The Stacks project

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.

Asssume $U, V$ 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 inverval $[a, b + \max (N(U), N(V), N(U \cap V))$. 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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