# The Stacks Project

## Tag 0AYM

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}

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.

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