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). $\qedsymbol$