Persistent Homology over Z_p

• Persistent homology modules over Z_p are graded modules over the polynomial ring Z_p[t] that capture the evolution of homological features via canonical barcode decompositions.
• The barcode theorem provides a unique decomposition into free and torsion components, ensuring robust feature extraction and stability under perturbations.
• Efficient computational methods, such as graded Smith normal form and matrix reduction, enable the practical application of these modules in analyzing high-dimensional noisy data.

Persistent homology modules over $\mathbb{Z}_p$ are central algebraic objects in topological data analysis (TDA), capturing the evolution of homological features across a filtration of a topological or combinatorial object, with computations performed in a finite field of prime order. The structure of these modules admits a canonical decomposition into interval modules—equivalently, barcodes—rooted in both algebraic and categorical principles. The theory is underpinned by the graded module structure over the principal ideal domain $\mathbb{Z}_p[t]$, producing a fully classifiable family that is computationally tractable, stable under perturbation, and adaptable to torsion-free analysis via strategic choice of $p$.

1. Formal Definition and Module Structure

Let $(\mathbb{R}_+, \leq)$ denote the poset of nonnegative real numbers, and $\mathrm{Vect}(\mathbb{Z}_p)$ the category of finite-dimensional $\mathbb{Z}_p$-vector spaces. A (one-parameter) persistence module $M$ over $\mathbb{Z}_p$ is a functor:

$M : \mathbb{R}_+ \to \mathrm{Vect}(\mathbb{Z}_p)$

assigning to each $r\in\mathbb{R}_+$ a vector space $M(r)$ and to each $r\leq s$ a linear map $M(r\leq s): M(r)\to M(s)$, satisfying the natural compatibilities. Restricting to discrete parameters $A=\mathbb{N}$, $M$ is equivalently an $\mathbb{N}$-graded module over the monoid algebra $\mathbb{Z}_p[t]$ (with $t$ in degree $+1$), with

$M_* = \bigoplus_{i\in\mathbb{N}} M(i)$

and $t$ acting by the transition maps $M(i\leq i+1)$, endowing $M_*$ with a graded $\mathbb{Z}_p[t]$-module structure. This categorical–algebraic equivalence provides the setting for structural analysis (Carlsson, 2020, Ranoa, 2024).

2. Structural Classification: The Barcode Theorem

The polynomial ring $\mathbb{Z}_p[t]$ is a principal ideal domain (PID), so the classical structure theorem for finitely generated modules applies. Any finitely presented, $\mathbb{N}$-graded module $M$ over $\mathbb{Z}_p[t]$ admits a canonical decomposition:

$$M \cong \bigoplus_{i=1}^m \mathbb{Z}_p[t](-a_i) \oplus \bigoplus_{j=1}^\ell \mathbb{Z}_p[t]/(t^{b_j - a_j})(-a_j)$$

with unique invariants up to permutation. Here, the grading shift $(-a)$ positions the generator in degree $a$. Algebraically, the free summands $\mathbb{Z}_p[t](-a_i)$ correspond to homology classes that appear at index $a_i$ and persist forever ("infinite bars"), while the torsion summands $\mathbb{Z}_p[t]/(t^{b_j-a_j})(-a_j)$ encode features alive for the interval $[a_j, b_j)$ (Corbet et al., 2017, Skraba et al., 2013).

Functorially, this matches the decomposition into interval modules $I[a,b)$:

$M \cong \bigoplus_{k=1}^n I[a_k, b_k)$

with $I[a, b)(r) = \mathbb{Z}_p$ if $a \leq r < b$, zero otherwise, and all maps in the interval are identity morphisms. The barcode

$\mathcal{B}(M) = \{ [a_k, b_k) \}_{k=1}^n$

completely determines the isomorphism class of $M$, and is unique up to permutation (Ranoa, 2024, Carlsson, 2020).

3. Computational Methods: Graded Smith Normal Form and Matrix Reduction

To extract barcodes in practice, the $\mathbb{Z}_p[t]$-module structure is realized algorithmically via graded Smith normal form (GSNF) or, equivalently, via the classical persistence matrix reduction over $\mathbb{Z}_p$.

For chain complexes with filtration indexed by $0\leq d \leq N$, simplices are assigned degrees, and boundaries are represented as matrices with polynomial shifts. The GSNF procedure diagonalizes these matrices, revealing the invariant factors—each a monomial $t^q$—thus detecting every interval $[a, a+q)$ (torsion) and $[a,\infty)$ (free part) (Skraba et al., 2013). Operations are restricted to maintain grading.

