In the paper ["Computing Persistent Homology"](https://doi.org/10.1007/s00454-004-1146-y) 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.



Correspondence between persistence module and graded module over $R[t]$
Recently, René Corbet and Michael Kerber posted the preprint ["The Representation Theorem of Persistent Homology Revisited and Generalized"](https://arxiv.org/abs/1707.08864) which gives a proof of this statement with elementary methods. Among other things, the authors also give a short discussion on your question.


