The Stacks project

73.19 The coherator revisited

In Section 73.11 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 algebraic spaces have such a right adjoint.

Lemma 73.19.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The inclusion functor $D_\mathit{QCoh}(\mathcal{O}_ X) \to D(\mathcal{O}_ X)$ has a right adjoint.

First proof. We will use the induction principle in Lemma 73.9.3 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 73.11 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 algebraic spaces, see Lemma 73.11.3. Thus we see that it suffices to show: if $(U \subset X, f : V \to X)$ is an elementary distinguished square with $U$ quasi-compact and $V$ affine and the lemma holds for $U$, $V$, and $U \times _ X V$, 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 13.39.5. Consider the distinguished triangle

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

in $D(\mathcal{O}_ X)$ of Lemma 73.10.2. By Derived Categories, Lemma 13.39.4 it suffices to construct the desired distinguished triangles for $Rj_{U, *}E|_ U$, $Rj_{V, *}E|_ V$, and $Rj_{U \times _ X V, *}E|_{U \times _ X V}$. This reduces us to the statement discussed in the next paragraph.

Let $j : U \to X$ be an étale morphism corresponding with $U$ quasi-compact and quasi-separated and the lemma is true for $U$. 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 13.39.5 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 73.6.1. 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 73.5.5. 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 73.5.3). On the other hand, $D_\mathit{QCoh}(\mathcal{O}_ X)$ is compactly generated because it has a perfect generator Theorem 73.15.4 and because perfect objects are compact by Proposition 73.16.1. $\square$

Lemma 73.19.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. 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 73.6.1. The statement is true because $Lf^*$ similarly maps $D_\mathit{QCoh}(\mathcal{O}_ Y)$ into $D_\mathit{QCoh}(\mathcal{O}_ X)$ (Lemma 73.5.5) 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 73.19.3. Let $S$ be a scheme. Let $(U \subset X, f : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Assume $X$, $U$, $V$ are quasi-compact and quasi-separated. By Lemma 73.19.1 the functors $DQ_ X$, $DQ_ U$, $DQ_ V$, $DQ_{U \times _ X V}$ exist. Moreover, there is a canonical distinguished triangle

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

for any $K \in D(\mathcal{O}_ X)$. This follows by applying the exact functor $DQ_ X$ to the distinguished triangle of Lemma 73.10.2 and using Lemma 73.19.2 three times.

Lemma 73.19.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. The functor $DQ_ X$ of Lemma 73.19.1 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 étale over $X$ 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 Lemma 73.9.3.

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 Lemma 73.11.3.

Let $(U \subset W, f : V \to W)$ be an elementary distinguished square with $W$ quasi-compact and quasi-separated, $U \subset W$ quasi-compact open, $V$ affine such that the lemma holds for $U$, $V$, and $U \times _ W V$. Say with integers $N(U)$, $N(V)$, and $N(U \times _ W V)$. Now suppose $K$ is in $D(\mathcal{O}_ X)$ with $H^ i(W, K) = 0$ for all affine $W$ étale over $X$ and all $i \not\in [a, b]$. Then $K|_ U$, $K|_ V$, $K|_{U \times _ W V}$ have the same property. Hence we see that $RQ_ U(K|_ U)$ and $RQ_ V(K|_ V)$ and $RQ_{U \cap V}(K|_{U \times _ W V})$ have vanishing cohomology sheaves outside the inverval $[a, b + \max (N(U), N(V), N(U \times _ W V))$. Since the functors $Rj_{U, *}$, $Rj_{V, *}$, $Rj_{U \times _ W V, *}$ have finite cohomological dimension on $D_\mathit{QCoh}$ by Lemma 73.6.1 we see that there exists an $N$ such that $Rj_{U, *}DQ_ U(K|_ U)$, $Rj_{V, *}DQ_ V(K|_ V)$, and $Rj_{U \cap V, *}DQ_{U \times _ W V}(K|_{U \times _ W V})$ have vanishing cohomology sheaves outside the interval $[a, b + N]$. Then finally we conclude by the distinguished triangle of Remark 73.19.3. $\square$

Example 73.19.5. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $(\mathcal{F}_ n)$ be an inverse system of quasi-coherent sheaves on $X$. 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 $U$ étale over $X$ 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 $D^ b_\mathit{QCoh}(\mathcal{O}_ X)$ and has vanishing cohomology sheaves in negative degrees by Lemma 73.19.4.

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

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CR3. Beware of the difference between the letter 'O' and the digit '0'.