The Stacks project

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.

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.

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

\[ 0 \to \mathcal{O}_ X(a) \to \ldots \to \mathcal{O}_ X(a + i)^{\oplus {n + 1 \choose i}} \to \ldots \to \mathcal{O}_ X(a + n + 1) \to 0 \]

of vector bundles on $X$.

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

\[ E = \mathcal{O}_ X \oplus \mathcal{O}_ X(-1) \oplus \ldots \oplus \mathcal{O}_ X(-n) \]

is a generator (Derived Categories, Definition 13.36.3) of $D_\mathit{QCoh}(X)$.

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

  1. for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $Rf_*K = 0$ if and only if $K = 0$,

  2. $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ reflects isomorphisms, and

  3. $Lf^*E$ 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

  1. if $X$ is a quasi-affine scheme then $\mathcal{O}_ X$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$,

  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 $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.16.3.

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