Lemma 10.134.8 (Flat base change). Let $R \to S$ be a ring map. Let $\alpha : P \to S$ be a presentation. Let $R \to R'$ be a flat ring map. Let $\alpha ' : P \otimes _ R R' \to S' = S \otimes _ R R'$ be the induced presentation. Then $\mathop{N\! L}\nolimits (\alpha ) \otimes _ R R' = \mathop{N\! L}\nolimits (\alpha ) \otimes _ S S' = \mathop{N\! L}\nolimits (\alpha ')$. In particular, the canonical map

$\mathop{N\! L}\nolimits _{S/R} \otimes _ S S' \longrightarrow \mathop{N\! L}\nolimits _{S \otimes _ R R'/R'}$

is a homotopy equivalence if $R \to R'$ is flat.

Proof. This is true because $\mathop{\mathrm{Ker}}(\alpha ') = R' \otimes _ R \mathop{\mathrm{Ker}}(\alpha )$ since $R \to R'$ is flat. $\square$

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