The Stacks project

73.27 Other applications

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

Lemma 73.27.1. 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 21.46.4 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 73.9.3.

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 consquence of the Mayer-Vietoris sequence, see Lemma 73.10.5. $\square$

Lemma 73.27.2. 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 73.17.3 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 73.27.1. $\square$

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 étale over $X$. This follows from Lemma 73.27.1 applied to the structure sheaf. $\square$

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