# 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.

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.

## Author: mathtuition88

https://mathtuition88.com/

This site uses Akismet to reduce spam. Learn how your comment data is processed.