**Proof.**
By Lemma 36.22.1 we have

\[ R\Gamma (U, Lf^*K) = R\Gamma (X, Rf_*\mathcal{O}_ U \otimes ^\mathbf {L} K) \]

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 20.47.5.

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.
$\square$

## Comments (0)