The Stacks project

73.28 Detecting Boundedness

In this section, we show that compact generators of $D_\mathit{QCoh}$ of a quasi-compact, quasi-separated scheme, as constructed in Section 73.15, have a special property. We recommend reading that section first as it is very similar to this one.

Lemma 73.28.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P \in D_{perf}(\mathcal{O}_ X)$ and $E \in D_{\mathit{QCoh}}(\mathcal{O}_ X)$. Let $a \in \mathbf{Z}$. The following are equivalent

  1. $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], E) = 0$ for $i \gg 0$, and

  2. $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], \tau _{\geq a} E) = 0$ for $i \gg 0$.

Proof. Using the triangle $ \tau _{< a} E \to E \to \tau _{\geq a} E \to $ we see that the equivalence follows if we can show

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], \tau _{< a} E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P, (\tau _{< a} E)[i]) = 0 \]

for $i \gg 0$. As $P$ is perfect this is true by Lemma 73.17.2. $\square$

Lemma 73.28.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $P \in D_{perf}(\mathcal{O}_ X)$ and $E \in D_{\mathit{QCoh}}(\mathcal{O}_ X)$. Let $a \in \mathbf{Z}$. The following are equivalent

  1. $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], E) = 0$ for $i \ll 0$, and

  2. $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], \tau _{\leq a} E) = 0$ for $i \ll 0$.

Proof. Using the triangle $ \tau _{\leq a} E \to E \to \tau _{> a} E \to $ we see that the equivalence follows if we can show

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], \tau _{> a} E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P, (\tau _{> a} E)[i]) = 0 \]

for $i \ll 0$. As $P$ is perfect this is true by Lemma 73.17.2. $\square$

Proposition 73.28.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G \in D_{perf}(\mathcal{O}_ X)$ be a perfect complex which generates $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent

  1. $E \in D^-_\mathit{QCoh}(\mathcal{O}_ X)$,

  2. $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(G[-i], E) = 0$ for $i \gg 0$,

  3. $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(G, E) = 0$ for $i \gg 0$,

  4. $R\mathop{\mathrm{Hom}}\nolimits _ X(G, E)$ is in $D^-(\mathbf{Z})$,

  5. $H^ i(X, G^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E) = 0$ for $i \gg 0$,

  6. $R\Gamma (X, G^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E)$ is in $D^-(\mathbf{Z})$,

  7. for every perfect object $P$ of $D(\mathcal{O}_ X)$

    1. the assertions (2), (3), (4) hold with $G$ replaced by $P$, and

    2. $H^ i(X, P \otimes _{\mathcal{O}_ X}^\mathbf {L} E) = 0$ for $i \gg 0$,

    3. $R\Gamma (X, P \otimes _{\mathcal{O}_ X}^\mathbf {L} E)$ is in $D^-(\mathbf{Z})$.

Proof. Assume (1). Since $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(G[-i], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(G, E[i])$ we see that this is zero for $i \gg 0$ by Lemma 73.17.2. This proves that (1) implies (2).

Parts (2), (3), (4) are equivalent by the discussion in Cohomology on Sites, Section 21.35. Part (5) and (6) are equivalent as $H^ i(X, -) = H^ i(R\Gamma (X, -))$ by definition. The equivalent conditions (2), (3), (4) are equivalent to the equivalent conditions (5), (6) by Cohomology on Sites, Lemma 21.46.4 and the fact that $(G[-i])^\vee = G^\vee [i]$.

It is clear that (7) implies (2). Conversely, let us prove that the equivalent conditions (2) – (6) imply (7). Recall that $G$ is a classical generator for $D_{perf}(\mathcal{O}_ X)$ by Remark 73.16.2. For $P \in D_{perf}(\mathcal{O}_ X)$ let $T(P)$ be the assertion that $R\mathop{\mathrm{Hom}}\nolimits _ X(P, E)$ is in $D^-(\mathbf{Z})$. Clearly, $T$ is inherited by direct sums, satisfies the 2-out-of-three property for distinguished triangles, is inherited by direct summands, and is perserved by shifts. Hence by Derived Categories, Remark 13.36.7 we see that (4) implies $T$ holds on all of $D_{perf}(\mathcal{O}_ X)$. The same argument works for all other properties, except that for property (7)(b) and (7)(c) we also use that $P \mapsto P^\vee $ is a self equivalence of $D_{perf}(\mathcal{O}_ X)$. Small detail omitted.

We will prove the equivalent conditions (2) – (7) imply (1) using the induction principle of Lemma 73.9.3.

