The Stacks project

Lemma 35.3.7. Let $R \to A$ be a faithfully flat ring map. Let $(N, \varphi )$ be a descent datum. Then $(N, \varphi )$ is effective if and only if the canonical map

\[ A \otimes _ R H^0(s(N_\bullet )) \longrightarrow N \]

is an isomorphism.

Proof. If $(N, \varphi )$ is effective, then we may write $N = A \otimes _ R M$ with $\varphi = can$. It follows that $H^0(s(N_\bullet )) = M$ by Lemmas 35.3.3 and 35.3.6. Conversely, suppose the map of the lemma is an isomorphism. In this case set $M = H^0(s(N_\bullet ))$. This is an $R$-submodule of $N$, namely $M = \{ n \in N \mid 1 \otimes n = \varphi (n \otimes 1)\} $. The only thing to check is that via the isomorphism $A \otimes _ R M \to N$ the canonical descent data agrees with $\varphi $. We omit the verification. $\square$


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