The Stacks project

21.43 Modules on a category

The material in this section will be used to define a variant of the derived category of quasi-coherent modules on a stack in groupoids over the category of schemes. See Sheaves on Stacks, Section 95.26.

Let $\mathcal{C}$ be a category. We think of $\mathcal{C}$ as a site with the chaotic topology. As in Example 21.42.2 we let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. In other words, $\mathcal{O}$ is a presheaf of rings on the category $\mathcal{C}$, see Categories, Definition 4.3.3.

Definition 21.43.1. In the situation above, we denote $\mathit{QC}(\mathcal{C}, \mathcal{O})$ or simply $\mathit{QC}(\mathcal{O})$ the full subcategory of $D(\mathcal{O}) = D(\mathcal{C}, \mathcal{O})$ consisting of objects $K$ such that for all $U \to V$ in $\mathcal{C}$ the canonical map

\[ R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \longrightarrow R\Gamma (U, K) \]

is an isomorphism in $D(\mathcal{O}(U))$.

Lemma 21.43.2. In the situation above, the subcategory $\mathit{QC}(\mathcal{O})$ is a strictly full, saturated, triangulated subcategory of $D(\mathcal{O})$ preserved by arbitrary direct sums.

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. $\square$

Lemma 21.43.3. In the situation above, suppose that $M$ is an object of $\mathit{QC}(\mathcal{O})$ and $b \in \mathbf{Z}$ such that $H^ i(M) = 0$ for all $i > b$. Then $H^ b(M)$ is a quasi-coherent module on $(\mathcal{C}, \mathcal{O})$ in the sense of Modules on Sites, Definition 18.23.1.

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

\[ H^ p(M)(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \to H^ b(M)(U) \]

is an isomorphism. We are given that the map

\[ R\Gamma (V, M) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \to R\Gamma (U, M) \]

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

Lemma 21.43.4. In the situation above, suppose that $\mathcal{C}$ has a final object $X$. Set $R = \mathcal{O}(X)$ and denote $f : (\mathcal{C}, \mathcal{O}) \to (pt, R)$ the obvious morphism of sites. Then $\mathit{QC}(\mathcal{O}) = D(R)$ given by $Lf^*$ and $Rf_*$.

Proof. Omitted. $\square$

Lemma 21.43.5. In the situation above, suppose that $K$ is an object of $\mathit{QC}(\mathcal{O})$ and $M$ arbitrary in $D(\mathcal{O})$. For every object $U$ of $\mathcal{C}$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(K|_ U, M|_ U) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U)}(R\Gamma (U, K), R\Gamma (U, M)) \]

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 $Rf_*(-) = R\Gamma (U, -)$. $\square$

Let $(\mathcal{C}, \mathcal{O})$ be as above. For a complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules we define the size $|\mathcal{F}^\bullet |$ of $\mathcal{F}^\bullet $ as

\[ |\mathcal{F}^\bullet | = \left| \coprod \nolimits _{i \in \mathbf{Z},\ U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} \mathcal{F}^ i(U) \right| \]

For an object $K$ of $D(\mathcal{O})$ we define the size $|K|$ of $K$ to be the cardinal

\[ |K| = \min \left\{ \left| \mathcal{F}^\bullet \right| \text{ where }\mathcal{F}^\bullet \text{ represents }K \right\} \]

By properties of cardinals the minimum exists.

Lemma 21.43.6. In the situation above, there exists a cardinal $\kappa $ with the following property: given a complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules and subsets $\Omega ^ i_ U \subset \mathcal{F}^ i(U)$ there exists a subcomplex $\mathcal{H}^\bullet \subset \mathcal{F}^\bullet $ with $\Omega ^ i_ U \subset \mathcal{H}^ i(U)$ and $|\mathcal{H}^\bullet | \leq \max (\kappa , |\bigcup \Omega ^ i_ U|)$.

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. $\square$

Lemma 21.43.7. In the situation above, there exists a cardinal $\kappa $ with the following property: given a complex $\mathcal{F}^\bullet $ of $\mathcal{O}$-modules representing an object $K$ of $D(\mathcal{O})$ there exists a subcomplex $\mathcal{H}^\bullet \subset \mathcal{F}^\bullet $ such that $\mathcal{H}^\bullet $ represents $K$ and such that $|\mathcal{H}^\bullet | \leq \max (\kappa , |K|)$.

