Decomposition of pointwise finite-dimensional persistence modules

By W. Crawley‐Boevey. 2012

304 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6b5a94032634e9b9fe6bbd960edbec2aeef04cb7

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In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Statistical topological data analysis using persistence landscapes (2012, background): “Assume that λ ( X ) satisﬁes the conditions of Corollary 3.6 and that Y is a corresponding real random variable as deﬁned in (3.1).”
The action spectrum and $$C^0$$ symplectic topology (2018, background): “The structure theorem for persistence modules [7] states that for every persistence module Qt there exists a unique finite barcode B(Q) = {(Ij ,mj)}16j6N such that Q is isomorphic to ⊕ N j=1Q(Ij) mj .”
Bounds on spectral norms and barcodes (2018, background): “For detailed treatment of these topics see [12, 22, 23, 34, 49, 121].”
Persistence modules with operators in Morse and Floer theory (2017, background): “For detailed treatment of these topics see [8, 9, 12,17,34,62].”
The observable structure of persistence modules (2014, methodology): “In this subsection we prove a mild generalization of the main result of [5].”
Cotorsion torsion triples and the representation theory of filtered hierarchical clustering (2019, background): “What makes such functors so useful is that they decompose as a direct sum of persistence modules kI (see the beginning Section 2 for the precise definition), where I ⊆ [0,∞) is an interval [9].”