{
  "abstract": "Let $R$ be a complete noetherian local ring of prime characteristic $p$, and let $R^\\infty$ denote its perfect closure. We prove that a finitely generated \\(R\\)-module $N$ has finite injective dimension if and only if $\\operatorname{Ext}_R^i(R^\\infty, N) = 0$ for all $i > 0$. As a Gorenstein counterpart, we show that finiteness of the Gorenstein injective (or projective) dimension of $R^\\infty$ forces $R$ to be Gorenstein, and we relate this to a non noetherian Cohen factorization of $R \\to R^\\infty$. Applications include preservation of Gorensteinness under weakly etale extensions, structural results on F-coherent and weakly F-nilpotent rings, along with the ascent of Frobenius closure of a parameter ideal to its powers. Assuming $\\operatorname{Ext}^{i}_{R}(\\frac{R}{\\mathfrak m},R^{\\infty})=0$ for some $i>\\dim(R)$, we show $R$ is regular. This has some applications. We study the rationality problem of Hilbert--Kunz multiplicity by linking it to $R^\\infty$. Recall that the global dimension can be viewed as a uniform bound on the injective dimension of modules. Finally, we determine $\\operatorname{gldim}(R^\\infty)$ by presenting a new bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.30416",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohsen Asgharzadeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6p168141yz4h2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.30416",
  "fulltext_url": null,
  "id": "6p168141yz4h2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.30416",
  "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"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7166654821",
  "pdf_url": "https://arxiv.org/pdf/2606.30416",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6p168141yz4h2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2606.30416",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.30416"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Perfect closure detects injective dimension",
  "title_text": "Perfect closure detects injective dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/6p168141yz4h2",
  "venue": "arXiv",
  "versions": []
}