Lemma 15.66.18. Given ring maps $R \to A \to B$ with $A \to B$ faithfully flat and $K \in D(A)$ the tor amplitude of $K$ over $R$ is the same as the tor amplitude of $K \otimes _ A^\mathbf {L} B$ over $R$.
Proof. This is true because for an $R$-module $M$ we have $H^ i(K \otimes _ R^\mathbf {L} M) \otimes _ A B = H^ i((K \otimes _ A^\mathbf {L} B) \otimes _ R^\mathbf {L} M)$ for all $i$. Namely, represent $K$ by a complex $K^\bullet $ of $A$-modules and choose a free resolution $F^\bullet \to M$. Then we have the equality
\[ \text{Tot}(K^\bullet \otimes _ A B \otimes _ R F^\bullet ) = \text{Tot}(K^\bullet \otimes _ R F^\bullet ) \otimes _ A B \]
The cohomology groups of the left hand side are $H^ i((K \otimes _ A^\mathbf {L} B) \otimes _ R^\mathbf {L} M)$ and on the right hand side we obtain $H^ i(K \otimes _ R^\mathbf {L} M) \otimes _ A B$. $\square$
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