The Stacks project

Proof. Pullback preserves “tor amplitude in $[a, b]$” by Derived Categories of Spaces, Lemma 75.20.7 Observe that $Y_ i$ and $X$ are tor independent over $Y$ as $Y_ i \to Y$ is flat. Let us assume (2) and prove (1). We can compute tor dimension at stalks, see Cohomology on Sites, Lemma 21.46.10 and Properties of Spaces, Theorem 66.19.12. Let $\overline{x}$ be a geometric point of $X$. Choose an $i$ and a geometric point $\overline{x}_ i$ in $X_ i$ with image $\overline{x}$ in $X$. Then

Let $\overline{y}_ i$ in $Y_ i$ and $\overline{y}$ in $Y$ be the image of $\overline{x}_ i$ and $\overline{x}$. Since $X$ and $Y_ i$ are tor independent over $Y$, we can apply More on Algebra, Lemma 15.61.2 to see that the right hand side of the displayed formula is equal to $E_{\overline{x}} \otimes _{\mathcal{O}_{Y, \overline{y}}}^\mathbf {L} \mathcal{O}_{Y_ i, \overline{y}_ i}$ in $D(\mathcal{O}_{Y_ i, \overline{y}_ i})$. Since we have assume the tor amplitude of this is in $[a, b]$, we conclude that the tor amplitude of $E_{\overline{x}}$ in $D(\mathcal{O}_{Y, \overline{y}})$ is in $[a, b]$ by More on Algebra, Lemma 15.66.17. Thus (1) follows.

Using some elementary topology the case “locally finite tor dimension” follows too. $\square$