In the paper "Computing Persistent Homology" by Zomorodian and Carlsson, it is stated as Theorem 3.1 that:

The correspondence $\alpha$ defines an equivalence of categories between the category of persistence modules of finite type over $R$ and the category of finitely generated non-negatively graded modules over $R[t]$.

The proof is just one line: "The proof is the Artin-Rees theory in commutative algebra".

I am curious exactly which part of Artin-Rees theory did they use to conclude that? I know there is a Artin-Rees lemma.

Thanks for any help or references.