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}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 0BQQ

    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 lower-right corner).

    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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?