bibkey: carlsson2008zigzag authors: Gunnar Carlsson; Vin de Silva year: 2008 title: “Zigzag Persistence” doi: null url: https://arxiv.org/abs/0812.0197v1 claim: “Proposition 3.11 constructs complementary summands for induced subspaces of filtered vector spaces; the endomorphism remark following Lemma 3.18 gives unique reconstruction from a terminal component preserving the induced filtration.” strata_touched:

• D5/S3/HomologicalAlgebra/FilteredVectorSpaceComplement
• D5/S3/HomologicalAlgebra/Persistence/ZigzagNaturalLift license: citation-only triage: anchor

Zigzag persistence

Gunnar Carlsson and Vin de Silva, “Zigzag Persistence,” arXiv:0812.0197v1 (2008). The paper develops the representation-theoretic and algorithmic foundations of persistence for diagrams whose arrows may point in either direction.

Verified locator

The checked primary version is https://arxiv.org/abs/0812.0197v1. Proposition 3.11 and its proof, on PDF pages 13–14, construct a complementary summand for every induced subspace of a filtered vector space by successively extending complements inside its increasing layers.

Lemma 3.18 and the endomorphism remark immediately following it are on PDF pages 16–17. That remark is the literature source for the terminal-filtration reconstruction formalized in ZigzagNaturalLift.

Mathematical scope

The paper works over a field with finite-dimensional vector spaces. Its filtered spaces start at zero, and the complement construction takes place inside the terminal layer. FilteredVectorSpaceComplement proves a broader statement over any division ring, without a dimension bound or prescribed chain endpoints: one complement of an arbitrary submodule in the ambient space splits every layer of a finite increasing chain, including an empty chain and repeated layers. This is a proved generalization of the recursive construction, not an attribution of the exact broader statement to the paper.

The theorem in ZigzagNaturalLift makes the predecessor reconstruction explicit for a finite actual oriented path, allows a single vertex with zero edges (n = 0) and arbitrary characteristic, and does not require finite-dimensional vertex spaces. The note attests the cited reconstruction principle; it does not claim that the paper states the repository’s exact Lean formulation or that the formal development proves the paper’s full interval-classification results.