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36.21 The coherator revisited

In Section 36.7 we constructed and studied the right adjoint RQ_ X to the canonical functor D(\mathit{QCoh}(\mathcal{O}_ X)) \to D(\mathcal{O}_ X). It was constructed as the right derived extension of the coherator Q_ X : \textit{Mod}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ X). In this section, we study when the inclusion functor

D_\mathit{QCoh}(\mathcal{O}_ X) \longrightarrow D(\mathcal{O}_ X)

has a right adjoint. If this right adjoint exists, we will denote1 it

DQ_ X : D(\mathcal{O}_ X) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X)

It turns out that quasi-compact and quasi-separated schemes have such a right adjoint.

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

E' \to E \to K \to E'[1]

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

E \to Rj_{U, *}E|_ U \oplus Rj_{V, *}E|_ V \to Rj_{U \cap V, *}E|_{U \cap V} \to E[1]

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

E' \to Rj_*L \to K \to E'[1]

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

L' \to L \to Q \to L'[1]

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

Rj_*L' \to Rj_*L \to Rj_*Q \to Rj_*L'[1]

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

\mathop{\mathrm{Hom}}\nolimits (M, Rj_*Q) = \mathop{\mathrm{Hom}}\nolimits (Lj^*M, Q) = 0

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

Lemma 36.21.2. Let f : X \to Y be a quasi-compact and quasi-separated morphism of schemes. If the right adjoints DQ_ X and DQ_ Y of the inclusion functors D_\mathit{QCoh}\to D exist for X and Y, then

Rf_* \circ DQ_ X = DQ_ Y \circ Rf_*

Proof. The statement makes sense because Rf_* sends D_\mathit{QCoh}(\mathcal{O}_ X) into D_\mathit{QCoh}(\mathcal{O}_ Y) by Lemma 36.4.1. The statement is true because Lf^* similarly maps D_\mathit{QCoh}(\mathcal{O}_ Y) into D_\mathit{QCoh}(\mathcal{O}_ X) (Lemma 36.3.8) and hence both Rf_* \circ DQ_ X and DQ_ Y \circ Rf_* are right adjoint to Lf^* : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D(\mathcal{O}_ X). \square

Remark 36.21.3. Let X be a quasi-compact and quasi-separated scheme. Let X = U \cup V with U and V quasi-compact open. By Lemma 36.21.1 the functors DQ_ X, DQ_ U, DQ_ V, DQ_{U \cap V} exist. Moreover, there is a canonical distinguished triangle

DQ_ X(K) \to Rj_{U, *}DQ_ U(K|_ U) \oplus Rj_{V, *}DQ_ V(K|_ V) \to Rj_{U \cap V, *}DQ_{U \cap V}(K|_{U \cap V}) \to

for any K \in D(\mathcal{O}_ X). This follows by applying the exact functor DQ_ X to the distinguished triangle of Cohomology, Lemma 20.33.2 and using Lemma 36.21.2 three times.

Lemma 36.21.4. Let X be a quasi-compact and quasi-separated scheme. The functor DQ_ X of Lemma 36.21.1 has the following boundedness property: there exists an integer N = N(X) such that, if K in D(\mathcal{O}_ X) with H^ i(U, K) = 0 for U affine open in X and i \not\in [a, b], then the cohomology sheaves H^ i(DQ_ X(K)) are zero for i \not\in [a, b + N].

Proof. We will prove this using the induction principle of Cohomology of Schemes, Lemma 30.4.1.

If X is affine, then the lemma is true with N = 0 because then RQ_ X = DQ_ X is given by taking the complex of quasi-coherent sheaves associated to R\Gamma (X, K). See Lemmas 36.3.5 and 36.7.3.

Assume U, V are quasi-compact open in X and the lemma holds for U, V, and U \cap V. Say with integers N(U), N(V), and N(U \cap V). Now suppose K is in D(\mathcal{O}_ X) with H^ i(W, K) = 0 for all affine open W \subset X and all i \not\in [a, b]. Then K|_ U, K|_ V, K|_{U \cap V} have the same property. Hence we see that RQ_ U(K|_ U) and RQ_ V(K|_ V) and RQ_{U \cap V}(K|_{U \cap V}) have vanishing cohomology sheaves outside the interval [a, b + \max (N(U), N(V), N(U \cap V)). Since the functors Rj_{U, *}, Rj_{V, *}, Rj_{U \cap V, *} have finite cohomological dimension on D_\mathit{QCoh} by Lemma 36.4.1 we see that there exists an N such that Rj_{U, *}DQ_ U(K|_ U), Rj_{V, *}DQ_ V(K|_ V), and Rj_{U \cap V, *}DQ_{U \cap V}(K|_{U \cap V}) have vanishing cohomology sheaves outside the interval [a, b + N]. Then finally we conclude by the distinguished triangle of Remark 36.21.3. \square

Example 36.21.5. Let X be a quasi-compact and quasi-separated scheme. Let (\mathcal{F}_ n) be an inverse system of quasi-coherent sheaves. Since DQ_ X is a right adjoint it commutes with products and therefore with derived limits. Hence we see that

DQ_ X(R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n) = (R\mathop{\mathrm{lim}}\nolimits \text{ in }D_\mathit{QCoh}(\mathcal{O}_ X))(\mathcal{F}_ n)

where the first R\mathop{\mathrm{lim}}\nolimits is taken in D(\mathcal{O}_ X). In fact, let's write K = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n for this. For any affine open U \subset X we have

H^ i(U, K) = H^ i(R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n)) = H^ i(R\mathop{\mathrm{lim}}\nolimits R\Gamma (U, \mathcal{F}_ n)) = H^ i(R\mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{F}_ n))

since cohomology commutes with derived limits and since the quasi-coherent sheaves \mathcal{F}_ n have no higher cohomology on affines. By the computation of R\mathop{\mathrm{lim}}\nolimits in the category of abelian groups, we see that H^ i(U, K) = 0 unless i \in [0, 1]. Then finally we conclude that the R\mathop{\mathrm{lim}}\nolimits in D_\mathit{QCoh}(\mathcal{O}_ X), which is DQ_ X(K) by the above, is in D^ b_\mathit{QCoh}(\mathcal{O}_ X) by Lemma 36.21.4.

[1] This is probably nonstandard notation. However, we have already used Q_ X for the coherator and RQ_ X for its derived extension.

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