Ideal sheaf
Encyclopedia
In algebraic geometry
and other areas of mathematics
, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal
in a ring
. The ideal sheaves on a geometric object are closely connected to its subspaces.
and A a sheaf
of rings on X. (In other words, (X, A) is a ringed space
.) An ideal sheaf J in A is a subobject
of A in the category
of sheaves of A-modules, i.e., a subsheaf
of A viewed as a sheaf of abelian groups such that
for all open subsets U of X.
, the importance of ideal sheaves lies mainly in the correspondence between closed subschemes and quasi-coherent ideal sheaves. Consider a scheme X and a quasi-coherent ideal sheaf J in OX. Then, the support Z of OX/J is a closed subspace of X, and (Z, OX/J) is a scheme (both assertions can be checked locally). It is called the closed subscheme of X defined by J. Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map
is surjective on the stalks. Then, the kernel J of i# is a quasi-coherent ideal sheaf, and i induces an isomorphism from Z onto the closed subscheme defined by J.
A particular case of this correspondence is the unique reduced subscheme Xred of X having the same underlying space, which is defined by the nilradical of OX (defined stalk-wise, or on open affine charts).
For a morphism f: X → Y and a closed subscheme Y′ ⊆ Y defined by an ideal sheaf J, the preimage Y′ ×Y X is defined by the ideal sheaf
The pull-back of an ideal sheaf J to the subscheme Z defined by J contains important information, it is called the conormal bundle of Z. For example, the sheaf of Kähler differential
s may be defined as the pull-back of the ideal sheaf defining the diagonal X → X × X to X. (Assume for simplicity that X is separated so that the diagonal is a closed immersion.)
s, the Oka-Cartan theorem states that a closed subset A of a complex space is analytic if and only if the ideal sheaf of functions vanishing on A is coherent
. This ideal sheaf also gives A the structure of a reduced closed complex subspace.
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 other areas of mathematics
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 ideal sheaf (or sheaf of ideals) is the global analogue of an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. The ideal sheaves on a geometric object are closely connected to its subspaces.
Definition
Let X be a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
and A a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of rings on X. (In other words, (X, A) is 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...
.) An ideal sheaf J in A is a subobject
Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory...
of A in the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of sheaves of A-modules, i.e., a subsheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of A viewed as a sheaf of abelian groups such that
- Γ(U, A) · Γ(U, J) ⊆ Γ(U, J)
for all open subsets U of X.
General Properties
- If f: A → B is a homomorphism between two sheaves of rings on the same space X, the kernel of f is an ideal sheaf in A.
- Conversely, for any ideal sheaf J in a sheaf of rings A, there is a natural structure of a sheaf of rings on the quotient sheaf A/J. Note that the canonical map
-
- Γ(U, A)/Γ(U, J) → Γ(U, A/J)
- for open subsets U is injective, but not surjective in general. (See sheaf cohomologySheaf cohomologyIn mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
.)
Algebraic Geometry
In the context of schemesScheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, the importance of ideal sheaves lies mainly in the correspondence between closed subschemes and quasi-coherent ideal sheaves. Consider a scheme X and a quasi-coherent ideal sheaf J in OX. Then, the support Z of OX/J is a closed subspace of X, and (Z, OX/J) is a scheme (both assertions can be checked locally). It is called the closed subscheme of X defined by J. Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map
- i#: OX → i⋆OZ
is surjective on the stalks. Then, the kernel J of i# is a quasi-coherent ideal sheaf, and i induces an isomorphism from Z onto the closed subscheme defined by J.
A particular case of this correspondence is the unique reduced subscheme Xred of X having the same underlying space, which is defined by the nilradical of OX (defined stalk-wise, or on open affine charts).
For a morphism f: X → Y and a closed subscheme Y′ ⊆ Y defined by an ideal sheaf J, the preimage Y′ ×Y X is defined by the ideal sheaf
- f⋆(J)OX = im(f⋆J → OX).
The pull-back of an ideal sheaf J to the subscheme Z defined by J contains important information, it is called the conormal bundle of Z. For example, the sheaf of Kähler differential
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
s may be defined as the pull-back of the ideal sheaf defining the diagonal X → X × X to X. (Assume for simplicity that X is separated so that the diagonal is a closed immersion.)
Analytic Geometry
In the theory of complex spaceComplex space
In mathematics, n-dimensional complex space is a multi-dimensional generalisation of the complex numbers, which have both real and imaginary parts or dimensions...
s, the Oka-Cartan theorem states that a closed subset A of a complex space is analytic if and only if the ideal sheaf of functions vanishing on A is coherent
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...
. This ideal sheaf also gives A the structure of a reduced closed complex subspace.