The Stacks project

Proof. Combine Lemma 36.30.1 with More on Algebra, Lemmas 15.64.18 and 15.74.2. $\square$

Proof. Using Cohomology, Lemma 20.50.5 we see that it suffices to prove the result for the groups $H^ i(U, K \otimes ^\mathbf {L} E)$. We will use the induction principle to prove the lemma, see Cohomology of Schemes, Lemma 30.4.1.

First we show that it holds when $U = \mathop{\mathrm{Spec}}(A)$ is affine. Namely, we can represent $K$ by a complex of $A$-modules $K^\bullet $ and $E$ by a finite complex of finite projective $A$-modules $P^\bullet $. See Lemmas 36.3.5 and 36.10.7 and our definition of perfect complexes of $A$-modules (More on Algebra, Definition 15.74.1). Then $(E \otimes ^\mathbf {L} K)|_ U$ is represented by the total complex associated to the double complex $P^\bullet \otimes _ A K^\bullet $ (Lemma 36.3.9). Using induction on the length of the complex $P^\bullet $ (or using a suitable spectral sequence) we see that it suffices to show that $H^ i(P^ a \otimes _ A K^\bullet )$ is countable for each $a$. Since $P^ a$ is a direct summand of $A^{\oplus n}$ for some $n$ this follows from the assumption that the cohomology group $H^ i(K^\bullet )$ is countable.

To finish the proof it suffices to show: if $U = V \cup W$ and the result holds for $V$, $W$, and $V \cap W$, then the result holds for $U$. This is an immediate consequence of the Mayer-Vietoris sequence, see Cohomology, Lemma 20.33.4. $\square$

Proof. If (1) is true, then (2) is true because homotopy colimits commutes with taking cohomology sheaves (by Derived Categories, Lemma 13.33.8) and because a perfect complex is locally isomorphic to a finite complex of finite free $\mathcal{O}_ X$-modules and therefore satisfies (2) by assumption on $X$.

Assume (2). Choose a K-injective complex $\mathcal{K}^\bullet $ representing $K$. Choose a perfect generator $E$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ and represent it by a K-injective complex $\mathcal{I}^\bullet $. According to Theorem 36.18.2 and its proof there is an equivalence of triangulated categories $F : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A, \text{d})$ where $(A, \text{d})$ is the differential graded algebra

which maps $K$ to the differential graded module

Note that $H^ i(A) = \mathop{\mathrm{Ext}}\nolimits ^ i(E, E)$ and $H^ i(M) = \mathop{\mathrm{Ext}}\nolimits ^ i(E, K)$. Moreover, since $F$ is an equivalence it and its quasi-inverse commute with homotopy colimits. Therefore, it suffices to write $M$ as a homotopy colimit of compact objects of $D(A, \text{d})$. By Differential Graded Algebra, Lemma 22.38.3 it suffices show that $\mathop{\mathrm{Ext}}\nolimits ^ i(E, E)$ and $\mathop{\mathrm{Ext}}\nolimits ^ i(E, K)$ are countable for each $i$. This follows from Lemma 36.33.2. $\square$

Proof. By Lemma 36.22.1 we have

functorially in $K$. Observe that $R\Gamma (X, -)$ commutes with homotopy colimits because it commutes with direct sums by Lemma 36.4.5. Similarly, $- \otimes ^\mathbf {L} K$ commutes with derived colimits because $- \otimes ^\mathbf {L} K$ commutes with direct sums (because direct sums in $D(\mathcal{O}_ X)$ are given by direct sums of representing complexes). Hence to prove (2) it suffices to write $Rf_*\mathcal{O}_ U = \text{hocolim} E_ n$ for a system of perfect objects $E_ n$ of $D(\mathcal{O}_ X)$. Once this is done we obtain (1) by setting $L_ n = E_ n^\vee $, see Cohomology, Lemma 20.50.5.

Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i$ of finite type over $\mathbf{Z}$. By Limits, Lemma 32.10.1 we can find an $i$ and morphisms $U_ i \to X_ i \to \mathop{\mathrm{Spec}}(A_ i)$ of finite presentation whose base change to $\mathop{\mathrm{Spec}}(A)$ recovers $U \to X \to \mathop{\mathrm{Spec}}(A)$. After increasing $i$ we may assume that $f_ i : U_ i \to X_ i$ is flat, see Limits, Lemma 32.8.7. By Lemma 36.22.5 the derived pullback of $Rf_{i, *}\mathcal{O}_{U_ i}$ by $g : X \to X_ i$ is equal to $Rf_*\mathcal{O}_ U$. Since $Lg^*$ commutes with derived colimits, it suffices to prove what we want for $f_ i$. Hence we may assume that $U$ and $X$ are of finite type over $\mathbf{Z}$.

Assume $f : U \to X$ is a morphism of schemes of finite type over $\mathbf{Z}$. To finish the proof we will show that $Rf_*\mathcal{O}_ U$ is a homotopy colimit of perfect complexes. To see this we apply Lemma 36.33.3. Thus it suffices to show that $R^ if_*\mathcal{O}_ U$ has countable sets of sections over affine opens. This follows from Lemma 36.33.2 applied to the structure sheaf. $\square$