Picard group

Summary

The Picard group of a ringed space $X$ is the group of isomorphism classes of invertible sheaves on $X$. Its group law is the tensor product. It is a basic invariant of a scheme, and it measures the line bundles available for maps to projective space. For the spectrum of a Dedekind domain it is the ideal class group. For projective space $\mathbb{P}^n$ over a field, with $n \ge 1$, it is $\Z$, generated by $\mathcal{O}(1)$.