103.5 Derived categories of quasi-coherent modules

Let $\mathcal{X}$ be an algebraic stack. As the inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ isn't exact, we cannot define $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ as the full subcategory of $D(\mathcal{O}_\mathcal {X})$ consisting of complexes with quasi-coherent cohomology sheaves. Instead we define the derived category of quasi-coherent modules as a quotient by analogy with Cohomology of Stacks, Remark 102.10.7.

Recall that $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$ denotes the full subcategory of locally quasi-coherent $\mathcal{O}_\mathcal {X}$-modules with the flat base change property, see Cohomology of Stacks, Section 102.8. We will abbreviate

$D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) = D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})}(\mathcal{O}_\mathcal {X})$

From Derived Categories, Lemma 13.17.1 and Cohomology of Stacks, Proposition 102.8.1 part (2) we deduce that $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ is a strictly full, saturated triangulated subcategory of $D(\mathcal{O}_\mathcal {X})$.

Let $\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$ denote the full subcategory of parasitic $\mathcal{O}_\mathcal {X}$-modules, see Cohomology of Stacks, Section 102.9. Let us abbreviate

$D_{\textit{Parasitic}}(\mathcal{O}_\mathcal {X}) = D_{\textit{Parasitic}(\mathcal{O}_\mathcal {X})}(\mathcal{O}_\mathcal {X})$

As before this is a strictly full, saturated triangulated subcategory of $D(\mathcal{O}_\mathcal {X})$ since $\textit{Parasitic}(\mathcal{O}_\mathcal {X})$ is a Serre subcategory of $\textit{Mod}(\mathcal{O}_\mathcal {X})$, see Cohomology of Stacks, Lemma 102.9.2.

The intersection of the weak Serre subcategories $\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ of $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is another one. Let us similarly abbreviate

\begin{align*} D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) & = D_{\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})}(\mathcal{O}_\mathcal {X}) \\ & = D_{\textit{Parasitic}}(\mathcal{O}_\mathcal {X}) \cap D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \end{align*}

As before this is a strictly full, saturated triangulated subcategory of $D(\mathcal{O}_\mathcal {X})$. Hence a fortiori it is a strictly full, saturated triangulated subcategory of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$.

Definition 103.5.1. Let $\mathcal{X}$ be an algebraic stack. With notation as above we define the derived category of $\mathcal{O}_\mathcal {X}$-modules with quasi-coherent cohomology sheaves as the Verdier quotient1

$D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) = D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})/ D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$

The Verdier quotient is defined in Derived Categories, Section 13.6. A morphism $a : E \to E'$ of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ becomes an isomorphism in $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ if and only if the cone $C(a)$ has parasitic cohomology sheaves, see Derived Categories, Lemma 13.6.10.

Consider the functors

$D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \xrightarrow {H^ i} \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \xrightarrow {Q} \mathit{QCoh}(\mathcal{O}_\mathcal {X})$

Note that $Q$ annihilates the subcategory $\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, see Cohomology of Stacks, Lemma 102.10.2. By Derived Categories, Lemma 13.6.8 we obtain a cohomological functor

103.5.1.1
$$\label{stacks-perfect-equation-Hi-quasi-coherent} H^ i : D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {X})$$

Moreover, note that $E \in D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is zero if and only if $H^ i(E) = 0$ for all $i \in \mathbf{Z}$ since the kernel of $Q$ is exactly equal to $\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ by Cohomology of Stacks, Lemma 102.10.2.

Note that the categories $\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ are also weak Serre subcategories of the abelian category $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ of modules in the étale topology, see Cohomology of Stacks, Proposition 102.8.1 and Lemma 102.9.2. Hence the statement of the following lemma makes sense.

