The Stacks project

Proof. Let $U$ be an object of $\mathcal{C}$. Since the topology on $\mathcal{C}$ is chaotic, the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is exact and commutes with direct sums. Hence the exact functor $K \mapsto R\Gamma (U, K)$ is computed by representing $K$ by any complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules and taking $\mathcal{F}^\bullet (U)$. Thus $R\Gamma (U, -)$ commutes with direct sums, see Injectives, Lemma 19.13.4. Similarly, given a morphism $U \to V$ of $\mathcal{C}$ the derived tensor product functor $- \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) : D(\mathcal{O}(V)) \to D(\mathcal{O}(U))$ is exact and commutes with direct sums. The lemma follows from these observations in a straightforward manner; details omitted. $\square$

Proof. By Modules on Sites, Lemma 18.24.2 it suffices to show that for every morphism $U \to V$ of $\mathcal{C}$ the map

is an isomorphism. We are given that the map

is an isomorphism. Thus the result by the Tor spectral sequence for example. Details omitted. $\square$

Proof. Omitted. $\square$

Proof. We may replace $\mathcal{C}$ by $\mathcal{C}/U$. Thus we may assume $U = X$ is a final object of $\mathcal{C}$. By Lemma 21.43.4 we see that $K = Lf^*P$ where $P = R\Gamma (U, K) = R\Gamma (X, K) = Rf_*K$. Thus the result because $Lf^*$ is the left adjoint to $Rf_*(-) = R\Gamma (U, -)$. $\square$

Proof. Define $\mathcal{H}^ i(U)$ to be the $\mathcal{O}(U)$-submodule of $\mathcal{F}^ i(U)$ generated by the images of $\Omega ^ i_ V$ and $\text{d}(\Omega ^{i - 1}_ U)$ by restriction along any morphism $f : U \to V$. The cardinality of $\mathcal{H}^ i(U)$ is bounded by the maximum of $\aleph _0$, the cardinality of the $\mathcal{O}(U)$, the cardinality of $\text{Arrows}(\mathcal{C})$, and $|\bigcup \Omega ^ i_ U|$. Details omitted. $\square$

Proof. First, for every $i$ and $U$ we choose a subset $\Omega ^ i_ U \subset \mathop{\mathrm{Ker}}(\text{d} : \mathcal{F}^ i(U) \to \mathcal{F}^{i + 1}(U))$ mapping bijectively onto $H^ i(K)(U) = H^ i(\mathcal{F}^\bullet (U))$. Hence $|\Omega ^ i_ U| \leq |K|$ as we may represent $K$ by a complex whose size is $|K|$. Applying Lemma 21.43.6 we find a subcomplex $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet $ of size at most $\max (\kappa , |K|)$ containing $\Omega ^ i_ U$ and hence such that $H^ i(\mathcal{S}^\bullet ) \to H^ i(\mathcal{F}^\bullet )$ is a surjection of sheaves.

We are going to inductively construct subcomplexes

of size $\leq \max (\kappa , |K|)$ such that the kernel of $H^ i(\mathcal{S}_ n^\bullet ) \to H^ i(\mathcal{F}^\bullet )$ is the same as the kernel of $H^ i(\mathcal{S}_ n^\bullet ) \to H^ i(\mathcal{S}_{n + 1}^\bullet )$. Once this is done we can take $\mathcal{H}^\bullet = \bigcup \mathcal{S}_ n^\bullet $ as our solution.

Construction of $\mathcal{S}_{n + 1}^\bullet $ given $\mathcal{S}_ n^\bullet $. For ever $U$ and $i$ let $\Omega ^{i - 1}_ U \subset \mathcal{F}^{i - 1}(U)$ be a subset such that $\text{d} : \mathcal{F}^{i - 1}(U) \to \mathcal{F}^ i(U)$ maps $\Omega ^{i - 1}_ U$ bijectively onto

Observe that $|\Omega ^ i_ U| \leq |K|$ because $\mathcal{S}_ n^ i(U)$ is so bounded. Then we get $\mathcal{S}_{n + 1}^\bullet $ by an application of Lemma 21.43.6 to the subsets

and everything is clear. $\square$

Proof. Let $\kappa $ be an upper bound for the following set of cardinals:

1. $|\coprod _ V j_{U!}\mathcal{O}_ U(V)|$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$,

2. the cardinals $\kappa (\mathcal{O}(V) \to \mathcal{O}(U))$ found in More on Algebra, Lemma 15.102.5 for all morphisms $U \to V$ in $\mathcal{C}$,

3. the cardinal found in Lemma 21.43.7.

We claim that for any complex $\mathcal{F}^\bullet $ representing an object of $\mathit{QC}(\mathcal{O})$ and any subcomplex $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet $ with $|\mathcal{S}^\bullet | \leq \kappa $ there exists a subcomplex $\mathcal{H}^\bullet $ of $\mathcal{F}^\bullet $ containing $\mathcal{S}^\bullet $ such that $\mathcal{H}^\bullet $ represents an object of $\mathit{QC}(\mathcal{O})$ and such that $|\mathcal{H}^\bullet | \leq \kappa $. In the next two paragraphs we show that the claim implies the lemma.

