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closed immersion

DEFINITION 8.2   Let $ X$ be a scheme. Let $ \mathcal I$ be a quasi coherent sheaf of ideal of $ \mathcal{O}_X$ . Then the scheme $ \operatorname{Spec}(\mathcal{O}_X/\mathcal I)$ (which is affine over $ X$ ) is called a closed subscheme of $ X$ . We often call it $ V(\mathcal I)$ .

DEFINITION 8.3   A morphism $ X\to Y$ of schemes is a closed immersion if there exists a sheaf of ideal $ \mathcal I$ of $ \mathcal{O}_Y$ such that $ f$ induces an isomorphism $ X\to V(\mathcal I)$ of schemes.

PROPOSITION 8.4   Affine morphisms and closed immersions are stable under base extension. That is, if we are given morphisms $ f:X\to S$ and $ g:T\to S$ of schemes and
  1. if $ f$ is an affine morphism, then $ f_T:X_T=X\times_S T \to T$ (``projection on the second variable'') is also an affine morphism.
  2. if $ f$ is a closed immersion, then $ f_T:X_T \to T$ is also a closed immersion.

PROOF.. (1): We may assume $ X=\operatorname{Spec}(\mathcal A)$ . Then we may verify immediately that $ X_T =\operatorname{Spec}(g^* \mathcal A)$ . This argument also proves (2). $ \qedsymbol$


next up previous
Next: differential calculus of schemes Up: Various kind of morphisms. Previous: affine morphisms
2007-12-11