Lemma 34.3.8. Let $R \to A$ be a faithfully flat ring map, and let $R \to R'$ be faithfully flat. Set $A' = R' \otimes _ R A$. If all descent data for $R' \to A'$ are effective, then so are all descent data for $R \to A$.

**Proof.**
Let $(N, \varphi )$ be a descent datum for $R \to A$. Set $N' = R' \otimes _ R N = A' \otimes _ A N$, and denote $\varphi ' = \text{id}_{R'} \otimes \varphi $ the base change of the descent datum $\varphi $. Then $(N', \varphi ')$ is a descent datum for $R' \to A'$ and $H^0(s(N'_\bullet )) = R' \otimes _ R H^0(s(N_\bullet ))$. Moreover, the map $A' \otimes _{R'} H^0(s(N'_\bullet )) \to N'$ is identified with the base change of the $A$-module map $A \otimes _ R H^0(s(N)) \to N$ via the faithfully flat map $A \to A'$. Hence we conclude by Lemma 34.3.7.
$\square$

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