Lemma 103.5.2. Let $\mathcal{X}$ be an algebraic stack. Abbreviate $\mathcal{P}_\mathcal {X} = \textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. The comparison morphism $\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_{\acute{e}tale}$ induces a commutative diagram

$\xymatrix{ D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \ar[r] & D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \ar[r] & D(\mathcal{O}_\mathcal {X}) \\ D_{\mathcal{P}_\mathcal {X}}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \ar[r] \ar[u]^{\epsilon ^*} & D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \ar[r] \ar[u]^{\epsilon ^*} & D(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \ar[u]^{\epsilon ^*} }$

Moreover, the left two vertical arrows are equivalences of triangulated categories, hence we also obtain an equivalence

$D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})} (\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) / D_{\mathcal{P}_\mathcal {X}}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$

Proof. Since $\epsilon ^*$ is exact it is clear that we obtain a diagram as in the statement of the lemma. We will show the middle vertical arrow is an equivalence by applying Cohomology on Sites, Lemma 21.29.1 to the following situation: $\mathcal{C} = \mathcal{X}$, $\tau = fppf$, $\tau ' = {\acute{e}tale}$, $\mathcal{O} = \mathcal{O}_\mathcal {X}$, $\mathcal{A} = \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, and $\mathcal{B}$ is the set of objects of $\mathcal{X}$ lying over affine schemes. To see the lemma applies we have to check conditions (1), (2), (3), (4). Conditions (1) and (2) are clear from the discussion above (explicitly this follows from Cohomology of Stacks, Proposition 102.8.1). Condition (3) holds because every scheme has a Zariski open covering by affines. Condition (4) follows from Descent, Lemma 35.12.4.

We omit the verification that the equivalence of categories $\epsilon ^* : D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \to D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ induces an equivalence of the subcategories of complexes with parasitic cohomology sheaves. $\square$

Let $\mathcal{X}$ be an algebraic stack. By Cohomology of Stacks, Lemma 102.16.4 the category of quasi-coherent modules $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ forms a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and the category of quasi-coherent modules $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$ forms a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$. Thus we can consider

$D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) = D_{\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})}( \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \subset D(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$

and similarly

$D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}}) = D_{\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})}( \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \subset D(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$

As above these are strictly full, saturated triangulated subcategories. It turns out that $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is equivalent to either of these.

Lemma 103.5.3. Let $\mathcal{X}$ be an algebraic stack. Set $\mathcal{P}_\mathcal {X} = \textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

1. Let $\mathcal{F}^\bullet$ be an object of $D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})} (\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$. With $g$ as in Cohomology of Stacks, Lemma 102.14.2 for the lisse-étale site we have

1. $g^*\mathcal{F}^\bullet$ is in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$,

2. $g^*\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet$ is in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$,

3. $Lg_!\mathcal{H}^\bullet$ is in $D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ for $\mathcal{H}^\bullet$ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$, and

4. the functors $g^*$ and $Lg_!$ define mutually inverse functors

$\xymatrix{ D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^*} & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \ar@<1ex>[l]^-{Lg_!} }$
2. Let $\mathcal{F}^\bullet$ be an object of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. With $g$ as in Cohomology of Stacks, Lemma 102.14.2 for the flat-fppf site we have

1. $g^*\mathcal{F}^\bullet$ is in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$,

2. $g^*\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet$ is in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$,

3. $Lg_!\mathcal{H}^\bullet$ is in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ for $\mathcal{H}^\bullet$ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$, and

4. the functors $g^*$ and $Lg_!$ define mutually inverse functors

$\xymatrix{ D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^*} & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{Lg_!} }$

Proof. The functor $g^* = g^{-1}$ is exact, hence (1)(a), (2)(a), (1)(b), and (2)(b) follow from Cohomology of Stacks, Lemmas 102.16.3 and 102.14.6.

Proof of (1)(c) and (2)(c). The construction of $Lg_!$ in Lemma 103.3.1 (via Cohomology on Sites, Lemma 21.37.2 which in turn uses Derived Categories, Proposition 13.29.2) shows that $Lg_!$ on any object $\mathcal{H}^\bullet$ of $D(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ is computed as

$Lg_!\mathcal{H}^\bullet = \mathop{\mathrm{colim}}\nolimits g_!\mathcal{K}_ n^\bullet = g_! \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet$

(termwise colimits) where the quasi-isomorphism $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{H}^\bullet$ induces quasi-isomorphisms $\mathcal{K}_ n^\bullet \to \tau _{\leq n} \mathcal{H}^\bullet$. Since the inclusion functors

$\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \quad \text{and}\quad \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$

are compatible with filtered colimits we see that it suffices to prove (c) on bounded above complexes $\mathcal{H}^\bullet$ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$. In this case to show that $H^ n(Lg_!\mathcal{H}^\bullet )$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ we can argue by induction on the integer $m$ such that $\mathcal{H}^ i = 0$ for $i > m$. If $m < n$, then $H^ n(Lg_!\mathcal{H}^\bullet ) = 0$ and the result holds. In general consider the distinguished triangle

