Proposition 36.40.6. Let $X$ be a quasi-compact and quasi-separated scheme. 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 36.18.2. This proves that (1) implies (2).

Parts (2), (3), (4) are equivalent by the discussion in Cohomology, Section 20.41. 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, Lemma 20.47.5 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 36.17.2. 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 Cohomology of Schemes, Lemma 30.4.1.

First, we prove (2) – (7) $\Rightarrow$ (1) if $X$ is affine. Let $P = \mathcal{O}_ X$. From (7) we obtain $H^ i (X, E) = 0$ for $i \ll 0$. Hence (1) follows since $E$ is determined by $R\Gamma (X, E)$, see Lemma 36.3.5.

Now assume $X = U \cup V$ with $U$ a quasi-compact open of $X$ and $V$ an affine open, and assume the implication (2) – (7) $\Rightarrow$ (1) is known for the schemes $U$, $V$, and $U \cap V$. Suppose $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$ satisfies (2) – (7). By Lemma 36.15.1 and Theorem 36.15.3 there exists a perfect complex $Q$ on $X$ such that $Q|_ U$ generates $D_\mathit{QCoh}(\mathcal{O}_ U)$. Let $f_1, \dots , f_ r \in \Gamma (V, \mathcal{O}_ V)$ be such that $V \setminus U = V(f_1, \dots , f_ r)$ as subsets of $V$. Let $K \in D_{perf}(\mathcal{O}_ V)$ be the object corresponding to the Koszul complex on $f_1, \dots , f_ r$. Let $K' \in D_{perf}(\mathcal{O}_ X)$ be

36.40.6.1
\begin{equation} \label{perfect-equation-detecting-bounded-below} K' = R (V \to X)_* K = R (V \to X)_! K, \end{equation}

see Cohomology, Lemmas 20.33.6 and 20.46.10. This is a perfect complex on $X$ supported on the closed set $X \setminus U \subset V$ and isomorphic to $K$ on $V$. 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 the second description of $K'$ in (36.40.6.1) we have

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

for $i \ll 0$. Therefore, we may apply Lemma 36.40.2 to $E|_ V$ to obtain an integer $a$ such that $\tau _{\leq a}(E|_ V) = \tau _{\leq a} R(U \cap V \to V)_*(E|_{U \cap V})$. 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 36.40.4 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 bounded below objects (see Cohomology, Section 20.3) we get $\tau _{\leq a} E \in D^+_\mathit{QCoh}(\mathcal{O}_ X)$. Thus $E \in D^+_\mathit{QCoh}(\mathcal{O}_ X)$. $\square$

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