Decomposition of persistence modules
HTML articles powered by AMS MathViewer
- by Magnus Bakke Botnan and William Crawley-Boevey PDF
- Proc. Amer. Math. Soc. 148 (2020), 4581-4596 Request permission
Abstract:
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 result we give new, short proofs of fundamental structure theorems for persistence modules.References
- Gorô Azumaya, Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem, Nagoya Math. J. 1 (1950), 117–124. MR 37832
- Magnus Bakke Botnan, Interval decomposition of infinite zigzag persistence modules, Proc. Amer. Math. Soc. 145 (2017), no. 8, 3571–3577. MR 3652808, DOI 10.1090/proc/13465
- Magnus Bakke Botnan and Michael Lesnick, Algebraic stability of zigzag persistence modules, Algebr. Geom. Topol. 18 (2018), no. 6, 3133–3204. MR 3868218, DOI 10.2140/agt.2018.18.3133
- Gunnar Carlsson and Afra Zomorodian, The theory of multidimensional persistence, Discrete Comput. Geom. 42 (2009), no. 1, 71–93. MR 2506738, DOI 10.1007/s00454-009-9176-0
- Gunnar Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 2, 255–308. MR 2476414, DOI 10.1090/S0273-0979-09-01249-X
- Jérémy Cochoy and Steve Oudot, Decomposition of exact pfd persistence bimodules, Discrete Comput. Geom. 63 (2020), no. 2, 255–293. MR 4057439, DOI 10.1007/s00454-019-00165-z
- William Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22 (1994), no. 5, 1641–1674. MR 1264733, DOI 10.1080/00927879408824927
- William Crawley-Boevey, Decomposition of pointwise finite-dimensional persistence modules, J. Algebra Appl. 14 (2015), no. 5, 1550066, 8. MR 3323327, DOI 10.1142/S0219498815500668
- P. Gabriel and A. V. Roĭter, Representations of finite-dimensional algebras, Algebra, VIII, Encyclopaedia Math. Sci., vol. 73, Springer, Berlin, 1992, pp. 1–177. With a chapter by B. Keller. MR 1239447
- Michael Höppner, A note on the structure of injective diagrams, Manuscripta Math. 44 (1983), no. 1-3, 45–50. MR 709843, DOI 10.1007/BF01166072
- Michael Lesnick and Matthew Wright, Interactive visualization of 2-d persistence modules, arXiv preprint arXiv:1512.00180 (2015).
- Steve Y. Oudot, Persistence theory: from quiver representations to data analysis, Mathematical Surveys and Monographs, vol. 209, American Mathematical Society, Providence, RI, 2015. MR 3408277, DOI 10.1090/surv/209
- Claus Michael Ringel, Introduction to representation theory of finite dimensional algebras, 2014 (accessed August 19 2018).
- Claus Michael Ringel, Representation theory of Dynkin quivers. Three contributions, Front. Math. China 11 (2016), no. 4, 765–814. MR 3531031, DOI 10.1007/s11464-016-0548-5
Additional Information
- Magnus Bakke Botnan
- Affiliation: Department of Mathematics, VU University Amsterdam, The Netherlands
- MR Author ID: 1002262
- Email: m.b.botnan@vu.nl
- William Crawley-Boevey
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
- MR Author ID: 230720
- Email: wcrawley@math.uni-bielefeld.de
- Received by editor(s): November 21, 2018
- Received by editor(s) in revised form: January 16, 2019, and June 24, 2019
- Published electronically: August 14, 2020
- Additional Notes: The first author was supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”.
The second author was supported by the Alexander von Humboldt Foundation in the framework of an Alexander von Humboldt Professorship endowed by the German Federal Ministry of Education and Research. - Communicated by: Jerzy Weyman
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4581-4596
- MSC (2020): Primary 16G20; Secondary 55N31
- DOI: https://doi.org/10.1090/proc/14790
- MathSciNet review: 4143378