Lemma 15.75.4. Let $R' \to R$ be a surjective ring map whose kernel is a nilpotent ideal. Let $K' \in D(R')$ and set $K = K' \otimes _{R'}^\mathbf {L} R$. Then $K$ is pseudo-coherent if and only if $K'$ is pseudo-coherent.

Proof. One direction follows from Lemma 15.64.12. For the other direction, assume $K$ is pseudo-coherent. Then by Lemma 15.64.5 we can represent $K$ by a bounded above complex $E^\bullet$ of finite free $R$-modules. By Lemma 15.75.3 we can represent $K'$ by a bounded above complex $P^\bullet$ of projective $R'$-modules such that $P^ n \otimes _{R'} R = E^ n$. By Nakayama's lemma we see that $P^ n$ is finite free and we conclude that $K'$ is pseudo-coherent as well. $\square$

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