Proving that sheaf of ideals of a closed subscheme is quasi-coherent

Question

Let $X$ be a scheme and $Y$ be a closed subscheme, where we have $(\iota, \pi): (Y, O_Y) \to (X, O_X)$. I want to prove that the sheaf of ideals of $Y$, which is the kernel of $O_X \to \iota_* O_Y$ is quasi-coherent. I know that the kernel of quasi-coherent sheaves is quasi-coherent and the structure sheaf $O_X$ is quasi-coherent. It remains to show that $\iota_* O_Y$ is quasi-coherent. Any explanation about this last point would be appreciated. Thank you!

Answer 1

As said in the comment by @Mindlack, we can assume $X$ is affine, say $X=\operatorname{Spec}(A)$, where $A$ is a ring. Then, $Y$ being a closed subspace of $X$, corresponds to $V(I)$, where $I\triangleleft A$ is an ideal. And $\iota$ corresponds to the ring map $\pi:A\rightarrow A/I$.

In this case, $\iota_*(\mathcal O_Y)$ is just $\widetilde{(A/I)}$ viewed as an $\mathcal O_X$-module, hence quasi-coherent.

Proof:

We shall show that for every $f\in A$, we have $\iota_*(\mathcal O_Y)(D(f))=(A/I)\otimes_AA_f$. This will show that $\iota_*(\mathcal O_Y)$ is just $\widetilde{(A/I)}$ viewed as an $\mathcal O_X$-module. Here $D(f)$ is the open subset of $\operatorname{Spec}(A)$ consisting of primes of $A$ which do not contain $f$.

Note that $\iota^{-1}(D(f))=\left\{\mathfrak p\triangleleft A/I\mid f\not\in\pi^{-1}(\mathfrak p)\right\}=\left\{\mathfrak p\triangleleft A/I\mid \pi(f)\not\in\mathfrak p\right\}=D(\pi(f))$.

So by definition $\iota_*(\mathcal O_Y)(D(f))=\mathcal O_Y(\iota^{-1}(D(f)))=\mathcal O_Y(D(\pi(f)))=(A/I)_{\pi(f)}$.

Then define a module morphism $A/I\otimes_AA_f\rightarrow(A/I)_{\pi(f)}$ by sending $x\otimes_Aa/f^n$ to $ax/\pi(f)^n$. This is a module isomorphism, so indeed $\iota_*(\mathcal O_Y)(D(f))=(A/I)\otimes_AA_f$.

$\square$

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Note:

Notice that if two sheaves $F,G$ satisfy $F(D(f)=G(D(f)),\,\forall f\in A$, then $F=G$.

To see this, note that by the definition of the zariski topology on $X=\operatorname{Spec}(A)$, each open $U$ can be written as $\displaystyle\bigcup_{i\in I}D(f_i)$, where $I$ is some index set. By the sheaf property, we have an exact sequence

$$ 0\rightarrow F(U)\rightarrow\bigoplus_{i\in I}F(D(f_i))\rightarrow\bigoplus_{i,j\in I}F(D(f_i)\cap D(f_j))= \bigoplus_{i,j\in I}F(D(f_if_j)). $$ Similarly we have an exact sequence $$ 0\rightarrow G(U)\rightarrow\bigoplus_{i\in I}G(D(f_i))\rightarrow \bigoplus_{i,j\in I}G(D(f_if_j)). $$

In the above two sequences, the last two terms are equal by our assumption, so $F(U)=G(U)$. Thus $F(U)=G(U)$, for every open subset $U$ of $X$. Therefore $F=G$.

As a consequence, we see that $\iota_*(\mathcal O_Y)(D(f))=\widetilde{(A/I)}(D(f)),\,\forall f\in A$, and hence $\iota_*(\mathcal O_Y)=\widetilde{(A/I)}$.

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In a similar vein we can show that the direct image of a quasi-coherent sheaf for a closed immersion is still quasi-coherent.

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Hope this helps.