The Stacks project

Proof. To make sense of parts (2), (3), (4) of the lemma, observe that the object $E|_ U$ of $D_\mathit{QCoh}(\mathcal{O}_ U)$ corresponds to an object $E_0$ of $D_\mathit{QCoh}(\mathcal{O}_{U_0})$ where $U_0$ denotes the scheme underlying $U$, see Derived Categories of Spaces, Lemma 75.4.2. Moreover, in this case $E_0$ is pseudo-coherent if and only if $E|_ U$ is pseudo-coherent, see Derived Categories of Spaces, Lemma 75.13.2. Also, $E|_ U$ locally has finite tor dimension over $f^{-1}\mathcal{O}_ Y|_ U = g^{-1}\mathcal{O}_ V$ if and only if $E_0$ locally has finite tor dimension over $g_0^{-1}\mathcal{O}_{V_0}$ by Derived Categories of Spaces, Lemma 75.13.4. Here $g_0 : U_0 \to V_0$ is the morphism of schemes representing $g : U \to V$ (notation as in Derived Categories of Spaces, Remark 75.6.3). Finally, observe that “being pseudo-coherent” is étale local and of course “having locally finite tor dimension” is étale local. Thus we see that it suffices to check $Y$-perfectness étale locally and by the above discussion we see that (1) implies (2) and (3) implies (1). Since part (4) is equivalent to (2) and (3) by Derived Categories of Schemes, Lemma 36.35.3 the proof is complete. $\square$

Proof. This follows from Cohomology on Sites, Lemmas 21.45.4, 21.45.6, 21.46.6, and 21.46.8. $\square$

Proof. Reduce to the case of schemes using Lemma 76.52.3 and then apply Derived Categories of Schemes, Lemma 36.35.5. $\square$

Proof. Reduce to the case of schemes using Lemma 76.52.3 and then apply Derived Categories of Schemes, Lemma 36.35.6. $\square$

Proof. For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that

is an isomorphism where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for each $U_0$.

Suppose that $(U_0 \subset W_0, V_0 \to W_0)$ is an elementary distinguished square in $(X_0)_{spaces, {\acute{e}tale}}$ and $P$ holds for $U_0, V_0, U_0 \times _{W_0} V_0$. Then $P$ holds for $W_0$ by Mayer-Vietoris for hom in the derived category, see Derived Categories of Spaces, Lemma 75.10.4.

We first consider $U_0 = W_0 \times _{Y_0} X_0$ with $W_0$ a quasi-compact and quasi-separated object of $(Y_0)_{spaces, {\acute{e}tale}}$. By the induction principle of Derived Categories of Spaces, Lemma 75.9.3 applied to these $W_0$ and the previous paragraph, we find that it is enough to prove $P$ for $U_0 = W_0 \times _{Y_0} X_0$ with $W_0$ affine. In other words, we have reduced to the case where $Y_0$ is affine. Next, we apply the induction principle again, this time to all quasi-compact and quasi-separated opens of $X_0$, to reduce to the case where $X_0$ is affine as well.

If $X_0$ and $Y_0$ are affine, then we are back in the case of schemes which is proved in Derived Categories of Schemes, Lemma 36.35.8. The reader may use Derived Categories of Spaces, Lemmas 75.13.6, 75.4.2, 75.13.2, and 75.13.4 to accomplish the translation of the statement into a statement involving only schemes and derived categories of modules on schemes. $\square$

Proof. For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that the functor

is an equivalence where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We observe that we already know this functor is fully faithful by Lemma 76.52.8. Thus it suffices to prove essential surjectivity.

