Persistence module and Graded Module

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 R=\bigoplus_i R_i (a direct sum of abelian groups R_i) such that R_iR_j\subset R_{i+j} for all i, j.

A graded ring R is called non-negatively graded if R_n=0 for all n\leq 0. Elements of any factor R_n of the decomposition are called homogenous elements of degree n.

Polynomial ring with standard grading:
We may grade the polynomial ring R[t] non-negatively with the standard grading R_n=Rt^n for all n\geq 0.

Graded module:
A graded module is a left module M over a graded ring R such that M=\bigoplus_i M_i and R_iM_j\subseteq M_{i+j}.

Let R be a commutative ring with unity. Let \mathscr{M}=\{M^i,\varphi^i\}_{i\geq 0} be a persistence module over R.

We now equip R[t] with the standard grading and define a graded module over R[t] by \displaystyle \alpha(\mathscr{M})=\bigoplus_{i=0}^\infty M^i where the R-module structure is the sum of the structures on the individual components. That is, for all r\in R, \displaystyle r\cdot (m^0,m^1,m^2,\dots)=(rm^0,rm^1,rm^2,\dots).

The action of t is given by \displaystyle t\cdot (m^0, m^1, m^2,\dots)=(0,\varphi^0(m^0),\varphi^1(m^1),\varphi^2(m^2),\dots).
That is, t shifts elements of the module up in the gradation.

Source: “Computing Persistent Homology” by Zomorodian and Carlsson.


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