$\tau _{\leq m - 1}\mathcal{H}^\bullet \to \mathcal{H}^\bullet \to H^ m(\mathcal{H}^\bullet )[-m] \to \ldots$

(Derived Categories, Remark 13.12.4) and apply the functor $Lg_!$. Since $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ is a weak Serre subcategory of the module category it suffices to prove (c) for two out of three. We have the result for $Lg_!\tau _{\leq m - 1}\mathcal{H}^\bullet$ by induction and we have the result for $Lg_!H^ m(\mathcal{H}^\bullet )[-m]$ by Lemma 103.3.3. Whence (c) holds.

Let us prove (2)(d). By (2)(a) and (2)(b) the functor $g^{-1} = g^*$ induces a functor

$c : D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$

see Derived Categories, Lemma 13.6.8. Thus we have the following diagram of triangulated categories

$\xymatrix{ D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \ar[rd]^{g^{-1}} \ar[rr]_ q & & D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar[ld]^ c \\ & D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \ar@<1ex>[lu]^{Lg_!} }$

where $q$ is the quotient functor, the inner triangle is commutative, and $g^{-1}Lg_! = \text{id}$. For any object of $E$ of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ the map $a : Lg_!g^{-1}E \to E$ maps to a quasi-isomorphism in $D(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Hence the cone on $a$ maps to zero under $g^{-1}$ and by (2)(b) we see that $q(a)$ is an isomorphism. Thus $q \circ Lg_!$ is a quasi-inverse to $c$.

In the case of the lisse-étale site exactly the same argument as above proves that

$D_{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) / D_{\mathcal{P}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$

is equivalent to $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$. Applying the last equivalence of Lemma 103.5.2 finishes the proof. $\square$

The following lemma tells us that the quotient functor $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \to D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has a left adjoint. See Remark 103.5.5.

Lemma 103.5.4. Let $\mathcal{X}$ be an algebraic stack. Let $E$ be an object of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. There exists a canonical distinguished triangle

$E' \to E \to P \to E'[1]$

in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ such that $P$ is in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}} (\mathcal{O}_\mathcal {X})$ and

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {X})}(E', P') = 0$

for all $P'$ in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$.

Proof. Consider the morphism of ringed topoi $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat, fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ studied in Cohomology of Stacks, Section 102.14. Set $E' = Lg_!g^*E$ and let $P$ be the cone on the adjunction map $E' \to E$, see Lemma 103.3.1 part (4). By Lemma 103.5.3 parts (2)(a) and (2)(c) we have that $E'$ is in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. Hence also $P$ is in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. The map $g^*E' \to g^*E$ is an isomorphism as $g^*Lg_! = \text{id}$ by Lemma 103.3.1 part (4). Hence $g^*P = 0$ and whence $P$ is an object of $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ by Lemma 103.5.3 part (2)(b). Finally, for $P'$ in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ we have

$\mathop{\mathrm{Hom}}\nolimits (E', P') = \mathop{\mathrm{Hom}}\nolimits (Lg_!g^*E, P') = \mathop{\mathrm{Hom}}\nolimits (g^*E, g^*P') = 0$

as $g^*P' = 0$ by Lemma 103.5.3 part (2)(b). The distinguished triangle $E' \to E \to P \to E'[1]$ is canonical (more precisely unique up to isomorphism of triangles induces the identity on $E$) by the discussion in Derived Categories, Section 13.40. $\square$

Remark 103.5.5. The result of Lemma 103.5.4 tells us that

$D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \subset D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$

is a left admissible subcategory, see Derived Categories, Section 13.40. In particular, if $\mathcal{A} \subset D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ denotes its left orthogonal, then Derived Categories, Proposition 13.40.10 implies that $\mathcal{A}$ is right admissible in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ and that the composition

$\mathcal{A} \longrightarrow D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$

is an equivalence. This means that we can view $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ as a strictly full saturated triangulated subcategory of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ and also of $D(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$.

[1] This definition is different from the one in the literature, see [6.3, olsson_sheaves], but it agrees with that definition by Lemma 103.5.3.

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