The Stacks project

Proof. Being pseudo-coherent is a local property and “locally having finite tor dimension” is a local property. Hence this lemma immediately reduces to the statement: if $X$ and $S$ are affine, then $E$ is $S$-perfect if and only if $K = R\Gamma (X, E)$ is $\mathcal{O}_ S(S)$-perfect. Say $X = \mathop{\mathrm{Spec}}(A)$, $S = \mathop{\mathrm{Spec}}(R)$ and $E$ corresponds to $K \in D(A)$, i.e., $K = R\Gamma (X, E)$, see Lemma 36.3.5.

Observe that $K$ is $R$-perfect if and only if $K$ is pseudo-coherent and has finite tor dimension as a complex of $R$-modules (More on Algebra, Definition 15.83.1). By Lemma 36.10.2 we see that $E$ is pseudo-coherent if and only if $K$ is pseudo-coherent. By Lemma 36.10.5 we see that $E$ has finite tor dimension over $f^{-1}\mathcal{O}_ S$ if and only if $K$ has finite tor dimension as a complex of $R$-modules. $\square$

Proof. This follows from Cohomology, Lemmas 20.47.4, 20.47.6, 20.48.6, and 20.48.8. $\square$

Proof. First proof: reduce to the affine case using Lemma 36.35.3 and then apply More on Algebra, Lemma 15.83.3. $\square$

Proof. First proof: reduce to the affine case using Lemma 36.35.3 and then apply More on Algebra, Lemma 15.83.5.

Second proof: $L(g')^*K$ is pseudo-coherent by Cohomology, Lemma 20.47.3 and the bounded tor dimension property follows from Lemma 36.22.8. $\square$

Proof. For every quasi-compact open $U_0 \subset X_0$ consider the condition $P$ that

is an isomorphism where $U = f_0^{-1}(U_0)$ and $U_ i = f_{i0}^{-1}(U_0)$. If $P$ holds for $U_0$, $V_0$ and $U_0 \cap V_0$, then it holds for $U_0 \cup V_0$ by Mayer-Vietoris for hom in the derived category, see Cohomology, Lemma 20.33.3.

Denote $\pi _0 : X_0 \to S_0$ the given morphism. Then we can first consider $U_0 = \pi _0^{-1}(W_0)$ with $W_0 \subset S_0$ quasi-compact open. By the induction principle of Cohomology of Schemes, Lemma 30.4.1 applied to quasi-compact opens of $S_0$ and the remark above, we find that it is enough to prove $P$ for $U_0 = \pi _0^{-1}(W_0)$ with $W_0$ affine. In other words, we have reduced to the case where $S_0$ 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 $S_0$ are affine, the result follows from More on Algebra, Lemma 15.83.7. Namely, by Lemmas 36.10.1 and 36.3.5 the statement is translated into computations of homs in the derived categories of modules. Then Lemma 36.10.2 shows that the complex of modules corresponding to $K_0$ is pseudo-coherent. And Lemma 36.10.5 shows that the complex of modules corresponding to $L_0$ has finite tor dimension over $\mathcal{O}_{S_0}(S_0)$. Thus the assumptions of More on Algebra, Lemma 15.83.7 are satisfied and we win. $\square$

Proof. For every quasi-compact open $U_0 \subset X_0$ consider the condition $P$ that the functor

is an equivalence where $U = f_0^{-1}(U_0)$ and $U_ i = f_{i0}^{-1}(U_0)$. We observe that we already know this functor is fully faithful by Lemma 36.35.8. Thus it suffices to prove essential surjectivity.

Suppose that $P$ holds for quasi-compact opens $U_0$, $V_0$ of $X_0$. We claim that $P$ holds for $U_0 \cup V_0$. We will use the notation $U_ i = f_{i0}^{-1}U_0$, $U = f_0^{-1}U_0$, $V_ i = f_{i0}^{-1}V_0$, and $V = f_0^{-1}V_0$ and we will abusively use the symbol $f_ i$ for all the morphisms $U \to U_ i$, $V \to V_ i$, $U \cap V \to U_ i \cap V_ i$, and $U \cup V \to U_ i \cup V_ i$. Suppose $E$ is an $S$-perfect object of $D(\mathcal{O}_{U \cup V})$. Goal: show $E$ is in the essential image of the functor. By assumption, we can find $i \geq 0$, an $S_ i$-perfect object $E_{U, i}$ on $U_ i$, an $S_ i$-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 $c : E_{U, i}|_{U_ i \cap V_ i} \to E_{V, i}|_{U_ i \cap V_ i}$ which pulls back to the identifications

Apply Cohomology, Lemma 20.45.1 to get an object $E_ i$ on $U_ i \cup V_ i$ 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 $S_ i$-perfect (because being relatively perfect is a 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 36.35.8 using the induction principle (Cohomology of Schemes, Lemma 30.4.1) we reduce to the case where both $X_0$ and $S_0$ are affine. (First work with opens in $S_0$ to reduce to $S_0$ affine, then work with opens in $X_0$ to reduce to $X_0$ affine.) In the affine case the result follows from More on Algebra, Lemma 15.83.7. The translation into algebra is done by Lemma 36.35.3. $\square$

Proof. The statement on base change is Lemma 36.22.5. Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce to the case where $S$ is Noetherian affine by a limit argument.

The question is local on $S$, hence we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits, Lemma 32.10.1 there exists an $i$ and a scheme $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits, Lemmas 32.13.1 and 32.8.7 we may assume $X_ i$ is proper and flat over $R_ i$. By Lemma 36.35.9 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 Lemma 36.27.1 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 $S$ is quasi-compact and quasi-separated, the same is true for $X$. By Lemma 36.19.1 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, Lemma 20.50.5). We obtain an inverse system $\ldots \to K_3^\vee \to K_2^\vee \to K_1^\vee $ of perfect objects. By Lemma 36.35.5 we see that $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ is $S$-perfect. Thus we may apply Lemma 36.35.10 to $K_ n^\vee \otimes _{\mathcal{O}_ X} E$ and we obtain an inverse system

of perfect complexes on $S$ 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 Lemma 36.22.8. 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

Take $S'$ equal to the spectrum of the residue field at a point $s \in S$ and pull back to see that $C_ n \otimes _{\mathcal{O}_ S}^\mathbf {L} \kappa (s)$ has nonzero cohomology only in degrees $\geq n + a$. By More on Algebra, Lemma 15.75.6 we see that the perfect complex $C_ n$ has tor amplitude in $[n + a, m_ n]$ for some integer $m_ n$. 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 Lemma 36.19.1. Since we have

(Cohomology, Lemma 20.51.1) and since $R\mathop{\mathrm{lim}}\nolimits $ commutes with $Rf_*$ we find that

This proves the formula over $S$. 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 36.35.3. Say $X \to S$ corresponds to $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(R)$ and $E$ corresponds to $K$ in $D(A)$. If $s$ corresponds to the prime $\mathfrak p \subset R$, then $L(X_ s \to X)^*E$ corresponds to $K \otimes _ R^\mathbf {L} \kappa (\mathfrak p)$ as $R \to A$ is flat, see for example Lemma 36.22.5. Thus we see that our lemma follows from the corresponding algebra result, see More on Algebra, Lemma 15.83.10. $\square$