The Stacks project

Proof. Let $M$ be a differential graded $E$-module. For every object $U$ of $\mathcal{C}$ the complex $K^\bullet (U)$ is a left differential graded $E$-module as well as a right $\mathcal{O}(U)$-module. The actions commute, so we have a bimodule. Thus, by the constructions in Sections 22.12 and 22.28 we can form the tensor product

which is a differential graded $\mathcal{O}(U)$-module, i.e., a complex of $\mathcal{O}(U)$-modules. This construction is functorial with respect to $U$, hence we can sheafify to get a complex of $\mathcal{O}$-modules which we denote

Moreover, for each $U$ the construction determines a functor $\text{Mod}^{dg}_{(E, \text{d})} \to \text{Comp}^{dg}(\mathcal{O}(U))$ of differential graded categories by Lemma 22.29.1. It is therefore clear that we obtain a functor as stated in the lemma. $\square$

Proof. The functor induces a functor between homotopy categories by Lemma 22.26.5. We have to show that $- \otimes _ E K^\bullet $ transforms distinguished triangles into distinguished triangles. Suppose that $0 \to K \to L \to M \to 0$ is an admissible short exact sequence of differential graded $E$-modules. Let $s : M \to L$ be a graded $E$-module homomorphism which is left inverse to $L \to M$. Then $s$ defines a map $M \otimes _ E K^\bullet \to L \otimes _ E K^\bullet $ of graded $\mathcal{O}$-modules (i.e., respecting $\mathcal{O}$-module structure and grading, but not differentials) which is left inverse to $L \otimes _ E K^\bullet \to M \otimes _ E K^\bullet $. Thus we see that

is a termwise split short exact sequences of complexes, i.e., a defines a distinguished triangle in $K(\mathcal{O})$. $\square$

Proof. We will use Derived Categories, Lemma 13.14.15 to prove this. As our collection $\mathcal{P}$ of objects we will use the objects with property (P). Property (1) was shown in Lemma 22.20.4. Property (2) holds because if $s : P \to P'$ is a quasi-isomorphism of modules with property (P), then $s$ is a homotopy equivalence by Lemma 22.22.3. $\square$

Proof. The right hand side of the arrow is the global derived hom introduced in Cohomology on Sites, Section 21.36 which has the correct cohomology modules. For the left hand side we think of $M$ as a $(R, A)$-bimodule and we have the derived $\mathop{\mathrm{Hom}}\nolimits $ introduced in Section 22.31 which also has the correct cohomology modules. To prove the lemma we may assume $M$ and $N$ are differential graded $E$-modules with property (P); this does not change the left hand side of the arrow by Lemma 22.31.3. By Lemma 22.31.5 this means that the left hand side of the arrow becomes $\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(M, N)$. In Lemmas 22.35.1, 22.35.2, and 22.35.3 we have constructed a functor

of differential graded categories and we have shown that $- \otimes _ E^\mathbf {L} K^\bullet $ is computed by evaluating this functor on differential graded $E$-modules with property (P). Hence we obtain a map of complexes of $R$-modules

For any complexes of $\mathcal{O}$-modules $\mathcal{F}^\bullet $, $\mathcal{G}^\bullet $ there is a canonical map

Combining these maps we obtain the desired map of the lemma. $\square$

Proof. The statement means that we have

bifunctorially in $M$ and $L^\bullet $. To see this we may replace $M$ by a differential graded $E$-module $P$ with property (P). We also may replace $L^\bullet $ by a K-injective complex of $\mathcal{O}$-modules $I^\bullet $. The computation of the derived functors given in the lemmas referenced in the statement combined with Lemma 22.22.3 translates the above into

where $\mathcal{B} = \text{Comp}^{dg}(\mathcal{O})$. There is an evaluation map from right to left functorial in $P$ and $I^\bullet $ (details omitted). Choose a filtration $F_\bullet $ on $P$ as in the definition of property (P). By Lemma 22.20.1 and the fact that both sides of the equation are homological functors in $P$ on $K(\text{Mod}_{(E, \text{d})})$ we reduce to the case where $P$ is replaced by the differential graded $E$-module $\bigoplus F_ iP$. Since both sides turn direct sums in the variable $P$ into direct products we reduce to the case where $P$ is one of the differential graded $E$-modules $F_ iP$. Since each $F_ iP$ has a finite filtration (given by admissible monomorphisms) whose graded pieces are graded projective $E$-modules we reduce to the case where $P$ is a graded projective $E$-module. In this case we clearly have

as graded $\mathbf{Z}$-modules (because this statement reduces to the case $P = E[k]$ where it is obvious). As the isomorphism is compatible with differentials we conclude. $\square$

Proof. Because our functor has a left adjoint given by $R\mathop{\mathrm{Hom}}\nolimits (K^\bullet , -)$ by Lemma 22.35.5 it suffices to show for a differential graded $E$-module $M$ that the map

is an isomorphism. We may assume that $M = P$ is a differential graded $E$-module which has property (P). Since $K^\bullet $ defines a compact object, we reduce using Lemma 22.20.1 to the case where $P$ has a finite filtration whose graded pieces are direct sums of $E[k]$. Again using compactness we reduce to the case $P = E[k]$. The assumption on $K^\bullet $ is that the result holds for these. $\square$