The Stacks project

Lemma 16.2.7. Let $R \to A$ be a ring map of finite presentation. Let $R \to R'$ be a ring map. If $a \in A$ is elementary, resp. strictly standard in $A$ over $R$, then $a \otimes 1$ is elementary, resp. strictly standard in $A \otimes _ R R'$ over $R'$.

Proof. If $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ is a presentation of $A$ over $R$, then $A \otimes _ R R' = R'[x_1, \ldots , x_ n]/(f'_1, \ldots , f'_ m)$ is a presentation of $A \otimes _ R R'$ over $R'$. Here $f'_ j$ is the image of $f_ j$ in $R'[x_1, \ldots , x_ n]$. Hence the result follows from the definitions. $\square$

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