# The Stacks Project

## Tag 0BQQ

### 35.15. 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.

Lemma 35.15.1. Let $X$ be a scheme and $\mathcal{L}$ an ample invertible $\mathcal{O}_X$-module. If $K$ is a nonzero object of $D_\textit{QCoh}(\mathcal{O}_X)$, 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.

Proof. Recall that as $X$ has an ample invertible sheaf, it is quasi-compact and separated (Properties, Definition 27.26.1 and Lemma 27.26.7). Thus we may apply Proposition 35.6.5 and represent $K$ by a complex $\mathcal{F}^\bullet$ of quasi-coherent modules. Pick any $p$ such that $\mathcal{H}^p = \mathop{\rm Ker}(\mathcal{F}^p \to \mathcal{F}^{p + 1})/ \mathop{\rm Im}(\mathcal{F}^{p - 1} \to \mathcal{F}^p)$ 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 27.17.2 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 35.15.2. Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. For every $a \in \mathbf{Z}$ there exists an exact complex $$0 \to \mathcal{O}_X(a) \to \ldots \to \mathcal{O}_X(a + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_X(a + n + 1) \to 0$$ of vectorbundles on $X$.

Proof. Recall that $\mathbf{P}^n_A$ is $\text{Proj}(A[X_0, \ldots, X_n])$, see Constructions, Definition 26.13.2. 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 15.27.2) 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 26.8.4 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 35.15.3. Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. Then $$E = \mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n)$$ is a generator (Derived Categories, Definition 13.33.2) of $D_\textit{QCoh}(X)$.

Proof. Let $K \in D_\textit{QCoh}(\mathcal{O}_X)$. Assume $\mathop{\rm Hom}\nolimits(E, K[p]) = 0$ for all $p \in \mathbf{Z}$. We have to show that $K = 0$. By Derived Categories, Lemma 13.33.3 we see that $\mathop{\rm Hom}\nolimits(E', K[p])$ is zero for all $E' \in \langle E \rangle$ and $p \in \mathbf{Z}$. By Lemma 35.15.2 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 $$\mathop{\rm Hom}\nolimits(\mathcal{O}_X(-m), K[p]) = H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(m)) = H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(1)^{\otimes m})$$ we conclude that $K = 0$ by Lemma 35.15.1. (This also uses that $\mathcal{O}_X(1)$ is an ample invertible sheaf on $X$ which follows from Properties, Lemma 27.26.12.) $\square$

Remark 35.15.4. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $E \in D_\textit{QCoh}(\mathcal{O}_Y)$ be a generator (see Theorem 35.14.3). Then the following are equivalent

1. for $K \in D_\textit{QCoh}(\mathcal{O}_X)$ we have $Rf_*K = 0$ if and only if $K = 0$,
2. $Rf_* : D_\textit{QCoh}(\mathcal{O}_X) \to D_\textit{QCoh}(\mathcal{O}_Y)$ reflects isomorphisms, and
3. $Lf^*E$ is a generator for $D_\textit{QCoh}(\mathcal{O}_X)$.

The equivalence between (1) and (2) is a formal consequence of the fact that $Rf_* : D_\textit{QCoh}(\mathcal{O}_X) \to D_\textit{QCoh}(\mathcal{O}_Y)$ 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 35.5.1) or if $f$ is an open immersion, or if $f$ is a composition of such. We conclude that

1. if $X$ is a quasi-affine scheme then $\mathcal{O}_X$ is a generator for $D_\textit{QCoh}(\mathcal{O}_X)$,
2. 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 $D_\textit{QCoh}(\mathcal{O}_X)$ by Lemma 35.15.3.

The code snippet corresponding to this tag is a part of the file perfect.tex and is located in lines 3167–3341 (see updates for more information).

\section{An example generator}
\label{section-example-generator}

\noindent
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.

