algebraic variety $ X $ has a natural algebraic structure (see Picard scheme). The reduced connected component of the zero of $ \operatorname{Pic}(X) $ is denoted by $ {\operatorname{Pic}^{0}}(X) $ and is called the Picard variety for $ X $; it is an algebraic group (an Abelian variety if $ X $ is a complete non-singular variety). The quotient group $ \operatorname{Pic}(X) / {\operatorname{Pic}^{0}}(X) $ is called the Néron–Severi group, and it has a finite number of generators; its rank is called the Picard number. In the complex case, where $ X $ is a smooth projective variety over $ \mathbb{C} $, the group $ {\operatorname{Pic}^{0}}(X) $ is isomorphic to the quotient group of the space $ {H^{0}}(X,\Omega_{X}) $ of holomorphic $ 1 $-forms on $ X $ by the lattice $ {H^{1}}(X,\mathbb{Z}) $.

References

[1] D. Mumford, “Lectures on curves on an algebraic surface”, Princeton Univ. Press (1966). MR0209285 Zbl 0187.42701

References

[a1] R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. MR0463157 Zbl 0367.14001
How to Cite This Entry:
Picard group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Picard_group&oldid=21907
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article