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.