The Stacks Project


Tag 0AYM

Chapter 30: Divisors > Section 30.15: Effective Cartier divisors on Noetherian schemes

On a projective scheme, every line bundle has a regular meromorphic section.

Lemma 30.15.12. Let $X$ be a Noetherian scheme which has an ample invertible sheaf. Then every invertible $\mathcal{O}_X$-module is isomorphic to $$ \mathcal{O}_X(D - D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')^{\otimes -1} $$ for some effective Cartier divisors $D, D'$ in $X$.

Proof. Let $x_1, \ldots, x_n$ be the associated points of $X$ (Lemma 30.2.5). Let $\mathcal{L}$ be an ample invertible sheaf. There exists an $n > 0$ and a section $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $X_s = \mathop{\rm Spec}(A)$ is affine and such that $x_i \in X_s$ for $i = 1, \ldots, n$ (Properties, Lemma 27.29.6). Let $\mathfrak p_1, \ldots, \mathfrak p_n \subset A$ be the prime ideals corresponding to $x_1, \ldots, x_n$.

Then $\mathcal{N}|_{X_s}$ corresponds to an invertible $A$-module $N$. Choose an element $t \in N$, $t \not \in \mathfrak p_iN$ for all $i$. Such an element exists. This is clear if $n = 1$. If $n > 1$ first rearrange the primes such that $\mathfrak p_i \not \subset \mathfrak p_n$ for all $i < n$. Then using induction choose an element $t \in N$ with $t \not \in \mathfrak p_i N$ for $i < n$. Then we are done if $t \not \in \mathfrak p_nN$. Otherwise, pick an $t' \in N$, $t' \not \in \mathfrak p_nN$ and $f_i \in \mathfrak p_i$, $f_i \not \in \mathfrak p_n$. The element $t + f_1 f_2 \ldots f_{n - 1}t'$ will be as desired.

By Properties, Lemma 27.17.2 we see that for some $e \geq 0$ the section $s^e|_U t$ extends to a global section $\tau$ of $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$. Thus both $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$ and $\mathcal{L}^{\otimes e}$ are invertible sheaves which have global sections which generate the stalks at the associated points of $X$. Thus these are regular sections by Lemma 30.15.1. Hence $\mathcal{L}^{\otimes e} \otimes \mathcal{N} \cong \mathcal{O}_X(D)$ and $\mathcal{L}^{\otimes e} \cong \mathcal{O}_X(D')$ for some effective Cartier divisors, see Lemma 30.14.10. $\square$

    The code snippet corresponding to this tag is a part of the file divisors.tex and is located in lines 2727–2739 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-projective-Noetherian-pic-effective-Cartier}
    \begin{slogan}
    On a projective scheme, every line bundle has a regular meromorphic section.
    \end{slogan}
    Let $X$ be a Noetherian scheme which has an ample invertible sheaf.
    Then every invertible $\mathcal{O}_X$-module is isomorphic to
    $$
    \mathcal{O}_X(D - D') =
    \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D')^{\otimes -1}
    $$
    for some effective Cartier divisors $D, D'$ in $X$.
    \end{lemma}
    
    \begin{proof}
    Let $x_1, \ldots, x_n$ be the associated points of $X$
    (Lemma \ref{lemma-finite-ass}). Let $\mathcal{L}$ be an ample invertible sheaf.
    There exists an $n > 0$ and a section
    $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $X_s = \Spec(A)$
    is affine and such that $x_i \in X_s$ for $i = 1, \ldots, n$
    (Properties, Lemma \ref{properties-lemma-ample-finite-set-in-principal-affine}).
    Let $\mathfrak p_1, \ldots, \mathfrak p_n \subset A$ be the
    prime ideals corresponding to $x_1, \ldots, x_n$.
    
    \medskip\noindent
    Then $\mathcal{N}|_{X_s}$ corresponds to an invertible $A$-module $N$.
    Choose an element $t \in N$, $t \not \in \mathfrak p_iN$ for all $i$.
    Such an element exists. This is clear if $n = 1$. If $n > 1$ first
    rearrange the primes such that $\mathfrak p_i \not \subset \mathfrak p_n$
    for all $i < n$. Then using induction choose an element
    $t \in N$ with $t \not \in \mathfrak p_i N$ for $i < n$.
    Then we are done if $t \not \in \mathfrak p_nN$. Otherwise, pick an
    $t' \in N$, $t' \not \in \mathfrak p_nN$ and $f_i \in \mathfrak p_i$,
    $f_i \not \in \mathfrak p_n$. The element $t + f_1 f_2 \ldots f_{n - 1}t'$
    will be as desired.
    
    \medskip\noindent
    By Properties, Lemma \ref{properties-lemma-invert-s-sections}
    we see that for some $e \geq 0$ the section $s^e|_U t$ extends to
    a global section $\tau$ of $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$.
    Thus both $\mathcal{L}^{\otimes e} \otimes \mathcal{N}$ and
    $\mathcal{L}^{\otimes e}$ are invertible sheaves which have global sections
    which generate the stalks at the associated points of $X$.
    Thus these are regular sections by
    Lemma \ref{lemma-regular-section-associated-points}.
    Hence $\mathcal{L}^{\otimes e} \otimes \mathcal{N} \cong \mathcal{O}_X(D)$
    and $\mathcal{L}^{\otimes e} \cong \mathcal{O}_X(D')$ for some
    effective Cartier divisors, see Lemma \ref{lemma-characterize-OD}.
    \end{proof}

    Comments (1)

    Comment #2599 by Rogier Brussee on June 5, 2017 a 11:16 am UTC

    Suggested slogan: On a projective scheme, every line bundle has a meromorphic section.

    Add a comment on tag 0AYM

    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?