Invertible sheaf: Difference between revisions
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{{Short description|Type of sheaf}} |
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In [[mathematics]], an '''invertible sheaf''' is a [[coherent sheaf]] ''S'' on a [[ringed space]] ''X'', for which there is an inverse ''T'' with respect to [[tensor product]] of ''O''<sub>''X''</sub>-modules. That is, we have |
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In [[mathematics]], an '''invertible sheaf''' is a [[sheaf (mathematics)|sheaf]] on a [[ringed space]] which has an inverse with respect to [[tensor product]] of sheaves of [[module (mathematics)|module]]s. It is the equivalent in [[algebraic geometry]] of the topological notion of a [[line bundle]]. Due to their interactions with [[Cartier divisor]]s, they play a central role in the study of [[algebraic varieties]]. |
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==Definition== |
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:''S'' ⊗ ''T'' |
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Let (''X'', ''O''<sub>''X''</sub>) be a ringed space. Isomorphism classes of sheaves of ''O''<sub>''X''</sub>-modules form a monoid under the operation of tensor product of ''O''<sub>''X''</sub>-modules. The [[identity element]] for this operation is ''O''<sub>''X''</sub> itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if ''L'' is a sheaf of ''O''<sub>''X''</sub>-modules, then ''L'' is called '''invertible''' if it satisfies any of the following equivalent conditions:<ref>EGA 0<sub>I</sub>, 5.4.</ref><ref>Stacks Project, tag 01CR, [https://stacks.math.columbia.edu/tag/01CR].</ref> |
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* There exists a sheaf ''M'' such that <math>L \otimes_{\mathcal{O}_X} M \cong \mathcal{O}_X</math>. |
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* The natural homomorphism <math>L \otimes_{\mathcal{O}_X} L^\vee \to \mathcal{O}_X</math> is an isomorphism, where <math>L^\vee</math> denotes the dual sheaf <math>\underline{\operatorname{Hom}}(L, \mathcal{O}_X)</math>. |
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* The functor from ''O''<sub>''X''</sub>-modules to ''O''<sub>''X''</sub>-modules defined by <math>F \mapsto F \otimes_{\mathcal{O}_X} L</math> is an equivalence of categories. |
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Every locally free sheaf of rank one is invertible. If ''X'' is a locally ringed space, then ''L'' is invertible if and only if it is locally free of rank one. Because of this fact, invertible sheaves are closely related to [[line bundle]]s, to the point where the two are sometimes conflated. |
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isomorphic to ''O''<sub>''X''</sub>, which acts as [[identity element]] for the tensor product. The most significant cases are those coming from [[algebraic geometry]] and [[complex manifold]] theory. The invertible sheaves in those theories are in effect the [[line bundle]]s appropriately formulated. |
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==Examples== |
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In fact, the abstract definition in [[scheme theory]] of invertible sheaf can be replaced by the condition of being ''locally free, of rank 1''. That is, the condition of a tensor inverse then implies, locally on ''X'', that ''S'' is the sheaf form of a free rank 1 module over a [[commutative ring]]. Examples come from [[fractional ideal]]s in [[algebraic number theory]], so that the definition captures that theory. More generally, when ''X'' is an [[affine scheme]] ''Spec(R)'', the invertible sheaves come from [[projective module]]s over ''R'', of rank 1. |
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Let ''X'' be an affine scheme {{math|Spec ''R''}}. Then an invertible sheaf on ''X'' is the sheaf associated to a rank one [[projective module]] over ''R''. For example, this includes [[fractional ideal]]s of [[algebraic number fields]], since these are rank one projective modules over the rings of integers of the number field. |
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==The Picard group== |
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{{main|Picard group}} |
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Quite generally, the isomorphism classes of invertible sheaves on ''X'' themselves form an [[abelian group]] under tensor product. This group generalises the [[ideal class group]]. In general it is written |
Quite generally, the isomorphism classes of invertible sheaves on ''X'' themselves form an [[abelian group]] under tensor product. This group generalises the [[ideal class group]]. In general it is written |
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The direct construction of invertible sheaves by means of data on ''X'' leads to the concept of [[Cartier divisor]]. |
The direct construction of invertible sheaves by means of data on ''X'' leads to the concept of [[Cartier divisor]]. |
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==See also== |
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{{algebra-stub}} |
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* [[Vector bundles in algebraic geometry]] |
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{{geometry-stub}} |
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* [[Line bundle]] |
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* [[First Chern class]] |
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* [[Picard group]] |
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* [[Birkhoff-Grothendieck theorem]] |
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==References== |
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{{Reflist}} |
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*{{EGA|book=I}} |
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[[Category:Geometry of divisors]] |
[[Category:Geometry of divisors]] |
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[[Category:Sheaf theory]] |
[[Category:Sheaf theory]] |
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[[he:אלומה הפיכה]] |
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[[pt:Feixe invertível]] |
Latest revision as of 19:02, 11 July 2023
In mathematics, an invertible sheaf is a sheaf on a ringed space which has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.
Definition
[edit]Let (X, OX) be a ringed space. Isomorphism classes of sheaves of OX-modules form a monoid under the operation of tensor product of OX-modules. The identity element for this operation is OX itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if L is a sheaf of OX-modules, then L is called invertible if it satisfies any of the following equivalent conditions:[1][2]
- There exists a sheaf M such that .
- The natural homomorphism is an isomorphism, where denotes the dual sheaf .
- The functor from OX-modules to OX-modules defined by is an equivalence of categories.
Every locally free sheaf of rank one is invertible. If X is a locally ringed space, then L is invertible if and only if it is locally free of rank one. Because of this fact, invertible sheaves are closely related to line bundles, to the point where the two are sometimes conflated.
Examples
[edit]Let X be an affine scheme Spec R. Then an invertible sheaf on X is the sheaf associated to a rank one projective module over R. For example, this includes fractional ideals of algebraic number fields, since these are rank one projective modules over the rings of integers of the number field.
The Picard group
[edit]Quite generally, the isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product. This group generalises the ideal class group. In general it is written
with Pic the Picard functor. Since it also includes the theory of the Jacobian variety of an algebraic curve, the study of this functor is a major issue in algebraic geometry.
The direct construction of invertible sheaves by means of data on X leads to the concept of Cartier divisor.
See also
[edit]- Vector bundles in algebraic geometry
- Line bundle
- First Chern class
- Picard group
- Birkhoff-Grothendieck theorem