The Stacks project

Lemma 46.3.20. Let $A \to A'$ be a ring map and let $F$ be a module-valued functor on $\textit{Alg}_ A$ such that

  1. the restriction $F'$ of $F$ to the category of $A'$-algebras is adequate, and

  2. for any $A$-algebra $B$ the sequence

    \[ 0 \to F(B) \to F(B \otimes _ A A') \to F(B \otimes _ A A' \otimes _ A A') \]

    is exact.

Then $F$ is adequate.

Proof. The functors $B \to F(B \otimes _ A A')$ and $B \mapsto F(B \otimes _ A A' \otimes _ A A')$ are adequate, see Lemmas 46.3.18 and 46.3.17. Hence $F$ as a kernel of a map of adequate functors is adequate, see Lemma 46.3.11. $\square$

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