The Stacks project

Lemma 55.8.1. Let $V_1 \to V_2$ be a closed immersion of algebraic schemes over $K$. If $X_2$ is a model for $V_2$, then the scheme theoretic image of $V_1 \to X_2$ is a model for $V_1$.

Proof. Using Morphisms, Lemma 29.6.3 and Example 29.6.4 this boils down to the following algebra statement. Let $A_1$ be a finite type $R$-algebra flat over $R$. Let $A_1 \otimes _ R K \to B_2$ be a surjection. Then $A_2 = A_1 / \mathop{\mathrm{Ker}}(A_1 \to B_2)$ is a finite type $R$-algebra flat over $R$ such that $B_2 = A_2 \otimes _ R K$. We omit the detailed proof; use More on Algebra, Lemma 15.22.11 to prove that $A_2$ is flat. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 55.8: Models

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Your email address will not be published.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 0C2S. Beware of the difference between the letter 'O' and the digit '0'.