# 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 sheaf*A*/*J*. Note that the canonical map

- Γ(
*U*,*A*)/Γ(*U*,*J*) → Γ(*U*,*A*/*J*)

- Γ(
- 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 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 map

*i*^{#}: 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*.[1]

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).[2]

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*)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.)[4]

## 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.

## References

- EGA I, 4.2.2 b)
- EGA I, 5.1
- EGA I, 4.4.5
- EGA IV, 16.1.2 and 16.3.1

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