# Invertible sheaf

A locally free sheaf of $ {\mathcal O} _ {X} $-
modules of rank 1 on a ringed space $ ( X , {\mathcal O} _ {X} ) $.
An equivalent definition is: A sheaf of $ {\mathcal O} _ {X} $-
modules that is locally isomorphic to the sheaf $ {\mathcal O} _ {X} $.
The invertible sheaves on $ X $,
considered up to isomorphism, form an Abelian group with respect to the operation of tensor multiplication over $ {\mathcal O} _ {X} $.
This group is called the Picard group of the space $ X $,
and is denoted by $ \mathop{\rm Pic} X $.
The inverse of a sheaf $ {\mathcal L} $
in this group is the sheaf $ {\mathcal L} ^ {-} 1 = fs {Hom } ( {\mathcal L} , {\mathcal O} _ {X} ) $
dual to $ {\mathcal L} $.
In the case when $ ( X , {\mathcal O} _ {X} ) $
is a scheme (in particular, an algebraic variety) or an analytic space, a sheaf of $ {\mathcal O} _ {X} $-
modules is invertible if and only if it is isomorphic to the sheaf of regular (respectively, analytic) sections of some algebraic (respectively, analytic) line bundle over $ X $.

Invertible sheaves on schemes are closely connected with divisors (cf. Divisor). With each Cartier divisor $ D $ on $ X $ is associated an invertible sheaf $ {\mathcal O} _ {X} ( D) $, thus defining an injective homomorphism $ \mathop{\rm Cl} X \rightarrow \mathop{\rm Pic} X $, where $ \mathop{\rm Cl} X $ is the group of classes of Cartier divisors on $ X $. For integral schemes $ X $ this homomorphism is an isomorphism.

On a projective scheme $ X $ Serre's twisted invertible sheaf $ {\mathcal O} _ {X} ( 1) = {\mathcal O} ( 1) $ can be defined. In fact, if an imbedding of the scheme $ X $ in a projective space $ P ^ {N} $ is given, then $ {\mathcal O} _ {X} $ corresponds to the class of a hyperplane section. In particular, if $ X = P ^ {N} ( k) $ is a projective space over a field $ k $, then the sheaf $ {\mathcal O} ( 1) $ is the direct image of the sheaf of linear functions on $ k ^ {N+} 1 $ under the natural mapping $ k ^ {N+} 1 \setminus \{ 0 \} \rightarrow P ^ {N} ( k) $. The system of homogeneous coordinates $ x _ {0} \dots x _ {n} $ in $ P ^ {N} ( k) $ can be identified with a basis for the space of sections $ \Gamma ( P ^ {N} , {\mathcal O} ( 1) ) $.

An invertible sheaf on a scheme $ X $ is related to rational mappings of $ X $ into projective spaces. Let $ {\mathcal L} $ be an invertible sheaf on a scheme and let $ s _ {0} \dots s _ {N} $ be sections of $ {\mathcal L} $ the values of which at any point $ x \in X $ generate the stalk $ {\mathcal L} _ {x} $ over $ {\mathcal O} _ {x} $. Then there exists a unique morphism $ \phi : X \rightarrow P ^ {N} ( k) $ such that $ \phi ^ {*} {\mathcal O} ( 1) \cong {\mathcal L} $ and $ \phi ^ {*} x _ {i} = s _ {i} $, where $ x _ {0} \dots x _ {N} $ are homogeneous coordinates in $ P ^ {N} ( k) $. An invertible sheaf on $ X $ is called very ample if there exists an imbedding $ \phi : X \rightarrow P ^ {N} ( k) $ such that $ \phi ^ {*} {\mathcal O} ( 1) \cong {\mathcal L} $. An invertible sheaf $ {\mathcal L} $ on $ X $ is called ample if there exists a positive integer $ n $ for which $ {\mathcal L} ^ {n} $ is very ample. On a Noetherian scheme $ X $ over $ k $ an invertible sheaf $ {\mathcal L} $ is ample if and only if for each coherent sheaf $ {\mathcal F} $ on $ X $ there exists an integer $ n _ {0} > 0 $ such that the sheaf $ {\mathcal F} \otimes {\mathcal L} ^ {n} $ is generated by its global sections for $ n \geq n _ {0} $.

If $ {\mathcal L} $ is an ample invertible sheaf on $ X $ corresponding to a divisor $ D $, then $ D $ is called an ample divisor. A Cartier divisor $ D $ on a scheme $ X $ that is proper and smooth over an algebraically closed field $ k $ is ample if and only if for each closed integral subscheme $ Y \subseteq X $ the intersection index $ D ^ {r} \cdot Y $ is positive, where $ r = \mathop{\rm dim} Y $( cf. Intersection index (in algebraic geometry)). For other criteria of ampleness see [5]. There is also a generalization of the concept of an ample divisor on subvarieties of large codimension [2].

The concepts of very ample and ample invertible sheaves can be carried over to the case of analytic spaces (for criteria for ampleness in this situation see Positive vector bundle).

#### References

[1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |

[2] | R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) MR0282977 Zbl 0208.48901 |

[3] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701 |

[4] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |

[5] | I.V. Dolgachev, "Abstract algebraic geometry" J. Soviet Math. , 2 : 3 (1974) pp. 264–303 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 10 (1972) pp. 47–112 Zbl 1068.14059 |

#### Comments

The definition of Serre's twisted invertible sheaf is not precise enough. There is an action of the multiplicative group $ k ^ \times $ on $ k ^ {N+} 1 \setminus \{ 0 \} $ which has $ P ^ {N} ( k) $ as its quotient. The direct image of the structure sheaf under the mapping $ k ^ {N+} 1 \setminus \{ 0 \} \rightarrow P ^ {N} ( k) $ splits into a direct sum of invertible sheaves $ {\mathcal O} ( n) $, $ n \in \mathbf Z $, such that $ k ^ \times $ acts on $ {\mathcal O} ( n) $ via the character $ t \rightarrow t ^ {n} $.

**How to Cite This Entry:**

Invertible sheaf.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Invertible_sheaf&oldid=47427