The Stacks project

Lemma 46.3.17. Let $A \to A'$ be a ring map. If $F$ is an adequate functor on $\textit{Alg}_ A$, then its restriction $F'$ to $\textit{Alg}_{A'}$ is adequate too.

Proof. Choose an exact sequence $0 \to F \to \underline{M} \to \underline{N}$. Then $F'(B') = F(B') = \mathop{\mathrm{Ker}}(M \otimes _ A B' \to N \otimes _ A B')$. Since $M \otimes _ A B' = M \otimes _ A A' \otimes _{A'} B'$ and similarly for $N$ we see that $F'$ is the kernel of $\underline{M \otimes _ A A'} \to \underline{N \otimes _ A A'}$. $\square$

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