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Effective Cartier divisors on a scheme are the same as invertible sheaves with fixed regular global section.

Lemma 31.14.10. Let $X$ be a scheme.

  1. If $D \subset X$ is an effective Cartier divisor, then the canonical section $1_ D$ of $\mathcal{O}_ X(D)$ is regular.

  2. Conversely, if $s$ is a regular section of the invertible sheaf $\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \subset X$ and a unique isomorphism $\mathcal{O}_ X(D) \to \mathcal{L}$ which maps $1_ D$ to $s$.

The constructions $D \mapsto (\mathcal{O}_ X(D), 1_ D)$ and $(\mathcal{L}, s) \mapsto Z(s)$ give mutually inverse maps

\[ \left\{ \begin{matrix} \text{effective Cartier divisors on }X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{isomorphism classes of pairs }(\mathcal{L}, s) \\ \text{consisting of an invertible } \mathcal{O}_ X\text{-module} \\ \mathcal{L}\text{ and a regular global section }s \end{matrix} \right\} \]

Proof. Omitted. $\square$

Comments (3)

Comment #877 by on

Suggested slogan: Effective Cartier divisors on a scheme are the same as invertible sheaves with fixed regular global section.

Comment #3582 by Laurent Moret-Bailly on

The right-hand side of the correspondence is to be understood as isomorphism classes (even though "isomorphisms are unique").

Comment #3706 by on

Yes of course you are right. Thanks very much. This is fixed here.

There are also:

  • 2 comment(s) on Section 31.14: Effective Cartier divisors and invertible sheaves

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