Invertible sheaf
Encyclopedia
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 OX-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.
S on a ringed space
X, for which there is an inverse T with respect to tensor product
of OX-modules, that is, we have
isomorphic to OX, 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.
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an invertible sheaf is a coherent sheaf
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
S on a ringed space
Ringed space
In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
X, for which there is an inverse T with respect to tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of OX-modules. It is the equivalent in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
of the topological notion of a line bundle
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 organising these...
. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.
Definition
An invertible sheaf is a coherent sheafCoherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
S on a ringed space
Ringed space
In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
X, for which there is an inverse T with respect to tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of OX-modules, that is, we have
isomorphic to OX, which acts as identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
for the tensor product. The most significant cases are those coming from algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
theory. The invertible sheaves in those theories are in effect the line bundle
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 organising these...
s appropriately formulated.
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
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
. Examples come from fractional ideal
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 domain are like ideals where denominators are allowed...
s in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, so that the definition captures that theory. More generally, when X is an affine scheme Spec(R), the invertible sheaves come from projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
s over R, of rank 1.
The Picard group
Quite generally, the isomorphism classes of invertible sheaves on X themselves form an abelian groupAbelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
under tensor product. This group generalises the ideal class group
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
. In general it is written
with Pic the Picard functor. Since it also includes the theory of the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
of an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
, 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
- Vector bundles in algebraic geometry
- Line bundleLine bundleIn 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 organising these...
- First Chern class
- Picard group
- Birkhoff-Grothendieck theorem