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

\[ \mathcal{S}^\bullet = \mathcal{S}_0^\bullet \subset \mathcal{S}_1^\bullet \subset \mathcal{S}_2^\bullet \subset \ldots \subset \mathcal{F}^\bullet \]

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

\[ \mathcal{S}_ n^ i(U) \cap \text{Im}(\text{d} : \mathcal{F}^{i - 1}(U)\to \mathcal{F}^ i(U)) \]

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

\[ \mathcal{S}^ i(U) \cup \Omega ^ i_ U \subset \mathcal{F}^ i(U) \]

and everything is clear. $\square$

Lemma 21.43.8. In the situation above, there exists a cardinal $\kappa $ with the following properties:

  1. for every nonzero object $K$ of $\mathit{QC}(\mathcal{O})$ there exists a nonzero morphism $E \to K$ of $\mathit{QC}(\mathcal{O})$ such that $|E| \leq \kappa $,

  2. for every morphism $\alpha : E \to \bigoplus _ n K_ n$ of $\mathit{QC}(\mathcal{O})$ such that $|E| \leq \kappa $, there exist morphisms $E_ n \to K_ n$ in $\mathit{QC}(\mathcal{O})$ with $|E_ n| \leq \kappa $ such that $\alpha $ factors through $\bigoplus E_ n \to \bigoplus K_ n$.

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

\[ \mathcal{S}^\bullet = \mathcal{S}_0^\bullet \subset \mathcal{S}_1^\bullet \subset \mathcal{S}_2^\bullet \subset \ldots \subset \mathcal{F}^\bullet \]

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

\[ \mathcal{H}^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \longrightarrow \mathcal{H}^\bullet (U) \]

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

\[ \xymatrix{ \mathcal{S}_ n^\bullet (V) \ar[r] \ar[d] & \mathcal{S}_ n^\bullet (U) \ar[d] \\ \mathcal{F}^\bullet (V) \ar[r] & \mathcal{F}^\bullet (U) } \]

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

\[ \mathcal{F}^\bullet (V) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \longrightarrow \mathcal{F}^\bullet (U) \]

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

\[ \mathcal{S}_ n^\bullet (V) \subset M^\bullet _ f \subset \mathcal{F}^\bullet (V) \quad \text{and}\quad \mathcal{S}_ n^\bullet (U) \subset N^\bullet _ f \subset \mathcal{F}^\bullet (U) \]

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

\[ \mathcal{S}_ n^ i(U) \amalg \coprod \nolimits _{f : U \to V} N^ i_ f \amalg \coprod \nolimits _{g : W \to U} M^ i_ g \subset \mathcal{F}^ i(U) \]

to find a subcomplex

\[ \mathcal{S}_ n^\bullet \subset \mathcal{S}_{n + 1}^\bullet \subset \mathcal{F}^\bullet \]

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. $\square$

Proposition 21.43.9. Let $\mathcal{C}$ be a category viewed as a site with the chaotic topology. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. With $\mathit{QC}(\mathcal{O})$ as in Definition 21.43.1 we have

  1. $\mathit{QC}(\mathcal{O})$ is a strictly full, saturated, triangulated subcategory of $D(\mathcal{O})$ preserved by arbitrary direct sums,

  2. any contravariant cohomological functor $H : \mathit{QC}(\mathcal{O}) \to \textit{Ab}$ which transforms direct sums into products is representable,

  3. any exact functor $F : \mathit{QC}(\mathcal{O}) \to \mathcal{D}$ of triangulated categories which transforms direct sums into direct sums has an exact right adjoint, and

  4. the inclusion functor $\mathit{QC}(\mathcal{O}) \to D(\mathcal{O})$ has an exact right adjoint.

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). $\square$

Let $u : \mathcal{C}' \to \mathcal{C}$ be a functor between categories. If we view $\mathcal{C}$ and $\mathcal{C}'$ as sites with the chaotic topology, then $u$ is a continuous and cocontinuous functor. Hence we obtain a morphism $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of topoi, see Sites, Lemma 7.21.1. Additionally, suppose given sheaves of rings $\mathcal{O}$ on $\mathcal{C}$ and $\mathcal{O}'$ on $\mathcal{C}'$ and a map $g^\sharp : g^{-1}\mathcal{O} \to \mathcal{O}'$. We denote the corresponding morphism of ringed topoi simply $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, see Modules on Sites, Section 18.7.

Lemma 21.43.10. Let $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be as above. Then the functor $Lg^* : D(\mathcal{O}) \to D(\mathcal{O}')$ maps $\mathit{QC}(\mathcal{O})$ into $\mathit{QC}(\mathcal{O}')$.

