The Stacks project

Lemma 15.80.6. Let $R \to A$ be a finite type ring map. Let $M$ be an $A$-module finitely presented relative to $R$. Let $A \to A'$ be a ring map of finite presentation. The $A'$-module $M \otimes _ A A'$ is finitely presented relative to $R$.

Proof. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Choose a presentation $A' = A[y_1, \ldots , y_ m]/(g_1, \ldots , g_ l)$. Pick $g'_ i \in R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]$ mapping to $g_ i$. Say

\[ R[x_1, \ldots , x_ n]^{\oplus s} \to R[x_1, \ldots , x_ n]^{\oplus t} \to M \to 0 \]

is a presentation of $M$ given by a matrix $(h_{ij})$. Then

\[ R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]^{\oplus s + tl} \to R[x_0, x_1, \ldots , x_ n, y_1, \ldots , y_ m]^{\oplus t} \to M \otimes _ A A' \to 0 \]

is a presentation of $M \otimes _ A A'$. Here the $t \times (s + lt)$ matrix defining the map has a first $t \times s$ block consisting of the matrix $h_{ij}$, followed by $l$ blocks of size $t \times t$ which are $g'_ iI_ t$. $\square$

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