# Acknowledgments

The authors are grateful to Steve Oudot, François Petit, Clément Levrard, Wolfgang Polonik and Mathieu Carrière for useful discussions. The authors are also grateful to the Inria Datashape team for comments on an early version of this work at the annual seminar and to the anonymous reviewer for their insightful comments that helped greatly improve the exposition.

# Appendix A

In this appendix, we prove that a finitely generated filtration has finitely presentable persistent homology. As explained in Section 7.1, this fact is well-known. Its proof is included for completeness. We follow the same notations and conventions as in Oudot and Scoccola (2021, Section 2).

**Lemma A** Let $\mathcal{F}$ be a finitely generated $m$-parameter filtration of a simplicial complex $\mathcal{K}$ and let $k \ge 0$. The $m$-parameter persistence module $H_k(\mathcal{F})$ is finitely presentable. In particular, its Hilbert function is finitely presented.

**Proof** Since $\mathcal{F}$ is finitely generated, the support of any $\sigma \in \mathcal{K}$ has a finite number of minimal elements. The set of these elements is called the *births* of $\sigma$ and denoted by $\text{birth}(\sigma)$. Since $\mathcal{K}$ is finite and $\mathcal{F}$ is finitely generated, there is a finite subset $G = I_1 \times \dots \times I_m \subseteq \mathbb{R}^m$ such that $\text{birth}(\sigma) \subseteq G$ for any $\sigma \in \mathcal{K}$.

Given a persistence module $M$ over $\mathbb{R}^m$, we denote by $r(M)$ its restriction to $G$. Given a persistence module $N$ over $G$, the *extension* of $N$ is the persistence module $e(N)$ over $\mathbb{R}^m$ defined by:

$$e(N)(t) = N(\max\{g \in G : g \le t\}).$$

This defines functors $r$ and $e$ between the category of persistence modules over $\mathbb{R}^m$ to the category of persistence modules over $G$ and conversely. It is an easy exercise to check that these functors are exact.

We prove that $H_k(\mathcal{F}) \simeq e \circ r(H_k(\mathcal{F}))$. It is well known—see Bauer and Scoccola (2022, Lemma 5) for a proof—that this implies that $H_k(\mathcal{F})$ is finitely presentable. Recall that *barcodes* of free persistence modules are defined in Oudot and Scoccola (2021, Section 2) and denote by $C_i(\mathcal{F})$ the free persistence module with barcode $\bigcup_{\sigma} \text{birth}(\sigma)$ where the union is taken over all simplices $\sigma \in \mathcal{K}$ of dimension $i$. Consider the diagram:

$$C_{k+1}(\mathcal{F}) \xrightarrow{\partial_k} C_k(\mathcal{F}) \xrightarrow{\partial_{k-1}} C_{k-1}(\mathcal{F}),$$

where the maps $\partial_k$ and $\partial_{k-1}$ are induced by the boundary operator from simplicial homology. The persistence module $H_k(\mathcal{F})$ is then the homology of the above diagram, i.e.,

$$H_k(\mathcal{F}) = \text{Im}(\partial_k)/\text{Ker}(\partial_{k-1}).$$

For any $i \in \mathbb{N}$, the definition of $G$ and of $C_i(\mathcal{F})$ implies that $C_i(\mathcal{F}) \simeq e \circ r(C_i(\mathcal{F}))$. The result then follows from the fact that $e \circ r$ is an exact functor and hence commutes with computing homology.
Datashape team for comments on an early version of this work at the annual seminar and to the anonymous reviewer for their insightful comments that helped greatly improve the exposition. The second named author was funded in part by EPSRC EP/R018472/1. For the purpose of Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

## Appendix A

In this appendix, we prove that a finitely generated filtration has finitely presentable persistent homology. As explained in Section 7.1, this fact is well-known. Its proof is included for completeness. We follow the same notations and conventions as in Oudot and Scoccola (2021, Section 2).

**Lemma A** Let $\mathcal{F}$ be a finitely generated m-parameter filtration of a simplicial complex $\mathcal{K}$ and let $k \ge 0$. The m-parameter persistence module $H_k(\mathcal{F})$ is finitely presentable. In particular, its Hilbert function is finitely presented.

