The Stacks project

Lemma 15.5.2. Let $R$ be a Noetherian ring. Let $I$ be a finite set. Suppose given a cartesian diagram

\[ \xymatrix{ \prod B_ i & \prod A_ i \ar[l]^{\prod \varphi _ i} \\ Q \ar[u]^{\prod \psi _ i} & P \ar[u] \ar[l] } \]

with $\psi _ i$ and $\varphi _ i$ surjective, and $Q$, $A_ i$, $B_ i$ of finite type over $R$. Then $P$ is of finite type over $R$.

Proof. Follows from Lemma 15.5.1 and induction on the size of $I$. Namely, let $I = I' \amalg \{ i_0\} $. Let $P'$ be the ring defined by the diagram of the lemma using $I'$. Then $P'$ is of finite type by induction hypothesis. Finally, $P$ sits in a fibre product diagram

\[ \xymatrix{ B_{i_0} & A_{i_0} \ar[l] \\ P' \ar[u] & P \ar[u] \ar[l] & } \]

to which the lemma applies. $\square$

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Your email address will not be published.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 08NI. Beware of the difference between the letter 'O' and the digit '0'.