The Picard group of a scheme 
, denoted 
, is the Abelian group
of isomorphism classes of line
bundles on 
under the tensor product. Its identity is the structure sheaf 
, and the inverse of a line bundle
is its dual (The Stacks Project Authors 2026). For a commutative
ring 
,
the same construction uses invertible 
-modules, which are locally free
of rank one.
The Picard group of an order 
in a number field 
is the Abelian group of equivalence classes of invertible fractional
ideals of 
,
with multiplication induced by ideal multiplication. Two invertible fractional
ideals are equivalent if one is a nonzero principal multiple of the other.
If 
is a maximal
order, i.e., the ring of integers of 
, then every nonzero fractional
ideal of 
is invertible and the Picard group of 
is the class group of 
. The group
order of the Picard group of 
is sometimes called the class
number of 
.
If 
is maximal, then the group
order of the Picard group is equal to the class number
of 
.
