Picard group of a ring

For a commutative unital ring $R$ , the Picard group $\operatorname{Pic}(R)$ is the set of isomorphism classes of invertible $R$ -modules, with operation

[L][M]=[L\otimes_R M].

The identity is $[R]$ , and $[L]^{-1}=[\operatorname{Hom}_R(L,R)]$ . Associativity and symmetry of the tensor product make this an abelian group.

Fractional ideals over a domain

If $R$ is a domain with fraction field $K$ , every invertible module can be embedded in $K$ after choosing a basis of its one-dimensional scalar extension. This identifies

\operatorname{Pic}(R)\cong \frac{\{\text{invertible fractional ideals of }R\}} {\{aR:a\in K^\times\}}.

Two such ideals are isomorphic as $R$ -modules exactly when one is a nonzero scalar multiple of the other. Changing the chosen embedding therefore does not change the class.

Orders and number rings

For a number-field order $\mathcal O$ , this is the group of invertible fractional ideals modulo principal ones. Noninvertible ideals are excluded.

For the maximal order $\mathcal O_K$ , all nonzero fractional ideals are invertible, and $\operatorname{Pic}(\mathcal O_K)$ is the existing ideal class group of $K$ .

References

The Stacks Project, Picard groups of rings, Definition 15.119.1, Lemma 15.119.2, and the Picard-group definition following the lemma.