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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
##### Keywords:
persistence modules; direct sums of interval modules
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##### References:
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