The Stacks project

Lemma 70.8.4. Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space over $S$. 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 $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Choose a dense affine open $U \subset X$ such that $\mathcal{L}|_ U$ is trivial. This is possible because $X$ has a dense open subspace which is a scheme, see Properties of Spaces, Proposition 65.13.3. Denote $s : \mathcal{O}_ U \to \mathcal{L}|_ U$ the trivialization. The complement of $U$ is an effective Cartier divisor $D$. We claim that for some $n > 0$ the map $s$ extends uniquely to a map

\[ s : \mathcal{O}_ X(-nD) \longrightarrow \mathcal{L} \]

The claim implies the lemma because it shows that $\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(nD)$ has a regular global section hence is isomorphic to $\mathcal{O}_ X(D')$ for some effective Cartier divisor $D'$ by Lemma 70.7.8. To prove the claim we may work ├ętale locally. Thus we may assume $X$ is an affine Noetherian scheme. Since $\mathcal{O}_ X(-nD) = \mathcal{I}^ n$ where $\mathcal{I} = \mathcal{O}_ X(-D)$ is the ideal sheaf of $D$ in $X$, this case follows from Cohomology of Schemes, Lemma 30.10.5. $\square$


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