# Decomposition of pointwise finite-dimensional persistence modules

@article{CrawleyBoevey2012DecompositionOP, title={Decomposition of pointwise finite-dimensional persistence modules}, author={William Crawley‐Boevey}, journal={arXiv: Representation Theory}, year={2012} }

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.

## 166 Citations

Decomposition of persistence modules

- Mathematics
- 2018

We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this…

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In this paper, we extend bottleneck stability to the setting of one dimensional constructible persistences module valued in any small abelian category.

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We define a persistent cohomology invariant called persistent cup-length which is able to extract non trivial information about the evolution of the cohomology ring structure across a filtration. We…

Exterior Critical Series of Persistence Modules

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This paper introduces a new discrete invariant: the exterior critical series, which is complete in the one-dimensional case and can be defined for multi-dimensional persistence modules, like the rank invariant.

Topological spaces of persistence modules and their properties

- MathematicsJ. Appl. Comput. Topol.
- 2018

This work considers various classes of persistence modules, including many of those that have been previously studied, and describes the relationships between them, and undertake a systematic study of the resulting topological spaces and their basic topological properties.

Interval Decomposition of Infinite Zigzag Persistence Modules

- Mathematics
- 2015

We show that every infinite zigzag persistence module decomposes into a direct sum of interval persistence modules.

J ul 2 02 1 The persistent cup-length invariant

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We define a persistent cohomology invariant called persistent cup-length which is able to extract non trivial information about the evolution of the cohomology ring structure across a filtration. We…

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A fractal dimension for a metric space based on the persistent homology of subsets of that space is introduced and hypotheses under which this dimension is comparable to the upper box dimension are exhibited.

Persistence modules with operators in Morse and Floer theory

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We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes…

The representation theorem of persistence revisited and generalized

- MathematicsJ. Appl. Comput. Topol.
- 2018

This work gives a more accurate statement of the original Representation Theorem and provides a complete and self-contained proof and generalizes the statement from the case of linear sequences of R- modules to R-modules indexed over more general monoids.

## References

SHOWING 1-9 OF 9 REFERENCES

Persistence stability for geometric complexes

- MathematicsArXiv
- 2012

The properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces are studied.

Computing persistent homology

- Mathematics, Computer ScienceSCG '04
- 2004

The homology of a filtered d-dimensional simplicial complex K is studied as a single algebraic entity and a correspondence is established that provides a simple description over fields that enables a natural algorithm for computing persistent homology over an arbitrary field in any dimension.

Proximity of persistence modules and their diagrams

- Mathematics, Computer ScienceSCG '09
- 2009

This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.

Decomposition of graded modules

- Mathematics
- 1985

In this paper, the primary objective is to obtain decomposition theorems for graded modules over the polynomial ring kfx], where k denotes a field. There is some overlap with recent work of Hoppner…

The indecomposable representations of the dihedral 2-groups

- Mathematics
- 1975

Let K be a field. We will give a complete list of the normal forms of pairs a, b of endomorphisms of a K-vector space such that a 2 b 2 = 0. Thus, we determine the modules over the ring R = K ( X ,…

Persistence Barcodes for Shapes

- MathematicsInt. J. Shape Model.
- 2005

The techniques combine the differentiating power of geometry with the classifying power of topology to obtain a shape descriptor, called a barcode, that is a finite union of intervals that reflects the geometric properties of shapes.

Barcodes: The persistent topology of data

- Mathematics
- 2007

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The…

Elements de geometrie algebrique III: Etude cohomologique des faisceaux coherents

- Mathematics
- 1961

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