Lemma 36.21.1. Let X be a quasi-compact and quasi-separated scheme. The inclusion functor D_\mathit{QCoh}(\mathcal{O}_ X) \to D(\mathcal{O}_ X) has a right adjoint DQ_ X.
First proof. We will use the induction principle as in Cohomology of Schemes, Lemma 30.4.1 to prove this. If D(\mathit{QCoh}(\mathcal{O}_ X)) \to D_\mathit{QCoh}(\mathcal{O}_ X) is an equivalence, then the lemma is true because the functor RQ_ X of Section 36.7 is a right adjoint to the functor D(\mathit{QCoh}(\mathcal{O}_ X)) \to D(\mathcal{O}_ X). In particular, our lemma is true for affine schemes, see Lemma 36.7.3. Thus we see that it suffices to show: if X = U \cup V is a union of two quasi-compact opens and the lemma holds for U, V, and U \cap V, then the lemma holds for X.
The adjoint exists if and only if for every object K of D(\mathcal{O}_ X) we can find a distinguished triangle
in D(\mathcal{O}_ X) such that E' is in D_\mathit{QCoh}(\mathcal{O}_ X) and such that \mathop{\mathrm{Hom}}\nolimits (M, K) = 0 for all M in D_\mathit{QCoh}(\mathcal{O}_ X). See Derived Categories, Lemma 13.40.7. Consider the distinguished triangle
in D(\mathcal{O}_ X) of Cohomology, Lemma 20.33.2. By Derived Categories, Lemma 13.40.5 it suffices to construct the desired distinguished triangles for Rj_{U, *}E|_ U, Rj_{V, *}E|_ V, and Rj_{U \cap V, *}E|_{U \cap V}. This reduces us to the statement discussed in the next paragraph.
Let j : U \to X be an open immersion corresponding with U a quasi-compact open for which the lemma is true. Let L be an object of D(\mathcal{O}_ U). Then there exists a distinguished triangle
in D(\mathcal{O}_ X) such that E' is in D_\mathit{QCoh}(\mathcal{O}_ X) and such that \mathop{\mathrm{Hom}}\nolimits (M, K) = 0 for all M in D_\mathit{QCoh}(\mathcal{O}_ X). To see this we choose a distinguished triangle
in D(\mathcal{O}_ U) such that L' is in D_\mathit{QCoh}(\mathcal{O}_ U) and such that \mathop{\mathrm{Hom}}\nolimits (N, Q) = 0 for all N in D_\mathit{QCoh}(\mathcal{O}_ U). This is possible because the statement in Derived Categories, Lemma 13.40.7 is an if and only if. We obtain a distinguished triangle
in D(\mathcal{O}_ X). Observe that Rj_*L' is in D_\mathit{QCoh}(\mathcal{O}_ X) by Lemma 36.4.1. On the other hand, if M in D_\mathit{QCoh}(\mathcal{O}_ X), then
because Lj^*M is in D_\mathit{QCoh}(\mathcal{O}_ U) by Lemma 36.3.8. This finishes the proof. \square
Second proof. The adjoint exists by Derived Categories, Proposition 13.38.2. The hypotheses are satisfied: First, note that D_\mathit{QCoh}(\mathcal{O}_ X) has direct sums and direct sums commute with the inclusion functor (Lemma 36.3.1). On the other hand, D_\mathit{QCoh}(\mathcal{O}_ X) is compactly generated because it has a perfect generator Theorem 36.15.3 and because perfect objects are compact by Proposition 36.17.1. \square
Comments (0)