First, we prove (2) – (7) $\Rightarrow $ (1) if $X$ is affine. This follows from the case of schemes, see Derived Categories of Schemes, Proposition 36.40.5.

Now assume $(U \subset X, j : V \to X)$ is an elementary distinguished square of quasi-compact and quasi-separated algebraic spaces over $S$ and assume the implication (2) – (7) $\Rightarrow $ (1) is known for $U$, $V$, and $U \times _ X V$. To finish the proof we have to show the implication (2) – (7) $\Rightarrow $ (1) for $X$. Suppose $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$ satisfies (2) – (7). By Lemma 73.15.3 and Theorem 73.15.4 there exists a perfect complex $Q$ on $X$ such that $Q|_ U$ generates $D_\mathit{QCoh}(\mathcal{O}_ U)$.

Say $V = \mathop{\mathrm{Spec}}(A)$. Let $Z \subset V$ be the reduced closed subscheme which is the inverse image of $X \setminus U$ and maps isomorphically to it (see Definition 73.9.1). This is a retrocompact closed subset of $V$. Choose $f_1, \ldots , f_ r \in A$ such that $Z = V(f_1, \ldots , f_ r)$. Let $K \in D(\mathcal{O}_ V)$ be the perfect object corresponding to the Koszul complex on $f_1, \ldots , f_ r$ over $A$. Note that since $K$ is supported on $Z$, the pushforward $K' = Rj_*K$ is a perfect object of $D(\mathcal{O}_ X)$ whose restriction to $V$ is $K$ (see Lemmas 73.14.3 and 73.10.7). By assumption, we know $R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(Q, E)$ and $R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K', E)$ are bounded above.

By Lemma 73.10.7 we have $K' = j_!K$ and hence

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K'[-i], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ V)}(K[-i], E|_ V) = 0 \]

for $i \gg 0$. Therefore, we may apply Derived Categories of Schemes, Lemma 36.40.1 to $E|_ V$ to obtain an integer $a$ such that $\tau _{\geq a}(E|_ V) = \tau _{\geq a} R (U \times _ X V \to V)_* (E|_{U \times _ X V})$. Then $\tau _{\geq a} E = \tau _{\geq a} R (U \to X)_* (E |_ U)$ (check that the canonical map is an isomorphism after restricting to $U$ and to $V$). Hence using Lemma 73.28.1 twice we see that

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(Q|_ U [-i], E|_ U) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(Q[-i], R (U \to X)_* (E|_ U)) = 0 \]

for $i \gg 0$. Since the Proposition holds for $U$ and the generator $Q|_ U$, we have $E|_ U \in D^-_\mathit{QCoh}(\mathcal{O}_ U)$. But then since the functor $R (U \to X)_*$ preserves $D^-_\mathit{QCoh}$ (by Lemma 73.6.1), we get $\tau _{\geq a}E \in D^-_\mathit{QCoh}(\mathcal{O}_ X)$. Thus $E \in D^-_\mathit{QCoh}(\mathcal{O}_ X)$. $\square$

Proposition 73.28.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $G \in D_{perf}(\mathcal{O}_ X)$ be a perfect complex which generates $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent

  1. $E \in D^+_\mathit{QCoh}(\mathcal{O}_ X)$,

  2. $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(G[-i], E) = 0$ for $i \ll 0$,

  3. $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(G, E) = 0$ for $i \ll 0$,

  4. $R\mathop{\mathrm{Hom}}\nolimits _ X(G, E)$ is in $D^+(\mathbf{Z})$,

  5. $H^ i(X, G^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E) = 0$ for $i \ll 0$,

  6. $R\Gamma (X, G^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E)$ is in $D^+(\mathbf{Z})$,

  7. for every perfect object $P$ of $D(\mathcal{O}_ X)$

    1. the assertions (2), (3), (4) hold with $G$ replaced by $P$, and

    2. $H^ i(X, P \otimes _{\mathcal{O}_ X}^\mathbf {L} E) = 0$ for $i \ll 0$,

    3. $R\Gamma (X, P \otimes _{\mathcal{O}_ X}^\mathbf {L} E)$ is in $D^+(\mathbf{Z})$.

Proof. Assume (1). Since $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(G[-i], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(G, E[i])$ we see that this is zero for $i \ll 0$ by Lemma 73.17.2. This proves that (1) implies (2).

