The Stacks project

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. $\square$

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. $\square$

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. $\square$

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

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

(Derived Categories, Remark 13.12.4) in $D(\mathcal{O}_ X)$. Since $H^{-n}(E)$ is quasi-coherent we have

for $i + n \geq N$ and

for $i + n + 1 \geq N$. We conclude that

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). $\square$

Proof. The functor $R\Gamma (X, -)$ gives a functor $D(\mathcal{O}_ X) \to D(A)$ and hence by restriction a functor

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

are adjoint (by Cohomology, Lemma 20.28.1). In particular we obtain canonical adjunction mappings

for $E$ in $D(\mathcal{O}_ X)$ and

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

(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

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

which proves that $H^0(b)$ is an isomorphism. $\square$

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. $\square$

Proof. Follows from Lemma 36.3.5 using that $R\Gamma (Y, Rf_*K) = R\Gamma (X, K)$ by Cohomology, Lemma 20.32.5. $\square$

Proof. We first prove the diagram

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. $\square$

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. $\square$