Suppose that $(U_0 \subset W_0, V_0 \to W_0)$ is an elementary distinguished square in $(X_0)_{spaces, {\acute{e}tale}}$ and $P$ holds for $U_0, V_0, U_0 \times _{W_0} V_0$. We claim that $P$ holds for $W_0$. We will use the notation $U_ i = X_ i \times _{X_0} U_0$, $U = X \times _{X_0} U_0$, and similarly for $V_0$ and $W_0$. We will abusively use the symbol $f_ i$ for all the morphisms $U \to U_ i$, $V \to V_ i$, $U \times _ W V \to U_ i \times _{W_ i} V_ i$, and $W \to W_ i$. Suppose $E$ is an $Y$-perfect object of $D(\mathcal{O}_ W)$. Goal: show $E$ is in the essential image of the functor. By assumption, we can find $i \geq 0$, an $Y_ i$-perfect object $E_{U, i}$ on $U_ i$, an $Y_ i$-perfect object $E_{V, i}$ on $V_ i$, and isomorphisms $Lf_ i^*E_{U, i} \to E|_ U$ and $Lf_ i^*E_{V, i} \to E|_ V$. Let

the maps adjoint to the isomorphisms $Lf_ i^*E_{U, i} \to E|_ U$ and $Lf_ i^*E_{V, i} \to E|_ V$. By fully faithfulness, after increasing $i$, we can find an isomorphism $c : E_{U, i}|_{U_ i \times _{W_ i} V_ i} \to E_{V, i}|_{U_ i \times _{W_ i} V_ i}$ which pulls back to the identifications

Apply Derived Categories of Spaces, Lemma 75.10.8 to get an object $E_ i$ on $W_ i$ and a map $d : E_ i \to Rf_{i, *}E$ which restricts to the maps $a$ and $b$ over $U_ i$ and $V_ i$. Then it is clear that $E_ i$ is $Y_ i$-perfect (because being relatively perfect is an étale local property) and that $d$ is adjoint to an isomorphism $Lf_ i^*E_ i \to E$.

By exactly the same argument as used in the proof of Lemma 76.52.8 using the induction principle (Derived Categories of Spaces, Lemma 75.9.3) we reduce to the case where both $X_0$ and $Y_0$ are affine: first work with quasi-compact and quasi-separated objects in $(Y_0)_{spaces, {\acute{e}tale}}$ to reduce to $Y_0$ affine, then work with quasi-compact and quasi-separated object in $(X_0)_{spaces, {\acute{e}tale}}$ to reduce to $X_0$ affine. In the affine case the result follows from the case of schemes which is Derived Categories of Schemes, Lemma 36.35.9. The translation into the case for schemes is done by Lemma 76.52.3. $\square$

Proof. The statement on base change is Derived Categories of Spaces, Lemma 75.21.4 (with $\mathcal{G}^\bullet $ equal to $\mathcal{O}_ X$ in degree $0$). Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce to the case where $Y$ is Noetherian affine by a limit argument.

The question is étale local on $Y$, hence we may assume $Y$ is affine. Say $Y = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits of Spaces, Lemma 70.7.1 there exists an $i$ and an algebraic space $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemmas 70.6.13 and 70.6.12 we may assume $X_ i$ is proper and flat over $R_ i$. By Lemma 76.52.9 we may assume there exists a $R_ i$-perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. Applying Derived Categories of Spaces, Lemma 75.22.1 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ and $E_ i$ and using the base change property already shown we obtain the result. $\square$

Proof. Since $Y$ is quasi-compact and quasi-separated, the same is true for $X$. By Derived Categories of Spaces, Lemma 75.18.1 we can write $K = \text{hocolim} K_ n$ with $K_ n$ perfect and $K_ n \to K$ inducing an isomorphism on truncations $\tau _{\geq -n}$. Let $K_ n^\vee $ be the dual perfect complex (Cohomology on Sites, Lemma 21.48.4). We obtain an inverse system $\ldots \to K_3^\vee \to K_2^\vee \to K_1^\vee $ of perfect objects. By Lemma 76.52.5 we see that $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ is $Y$-perfect. Thus we may apply Lemma 76.52.10 to $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ and we obtain an inverse system

of perfect complexes on $Y$ with

Moreover, the formation of these complexes commutes with any base change, namely $Lg^*M_ n = Rf'_*((L(g')^*K_ n)^\vee \otimes _{\mathcal{O}_{X'}}^\mathbf {L} L(g')^*E) = Rf'_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L(g')^*K_ n, L(g')^*E)$.

