Lemma 15.83.5. Let $R \to A$ be a flat ring map of finite presentation. Let $R \to R'$ be a ring map and set $A' = A \otimes _ R R'$. If $K \in D(A)$ is $R$-perfect, then $K \otimes _ A^\mathbf {L} A'$ is $R'$-perfect.

**Proof.**
By Lemma 15.64.12 we see that $K \otimes _ A^\mathbf {L} A'$ is pseudo-coherent. By Lemma 15.61.2 we see that $K \otimes _ A^\mathbf {L} A'$ is equal to $K \otimes _ R^\mathbf {L} R'$ in $D(R')$. Then we can apply Lemma 15.66.13 to see that $K \otimes _ R^\mathbf {L} R'$ in $D(R')$ has finite tor dimension.
$\square$

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