The Stacks project

Lemma 30.23.10. Let $f : X' \to X$ be a morphism of Noetherian schemes. Let $Z \subset X$ be a closed subscheme and denote $Z' = f^{-1}Z$ the scheme theoretic inverse image. Let $\mathcal{I} \subset \mathcal{O}_ X$, $\mathcal{I}' \subset \mathcal{O}_{X'}$ be the corresponding quasi-coherent sheaves of ideals. If $f$ is flat and the induced morphism $Z' \to Z$ is an isomorphism, then the pullback functor $f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', \mathcal{I}')$ (Lemma 30.23.9) is an equivalence.

Proof. If $X$ and $X'$ are affine, then this follows immediately from More on Algebra, Lemma 15.89.3. To prove it in general we let $Z_ n \subset X$, $Z'_ n \subset X'$ be the $n$th infinitesimal neighbourhoods of $Z$, $Z'$. The induced morphism $Z_ n \to Z'_ n$ is a homeomorphism on underlying topological spaces. On the other hand, if $z' \in Z'$ maps to $z \in Z$, then the ring map $\mathcal{O}_{X, z} \to \mathcal{O}_{X', z'}$ is flat and induces an isomorphism $\mathcal{O}_{X, z}/\mathcal{I}_ z \to \mathcal{O}_{X', z'}/\mathcal{I}'_{z'}$. Hence it induces an isomorphism $\mathcal{O}_{X, z}/\mathcal{I}_ z^ n \to \mathcal{O}_{X', z'}/(\mathcal{I}'_{z'})^ n$ for all $n \geq 1$ for example by More on Algebra, Lemma 15.89.2. Thus $Z'_ n \to Z_ n$ is an isomorphism of schemes. Thus $f^*$ induces an equivalence between the category of coherent $\mathcal{O}_ X$-modules annihilated by $\mathcal{I}^ n$ and the category of coherent $\mathcal{O}_{X'}$-modules annihilated by $(\mathcal{I}')^ n$, see Lemma 30.9.8. This clearly implies the lemma. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EHQ. Beware of the difference between the letter 'O' and the digit '0'.