Lemma 15.66.14. Let $A \to B$ be a flat ring map. Let $d \geq 0$. Let $M$ be an $A$-module of tor dimension $\leq d$. Then $M \otimes _ A B$ is a $B$-module of tor dimension $\leq d$.

Proof. Immediate consequence of Lemma 15.66.13 and the fact that $M \otimes _ A^{\mathbf{L}} B = M \otimes _ A B$ because $B$ is flat over $A$. $\square$

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