We show that the persistent homology of a filtered simplicial complex is the standard homology of a particular graded module over a polynomial ring.
First we review some definitions.
A graded ring is a ring (a direct sum of abelian groups ) such that for all , .
A graded ring is called non-negatively graded if for all . Elements of any factor of the decomposition are called homogenous elements of degree .
Polynomial ring with standard grading:
We may grade the polynomial ring non-negatively with the standard grading for all .
A graded module is a left module over a graded ring such that and .
Let be a commutative ring with unity. Let be a persistence module over .
We now equip with the standard grading and define a graded module over by where the -module structure is the sum of the structures on the individual components. That is, for all ,
The action of is given by
That is, shifts elements of the module up in the gradation.
Source: “Computing Persistent Homology” by Zomorodian and Carlsson.