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. $\square$

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