In standard persistence calculations over $\mathbb{Z}_p$, columns (simplices) are reduced by row operations mod $p$, yielding a unique birth–death matching of generators and relations:

• Each pair $(i, j)$ with simplex $j$ reducing to lowest pivot in row $i$ encodes a bar $[b(i), b(j))$.
• Unpaired generators correspond to infinite bars. This reduction, first systematized by Zomorodian–Carlsson and frequently implemented for $p=2$, is robust and efficient for high-throughput data (Ranoa, 2024).

4. Stability and Metric Properties of Barcodes

Barcodes, as multisets of intervals, are endowed with the bottleneck distance $W_\infty$, defined as the $L_\infty$-cost of the optimal matching between intervals (possibly pairing with diagonal elements for unmatched bars). Stability theorems assert that

• For metric spaces $X$, $Y$ and any $k$,

$$W_\infty(\mathcal{B}_k(X), \mathcal{B}_k(Y)) \leq d_{GH}(X, Y)$$

where $d_{GH}$ denotes the Gromov–Hausdorff distance.

• For tame functions $f, g$,

$$W_\infty(\mathcal{B}(f), \mathcal{B}(g)) \leq \|f-g\|_\infty$$

Both results hold for coefficients in $\mathbb{Z}_p$, since the algebraic theory over any field applies (Carlsson, 2020). The stability principles underwrite the reliability of persistent homology in data analysis, ensuring small input perturbations cause at most small changes in the extracted barcode.

5. Phenomena Related to Torsion and Prime Selection

When working over $\mathbb{Z}_p$, all homology is naturally a vector space, and torsion elements (i.e., elements killed by $n>1$ in integral homology) become invisible except when $n$ divides $p$. The Universal Coefficient Theorem yields

$$H_k(\Delta; \mathbb{Z}_p) \cong (H_k(\Delta; \mathbb{Z}) \otimes \mathbb{Z}_p) \oplus \mathrm{Tor}(H_{k-1}(\Delta; \mathbb{Z}), \mathbb{Z}_p)$$

For primes $p$ not dividing the torsion in $H_k(\Delta; \mathbb{Z})$ or $H_{k-1}(\Delta; \mathbb{Z})$, i.e., outside a finite set of "bad" primes determined by the Smith normal form of the boundary operators, all torsion disappears and persistent homology over $\mathbb{Z}_p$ recovers exactly the rank of the free part. Thus, the barcode over $\mathbb{Z}_p$ matches that over $\mathbb{Q}$ in these cases (Cavalcanti, 26 Dec 2025).

Computationally, one can analyze the Smith normal forms to pre-select a prime avoiding all torsion divisors, ensuring that persistent calculations are "torsion-free". This is particularly relevant in the geometric context of Finsler-TDA, where high-dimensional "fake" cycles are artifacts, and their algebraic detection via torsion must be suppressed for meaningful feature selection (Cavalcanti, 26 Dec 2025).

6. Practical Examples and Computational Implications

The fundamental decomposition supports efficient computation and direct interpretability. Typical examples include:

• A single vertex appearing at time $0$: barcode $[0,\infty)$.
• An edge appearing at time $0$ and removed at time $1$: barcode $[0,1)$.
• A 1-cycle forming at $2$ and filled at $5$: barcode $[2,5)$.

In noisy data settings—e.g., points sampled around a circle with added noise—persistent homology over $\mathbb{Z}_2$ produces barcodes with robust long bars reflecting true topological features (such as the $H_1$ bar for the main loop) and numerous short bars corresponding to noise or artifacts. Most persistent homology software defaults to $p=2$ for computational efficiency (Carlsson, 2020).

7. Connections, Limitations, and Advanced Modifications

While the barcodes over $\mathbb{Z}_p$ capture the essential structure of persistence modules in one-parameter filtrations, in multiparameter settings or over rings with non-trivial torsion (such as $\mathbb{Z}$), no such interval decomposition generally exists; the full classification becomes much subtler and is not covered by the same structure theorem (Corbet et al., 2017, Skraba et al., 2013).

In applications sensitive to torsion—such as certain geometric, combinatorial, or arithmetic contexts—it may be essential to analyze homology over several primes or perform integral computations with explicit torsion tracking, thus refining or extending the standard $\mathbb{Z}_p$-module analysis.

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References: (Carlsson, 2020, Corbet et al., 2017, Cavalcanti, 26 Dec 2025, Skraba et al., 2013, Ranoa, 2024)