Lemma 24.34.1. Let \mathcal{C}, \mathcal{O}, \mathcal{A} be as in Section 24.33. Let \mathcal{C}' \subset \mathcal{C} be a full subcategory with the following property: for every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) the category U/\mathcal{C}' of arrows U \to U' is cofiltered. Denote \mathcal{O}', \mathcal{A}' the restrictions of \mathcal{O}, \mathcal{A} to \mathcal{C}'. Then restrictions induces an equivalence \mathit{QC}(\mathcal{A}, \text{d}) \to \mathit{QC}(\mathcal{A}', \text{d}).
24.34 Differential graded modules on a category, bis
We develop a few more results on the notion of quasi-coherent modules introduced in Section 24.33.
Proof. We will construct a quasi-inverse of the functor. Namely, let M' be an object of \mathit{QC}(\mathcal{A}', \text{d}). We may represent M' by a good differential graded module \mathcal{M}', see Lemma 24.23.7. Then for every U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}') the differential graded \mathcal{A}'(U')-module \mathcal{M}'(U) is K-flat and graded flat and for every morphism U'_1 \to U'_2 of \mathcal{C}' the map
\mathcal{M}'(U'_2) \otimes _{\mathcal{A}'(U'_2)} \mathcal{A}'(U'_1) \longrightarrow \mathcal{M}'(U'_1)
is a quasi-isomorphism (as the source represents the derived tensor product). Consider the differential graded \mathcal{A}-module \mathcal{M} defined by the rule
\mathcal{M}(U) = \mathop{\mathrm{colim}}\nolimits _{U \to U' \in U/\mathcal{C}'} \mathcal{M}'(U') \otimes _{\mathcal{A}'(U')} \mathcal{A}(U)
This is a filtered colimit of complexes by our assumption in the lemma. Since M' is in \mathit{QC}(\mathcal{A}', \text{d}) all the transition maps in the system are quasi-isomorphisms. Since filtered colimits are exact, we see that \mathcal{M}(U) in D(\mathcal{A}(U), \text{d}) is isomorphic to \mathcal{M}'(U') \otimes _{\mathcal{A}'(U')} \mathcal{A}(U) for any morphism U \to U' with U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}').
We claim that \mathcal{M} is in \mathit{QC}(\mathcal{A}, \text{d}): namely, given U \to V in \mathcal{C} we choose a map V \to V' with V' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}'). By the above we see that the map \mathcal{M}(V) \to \mathcal{M}(U) is identified with the map
\mathcal{M}'(V') \otimes _{\mathcal{A}'(V')} \mathcal{A}(V) \longrightarrow \mathcal{M}'(V') \otimes _{\mathcal{A}'(V')} \mathcal{A}(U)
Since \mathcal{M'}(V') is K-flat as differential gradede \mathcal{A}'(V')-module, we conclude the claim is true.
The natural map \mathcal{M}|_{\mathcal{C}'} \to \mathcal{M}' is an isomorphism in D(\mathcal{A}', d) as follows immediately from the above.
Conversely, if we have an object E of \mathit{QC}(\mathcal{A}, \text{d}), then we represent it by a good differential graded module \mathcal{E}. Setting \mathcal{M}' = \mathcal{E}|_{\mathcal{C}'} (this is another good differential graded module) we see that there is a map
\mathcal{E} \to \mathcal{M}
which over U in \mathcal{C} is given by the map
\mathcal{E}(U) \longrightarrow \mathop{\mathrm{colim}}\nolimits _{U \to U' \in U/\mathcal{C}'} \mathcal{E}(U') \otimes _{\mathcal{A}'(U')} \mathcal{A}(U)
which is a quasi-isomorphism by the same reason. Thus restriction and the construction above are quasi-inverse functors as desired. \square
Lemma 24.34.2. Let \mathcal{C}, \mathcal{O} be as in Section 24.33. Let \varphi : \mathcal{A} \to \mathcal{B} be a homomorphism of differential graded \mathcal{O}-algebras which induces an isomorphism on cohomology sheaves, then the equivalence D(\mathcal{A}, \text{d}) \to D(\mathcal{B}, \text{d}) of Lemma 24.30.1 induces an equivalence \mathit{QC}(\mathcal{A}, \text{d}) \to \mathit{QC}(\mathcal{B}, \text{d}).
Proof. It suffices to show the following: given a morphism U \to V of \mathcal{C} and M in D(\mathcal{A}, \text{d}) the following are equivalent
R\Gamma (V, M) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}(U) \to \Gamma (U, M) is an isomorphism in D(\mathcal{A}(U), \text{d}), and
R\Gamma (V, M \otimes _\mathcal {A}^\mathbf {L} \mathcal{B}) \otimes _{\mathcal{B}(V)}^\mathbf {L} \mathcal{B}(U) \to \Gamma (U, M \otimes _\mathcal {A}^\mathbf {L} \mathcal{B}) is an isomorphism in D(\mathcal{B}(U), \text{d}).
Since the topology on \mathcal{C} is chaotic, this simply boils down to fact that \mathcal{A}(U) \to \mathcal{B}(U) and \mathcal{A}(V) \to \mathcal{B}(V) are quasi-isomorphisms. Details omitted. \square
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