{
  "abstract": "Matroid Stanley-Reisner rings admit essential deformation obstructions of arbitrarily high order over every field, already at corank three. For each $m\\ge3$, we construct a connected cosimple matroid and $m$ fine-homogeneous first-order classes whose proper subcollections integrate jointly and whose pairwise primary cup products vanish. A specified joint $(m-1)$-jet nevertheless has no lift to order $m$, even without preserving the grading. The construction rests on formulas for every multigraded component of the second cotangent cohomology $T^2$ and of its primary cup image, expressed through flats and parallel classes of the dual matroid. These formulas show that $T^2$ vanishes exactly when every connected component has corank at most two, or equivalently when there is no $U_{1,4}$-minor. This proves a conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello (2025) and characterizes formal smoothness of the affine deformation functor. For the duals of rank-three truncations of $U_{2,m}\\oplus U_{2,n}$, $m,n\\ge3$, each nonzero homogeneous class modulo the primary cup image is realized as a higher-order obstruction, and the quotient has an explicit presentation with $m+n-2$ minimal homogeneous generators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20649",
  "author_total": 1,
  "authors": [
    {
      "name": "Qihang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d39dy9jg8egz1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20649",
  "fulltext_url": null,
  "id": "d39dy9jg8egz1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20649",
  "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.CO"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7222677362",
  "pdf_url": "https://arxiv.org/pdf/2610.20649",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d39dy9jg8egz1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.20649",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20649"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cotangent cohomology and higher-order obstructions of matroids",
  "title_text": "Cotangent cohomology and higher-order obstructions of matroids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/d39dy9jg8egz1",
  "venue": "arXiv",
  "versions": []
}