Lemma 36.16.1. Let $X$ be a scheme and $\mathcal{L}$ an ample invertible $\mathcal{O}_ X$-module. If $K$ is a nonzero object of $D_\mathit{QCoh}(\mathcal{O}_ X)$, then for some $n \geq 0$ and $p \in \mathbf{Z}$ the cohomology group $H^ p(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{L}^{\otimes n})$ is nonzero.
36.16 An example generator
In this section we prove that the derived category of projective space over a ring is generated by a vector bundle, in fact a direct sum of shifts of the structure sheaf.
The following lemma says that $\bigoplus _{n \geq 0} \mathcal{L}^{\otimes -n}$ is a generator if $\mathcal{L}$ is ample.
Proof. Recall that as $X$ has an ample invertible sheaf, it is quasi-compact and separated (Properties, Definition 28.26.1 and Lemma 28.26.7). Thus we may apply Proposition 36.7.5 and represent $K$ by a complex $\mathcal{F}^\bullet $ of quasi-coherent modules. Pick any $p$ such that $\mathcal{H}^ p = \mathop{\mathrm{Ker}}(\mathcal{F}^ p \to \mathcal{F}^{p + 1})/ \mathop{\mathrm{Im}}(\mathcal{F}^{p - 1} \to \mathcal{F}^ p)$ is nonzero. Choose a point $x \in X$ such that the stalk $\mathcal{H}^ p_ x$ is nonzero. Choose an $n \geq 0$ and $s \in \Gamma (X, \mathcal{L}^{\otimes n})$ such that $X_ s$ is an affine open neighbourhood of $x$. Choose $\tau \in \mathcal{H}^ p(X_ s)$ which maps to a nonzero element of the stalk $\mathcal{H}^ p_ x$; this is possible as $\mathcal{H}^ p$ is quasi-coherent and $X_ s$ is affine. Since taking sections over $X_ s$ is an exact functor on quasi-coherent modules, we can find a section $\tau ' \in \mathcal{F}^ p(X_ s)$ mapping to zero in $\mathcal{F}^{p + 1}(X_ s)$ and mapping to $\tau $ in $\mathcal{H}^ p(X_ s)$. By Properties, Lemma 28.17.2 there exists an $m$ such that $\tau ' \otimes s^{\otimes m}$ is the image of a section $\tau '' \in \Gamma (X, \mathcal{F}^ p \otimes \mathcal{L}^{\otimes mn})$. Applying the same lemma once more, we find $l \geq 0$ such that $\tau '' \otimes s^{\otimes l}$ maps to zero in $\mathcal{F}^{p + 1} \otimes \mathcal{L}^{\otimes (m + l)n}$. Then $\tau ''$ gives a nonzero class in $H^ p(X, K \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{L}^{(m + l)n})$ as desired. $\square$
Lemma 36.16.2. Let $A$ be a ring. Let $X = \mathbf{P}^ n_ A$. For every $a \in \mathbf{Z}$ there exists an exact complex of vector bundles on $X$.
\[ 0 \to \mathcal{O}_ X(a) \to \ldots \to \mathcal{O}_ X(a + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_ X(a + n + 1) \to 0 \]
Proof. Recall that $\mathbf{P}^ n_ A$ is $\text{Proj}(A[X_0, \ldots , X_ n])$, see Constructions, Definition 27.13.2. Consider the Koszul complex
\[ K_\bullet = K_\bullet (A[X_0, \ldots , X_ n], X_0, \ldots , X_ n) \]
over $S = A[X_0, \ldots , X_ n]$ on $X_0, \ldots , X_ n$. Since $X_0, \ldots , X_ n$ is clearly a regular sequence in the polynomial ring $S$, we see that (More on Algebra, Lemma 15.30.2) that the Koszul complex $K_\bullet $ is exact, except in degree $0$ where the cohomology is $S/(X_0, \ldots , X_ n)$. Note that $K_\bullet $ becomes a complex of graded modules if we put the generators of $K_ i$ in degree $+i$. In other words an exact complex
\[ 0 \to S(-n - 1) \to \ldots \to S(-n - 1 + i)^{\oplus {n \choose i}} \to \ldots \to S \to S/(X_0, \ldots , X_ n) \to 0 \]
Applying the exact functor $\tilde{\ }$ functor of Constructions, Lemma 27.8.4 and using that the last term is in the kernel of this functor, we obtain the exact complex
\[ 0 \to \mathcal{O}_ X(-n - 1) \to \ldots \to \mathcal{O}_ X(-n - 1 + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_ X \to 0 \]
Twisting by the invertible sheaves $\mathcal{O}_ X(n + a)$ we get the exact complexes of the lemma. $\square$
Lemma 36.16.3. Let $A$ be a ring. Let $X = \mathbf{P}^ n_ A$. Then is a generator (Derived Categories, Definition 13.36.3) of $D_\mathit{QCoh}(X)$.
\[ E = \mathcal{O}_ X \oplus \mathcal{O}_ X(-1) \oplus \ldots \oplus \mathcal{O}_ X(-n) \]
Proof. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Assume $\mathop{\mathrm{Hom}}\nolimits (E, K[p]) = 0$ for all $p \in \mathbf{Z}$. We have to show that $K = 0$. By Derived Categories, Lemma 13.36.4 we see that $\mathop{\mathrm{Hom}}\nolimits (E', K[p])$ is zero for all $E' \in \langle E \rangle $ and $p \in \mathbf{Z}$. By Lemma 36.16.2 applied with $a = -n - 1$ we see that $\mathcal{O}_ X(-n - 1) \in \langle E \rangle $ because it is quasi-isomorphic to a finite complex whose terms are finite direct sums of summands of $E$. Repeating the argument with $a = -n - 2$ we see that $\mathcal{O}_ X(-n - 2) \in \langle E \rangle $. Arguing by induction we find that $\mathcal{O}_ X(-m) \in \langle E \rangle $ for all $m \geq 0$. Since
\[ \mathop{\mathrm{Hom}}\nolimits (\mathcal{O}_ X(-m), K[p]) = H^ p(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X(m)) = H^ p(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X(1)^{\otimes m}) \]
we conclude that $K = 0$ by Lemma 36.16.1. (This also uses that $\mathcal{O}_ X(1)$ is an ample invertible sheaf on $X$ which follows from Properties, Lemma 28.26.12.) $\square$
Remark 36.16.4. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ be a generator (see Theorem 36.15.3). Then the following are equivalent
for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $Rf_*K = 0$ if and only if $K = 0$,
$Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ reflects isomorphisms, and
$Lf^*E$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$.
The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is an exact functor of triangulated categories. Similarly, the equivalence between (1) and (3) follows formally from the fact that $Lf^*$ is the left adjoint to $Rf_*$. These conditions hold if $f$ is affine (Lemma 36.5.2) or if $f$ is an open immersion, or if $f$ is a composition of such. We conclude that
if $X$ is a quasi-affine scheme then $\mathcal{O}_ X$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$,
if $X \subset \mathbf{P}^ n_ A$ is a quasi-compact locally closed subscheme, then $\mathcal{O}_ X \oplus \mathcal{O}_ X(-1) \oplus \ldots \oplus \mathcal{O}_ X(-n)$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.16.3.
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