Lemma 36.3.1. Let $X$ be a scheme. Then $D_\mathit{QCoh}(\mathcal{O}_ X)$ has direct sums.
36.3 Derived category of quasi-coherent modules
In this section we discuss the relationship between quasi-coherent modules and all modules on a scheme $X$. A reference is [Appendix B, TT]. By the discussion in Schemes, Section 26.24 the embedding $\mathit{QCoh}(\mathcal{O}_ X) \subset \textit{Mod}(\mathcal{O}_ X)$ exhibits $\mathit{QCoh}(\mathcal{O}_ X)$ as a weak Serre subcategory of the category of $\mathcal{O}_ X$-modules. Denote
\[ D_\mathit{QCoh}(\mathcal{O}_ X) \subset D(\mathcal{O}_ X) \]
the subcategory of complexes whose cohomology sheaves are quasi-coherent, see Derived Categories, Section 13.17. Thus we obtain a canonical functor
36.3.0.1
\begin{equation} \label{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).
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 Schemes, Section 26.24. $\square$
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 36.3.2. Let $X$ be a scheme. 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 $\mathcal{B}$ be the set of affine opens of $X$. Since $H^ q(K_ n)$ is quasi-coherent we have $H^ p(U, H^ q(K_ n)) = 0$ for $U \in \mathcal{B}$ by 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 Schemes, Lemma 26.7.5. Part (1) follows from Cohomology, Lemma 20.37.11 whose conditions we have just verified. Parts (2) and (3) follow from Cohomology, Lemma 20.37.4. $\square$
The following lemma will help us to “compute” a right derived functor on an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$.
Lemma 36.3.3. Let $X$ be a scheme. 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 open subsets of $X$. Then $H^ p(U, \mathcal{H}^ i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$, and all $U \in \mathcal{B}$, see Cohomology of Schemes, Lemma 30.2.2. Thus the lemma follows from Cohomology, Lemma 20.37.9. $\square$
Lemma 36.3.4. Let $X$ be a scheme. Let $F : \textit{Mod}(\mathcal{O}_ X) \to \textit{Ab}$ be an additive functor and $N \geq 0$ an integer. Assume that
$F$ 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 36.3.3. 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). $\square$
The following lemma is the key ingredient to many of the results in this chapter.
Lemma 36.3.5. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. All the functors in the diagram are equivalences of triangulated categories. Moreover, for $E$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $H^0(X, E) = H^0(X, H^0(E))$.
\[ \xymatrix{ D(\mathit{QCoh}(\mathcal{O}_ X)) \ar[rr]_{(06VT)} & & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[ld]^{R\Gamma (X, -)} \\ & D(A) \ar[lu]^{\widetilde{\ \ }} } \]
Proof. The functor $R\Gamma (X, -)$ gives a functor $D(\mathcal{O}_ X) \to D(A)$ and hence by restriction a functor
36.3.5.1
\begin{equation} \label{perfect-equation-back} R\Gamma (X, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \longrightarrow D(A). \end{equation}
We will show this functor is quasi-inverse to (36.3.0.1) via the equivalence between quasi-coherent modules on $X$ and the category of $A$-modules.
Elucidation. Denote $(Y, \mathcal{O}_ Y)$ the one point space with sheaf of rings given by $A$. Denote $\pi : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ the obvious morphism of ringed spaces. Then $R\Gamma (X, -)$ can be identified with $R\pi _*$ and the functor (36.3.0.1) via the equivalence $\textit{Mod}(\mathcal{O}_ Y) = \text{Mod}_ A = \mathit{QCoh}(\mathcal{O}_ X)$ can be identified with $L\pi ^* = \pi ^* = \widetilde{\ }$ (see Modules, Lemma 17.10.5 and Schemes, Lemmas 26.7.1 and 26.7.5). Thus the functors
\[ \xymatrix{ D(A) \ar@<1ex>[r] & D(\mathcal{O}_ X) \ar@<1ex>[l] } \]
are adjoint (by Cohomology, Lemma 20.28.1). In particular we obtain canonical adjunction mappings
\[ a : \widetilde{R\Gamma (X, E)} \longrightarrow E \]
for $E$ in $D(\mathcal{O}_ X)$ and
\[ b : M^\bullet \longrightarrow R\Gamma (X, \widetilde{M^\bullet }) \]
for $M^\bullet $ a complex of $A$-modules.
Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. We may apply Lemma 36.3.4 to the functor $F(-) = \Gamma (X, -)$ with $N = 1$ by Cohomology of Schemes, Lemma 30.2.2. Hence
\[ H^0(R\Gamma (X, E)) = H^0(R\Gamma (X, \tau _{\geq 0}E)) = \Gamma (X, H^0(E)) \]
(the last equality by definition of the canonical truncation). Using this we will show that the adjunction mappings $a$ and $b$ induce isomorphisms $H^0(a)$ and $H^0(b)$. Thus $a$ and $b$ are quasi-isomorphisms (as the statement is invariant under shifts) and the lemma is proved.
In both cases we use that $\widetilde{\ }$ is an exact functor (Schemes, Lemma 26.5.4). Namely, this implies that
\[ H^0\left(\widetilde{R\Gamma (X, E)}\right) = \widetilde{H^0(R\Gamma (X, E))} = \widetilde{\Gamma (X, H^0(E))} \]
which is equal to $H^0(E)$ because $H^0(E)$ is quasi-coherent. Thus $H^0(a)$ is an isomorphism. For the other direction we have
\[ H^0(R\Gamma (X, \widetilde{M^\bullet })) = \Gamma (X, H^0(\widetilde{M^\bullet })) = \Gamma (X, \widetilde{H^0(M^\bullet )}) = H^0(M^\bullet ) \]
which proves that $H^0(b)$ is an isomorphism. $\square$
Lemma 36.3.6. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. If $K^\bullet $ is a K-flat complex of $A$-modules, then $\widetilde{K^\bullet }$ is a K-flat complex of $\mathcal{O}_ X$-modules.
Proof. By More on Algebra, Lemma 15.59.3 we see that $K^\bullet \otimes _ A A_\mathfrak p$ is a K-flat complex of $A_\mathfrak p$-modules for every $\mathfrak p \in \mathop{\mathrm{Spec}}(A)$. Hence we conclude from Cohomology, Lemma 20.26.4 (and Schemes, Lemma 26.5.4) that $\widetilde{K^\bullet }$ is K-flat. $\square$
Lemma 36.3.7. If $f : X \to Y$ is a morphism of affine schemes given by the ring map $A \to B$, then the diagram commutes.
\[ \xymatrix{ D(B) \ar[d] \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[d]^{Rf_*} \\ D(A) \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ Y) } \]
Proof. Follows from Lemma 36.3.5 using that $R\Gamma (Y, Rf_*K) = R\Gamma (X, K)$ by Cohomology, Lemma 20.32.5. $\square$
Lemma 36.3.8. Let $f : Y \to X$ be a morphism of schemes.
The functor $Lf^*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$.
If $X$ and $Y$ are affine and $f$ is given by the ring map $A \to B$, then the diagram
commutes.
\[ \xymatrix{ D(B) \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ Y) \\ D(A) \ar[r] \ar[u]^{- \otimes _ A^\mathbf {L} B} & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[u]_{Lf^*} } \]
Proof. We first prove the diagram
\[ \xymatrix{ D(B) \ar[r] & D(\mathcal{O}_ Y) \\ D(A) \ar[r] \ar[u]^{- \otimes _ A^\mathbf {L} B} & D(\mathcal{O}_ X) \ar[u]_{Lf^*} } \]
commutes. This is clear from Lemma 36.3.6 and the constructions of the functors in question. To see (1) let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To see that $Lf^*E$ has quasi-coherent cohomology sheaves we may work locally on $X$. Note that $Lf^*$ is compatible with restricting to open subschemes. Hence we can assume that $f$ is a morphism of affine schemes as in (2). Then we can apply Lemma 36.3.5 to see that $E$ comes from a complex of $A$-modules. By the commutativity of the first diagram of the proof the same holds for $Lf^*E$ and we conclude (1) is true. $\square$
Lemma 36.3.9. Let $X$ be a scheme.
For objects $K, L$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the derived tensor product $K \otimes ^\mathbf {L}_{\mathcal{O}_ X} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.
If $X = \mathop{\mathrm{Spec}}(A)$ is affine then
for any pair of complexes of $A$-modules $K^\bullet $, $M^\bullet $.
\[ \widetilde{M^\bullet } \otimes _{\mathcal{O}_ X}^\mathbf {L} \widetilde{K^\bullet } = \widetilde{M^\bullet \otimes _ A^\mathbf {L} K^\bullet } \]
Proof. The equality of (2) follows immediately from Lemma 36.3.6 and the construction of the derived tensor product. To see (1) let $K, L$ be objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To check that $K \otimes ^\mathbf {L} L$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we may work locally on $X$, hence we may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine. By Lemma 36.3.5 we may represent $K$ and $L$ by complexes of $A$-modules. Then part (2) implies the result. $\square$
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