\medskip\noindent
The following lemma says that $\bigoplus_{n \geq 0} \mathcal{L}^{\otimes -n}$
is a generator if $\mathcal{L}$ is ample.

\begin{lemma}
\label{lemma-nonzero-some-cohomology}
Let $X$ be a scheme and $\mathcal{L}$ an ample invertible
$\mathcal{O}_X$-module. If $K$ is a nonzero object of
$D_\QCoh(\mathcal{O}_X)$, 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.
\end{lemma}

\begin{proof}
Recall that as $X$ has an ample invertible sheaf, it is quasi-compact
and separated (Properties, Definition \ref{properties-definition-ample} and
Lemma \ref{properties-lemma-affine-s-opens-cover-quasi-separated}).
Thus we may apply
Proposition \ref{proposition-quasi-compact-affine-diagonal}
and represent $K$ by a complex $\mathcal{F}^\bullet$ of
quasi-coherent modules. Pick any $p$ such that
$\mathcal{H}^p = \Ker(\mathcal{F}^p \to \mathcal{F}^{p + 1})/ \Im(\mathcal{F}^{p - 1} \to \mathcal{F}^p)$ 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 \ref{properties-lemma-invert-s-sections}
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.
\end{proof}

\begin{lemma}
\label{lemma-construct-the-next-one}
Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. For every $a \in \mathbf{Z}$
there exists an exact complex
$$0 \to \mathcal{O}_X(a) \to \ldots \to \mathcal{O}_X(a + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_X(a + n + 1) \to 0$$
of vectorbundles on $X$.
\end{lemma}

\begin{proof}
Recall that $\mathbf{P}^n_A$ is $\text{Proj}(A[X_0, \ldots, X_n])$, see
Constructions, Definition \ref{constructions-definition-projective-space}.
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 \ref{more-algebra-lemma-regular-koszul-regular})
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 \ref{constructions-lemma-proj-sheaves} 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.
\end{proof}

\begin{lemma}
\label{lemma-generator-P1}
Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. Then
$$E = \mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n)$$
is a generator
(Derived Categories, Definition \ref{derived-definition-generators})
of $D_\QCoh(X)$.
\end{lemma}

\begin{proof}
Let $K \in D_\QCoh(\mathcal{O}_X)$. Assume
$\Hom(E, K[p]) = 0$ for all $p \in \mathbf{Z}$.
We have to show that $K = 0$.
By Derived Categories, Lemma
\ref{derived-lemma-right-orthogonal}
we see that $\Hom(E', K[p])$ is zero for all $E' \in \langle E \rangle$
and $p \in \mathbf{Z}$.
By Lemma \ref{lemma-construct-the-next-one}
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
$$\Hom(\mathcal{O}_X(-m), K[p]) = H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(m)) = H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(1)^{\otimes m})$$
we conclude that $K = 0$ by Lemma \ref{lemma-nonzero-some-cohomology}.
(This also uses that $\mathcal{O}_X(1)$ is an ample
invertible sheaf on $X$ which follows from
Properties, Lemma \ref{properties-lemma-open-in-proj-ample}.)
\end{proof}

\begin{remark}
\label{remark-pullback-generator}
Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes.
Let $E \in D_\QCoh(\mathcal{O}_Y)$ be a generator
(see Theorem \ref{theorem-bondal-van-den-Bergh}).
Then the following are equivalent
\begin{enumerate}
\item for $K \in D_\QCoh(\mathcal{O}_X)$ we have
$Rf_*K = 0$ if and only if $K = 0$,
\item $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$
reflects isomorphisms, and
\item $Lf^*E$ is a generator for $D_\QCoh(\mathcal{O}_X)$.
\end{enumerate}
The equivalence between (1) and (2) is a formal consequence of the fact that
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ 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 \ref{lemma-affine-morphism})
or if $f$ is an open immersion, or if $f$ is a composition of such.
We conclude that
\begin{enumerate}
\item if $X$ is a quasi-affine scheme then $\mathcal{O}_X$ is a generator
for $D_\QCoh(\mathcal{O}_X)$,
\item 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 $D_\QCoh(\mathcal{O}_X)$ by
Lemma \ref{lemma-generator-P1}.
\end{enumerate}
\end{remark}

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