Definition 75.5.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The derived category of $\mathcal{O}_ X$-modules with quasi-coherent cohomology sheaves is denoted $D_\mathit{QCoh}(\mathcal{O}_ X)$.
75.5 Derived category of quasi-coherent modules
Let $S$ be a scheme. Lemma 75.4.2 shows that the category $D_\mathit{QCoh}(\mathcal{O}_ S)$ can be defined in terms of complexes of $\mathcal{O}_ S$-modules on the scheme $S$ or by complexes of $\mathcal{O}$-modules on the small étale site of $S$. Hence the following definition is compatible with the definition in the case of schemes.
This makes sense by Properties of Spaces, Lemma 66.29.7 and Derived Categories, Section 13.17. Thus we obtain a canonical functor
75.5.1.1
\begin{equation} \label{spaces-perfect-equation-compare} D(\mathit{QCoh}(\mathcal{O}_ X)) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X) \end{equation}
see Derived Categories, Equation (13.17.1.1).
Observe that a flat morphism $f : Y \to X$ of algebraic spaces induces an exact functor $f^* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ Y)$, see Morphisms of Spaces, Lemma 67.30.9 and Modules on Sites, Lemma 18.31.2. In particular $Lf^* : D(\mathcal{O}_ X) \to D(\mathcal{O}_ Y)$ is computed on any representative complex (Derived Categories, Lemma 13.16.9). We will write $Lf^* = f^*$ when $f$ is flat and we have $H^ i(f^*E) = f^*H^ i(E)$ for $E$ in $D(\mathcal{O}_ X)$ in this case. We will use this often when $f$ is étale. Of course in the étale case the pullback functor is just the restriction to $Y_{\acute{e}tale}$, see Properties of Spaces, Equation (66.26.1.1).
Lemma 75.5.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\mathcal{O}_ X)$. The following are equivalent
$E$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$,
for every étale morphism $\varphi : U \to X$ where $U$ is an affine scheme $\varphi ^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$,
for every étale morphism $\varphi : U \to X$ where $U$ is a scheme $\varphi ^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$,
there exists a surjective étale morphism $\varphi : U \to X$ where $U$ is a scheme such that $\varphi ^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$, and
there exists a surjective étale morphism of algebraic spaces $f : Y \to X$ such that $Lf^*E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ Y)$.
Proof. This follows immediately from the discussion preceding the lemma and Properties of Spaces, Lemma 66.29.6. $\square$
Lemma 75.5.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $D_\mathit{QCoh}(\mathcal{O}_ X)$ has direct sums.
Proof. By Injectives, Lemma 19.13.4 the derived category $D(\mathcal{O}_ X)$ has direct sums and they are computed by taking termwise direct sums of any representatives. Thus it is clear that the cohomology sheaf of a direct sum is the direct sum of the cohomology sheaves as taking direct sums is an exact functor (in any Grothendieck abelian category). The lemma follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see Properties of Spaces, Lemma 66.29.7. $\square$
We will need some information on derived limits. We warn the reader that in the lemma below the derived limit will typically not be an object of $D_\mathit{QCoh}$.
Lemma 75.5.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $(K_ n)$ be an inverse system of $D_\mathit{QCoh}(\mathcal{O}_ X)$ with derived limit $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ in $D(\mathcal{O}_ X)$. Assume $H^ q(K_{n + 1}) \to H^ q(K_ n)$ is surjective for all $q \in \mathbf{Z}$ and $n \geq 1$. Then
$H^ q(K) = \mathop{\mathrm{lim}}\nolimits H^ q(K_ n)$,
$R\mathop{\mathrm{lim}}\nolimits H^ q(K_ n) = \mathop{\mathrm{lim}}\nolimits H^ q(K_ n)$, and
for every affine open $U \subset X$ we have $H^ p(U, \mathop{\mathrm{lim}}\nolimits H^ q(K_ n)) = 0$ for $p > 0$.
Proof. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ be the set of affine objects. Since $H^ q(K_ n)$ is quasi-coherent we have $H^ p(U, H^ q(K_ n)) = 0$ for $U \in \mathcal{B}$ by the discussion in Cohomology of Spaces, Section 69.3 and Cohomology of Schemes, Lemma 30.2.2. Moreover, the maps $H^0(U, H^ q(K_{n + 1})) \to H^0(U, H^ q(K_ n))$ are surjective for $U \in \mathcal{B}$ by similar reasoning. Part (1) follows from Cohomology on Sites, Lemma 21.23.12 whose conditions we have just verified. Parts (2) and (3) follow from Cohomology on Sites, Lemma 21.23.5. $\square$
Lemma 75.5.5. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. The functor $Lf^*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$.
