Lemma 25.4.5. Let $f : X \to Y$ be a closed immersion of locally ringed spaces. Let $\mathcal{I}$ be the kernel of the map $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$. Let $i : Z \to Y$ be the closed subspace of $Y$ associated to $\mathcal{I}$. There is a unique isomorphism $f' : X \cong Z$ of locally ringed spaces such that $f = i \circ f'$.

**Proof.**
Omitted.
$\square$

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