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.