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Decomposition of pointwise finite-dimensional persistence modules. (English) Zbl 1345.16015
From the introduction: We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on images and kernels.
We fix a base field \(k\) and a totally ordered indexing set \((R,<)\), and assume throughout that \(R\) has a countable subset which is dense in the order topology on \(R\). A persistence module \(V\) is a functor from \(R\), considered in the natural way as a category, to the category of vector spaces. Thus it consists of vector spaces \(V_t\) for \(t\in R\) and linear maps \(\rho_{ts}\colon V_s\to V_t\) for \(s\leq t\), satisfying \(\rho_{ts}\rho_{sr}=\rho_{tr}\) for all \(r\leq s\leq t\) and \(\rho_{tt}=1_{V_t}\) for all \(t\). We say that a persistence module \(V\) is pointwise finite-dimensional if all \(V_t\) are finite-dimensional. A subset \(I\subseteq R\) is an interval if it is non-empty and \(r\leq s\leq t\) with \(r,t\in I\) implies \(s\in I\). The corresponding interval module \(V=k_I\) is given by \(V_t=k\) for \(t\in I\), \(V_t=0\) for \(t\not\in I\) and \(\rho_{ts}=1\) for \(s,t\in I\) with \(s\leq t\).
Our aim is to give a short proof of the following result, which enables the use of barcodes for all pointwise finite-dimensional persistence modules.
Theorem 1.1. Any pointwise finite-dimensional persistence module is a direct sum of interval modules.
Theorem 1.2. Any persistence module with the descending chain condition on images and kernels is a direct sum of interval modules.

MSC:
16G20 Representations of quivers and partially ordered sets
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