{
  "abstract": "For $2 \\times 2$ matrices $A_1,\\dots,A_k$, write $[A_1,\\dots,A_k]$ for the left-normed iterated commutator $[\\dots[[A_1,A_2],A_3],\\dots,A_k]$, and $I_k$ for the ideal, in the $3k$-variable reduced-coordinate polynomial ring $R_k$, cutting out its vanishing locus. We prove, for every $k \\geq 2$ over any field of characteristic $\\neq 2$, and as four independently-established results rather than one bundled claim: $I_k$ has codimension 2; $I_k$ has exactly 3 minimal generators; $R_k / I_k$ is Cohen-Macaulay; and $I_k$ is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: $I_k = P_2 \\cap \\cdots \\cap P_k$ is an irredundant primary decomposition into exactly $k-1$ primes, following an explicit recursive block-involvement pattern. The proof identifies $I_k$ as the ideal of $2 \\times 2$ minors of an explicit $2 \\times 3$ matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05386",
  "author_total": 1,
  "authors": [
    {
      "name": "Jan Snellman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3cgtgsxpa1c19.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05386",
  "fulltext_url": null,
  "id": "3cgtgsxpa1c19",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05386",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7211899385",
  "pdf_url": "https://arxiv.org/pdf/2609.05386",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3cgtgsxpa1c19.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05386",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05386"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices",
  "title_text": "The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/3cgtgsxpa1c19",
  "venue": "arXiv",
  "versions": []
}