Definition 10.14.1. Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Let $R \to R'$ be any ring map. The *base change* of $\varphi $ by $R \to R'$ is the ring map $R' \to S \otimes _ R R'$. In this situation we often write $S' = S \otimes _ R R'$. The *base change* of the $S$-module $M$ is the $S'$-module $M \otimes _ R R'$.

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