The Stacks project

Lemma 15.81.12. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $A$-modules which is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R$. Let $R \to R'$ be a ring map such that $A$ and $R'$ are Tor independent over $R$. Set $A' = A \otimes _ R R'$. Then $K^\bullet \otimes _ A^{\mathbf{L}} A'$ is $m$-pseudo-coherent (resp. pseudo-coherent) relative to $R'$.

Proof. Choose a surjection $R[x_1, \ldots , x_ n] \to A$. Note that

\[ K^\bullet \otimes _ A^{\mathbf{L}} A' = K^\bullet \otimes _ R^{\mathbf{L}} R' = K^\bullet \otimes _{R[x_1, \ldots , x_ n]}^{\mathbf{L}} R'[x_1, \ldots , x_ n] \]

by Lemma 15.61.2 applied twice. Hence we win by Lemma 15.64.12. $\square$

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