Quasi-coherent ideals and subschemes Let $X$ be a scheme. Let $\mathcal{I}$ be a quasi-coherent ideal of $\mathcal{O}_X$.
Let $Y = Supp(\mathcal{O}_X/\mathcal{I})$.
Let $f\colon Y \rightarrow X$ be the canonical injection.
Then how do we prove that $(Y, f^{-1}(\mathcal{O}_X/\mathcal{I}))$ is a scheme and $f$ is a closed immersion?
 A: Let Spec $A$ be affine open in $X$.  The restriction of $\mathcal I$ to Spec $A$
is the sheafification of some ideal $I$ of $A$ (by quasi-coherence).  The sheaf
$\mathcal O_X/\mathcal I$ restricts to the sheaf associated to $A/I$, and so
its support, intersected with Spec $A$, is precisely the image of Spec $A/I$ in Spec $A$.  Furthermore, the restriction of the sheaf attached to $A/I$ to Spec $A/I$ is precisely the structure sheaf of Spec $A/I$.
(This is a special case of a more general fact: if $M$ is an $A$-module which
is annihilated by the ideal $I$ of $A$, then we can regard $M$ as an $A/I$-module too, and so we can get associated sheaves on both Spec $A$ and on Spec $A/I$.
These are canonically identified via $f^{-1}$ and $f_*$.)
Since Spec $A/I$, with its structure sheaf, is an open subset of $(Y,f^{-1}(\mathcal O_X/\mathcal I))$, and $Y$ is covered by such open sets, we see
that $(Y,(\mathcal O_X/\mathcal I))$ admits an open cover by affine schemes, and so is a scheme.
A: The following is a bit long for a comment.
Matt E wrote: "Furthermore, the restriction of the sheaf attached to $A/I$ to Spec $A/I$ is precisely the structure sheaf of Spec $A/I$."
Let me explain this.
We can assume $X =$ Spec $A$.
The canonical morphism $\mathcal{O}_X/\mathcal{I} \rightarrow f_*f^{-1}(\mathcal{O}_X/\mathcal{I})$ is an isomorphism by this result.
Hence $\Gamma(D(f), \mathcal{O}_X/\mathcal{I})$ is canonically isomorphic to $\Gamma(D(f) \cap Y, f^{-1}(\mathcal{O}_X/\mathcal{I}))$ for $f \in A$.
$\Gamma(D(f), \mathcal{O}_X/\mathcal{I}) = (A/I)_f = (A/I)_{\bar f}$, where $\bar f$ is the image of$f$ in $A/I$. $D(f) \cap Y = D(\bar f)$.
Hence $f^{-1}(\mathcal{O}_X/\mathcal{I})$ is the structure sheaf of Spec $A/I$.
