Lemma 75.20.7. Let $S$ be a scheme. Consider a cartesian square of algebraic spaces

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

over $S$. Assume $g$ and $f$ Tor independent.

If $E \in D(\mathcal{O}_ X)$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_ Y$-modules, then $L(g')^*E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_{Y'}$-modules.

If $\mathcal{G}$ is an $\mathcal{O}_ X$-module flat over $Y$, then $L(g')^*\mathcal{G} = (g')^*\mathcal{G}$.

**Proof.**
We can compute tor dimension at stalks, see Cohomology on Sites, Lemma 21.46.10 and Properties of Spaces, Theorem 66.19.12. If $\overline{x}'$ is a geometric point of $X'$ with image $\overline{x}$ in $X$, then

\[ (L(g')^*E)_{\overline{x}'} = E_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}}^\mathbf {L} \mathcal{O}_{X', \overline{x}'} \]

Let $\overline{y}'$ in $Y'$ and $\overline{y}$ in $Y$ be the image of $\overline{x}'$ and $\overline{x}$. Since $X$ and $Y'$ 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 $E_{\overline{x}} \otimes _{\mathcal{O}_{Y, \overline{y}}}^\mathbf {L} \mathcal{O}_{Y', \overline{y}'}$ in $D(\mathcal{O}_{Y', \overline{y}'})$. Thus (1) follows from More on Algebra, Lemma 15.66.13. To see (2) observe that flatness of $\mathcal{G}$ is equivalent to the condition that $\mathcal{G}[0]$ has tor amplitude in $[0, 0]$. Applying (1) we conclude.
$\square$

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