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".

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
sheaf cohomology.)

Algebraic Geometry

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 O_"X". Then, the support "Z" of O_"X"/"J" is a closed subspace of "X", and ("Z", O_"X"/"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
^#: O_"X" → "i"_⋆O_"Z"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". [EGA I, 4.2.2 b)]

A particular case of this correspondence is the unique reduced subscheme "X"_red 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 "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 [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 "X" → "X" × "X" 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 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.

References

* Éléments de géométrie algébrique
* H. Grauert, R. Remmert: "Coherent Analytic Sheaves". Springer-Verlag, Berlin 1984