The Stacks project

Lemma 73.27.3. Let $A$ be a ring. Let $f : U \to X$ be a flat morphism of algebraic spaces of finite presentation over $A$. Then

  1. there exists an inverse system of perfect objects $L_ n$ of $D(\mathcal{O}_ X)$ such that

    \[ R\Gamma (U, Lf^*K) = \text{hocolim}\ R\mathop{\mathrm{Hom}}\nolimits _ X(L_ n, K) \]

    in $D(A)$ functorially in $K$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$, and

  2. there exists a system of perfect objects $E_ n$ of $D(\mathcal{O}_ X)$ such that

    \[ R\Gamma (U, Lf^*K) = \text{hocolim}\ R\Gamma (X, E_ n \otimes ^\mathbf {L} K) \]

    in $D(A)$ functorially in $K$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. By Lemma 73.20.1 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 73.6.2. 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 on Sites, Lemma 21.46.4.

Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i$ of finite type over $\mathbf{Z}$. By Limits of Spaces, Lemma 68.7.1 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 of Spaces, Lemma 68.6.12. By Lemma 73.20.4 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 algebraic spaces 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 73.27.2. Thus it suffices to show that $R^ if_*\mathcal{O}_ U$ has countable sets of sections over affines ├ętale over $X$. This follows from Lemma 73.27.1 applied to the structure sheaf. $\square$


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