Persistence Interval

Next, we want to parametrize the isomorphism classes of the $F[t]$ -modules by suitable objects.

A $\mathcal{P}$ -interval is an ordered pair $(i,j)$ with $0\leq i<j\in\mathbb{Z}^\infty=\mathbb{Z}\cup\{+\infty\}$ .

We may associate a graded $F[t]$ -module to a set $\mathcal{S}$ of $\mathcal{P}$ -intervals via a bijection $Q$ . We define $\displaystyle Q(i,j)=\Sigma^i F[t]/(t^{j-i})$ for a $\mathcal{P}$ -interval $(i,j)$ . When $j=+\infty$ , we have $Q(i,+\infty)=\Sigma^iF[t]$ .

For a set of $\mathcal{P}$ -intervals $\mathcal{S}=\{(i_1,j_1),(i_2,j_2),\dots,(i_n,j_n)\}$ , we define $\displaystyle Q(\mathcal{S})=\bigoplus_{k=1}^n Q(i_k, j_k).$

We may now restate the correspondence as follows.

The correspondence $\mathcal{S}\to Q(\mathcal{S})$ defines a bijection between the finite sets of $\mathcal{P}$ -intervals and the finitely generated graded modules over the graded ring $F[t]$ .

Hence, the isomorphism classes of persistence modules of finite type over $F$ are in bijective correspondence with the finite sets of $\mathcal{P}$ -intervals.