Lemma 71.8.4. 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
for some effective Cartier divisors D, D' in X.
Lemma 71.8.4. 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
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 66.13.3. 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
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 71.7.8. To prove the claim we may work étale 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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