The Stacks project

75.27 Other applications

In this section we state and prove some results that can be deduced from the theory worked out above.

Lemma 75.27.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ such that the cohomology sheaves $H^ i(K)$ have countable sets of sections over affine schemes étale over $X$. Then for any quasi-compact and quasi-separated étale morphism $U \to X$ and any perfect object $E$ in $D(\mathcal{O}_ X)$ the sets

\[ H^ i(U, K \otimes ^\mathbf {L} E),\quad \mathop{\mathrm{Ext}}\nolimits ^ i(E|_ U, K|_ U) \]

are countable.

Proof. Using Cohomology on Sites, Lemma 21.48.4 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 Lemma 75.9.3.

When $U = \mathop{\mathrm{Spec}}(A)$ is affine the result follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.33.2.

To finish the proof it suffices to show: if $(U \subset W, V \to W)$ is an elementary distinguished triangle and the result holds for $U$, $V$, and $U \times _ W V$, then the result holds for $W$. This is an immediate consequence of the Mayer-Vietoris sequence, see Lemma 75.10.5. $\square$

Lemma 75.27.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Assume the sets of sections of $\mathcal{O}_ X$ over affines étale over $X$ are countable. Let $K$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent

  1. $K = \text{hocolim} E_ n$ with $E_ n$ a perfect object of $D(\mathcal{O}_ X)$, and

  2. the cohomology sheaves $H^ i(K)$ have countable sets of sections over affines étale over $X$.

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 75.17.3 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

\[ (A, \text{d}) = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)} (\mathcal{I}^\bullet , \mathcal{I}^\bullet ) \]

which maps $K$ to the differential graded module

\[ M = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)} (\mathcal{I}^\bullet , \mathcal{K}^\bullet ) \]

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 75.27.1. $\square$

Lemma 75.27.3. 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 75.20.1 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 75.6.2. 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 21.48.4.

Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i$ of finite type over $\mathbf{Z}$. By Limits of Spaces, Lemma 70.7.1 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 70.6.12. By Lemma 75.20.4 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 75.27.2. Thus it suffices to show that $R^ if_*\mathcal{O}_ U$ has countable sets of sections over affines étale over $X$. This follows from Lemma 75.27.1 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 0CRT. Beware of the difference between the letter 'O' and the digit '0'.