Parts (2), (3), (4) are equivalent by the discussion in Cohomology on Sites, Section 21.35. Part (5) and (6) are equivalent as $H^ i(X, -) = H^ i(R\Gamma (X, -))$ by definition. The equivalent conditions (2), (3), (4) are equivalent to the equivalent conditions (5), (6) by Cohomology on Sites, Lemma 21.46.4 and the fact that $(G[-i])^\vee = G^\vee [i]$.

It is clear that (7) implies (2). Conversely, let us prove that the equivalent conditions (2) – (6) imply (7). Recall that $G$ is a classical generator for $D_{perf}(\mathcal{O}_ X)$ by Remark 73.16.2. For $P \in D_{perf}(\mathcal{O}_ X)$ let $T(P)$ be the assertion that $R\mathop{\mathrm{Hom}}\nolimits _ X(P, E)$ is in $D^+(\mathbf{Z})$. Clearly, $T$ is inherited by direct sums, satisfies the 2-out-of-three property for distinguished triangles, is inherited by direct summands, and is perserved by shifts. Hence by Derived Categories, Remark 13.36.7 we see that (4) implies $T$ holds on all of $D_{perf}(\mathcal{O}_ X)$. The same argument works for all other properties, except that for property (7)(b) and (7)(c) we also use that $P \mapsto P^\vee $ is a self equivalence of $D_{perf}(\mathcal{O}_ X)$. Small detail omitted.

We will prove the equivalent conditions (2) – (7) imply (1) using the induction principle of Lemma 73.9.3.

First, we prove (2) – (7) $\Rightarrow $ (1) if $X$ is affine. This follows from the case of schemes, see Derived Categories of Schemes, Proposition 36.40.6.

Now assume $(U \subset X, j : V \to X)$ is an elementary distinguished square of quasi-compact and quasi-separated algebraic spaces over $S$ and assume the implication (2) – (7) $\Rightarrow $ (1) is known for $U$, $V$, and $U \times _ X V$. To finish the proof we have to show the implication (2) – (7) $\Rightarrow $ (1) for $X$. Suppose $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$ satisfies (2) – (7). By Lemma 73.15.3 and Theorem 73.15.4 there exists a perfect complex $Q$ on $X$ such that $Q|_ U$ generates $D_\mathit{QCoh}(\mathcal{O}_ U)$.

Say $V = \mathop{\mathrm{Spec}}(A)$. Let $Z \subset V$ be the reduced closed subscheme which is the inverse image of $X \setminus U$ and maps isomorphically to it (see Definition 73.9.1). This is a retrocompact closed subset of $V$. Choose $f_1, \ldots , f_ r \in A$ such that $Z = V(f_1, \ldots , f_ r)$. Let $K \in D(\mathcal{O}_ V)$ be the perfect object corresponding to the Koszul complex on $f_1, \ldots , f_ r$ over $A$. Note that since $K$ is supported on $Z$, the pushforward $K' = Rj_*K$ is a perfect object of $D(\mathcal{O}_ X)$ whose restriction to $V$ is $K$ (see Lemmas 73.14.3 and 73.10.7). By assumption, we know $R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(Q, E)$ and $R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K', E)$ are bounded below.

By Lemma 73.10.7 we have $K' = j_!K$ and hence

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K'[-i], E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ V)}(K[-i], E|_ V) = 0 \]

for $i \ll 0$. Therefore, we may apply Derived Categories of Schemes, Lemma 36.40.2 to $E|_ V$ to obtain an integer $a$ such that $\tau _{\leq a}(E|_ V) = \tau _{\leq a} R (U \times _ X V \to V)_* (E|_{U \times _ X V})$. Then $\tau _{\leq a} E = \tau _{\leq a} R (U \to X)_* (E |_ U)$ (check that the canonical map is an isomorphism after restricting to $U$ and to $V$). Hence using Lemma 73.28.2 twice we see that

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(Q|_ U [-i], E|_ U) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(Q[-i], R (U \to X)_* (E|_ U)) = 0 \]

for $i \ll 0$. Since the Proposition holds for $U$ and the generator $Q|_ U$, we have $E|_ U \in D^+_\mathit{QCoh}(\mathcal{O}_ U)$. But then since the functor $R (U \to X)_*$ preserves $D^+_\mathit{QCoh}$ (by Lemma 73.6.1), we get $\tau _{\leq a}E \in D^+_\mathit{QCoh}(\mathcal{O}_ X)$. Thus $E \in D^+_\mathit{QCoh}(\mathcal{O}_ X)$. $\square$


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 0GFE. Beware of the difference between the letter 'O' and the digit '0'.