Lemma 37.54.3 (Longke Tang). Let $f : X \to S$ be a surjective morphism of finite presentation. Let $M$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ S)$ such that $H^ i(M) = 0$ for $i > 0$. The following are equivalent
$M$ is isomorphic to a flat $\mathcal{O}_ S$-module placed in degree $0$,
$Lf^*M$ is isomorphic to a flat $\mathcal{O}_ X$-module placed in degree $0$.
Proof. We omit the proof that (1) implies (2). Assume (2). To prove (1) is a local question, hence we may assume that $S$ is affine. Choose a stratification $S \supset S_0 \supset S_1 \supset \ldots \supset S_ t = \emptyset $ as in Lemma 37.54.2. Set $T_ i = S_ i \setminus S_{i - 1}$ and $Y_ i = X \times _ S T_ i$. Denote $g_ i : T_ i \to S$ and $h_ i : Y_ i \to X$ the inclusion morphisms. Since $f$ is surjective, the morphisms $f_ i : X_ i \to T_ i$ are surjective and flat. Since $Lh_ i^* \circ Lf^* = Lf_ i^* \circ Lg_ i^*$ we conclude from (2) that $Lf_ i^* Lg_ i^*M$ is isomorphic to a flat $\mathcal{O}_{Y_ i}$-module placed in degree $0$. Since $f_ i$ is flat and surjective, we see that $Lg_ i^*M$ is isomorphic to a flat $\mathcal{O}_{T_ i}$-module placed in degree $0$.
We will prove this implies the result by induction on $t$. We will use the following notation: $S = \mathop{\mathrm{Spec}}(A)$ and $M$ is the object of $D_\mathit{QCoh}(\mathcal{O}_ S)$ corresponding to $N$ in $D(A)$, see Derived Categories of Schemes, Lemma 36.3.5.
Base case: $t = 1$. Write $S_0 = \mathop{\mathrm{Spec}}(A/I)$ where $I$ is a finite generated ideal. Then $I$ is nilpotent as $S_0 = S$ set theoretically. The assumption is that $N \otimes _ A^\mathbf {L} A/I$ has tor amplitude in $[0, 0]$. By More on Algebra, Lemma 15.67.20 the same is true for $N$.
Induction step. Assume $t > 1$. Write $S_{t - 1} = \mathop{\mathrm{Spec}}(A/I)$ where $I = (f_1, \ldots , f_ r)$ is a finitely generated ideal. We will argue by induction on $r$. Observe that $N_{f_ r} = N \otimes _ A^\mathbf {L} A_{f_ r}$ in $D(A_{f_ r})$ has tor amplitude in $[0, 0]$ by induction hypothesis (because the stratification on the principal open $D(f_ r)$ has length at most $t - 1$). On the other hand, consider $N' = N \otimes _ A^\mathbf {L} A/f_ rA$ in $D(A/f_ rA)$. Setting $S' = \mathop{\mathrm{Spec}}(A/f_ rA)$ we obtain a stratification
where $S' \cap S_{t - 1}$ is cut out by $r - 1$ equations in $S'$. Similarly to before, the derived pullbacks of $M$ to the parts $S' \cap S_ i \setminus S' \cap S_{i + 1}$ are of tor amplitude in $[0, 0]$. Whence by induction on $r$ we conclude that $N'$ has tor amplitude in $[0, 0]$. Then we finally conclude that $N$ has tor amplitude in $[0, 0]$ by More on Algebra, Lemma 15.103.7. $\square$
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