Next, we want to parametrize the isomorphism classes of the -modules by suitable objects.
A -interval is an ordered pair with .
We may associate a graded -module to a set of -intervals via a bijection . We define for a -interval . When , we have .
For a set of -intervals , we define
We may now restate the correspondence as follows.
The correspondence defines a bijection between the finite sets of -intervals and the finitely generated graded modules over the graded ring .
Hence, the isomorphism classes of persistence modules of finite type over are in bijective correspondence with the finite sets of -intervals.