As in (1) let $K$ be a nonzero object of $\mathit{QC}(\mathcal{O})$. Say $K$ is represented by the complex of $\mathcal{O}$-modules $\mathcal{F}^\bullet $. Then $H^ i(\mathcal{F}^\bullet )$ is nonzero for some $i$. Hence there exists an object $U$ of $\mathcal{C}$ and a section $s \in \mathcal{F}^ i(U)$ with $d(s) = 0$ which determines a nonzero section of $H^ i(\mathcal{F}^\bullet )$ over $U$. Then the image of $s : j_{U!}\mathcal{O}_ U[-i] \to \mathcal{F}^\bullet $ is a subcomplex $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet $ with $|\mathcal{S}^\bullet | \leq \kappa $. Applying the claim we get $\mathcal{H}^\bullet \to \mathcal{F}^\bullet $ in $\mathit{QC}(\mathcal{O})$ nonzero with $|\mathcal{H}^\bullet | \leq \kappa $. Thus (1) holds.

Let $\alpha : E \to \bigoplus K_ n$ be as in (2). Choose any complexes $\mathcal{K}_ n^\bullet $ representing $K_ n$. Then $\bigoplus \mathcal{K}_ n^\bullet $ represents $\bigoplus K_ n$. By the construction of the derived category we can represent $E$ by a complex $\mathcal{E}^\bullet $ such that $\alpha $ is represented by a morphism $a : \mathcal{E}^\bullet \to \bigoplus \mathcal{K}_ n^\bullet $ of complexes. By Lemma 21.43.7 and our choice of $\kappa $ above we may assume $|\mathcal{E}^\bullet | \leq \kappa $. By the claim we get subcomplexes $\mathcal{E}_ n^\bullet \subset \mathcal{K}_ n^\bullet $ representing objects $E_ n$ of $\mathit{QC}(\mathcal{O})$ with $|E_ n| \leq \kappa $ containing the image of $a_ n : \mathcal{E}^\bullet \to \mathcal{K}_ n^\bullet $ as desired.

Proof of the claim. Let $\mathcal{F}^\bullet $ be a complex representing an object of $\mathit{QC}(\mathcal{O})$ and let $\mathcal{S}^\bullet \subset \mathcal{F}^\bullet $ be a subcomplex of size $\leq \kappa $. We are going to inductively construct subcomplexes

of size $\leq \kappa $ such that for every morphism $f : U \to V$ of $\mathcal{C}$ and every $i \in \mathbf{Z}$

1. the kernel of the arrow $H^ i(\mathcal{S}_ n^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U)) \to H^ i(\mathcal{S}_ n^\bullet (U))$ maps to zero in $H^ i(\mathcal{S}_{n + 1}^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U))$,

2. the image of the arrow $H^ i(\mathcal{S}_ n^\bullet (U)) \to H^ i(\mathcal{S}_{n + 1}^\bullet (U))$ is contained in the image of $H^ i(\mathcal{S}_{n + 1}^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U)) \to H^ i(\mathcal{S}_{n + 1}^\bullet (U))$,

Once this is done we can set $\mathcal{H}^\bullet = \bigcup \mathcal{S}_ n^\bullet $. Namely, since derived tensor product and taking cohomology of complexes of modules over rings commute with filtered colimits, the conditions (1) and (2) together will guarantee that

is an isomorphism on cohomology in all degrees and hence an isomorphism in $D(\mathcal{O}(U))$ for all $f : U \to V$ in $\mathcal{C}$. Hence $\mathcal{H}^\bullet $ represents an object of $\mathit{QC}(\mathcal{O})$ as desired.

Construction of $\mathcal{S}_{n + 1}$ given $\mathcal{S}_ n$. For every morphism $f : U \to V$ of $\mathcal{C}$ we consider the commutative diagram

This is a diagram as in More on Algebra, Lemma 15.102.5 for the ring map $\mathcal{O}(V) \to \mathcal{O}(U)$, i.e., the bottom row induces an isomorphism

in $D(\mathcal{O}(U))$. Thus we may choose subcomplexes

as in More on Algebra, Lemma 15.102.5 and in particular we see that $|N^ i_ f|, |M^ i_ f| \leq \kappa $. Next, we apply Lemma 21.43.6 using the subsets

to find a subcomplex

with containing those subsets and such that $|\mathcal{S}_{n + 1}^\bullet | \leq \kappa $. Conditions (1) and (2) hold because the corresponding statements hold for $\mathcal{S}_ n^\bullet (V) \subset M^\bullet _ f$ and $\mathcal{S}_ n^\bullet (U) \subset N^\bullet _ f$ by the construction in More on Algebra, Lemma 15.102.5. Thus the proof is complete. $\square$

