The Stacks project

Lemma 36.33.4. Let $A$ be a ring. Let $X$ be a scheme of finite presentation over $A$. Let $f : U \to X$ be a flat morphism of finite presentation. 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 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)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CRR. Beware of the difference between the letter 'O' and the digit '0'.