In the book by Görtz-Wedhorn, the following definitions are given:
Let $Y$ be a scheme.
A closed subscheme of $Y$ is given by a closed subset $Z \subset Y$ and an ideal sheaf $\mathcal{I} \subset \mathcal{O}_Y$ such that $Z = \{x \in X \, : \, (\mathcal{O}_Y / \mathcal{I})_Z \neq 0\}$ and $(Z,(\mathcal{O}/\mathcal{I})|_{Z})$ is a scheme.
A morphism $f: X \to Y$ of schemes is called a closed immersion, if the underlying continuous map is a homeomorphism between $X$ and a closed subset of $Y$, and the sheaf homomorphism $f^{\flat}: \mathcal{O}_Y \to f_\ast \mathcal{O}_X$ is surjective.
Moreover, it's later mentioned casually that the condition $(Z,(\mathcal{O}/\mathcal{I})|_{Z})$ being a scheme in the first definition is non-trivial, and that every closed immersion induces a closed subscheme. I've been trying to show the latter, but a few questions have come up.
My approach:
First of all, I define $\mathcal{I} = \ker(f^{\flat}:\mathcal{O}_Y \to f_\ast \mathcal{O}_X)$, and I define $\mathcal{O}_Y / \mathcal{I}$ as the sheafification associated to the quotient presheaf. Following the definition, I define $Z = \text{supp}(\mathcal{O}_Y / \mathcal{I}) \subset Y$.
I claim that $(Z, \mathcal{O}_Z := (\mathcal{O}_Y / \mathcal{I})|_Z)$ is a scheme.
Then $(Z, \mathcal{O}_Z)$ would be a closed subscheme by definition, and moreover
- Being $f: X \to Y$ a closed embedding, $f(X) = \text{supp}(f_\ast \mathcal{O}_X) = Z$, since $(f_\ast \mathcal{O}_X)_{f(x)} \cong (\mathcal{O}_X)_x$ if $x \in X$, and $(f_\ast \mathcal{O}_X)_{y} = 0$ for $y \not \in f(X)$.
- I use the following fact, which I've proven: if $(\iota, \iota^{\flat}): (Z, \mathcal{O}_Z) \to (Y,\mathcal{O}_Y)$ is a closed immersion, then $(f,f^{\flat}): (X,\mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ factors through $(\iota, \iota^{\flat})$ if and only if $f^\flat$ factors through $\iota^{\flat}$. Then I can show that indeed I get a map of schemes $(g,g^{\flat}): (X, \mathcal{O}_X) \to (Z, \mathcal{O}_Z)$. Moreover, being $f^{\flat}$ surjective, the stalks of $\mathcal{O}_Z$ are isomorphic through $g^\flat$ to the ones of $g_\ast\mathcal{O}_X$, yielding that $(g, g^{\flat})$ is an isomorphism of schemes.
But why should $(Z,\text{supp}(\mathcal{O}_Y / \mathcal{I})|_Z)$ be a scheme (in particular if the scheme condition is non-trivial)? Also, is my approach correct? Thanks!