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YCor
YCor
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Correspondence between Persistence Modulepersistence module and Graded Modulegraded module over $R[t]$

formatting, change link to journal site since it seems to be open access
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j.c.
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In the paper "Computing Persistent Homology""Computing Persistent Homology" by Zomorodian and Carlsson (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.2471&rep=rep1&type=pdf), 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 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.

In the paper "Computing Persistent Homology" by Zomorodian and Carlsson (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.2471&rep=rep1&type=pdf), 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.

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.

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yoyostein
yoyostein
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Correspondence between Persistence Module and Graded Module over $R[t]$

In the paper "Computing Persistent Homology" by Zomorodian and Carlsson (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.2471&rep=rep1&type=pdf), 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.