As $K_ n \to K$ induces an isomorphism on $\tau _{\geq -n}$, we see that $K_ n \to K_{n + 1}$ induces an isomorphism on $\tau _{\geq -n}$. It follows that $K_{n + 1}^\vee \to K_ n^\vee $ induces an isomorphism on $\tau _{\leq n}$ as $K_ n^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, \mathcal{O}_ X)$. Suppose that $E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_ Y$-modules. Then the same is true after any base change, see Derived Categories of Spaces, Lemma 75.20.7. We find that $K_{n + 1}^\vee \otimes _{\mathcal{O}_ X} E \to K_ n^\vee \otimes _{\mathcal{O}_ X} E$ induces an isomorphism on $\tau _{\leq n + a}$ and the same is true after any base change. Applying the right derived functor $Rf_*$ we conclude the maps $M_{n + 1} \to M_ n$ induce isomorphisms on $\tau _{\leq n + a}$ and the same is true after any base change. Choose a distinguished triangle

Pick $y \in |Y|$. Choose an elementary étale neighbourhood $(U, u) \to (Y, y)$; this is possible by Decent Spaces, Lemma 68.11.4. Take $Y'$ equal to the spectrum of the residue field at $u$. Pull back to see that $C_ n|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \kappa (u)$ has nonzero cohomology only in degrees $\geq n + a$. By More on Algebra, Lemma 15.75.7 we see that the perfect complex $C_ n|_ U$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$ and after possibly shrinking $U$. Thus $C_ n$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$ (because $Y$ is quasi-compact). In particular, the dual perfect complex $C_ n^\vee $ has tor amplitude in $[-m_ n, -n - a]$.

Let $L_ n = M_ n^\vee $ be the dual perfect complex. The conclusion from the discussion in the previous paragraph is that $L_ n \to L_{n + 1}$ induces isomorphisms on $\tau _{\geq -n - a}$. Thus $L = \text{hocolim} L_ n$ is pseudo-coherent, see Derived Categories of Spaces, Lemma 75.18.1. Since we have

(Cohomology on Sites, Lemma 21.48.8) and since $R\mathop{\mathrm{lim}}\nolimits $ commutes with $Rf_*$ we find that

This proves the formula over $Y$. Since the construction of $M_ n$ is compatible with base chance, the formula continues to hold after any base change. $\square$

Proof. Since everything is local we immediately reduce to the case that $X$ and $S$ are affine, see Lemma 76.52.3. This case is handled by Derived Categories of Schemes, Lemma 36.35.13. $\square$

Proof. By Lemma 76.51.5, $K$ is pseudo-coherent relative to $A$. By Lemma 76.45.4, $K$ is pseudo-coherent in $D(\mathcal{O}_ X)$. By Derived Categories of Spaces, Proposition 75.29.4 we see that $K$ is in $D^-(\mathcal{O}_ X)$. Let $\mathfrak {p}$ be a prime ideal of $A$ and denote $i : Y \to X$ the inclusion of the scheme theoretic fibre over $\mathfrak {p}$, i.e., $Y$ is a scheme over $\kappa (\mathfrak p)$. By Lemma 76.52.13, we will be done if we can show $Li^*(K)$ is bounded below. Let $G \in D_{perf} (\mathcal{O}_ X)$ be a perfect complex which generates $D_\mathit{QCoh}(\mathcal{O}_ X)$, see Derived Categories of Spaces, Theorem 75.15.4. We have

The first equality uses that $Li^*$ preserves perfect objects and duals and Cohomology on Sites, Lemma 21.48.4; we omit some details. The second equality follows from Derived Categories of Spaces, Lemma 75.20.4 as $X$ is flat over $A$. It follows from our hypothesis that this is a perfect object of $D(\kappa (\mathfrak {p}))$. The object $Li^*(G) \in D_{perf}(\mathcal{O}_ Y)$ generates $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Derived Categories of Spaces, Remark 75.15.5. Hence Derived Categories of Spaces, Proposition 75.29.4 now implies that $Li^*(K)$ is bounded below and we win. $\square$