36.27 Producing perfect complexes

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

1. $E \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,

2. the support of $H^ i(E)$ is proper over $S$ for all $i$, and

3. $E$ 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 20.46.5. 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

$Rf_*(E) \otimes _{\mathcal{O}_ S}^\mathbf {L} \mathcal{F} = Rf_*\left(E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{F}\right) = Rf_*\left(E \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} f^{-1}\mathcal{F}\right)$

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

$\bigoplus \nolimits _{i = a, \ldots , b} (\mathcal{G}^ i)^{\oplus r_ i}$

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 20.47.5. 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 37.55.13.

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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