Proof. Part (1) is Lemma 21.43.2. Part (2) follows from Lemma 21.43.8 and Derived Categories, Lemma 13.39.1. Part (3) follows from Lemma 21.43.8 and Derived Categories, Proposition 13.39.2. Part (4) is a special case of (3). $\square$

Proof. Let $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ with image $U = u(U')$ in $\mathcal{C}$. Let $pt$ denote the category with a single object and a single morphism. Denote $(\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}'(U'))$ and $(\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U))$ the ringed topoi as indicated. Of course we identify the derived category of modules on these ringed topoi with $D(\mathcal{O}'(U'))$ and $D(\mathcal{O}(U))$. Then we have a commutative diagram of ringed topoi

Pullback along the lower horizontal morphism sends $K$ in $D(\mathcal{O})$ to $R\Gamma (U, K)$. Pullback by the left vertical arrow sends $M$ to $M \otimes _{\mathcal{O}(U)}^\mathbf {L} \mathcal{O}'(U')$. Going around the diagram either direction produces the same result (Lemma 21.18.3) and hence we conclude

Finally, let $f' : U' \to V'$ be a morphism in $\mathcal{C}'$ and denote $f = u(f') : U = u(U') \to V = u(V')$ the image in $\mathcal{C}$. If $K$ is in $\mathit{QC}(\mathcal{O})$ then we have

as desired. Here we have used the observation above both for $U'$ and $V'$. $\square$

Proof. Recall that $\mathit{QCoh}(\mathcal{O}) \subset \textit{Mod}(\mathcal{O})$ is a weak Serre subcategory under our assumptions, see Modules on Sites, Lemma 18.24.3. Thus taking the full subcategory

of $D(\mathcal{O})$ makes sense, see Derived Categories, Section 13.17. (Strictly speaking we don't need this in the proof of the lemma.)

Let $M$ be an object of $\mathit{QC}(\mathcal{O})$. Since for every morphism $U \to V$ in $\mathcal{C}$ the restriction map $\mathcal{O}(V) \to \mathcal{O}(U)$ is flat, we see that

and hence $H^ i(M)$ is quasi-coherent by Modules on Sites, Lemma 18.24.2. The first and last equality above follow from the fact that taking sections over an object of $\mathcal{C}$ is an exact functor due to the fact that the topology on $\mathcal{C}$ is chaotic.

Conversely, if $M$ is an object of $D_\mathit{QCoh}(\mathcal{O})$, then due to Modules on Sites, Lemma 18.24.2 we see that the map $R\Gamma (V, M) \to R\Gamma (U, M)$ induces isomorphisms $H^ i(M)(U) \to H^ i(M)(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U)$. Whence $R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \to R\Gamma (U, K)$ is an isomorphism in $D(\mathcal{O}(U))$ by the flatness of $\mathcal{O}(V) \to \mathcal{O}(U)$ and we conclude that $M$ is in $\mathit{QC}(\mathcal{O})$. $\square$

Proof. We will use the observations made in Section 21.27 without further mention. Since $R\epsilon _*$ is fully faithful and $\epsilon ^* \circ R\epsilon _* = \text{id}$, to prove the lemma it suffices to show that for $M$ in $\mathit{QC}(\mathcal{O})$ we have $R\epsilon _*(\epsilon ^*M) = M$. Condition (2) is exactly the condition needed to see this. Namely, we choose a K-flat complex $\mathcal{M}^\bullet $ of $\mathcal{O}$-modules with flat terms representing $M$. Then we see that $\epsilon ^*M$ is represented by the $\tau $-sheafification $(\mathcal{M}^\bullet )^\# $ of $\mathcal{M}^\bullet $. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. By Leray we get

The last equality since sheafification commutes with restriction to $\mathcal{C}/U$. As usual, denote $\mathcal{O}_ U$ the restriction of $\mathcal{O}$ to $\mathcal{C}/U$. Consider the map

of complexes of $\mathcal{O}_ U$-modules (in $\tau '$-topology). By our choice of $\mathcal{M}^\bullet $ the complex $\mathcal{M}^\bullet (U)$ is a K-flat complex of $\mathcal{O}(U)$-modules; see Lemma 21.18.1 and use that the inclusion of $U$ into $\mathcal{C}$ defines a morphism of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U)) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\tau '}), \mathcal{O})$. Since $M$ is in $\mathit{QC}(\mathcal{O})$ we conclude that the displayed arrow is a quasi-isomorphism. Since sheafification is exact, we see that the same remains true after sheafification. Hence

and assumption (2) tells us this is equal to $R\Gamma (U, M) = \mathcal{M}^\bullet (U)$ as desired. $\square$

Proof. We have

by adjunction. Hence the result by Lemma 21.43.5 and the fact that

by Leray. $\square$