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How far away are torsion-free rank 1 sheaves from the line bundles? Is there any condition that makes sure they are same? (for dimensions higher than 1). It is known that for a regular scheme of finite type over $\mathbb{Z}$ the Picard group is finitely generated. Is there something similar for torsion-free rank 1 sheaves? They do not form a group under tensor rank 1 sheaves but do form a monoid.

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    $\begingroup$ Any coherent sheaf of ideals on an integral scheme is rank one and torsion free. $\endgroup$ Jul 20, 2019 at 3:43
  • $\begingroup$ Oh Ok! Thanks! That was straightforward. $\endgroup$
    – user127776
    Jul 20, 2019 at 3:49

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