Ideal sheaf
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, (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
- Γ(U, A) · Γ(U, J) ⊆ Γ(U, J)
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 sheafA/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
- 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.[1]
A particular case of this correspondence is the unique
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[3]
- 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
Analytic geometry
In the theory of
References
- Éléments de géométrie algébrique
- H. Grauert, R. Remmert: Coherent Analytic Sheaves. Springer-Verlag, Berlin 1984