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.
Definition
Let
X be a topological space and
A a sheaf of rings on
X. (In other words, 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. In other words,
J is a sheaf of
A-submodules of
A.
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 cohomology.)
Algebraic geometry
In the context of schemes, the importance of ideal sheaves lies mainly in the correspondence between closed
and
quasi-coherent ideal sheaves. Consider a scheme
X and a quasi-coherent ideal sheaf
J in O
X. Then, the support
Z of O
X/
J is a closed subspace of
X, and is a scheme (both assertions can be checked locally). It is called the closed subscheme of
X defined by
J. Conversely, let 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 is a quasi-coherent ideal sheaf, and
i induces an isomorphism from
Z onto the closed subscheme defined by
J.
[EGA I, 4.2.2 b)]
A particular case of this correspondence is the unique reduced scheme subscheme Xred of X having the same underlying space, which is defined by the nilradical of O X (defined stalk-wise, or on open affine charts).[EGA I, 5.1]
For a morphism and a closed subscheme defined by an ideal sheaf J, the preimage is defined by the ideal sheaf[EGA I, 4.4.5]
- f*( J)O X = im( f* J → O X).
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 differentials may be defined as the pull-back of the ideal sheaf defining the diagonal to X. (Assume for simplicity that X is separated so that the diagonal is a closed immersion.)[EGA IV, 16.1.2 and 16.3.1]
Analytic geometry
In the theory of complex-analytic spaces, 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 sheaf. This ideal sheaf also gives
A the structure of a reduced closed complex subspace.
