Ideal sheaf


In algebraic geometry and other areas of mathematics, an ideal sheaf 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. 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

In the context of schemes, 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 is a scheme. 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.
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 differentials may be defined as the pull-back of the ideal sheaf defining the diagonal X → X × X to X.

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. This ideal sheaf also gives A the structure of a reduced closed complex subspace.