The following lemma is our main technical tool for producing perfect complexes. Later versions of this result will reduce to this by Noetherian approximation, see Section 36.30.
Lemma 36.27.1. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_ X)$ such that
$E \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,
the support of $H^ i(E)$ is proper over $S$ for all $i$, and
$E$ has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_ S)$.
Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$.
Proof. By Lemma 36.11.3 we see that $Rf_*E$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ S)$. Hence $Rf_*E$ is pseudo-coherent (Lemma 36.10.3). Hence it suffices to show that $Rf_*E$ has finite tor dimension, see Cohomology, Lemma 20.49.5. By Lemma 36.10.6 it suffices to check that $Rf_*(E) \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{F}$ has universally bounded cohomology for all quasi-coherent sheaves $\mathcal{F}$ on $S$. Bounded from above is clear as $Rf_*(E)$ is bounded from above. Let $T \subset X$ be the union of the supports of $H^ i(E)$ for all $i$. Then $T$ is proper over $S$ by assumptions (1) and (2), see Cohomology of Schemes, Lemma 30.26.6. In particular there exists a quasi-compact open $X' \subset X$ containing $T$. Setting $f' = f|_{X'}$ we have $Rf_*(E) = Rf'_*(E|_{X'})$ because $E$ restricts to zero on $X \setminus T$. Thus we may replace $X$ by $X'$ and assume $f$ is quasi-compact. Moreover, $f$ is quasi-separated by Morphisms, Lemma 29.15.7. Now
by Lemma 36.22.1 and Cohomology, Lemma 20.27.4. By assumption (3) the complex $E \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} f^{-1}\mathcal{F}$ has cohomology sheaves in a given finite range, say $[a, b]$. Then $Rf_*$ of it has cohomology in the range $[a, \infty )$ and we win. $\square$
Lemma 36.27.2. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_ X)$ be perfect. Let $\mathcal{G}^\bullet $ be a bounded complex of coherent $\mathcal{O}_ X$-modules flat over $S$ with support proper over $S$. Then $K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet )$ is a perfect object of $D(\mathcal{O}_ S)$.
Proof. The object $K$ is perfect by Lemma 36.27.1. We check the lemma applies: Locally $E$ is isomorphic to a finite complex of finite free $\mathcal{O}_ X$-modules. Hence locally $E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet $ is isomorphic to a finite complex whose terms are of the form
for some integers $a, b, r_ a, \ldots , r_ b$. This immediately implies the cohomology sheaves $H^ i(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G})$ are coherent. The hypothesis on the tor dimension also follows as $\mathcal{G}^ i$ is flat over $f^{-1}\mathcal{O}_ S$. $\square$
Lemma 36.27.3. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E \in D(\mathcal{O}_ X)$ be perfect. Let $\mathcal{G}^\bullet $ be a bounded complex of coherent $\mathcal{O}_ X$-modules flat over $S$ with support proper over $S$. Then $K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet )$ is a perfect object of $D(\mathcal{O}_ S)$.
Proof. Since $E$ is a perfect complex there exists a dual perfect complex $E^\vee $, see Cohomology, Lemma 20.50.5. Observe that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet ) = E^\vee \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{G}^\bullet $. Thus the perfectness of $K$ follows from Lemma 36.27.2. $\square$
We will generalize the following lemma to flat and proper morphisms over general bases in Lemma 36.30.4 and to perfect proper morphisms in More on Morphisms, Lemma 37.61.13.
Lemma 36.27.4. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a flat proper morphism of schemes. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$.
Proof. We claim that Lemma 36.27.1 applies. Conditions (1) and (2) are immediate. Condition (3) is local on $X$. Thus we may assume $X$ and $S$ affine and $E$ represented by a strictly perfect complex of $\mathcal{O}_ X$-modules. Since $\mathcal{O}_ X$ is flat as a sheaf of $f^{-1}\mathcal{O}_ S$-modules we find that condition (3) is satisfied. $\square$
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