Lemma 15.5.1. Let $R$ be a ring. Let $A \to B$ and $C \to B$ be $R$-algebra maps. Assume

1. $R$ is Noetherian,

2. $A$, $B$, $C$ are of finite type over $R$,

3. $A \to B$ is surjective, and

4. $B$ is finite over $C$.

Then $A \times _ B C$ is of finite type over $R$.

Proof. Set $D = A \times _ B C$. There is a commutative diagram

$\xymatrix{ 0 & B \ar[l] & A \ar[l] & I \ar[l] & 0 \ar[l] \\ 0 & C \ar[l] \ar[u] & D \ar[l] \ar[u] & I \ar[l] \ar[u] & 0 \ar[l] }$

with exact rows. Choose $y_1, \ldots , y_ n \in B$ which are generators for $B$ as a $C$-module. Choose $x_ i \in A$ mapping to $y_ i$. Then $1, x_1, \ldots , x_ n$ are generators for $A$ as a $D$-module. The map $D \to A \times C$ is injective, and the ring $A \times C$ is finite as a $D$-module (because it is the direct sum of the finite $D$-modules $A$ and $C$). Hence the lemma follows from the Artin-Tate lemma (Algebra, Lemma 10.51.7). $\square$

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