Proof. Choose a diagram
\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]
where $U$ and $V$ are schemes, the vertical arrows are étale, and $a$ is surjective. Since $a^* \circ Lf^* = Lh^* \circ b^*$ the result follows from Lemma 75.5.2 and the case of schemes which is Derived Categories of Schemes, Lemma 36.3.8. $\square$
Lemma 75.5.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For objects $K, L$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the derived tensor product $K \otimes ^\mathbf {L} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.
Proof. Let $\varphi : U \to X$ be a surjective étale morphism from a scheme $U$. Since $\varphi ^*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) = \varphi ^*K \otimes _{\mathcal{O}_ U}^\mathbf {L} \varphi ^*L$ we see from Lemma 75.5.2 that this follows from the case of schemes which is Derived Categories of Schemes, Lemma 36.3.9. $\square$
The following lemma will help us to “compute” a right derived functor on an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$.
Lemma 75.5.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Then the map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}E$ of Derived Categories, Remark 13.34.4 is an isomorphism1.
Proof. Denote $\mathcal{H}^ i = H^ i(E)$ the $i$th cohomology sheaf of $E$. Let $\mathcal{B}$ be the set of affine objects of $X_{\acute{e}tale}$. Then $H^ p(U, \mathcal{H}^ i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$, and all $U \in \mathcal{B}$ as $U$ is an affine scheme. See discussion in Cohomology of Spaces, Section 69.3 and Cohomology of Schemes, Lemma 30.2.2. Thus the lemma follows from Cohomology on Sites, Lemma 21.23.10 with $d = 0$. $\square$
Lemma 75.5.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $F : \textit{Mod}(\mathcal{O}_ X) \to \textit{Ab}$ be a functor and $N \geq 0$ an integer. Assume that
$F$ is left exact,
$F$ commutes with countable direct products,
$R^ pF(\mathcal{F}) = 0$ for all $p \geq N$ and $\mathcal{F}$ quasi-coherent.
Then for $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$
$H^ i(RF(\tau _{\leq a}E) \to H^ i(RF(E))$ is an isomorphism for $i \leq a$,
$H^ i(RF(E)) \to H^ i(RF(\tau _{\geq b - N + 1}E))$ is an isomorphism for $i \geq b$,
if $H^ i(E) = 0$ for $i \not\in [a, b]$ for some $-\infty \leq a \leq b \leq \infty $, then $H^ i(RF(E)) = 0$ for $i \not\in [a, b + N - 1]$.
Proof. Statement (1) is Derived Categories, Lemma 13.16.1.
Proof of statement (2). Write $E_ n = \tau _{\geq -n}E$. We have $E = R\mathop{\mathrm{lim}}\nolimits E_ n$, see Lemma 75.5.7. Thus $RF(E) = R\mathop{\mathrm{lim}}\nolimits RF(E_ n)$ in $D(\textit{Ab})$ by Injectives, Lemma 19.13.6. Thus for every $i \in \mathbf{Z}$ we have a short exact sequence
\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{i - 1}(RF(E_ n)) \to H^ i(RF(E)) \to \mathop{\mathrm{lim}}\nolimits H^ i(RF(E_ n)) \to 0 \]
see More on Algebra, Remark 15.86.10. To prove (2) we will show that the term on the left is zero and that the term on the right equals $H^ i(RF(E_{-b + N - 1})$ for any $b$ with $i \geq b$.
For every $n$ we have a distinguished triangle
\[ H^{-n}(E)[n] \to E_ n \to E_{n - 1} \to H^{-n}(E)[n + 1] \]
(Derived Categories, Remark 13.12.4) in $D(\mathcal{O}_ X)$. Since $H^{-n}(E)$ is quasi-coherent we have
\[ H^ i(RF(H^{-n}(E)[n])) = R^{i + n}F(H^{-n}(E)) = 0 \]
for $i + n \geq N$ and
\[ H^ i(RF(H^{-n}(E)[n + 1])) = R^{i + n + 1}F(H^{-n}(E)) = 0 \]
for $i + n + 1 \geq N$. We conclude that
\[ H^ i(RF(E_ n)) \to H^ i(RF(E_{n - 1})) \]
is an isomorphism for $n \geq N - i$. Thus the systems $H^ i(RF(E_ n))$ all satisfy the ML condition and the $R^1\mathop{\mathrm{lim}}\nolimits $ term in our short exact sequence is zero (see discussion in More on Algebra, Section 15.86). Moreover, the system $H^ i(RF(E_ n))$ is constant starting with $n = N - i - 1$ as desired.
Proof of (3). Under the assumption on $E$ we have $\tau _{\leq a - 1}E = 0$ and we get the vanishing of $H^ i(RF(E))$ for $i \leq a - 1$ from (1). Similarly, we have $\tau _{\geq b + 1}E = 0$ and hence we get the vanishing of $H^ i(RF(E))$ for $i \geq b + n$ from part (2). $\square$
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