mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an invertible sheaf is a

coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...

''S'' on a

ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...

''X'', for which there is an inverse ''T'' with respect to

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

of ''O''_''X''-modules. It is the equivalent in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

of the topological notion of a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

. Due to their interactions with

Cartier divisor In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mu ...

s, they play a central role in the study of

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

Definition

An invertible sheaf is a

locally free sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...

''S'' on a

''X'', for which there is an inverse ''T'' with respect to

of ''O''_''X''-modules, that is, we have :

S \otimes T\

isomorphic to ''O''_''X'', which acts as

identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

for the tensor product. The most significant cases are those coming from

and

complex geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...

. For spaces such as (locally)

Noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thu ...

s or

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

s, one can actually replace 'locally free' by 'coherent' in the definition. The invertible sheaves in those theories are in effect the

s appropriately formulated. In fact, the abstract definition in

scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...

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 In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

. Examples come from

fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral doma ...

s in

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, so that the definition captures that theory. More generally, when ''X'' is an

affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...

''Spec(R)'', the invertible sheaves come from projective modules over ''R'', of rank 1.

The Picard group

Quite generally, the isomorphism classes of invertible sheaves on ''X'' themselves form an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...

under tensor product. This group generalises the

ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...

. In general it is written :

\mathrm(X)\

with ''Pic'' the

Picard functor In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ...

. Since it also includes the theory of the

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...

of an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

, 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

References

*Section 0.5.4 of {{EGA, book=I Geometry of divisors Sheaf theory

Definition

The Picard group

See also

References