The Stacks project

Lemma 61.6.5. Let $A \to B$ be ring map such that

  1. $A \to B$ identifies local rings,

  2. the topological spaces $\mathop{\mathrm{Spec}}(B)$, $\mathop{\mathrm{Spec}}(A)$ are w-local, and

  3. $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is w-local.

Then $A \to B$ is ind-Zariski.

Proof. Set $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(B)$. Let $X_0 \subset X$ and $Y_0 \subset Y$ be the set of closed points. Let $A \to A'$ be the ind-Zariski morphism of affine schemes such that with $X' = \mathop{\mathrm{Spec}}(A')$ the diagram

\[ \xymatrix{ X' \ar[r] \ar[d] & \pi _0(X') \ar[d] \\ X \ar[r] & \pi _0(X) } \]

is cartesian in the category of topological spaces and such that $\pi _0(X') = \pi _0(Y)$ as spaces over $\pi _0(X)$, see Lemma 61.6.2. By Lemma 61.2.5 we see that $X'$ is w-local and the set of closed points $X'_0 \subset X'$ is the inverse image of $X_0$.

We obtain a continuous map $Y \to X'$ of underlying topological spaces over $X$ identifying $\pi _0(Y)$ with $\pi _0(X')$. By Lemma 61.3.8 (and Lemma 61.4.6) this corresponds to a morphism of affine schemes $Y \to X'$ over $X$. Since $Y \to X$ maps $Y_0$ into $X_0$ we see that $Y \to X'$ maps $Y_0$ into $X'_0$, i.e., $Y \to X'$ is w-local. By Lemma 61.6.4 we see that $Y \cong X'$ and we win. $\square$

Comments (2)

Comment #5930 by Mingchen on

There is a typo in the proof: "this is corresponds", remove "is".

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Your email address will not be published.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 097F. Beware of the difference between the letter 'O' and the digit '0'.