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Picard groups of finite disjoint unions
PDF p. 13: Picard group of a smooth affine curve
Statement
For a finite disjoint union of open and closed subschemes , restriction gives an isomorphism . The empty union has the zero Picard group.
A line bundle is a sheaf of modules locally isomorphic to the structure sheaf. Its isomorphism classes form the Picard group under tensor product.
Facts & Assumptions
The Picard group is the group of line-bundle isomorphism classes under tensor product (Picard group of a scheme).
Given: A finite disjoint union as stated.
Goal: Establish the restriction isomorphism.
Proof
A line bundle on restricts to one on each . Conversely, sheaves on the disjoint open pieces glue uniquely because all pairwise intersections are empty; gluing line bundles gives a line bundle, since local freeness is checked on these open pieces. Restriction and gluing are inverse up to isomorphism, also for the empty union.
Restriction commutes with tensor products, and gluing does so componentwise. The bijection of step 1.1 is therefore a group isomorphism by [F1]. For the empty scheme there is one line-bundle class, the identity.
Depends on
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