The Stacks project

Lemma 75.13.4. Let $f : X \to Y$ be a morphism of schemes. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then

  1. $E$ as an object of $D(f^{-1}\mathcal{O}_ Y)$ has tor amplitude in $[a, b]$ if and only if $\epsilon ^*E$ has tor amplitude in $[a, b]$ as an object of $D(f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}})$.

  2. $E$ locally has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ Y)$ if and only if $\epsilon ^*E$ locally has finite tor dimension as an object of $D(f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}})$.

Here $\epsilon $ is as in (75.4.0.1).

Proof. The easy direction in (1) follows from Cohomology on Sites, Lemma 21.46.5. Let $x \in X$ be a point and let $\overline{x}$ be a geometric point lying over $x$. Let $y = f(x)$ and denote $\overline{y}$ the geometric point of $Y$ coming from $\overline{x}$. Then $(f^{-1}\mathcal{O}_ Y)_ x = \mathcal{O}_{Y, y}$ (Sheaves, Lemma 6.21.5) and

\[ (f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}})_{\overline{x}} = \mathcal{O}_{Y_{\acute{e}tale}, \overline{y}} = \mathcal{O}_{Y, y}^{sh} \]

is the strict henselization (by √Čtale Cohomology, Lemmas 59.36.2 and 59.33.1). Since the stalk of $\mathcal{O}_{X_{\acute{e}tale}}$ at $X$ is $\mathcal{O}_{X, x}^{sh}$ we obtain

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

by transitivity of pullbacks. If $\epsilon ^*E$ has tor amplitude in $[a, b]$ as a complex of $f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}}$-modules, then $(\epsilon ^*E)_{\overline{x}}$ has tor amplitude in $[a, b]$ as a complex of $\mathcal{O}_{Y, y}^{sh}$-modules (because taking stalks is a pullback and lemma cited above). By More on Flatness, Lemma 38.2.6 we find the tor amplitude of $(\epsilon ^*E)_{\overline{x}}$ as a complex of $\mathcal{O}_{Y, y}$-modules is in $[a, b]$. Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}$ is faithfully flat (More on Algebra, Lemma 15.45.1) and since $(\epsilon ^*E)_{\overline{x}} = E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{O}_{X, x}^{sh}$ we may apply More on Algebra, Lemma 15.66.18 to conclude the tor amplitude of $E_ x$ as a complex of $\mathcal{O}_{Y, y}$-modules is in $[a, b]$. By Cohomology, Lemma 20.48.5 we conclude that $E$ as an object of $D(f^{-1}\mathcal{O}_ Y)$ has tor amplitude in $[a, b]$. This gives the reverse implication in (1). Part (2) follows formally from (1). $\square$


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