---
tag: 3Z9DCWYXQZ
kind: concept
name: Picard group
msc: [14C22, 14C20]
sources:
  - url: https://en.wikipedia.org/wiki/Picard_group
same_as:
  wikidata: Q2042963
  wikipedia: 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 $\mathcal{O}(1)$.

## Explanations

### L1 human

Line bundles can be multiplied by the tensor product, and each line bundle has an inverse, its dual.
So the line bundles on a space, up to isomorphism, form a group: the Picard group.
On affine space every line bundle is trivial, so the Picard group is trivial.
On projective space every line bundle is a power of one basic bundle, so the Picard group is the integers.