{
  "abstract": "Let $(R,\\mathfrak{m},k)$ be a commutative Noetherian ring which is either local or positively graded over the field $k$. We provide a method based on liftings for constructing free coresolutions of $k$ over $R$. We apply this method to twisted trivial extensions of the node $k[u,v]/(uv)$ to build a family of positively graded $k$-algebras of dimension $1$ over which $k$ is an infinite graded syzygy, resolving an open question of Iyengar in the negative. We also prove a formula for alternating sums of lengths of Ext modules when $R$ is positively graded, and use it to show that a graded free coresolution of $k$ cannot dualize to a resolution unless $R$ is Artinian Gorenstein.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38393",
  "author_total": 2,
  "authors": [
    {
      "name": "Jian Liu"
    },
    {
      "name": "Justin Lyle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/14znvm505zcn4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38393",
  "fulltext_url": null,
  "id": "14znvm505zcn4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38393",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38393",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/14znvm505zcn4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38393",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38393"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The residue field as an infinite syzygy",
  "title_text": "The residue field as an infinite syzygy",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/14znvm505zcn4",
  "venue": "arXiv",
  "versions": []
}