In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an invertible sheaf is a
sheaf on a
ringed space that has an inverse with respect to
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of
sheaves of modules. It is the equivalent in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
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 organis ...
. Due to their interactions with
Cartier divisors, they play a central role in the study of
algebraic varieties.
Definition
Let (''X'', ''O''
''X'') be a ringed space. Isomorphism classes of sheaves of ''O''
''X''-modules form a
monoid under the operation of tensor product of ''O''
''X''-modules. The
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
for this operation is ''O''
''X'' itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if ''L'' is a sheaf of ''O''
''X''-modules, then ''L'' is called invertible if it satisfies any of the following equivalent conditions:
[ Stacks Project, tag 01CR]
* There exists a sheaf ''M'' such that
.
* The natural homomorphism
is an isomorphism, where
denotes the dual sheaf
.
* The functor from ''O''
''X''-modules to ''O''
''X''-modules defined by
is an
equivalence of categories
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...
.
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
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 organis ...
s, to the point where the two are sometimes conflated.
Examples
Let ''X'' be an affine scheme . Then an invertible sheaf on ''X'' is the sheaf associated to a rank one
projective module over ''R''. For example, this includes
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 do ...
s of
algebraic number fields, since these are rank one projective modules over the rings of integers of the number field.
The Picard group
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 mathematics, the ideal class group (or class group) of an algebraic number field K is the quotient group J_K/P_K where J_K is the group of fractional ideals of the ring of integers of K, and P_K is its subgroup of principal ideals. The ...
. In general it is written
:
with ''Pic'' the
Picard functor. Since it also includes the theory of the
Jacobian variety 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 cu ...
, 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
*
First Chern class
*
Birkhoff-Grothendieck theorem
References
*{{EGA, book=I
Geometry of divisors
Sheaf theory