Ideal Sheaf - Algebraic Geometry

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.

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).

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

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.)