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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