The Stacks project

36.33 Other applications

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

Lemma 36.33.1. Let $R$ be a coherent ring. Let $X$ be a scheme of finite presentation over $R$. Let $\mathcal{G}$ be an $\mathcal{O}_ X$-module of finite presentation, flat over $R$, with support proper over $R$. Then $H^ i(X, \mathcal{G})$ is a coherent $R$-module.

Lemma 36.33.2. Let $X$ be a quasi-compact and quasi-separated scheme. 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 opens. Then for any quasi-compact open $U \subset X$ 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, Lemma 20.47.5 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 Cohomology of Schemes, Lemma 30.4.1.

First we show that it holds when $U = \mathop{\mathrm{Spec}}(A)$ is affine. Namely, we can represent $K$ by a complex of $A$-modules $K^\bullet $ and $E$ by a finite complex of finite projective $A$-modules $P^\bullet $. See Lemmas 36.3.5 and 36.10.7 and our definition of perfect complexes of $A$-modules (More on Algebra, Definition 15.73.1). Then $(E \otimes ^\mathbf {L} K)|_ U$ is represented by the total complex associated to the double complex $P^\bullet \otimes _ A K^\bullet $ (Lemma 36.3.9). Using induction on the length of the complex $P^\bullet $ (or using a suitable spectral sequence) we see that it suffices to show that $H^ i(P^ a \otimes _ A K^\bullet )$ is countable for each $a$. Since $P^ a$ is a direct summand of $A^{\oplus n}$ for some $n$ this follows from the assumption that the cohomology group $H^ i(K^\bullet )$ is countable.

To finish the proof it suffices to show: if $U = V \cup W$ and the result holds for $V$, $W$, and $V \cap W$, then the result holds for $U$. This is an immediate consquence of the Mayer-Vietoris sequence, see Cohomology, Lemma 20.33.4. $\square$

Lemma 36.33.3. Let $X$ be a quasi-compact and quasi-separated scheme such that the sets of sections of $\mathcal{O}_ X$ over affine opens 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 affine opens.

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 36.18.3 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 36.33.2. $\square$

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. $\square$

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