**Proof** Since $\mathcal{F}$ is finitely generated, the support of any $\sigma \in \mathcal{K}$ has a finite number of minimal elements. The set of these elements is called the *births* of $\sigma$ and denoted by $\text{birth}(\sigma)$. Since $\mathcal{K}$ is finite and $\mathcal{F}$ is finitely generated, there is a finite subset $G = I_1 \times \dots \times I_m \subseteq \mathbb{R}^m$ such that $\text{birth}(\sigma) \subseteq G$ for any $\sigma \in \mathcal{K}$.

Given a persistence module $M$ over $\mathbb{R}^m$, we denote by $r(M)$ its restriction to $G$. Given a persistence module $N$ over $G$, the *extension* of $N$ is the persistence module $e(N)$ over $\mathbb{R}^m$ defined by:

$$e(N)(t) = N(\max\{g \in G : g \le t\}).$$

This defines functors $r$ and $e$ between the category of persistence modules over $\mathbb{R}^m$ to the category of persistence modules over $G$ and conversely. It is an easy exercise to check that these functors are exact.

We prove that $H_k(\mathcal{F}) \simeq e \circ r(H_k(\mathcal{F}))$. It is well known—see Bauer and Scoccola (2022, Lemma 5) for a proof—that this implies that $H_k(\mathcal{F})$ is finitely presentable. Recall that *barcodes* of free persistence modules are defined in Oudot and Scoccola (2021, Section 2) and denote by $C_i(\mathcal{F})$ the free persistence module with barcode $\bigcup_{\sigma} \text{birth}(\sigma)$ where the union is taken over all simplices $\sigma \in \mathcal{K}$ of dimension $i$. Consider the diagram:

$$C_{k+1}(\mathcal{F}) \xrightarrow{\partial_k} C_k(\mathcal{F}) \xrightarrow{\partial_{k-1}} C_{k-1}(\mathcal{F}),$$

where the maps $\partial_k$ and $\partial_{k-1}$ are induced by the boundary operator from simplicial homology. The persistence module $H_k(\mathcal{F})$ is then the homology of the above diagram, i.e.,

$$H_k(\mathcal{F}) = \text{Im}(\partial_k)/\text{Ker}(\partial_{k-1}).$$

For any $i \in \mathbb{N}$, the definition of $G$ and of $C_i(\mathcal{F})$ implies that $C_i(\mathcal{F}) \simeq e \circ r(C_i(\mathcal{F}))$. The result then follows from the fact that $e \circ r$ is an exact functor and hence commutes with computing homology.

We are left to prove that the Hilbert function of $H_k(\mathcal{F})$ is finitely presented. Since the persistence module $H_k(\mathcal{F})$ is finitely presentable, it admits a finite free resolution:

$$0 \to F_m \to \dots \to F_0 \to H_k(\mathcal{F}) \to 0. \qquad (A.1)$$
We are left to prove that the Hilbert function of $H_k(\mathcal{F})$ is finitely presented. Since the persistence module $H_k(\mathcal{F})$ is finitely presentable, it admits a finite free resolution:

$$0 \to F_m \to \dots \to F_0 \to H_k(\mathcal{F}) \to 0. \qquad (A.1)$$

See for instance Botnan and Lesnick (2022, Section 7.2) for more details. Each free module with barcode $\mathcal{B}(F_i)$ has a finitely presented Hilbert function:

$$\mathrm{Hil}(F_i) = \sum_{t \in \mathcal{B}(F_i)} 1_{Q_t}.$$

Now, exactness of the sequence (A.1) ensures that:

$$\mathrm{Hil}(H_k(\mathcal{F})) = \sum_{i=0}^{m} (-1)^i \mathrm{Hil}(F_i),$$

hence the result.

<figure><img src="image_14.png" /></figure>

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See for instance Botnan and Lesnick (2022, Section 7.2) for more details. Each free module with barcode $\mathcal{B}(F_i)$ has a finitely presented Hilbert function:

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<figure><img src="image_15.png" /></figure>

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