<p>The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of finite-dimensional vector spaces admits an interval decomposition — that is, a decomposition as a direct sum of simpler components called interval modules. This result fails if we replace vector spaces with modules over more general coefficient rings. To address this problem, we introduce an algorithm to determine whether or not a persistence module of pointwise free and finitely-generated modules over a principal ideal domain (PID) splits as a direct sum of interval submodules. If one exists, our algorithm outputs an interval decomposition. When considering persistence modules with coefficients in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10208_2025_9732_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10208_2025_9732_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}[x]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, our algorithm computes an interval decomposition in polynomial time. This is the first algorithm with these properties of which we are aware. We also show that a persistence module of pointwise free and finitely-generated modules over a PID splits as a direct sum of interval submodules if and only if the cokernel of every structure map is free. This result underpins the formulation of our algorithm. It also complements prior findings by Obayashi and Yoshiwaki regarding persistent homology, including a criterion for field independence and an algorithm to decompose persistence homology modules.</p>

Interval Decomposition of Persistence Modules over a Principal Ideal Domain

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发表日期：

2025年9月30日

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Jiajie Luo,
Gregory Henselman-Petrusek

摘要

The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of finite-dimensional vector spaces admits an interval decomposition — that is, a decomposition as a direct sum of simpler components called interval modules. This result fails if we replace vector spaces with modules over more general coefficient rings. To address this problem, we introduce an algorithm to determine whether or not a persistence module of pointwise free and finitely-generated modules over a principal ideal domain (PID) splits as a direct sum of interval submodules. If one exists, our algorithm outputs an interval decomposition. When considering persistence modules with coefficients in $\mathbb {Z}$ $Z$ or $\mathbb {Q}[x]$ $Q [x]$ , our algorithm computes an interval decomposition in polynomial time. This is the first algorithm with these properties of which we are aware. We also show that a persistence module of pointwise free and finitely-generated modules over a PID splits as a direct sum of interval submodules if and only if the cokernel of every structure map is free. This result underpins the formulation of our algorithm. It also complements prior findings by Obayashi and Yoshiwaki regarding persistent homology, including a criterion for field independence and an algorithm to decompose persistence homology modules.