Lemma 38.2.6. Let $S$ be a scheme and $s \in S$ a point. Denote $\mathcal{O}_{S, s}^ h$ (resp. $\mathcal{O}_{S, s}^{sh}$) the henselization (resp. strict henselization), see Algebra, Definition 10.155.3. Let $M^{sh}$ be an object of $D(\mathcal{O}_{S, s}^{sh})$. Let $a, b \in \mathbf{Z}$. The following are equivalent

1. $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$,

2. $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^ h$, and

3. $M^{sh}$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^{sh}$.

If $M^{sh} = M^ h \otimes _{\mathcal{O}_{S, s}^ h}^\mathbf {L} \mathcal{O}_{S, s}^{sh}$ for $M^ h \in D(\mathcal{O}_{S, s}^ h)$ this is also equivalent to

1. $M^ h$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$, and

2. $M^ h$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}^ h$.

If $M^ h = M \otimes _{\mathcal{O}_{S, s}}^\mathbf {L} \mathcal{O}_{S, s}^ h$ for $M \in D(\mathcal{O}_{S, s})$ this is also equivalent to

1. $M$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{S, s}$.

Proof. By More on Algebra, Lemma 15.45.1 the local ring maps $\mathcal{O}_{S, s} \to \mathcal{O}_{S, s}^ h \to \mathcal{O}_{S, s}^{sh}$ are faithfully flat. Hence (3) $\Rightarrow$ (2) $\Rightarrow$ (1) and (5) $\Rightarrow$ (4) follow from More on Algebra, Lemma 15.66.11. By faithful flatness the equivalences (6) $\Leftrightarrow$ (5) and (5) $\Leftrightarrow$ (3) follow from More on Algebra, Lemma 15.66.17. Thus it suffices to show that (1) $\Rightarrow$ (3), (2) $\Rightarrow$ (3), and (4) $\Rightarrow$ (5).

Assume (1). In particular $M^{sh}$ has vanishing cohomology in degrees $< a$ and $> b$. Hence we can represent $M^{sh}$ by a complex $P^\bullet$ of free $\mathcal{O}_{X, x}^{sh}$-modules with $P^ i = 0$ for $i > b$ (see for example the very general Derived Categories, Lemma 13.15.4). Note that $P^ n$ is flat over $\mathcal{O}_{S, s}$ for all $n$. Consider $\mathop{\mathrm{Coker}}(d_ P^{a - 1})$. By More on Algebra, Lemma 15.66.2 this is a flat $\mathcal{O}_{S, s}$-module. Hence by Lemma 38.2.5 this is a flat $\mathcal{O}_{S, s}^{sh}$-module. Thus $\tau _{\geq a}P^\bullet$ is a complex of flat $\mathcal{O}_{S, s}^{sh}$-modules representing $M^{sh}$ in $D(\mathcal{O}_{S, s}^{sh}$ and we find that $M^{sh}$ has tor amplitude in $[a, b]$, see More on Algebra, Lemma 15.66.3. Thus (1) implies (3). Of course this implies also (2) $\Rightarrow$ (3) by replacing $\mathcal{O}_{S, s}$ by $\mathcal{O}_{S, s}^ h$. The same argument applies to prove (4) $\Rightarrow$ (5). $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).