Lemma 35.3.7 (039W)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.3: Descent for modules
Lemma 35.3.7 (cite)

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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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