Lemma 15.83.8. Let $R' \to A'$ be a flat ring map of finite presentation. Let $R' \to R$ be a surjective ring map whose kernel is a nilpotent ideal. Set $A = A' \otimes _{R'} R$. Let $K' \in D(A')$ and set $K = K' \otimes _{A'}^\mathbf {L} A$ in $D(A)$. If $K$ is $R$-perfect, then $K'$ is $R'$-perfect.

**Proof.**
Observe that $A' \to A$ has nilpotent kernel and that by flatness of $R' \to A'$ we have $K = K' \otimes _{R'}^\mathbf {L} R$ (see Section 15.61). Hence the lemma follows by combining Lemmas 15.75.4 and 15.66.20.
$\square$

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