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

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}'(U')) \ar[rr]_{U'} \ar[d] & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \ar[d]^ g \\ (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U)) \ar[rr]^ U & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) } \]

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

\[ R\Gamma (U', Lg^*K) = R\Gamma (U, K) \otimes _{\mathcal{O}(U)}^\mathbf {L} \mathcal{O}'(U') \]

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

\begin{align*} R\Gamma (V', Lg^*K) \otimes _{\mathcal{O}'(V')}^\mathbf {L} \mathcal{O}'(U') & = R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}'(V') \otimes _{\mathcal{O}'(V')}^\mathbf {L} \mathcal{O}'(U') \\ & = R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}'(U') \\ & = R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \otimes _{\mathcal{O}(U)}^\mathbf {L} \mathcal{O}'(U') \\ & = R\Gamma (U, K) \otimes _{\mathcal{O}(U)}^\mathbf {L} \mathcal{O}'(U') \\ & = R\Gamma (U', Lg^*K) \end{align*}

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

Lemma 21.43.11. Let $\mathcal{C}$ be a category viewed as a site with the chaotic topology. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Assume for all $U \to V$ in $\mathcal{C}$ the restriction map $\mathcal{O}(V) \to \mathcal{O}(U)$ is a flat ring map. Then $\mathit{QC}(\mathcal{O})$ agrees with the subcategory $D_\mathit{QCoh}(\mathcal{O}) \subset D(\mathcal{O})$ of complexes whose cohomology sheaves are quasi-coherent.

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

\[ D_\mathit{QCoh}(\mathcal{O}) = D_{\mathit{QCoh}(\mathcal{O})}(\textit{Mod}(\mathcal{O})) \]

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

\begin{align*} H^ i(M)(U) & = H^ i(R\Gamma (U, M)) \\ & = H^ i(R\Gamma (V, M) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U)) \\ & = H^ i(R\Gamma (V, M)) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \\ & = H^ i(M)(V) \otimes _{\mathcal{O}(V)} \mathcal{O}(U) \end{align*}

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 $\mathit{QC}(\mathcal{O})$. $\square$

Lemma 21.43.12. Let $\epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '})$ be as in Section 21.27. Assume

  1. $\tau '$ is the chaotic topology on the category $\mathcal{C}$,

  2. for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and all K-flat complexes of $\mathcal{O}(U)$-modules $M^\bullet $ the map

    \[ M^\bullet \longrightarrow R\Gamma ((\mathcal{C}/U)_\tau , (M^\bullet \otimes _{\mathcal{O}(U)} \mathcal{O}_ U)^\# ) \]

    is a quasi-isomorphism (see proof for an explanation).

Then $\epsilon ^*$ and $R\epsilon _*$ define mutually quasi-inverse equivalences between $\mathit{QC}(\mathcal{O})$ and the full subcategory of $D(\mathcal{C}_\tau , \mathcal{O}_\tau )$ consisting of objects $K$ such that $R\epsilon _*K$ is in $\mathit{QC}(\mathcal{O})$1.

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

\[ R\Gamma (U, R\epsilon _*(\epsilon ^*M)) = R\Gamma ((\mathcal{C}/U)_\tau , (\mathcal{M}^\bullet )^\# |_{\mathcal{C}/U}) = R\Gamma ((\mathcal{C}/U)_\tau , (\mathcal{M}^\bullet |_{\mathcal{C}/U})^\# ) \]

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

\[ \mathcal{M}^\bullet (U) \otimes _{\mathcal{O}(U)} \mathcal{O}_ U \longrightarrow \mathcal{M}^\bullet |_{\mathcal{C}/U} \]

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

\[ R\Gamma (U, R\epsilon _*(\epsilon ^*M)) = R\Gamma ((\mathcal{C}/U)_\tau , (M^\bullet \otimes _{\mathcal{O}(U)} \mathcal{O}_ U)^\# ) \]

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

Lemma 21.43.13. Notation and assumptions as in Lemma 21.43.12. Suppose that $K$ is an object of $\mathit{QC}(\mathcal{O})$ and $M$ arbitrary in $D(\mathcal{O}_\tau )$. For every object $U$ of $\mathcal{C}$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D((\mathcal{O}_ U)_\tau )}(\epsilon ^*K|_ U, M|_ U) = R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}(U)}(R\Gamma (U, K), R\Gamma (U, M)) \]

Proof. We have

\[ \mathop{\mathrm{Hom}}\nolimits _{D((\mathcal{O}_ U)_\tau )}(\epsilon ^*K|_ U, M|_ U) = \mathop{\mathrm{Hom}}\nolimits _{D((\mathcal{O}_ U)_{\tau '})}(K|_ U, R\epsilon _*M|_ U) \]

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

\[ R\Gamma (U, M) = R\Gamma (U, R\epsilon _*M) \]

by Leray. $\square$

[1] This means that $R\Gamma (V, K) \otimes _{\mathcal{O}(V)}^\mathbf {L} \mathcal{O}(U) \to R\Gamma (U, K)$ is an isomorphism for all $U \to V$ in $\mathcal{C}$.

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