{
  "abstract": "Let $π:S\\twoheadrightarrow R$ be a split ring epimorphism with nilpotent kernel $J$, and let $R\\hookrightarrow S$ be a fixed ring-theoretic section. We prove a change-of-rings formula which is considerably stronger than an equality of global invariants: if $X$ is a left $S$-module satisfying $\\Tor_i^S(R,X)=0$ for every $i>0$, then \\[ \\pd_S X=\\pd_R(R\\otimes_S X) \\] in $\\mathbb N\\cup\\{0,\\infty\\}$. In particular, for every flat left $S$-module $F$, \\[ \\pd_SF=\\pd_R(F/JF), \\] and $F/JF$ is flat over $R$. Reduction and induction of scalars therefore show that the same numerical set of projective dimensions is attained by flat modules over $S$ and over $R$. Consequently, left $n$-perfectness is invariant under every split-by-nilpotent extension. The right-handed analogue holds as well. For a commutative ring $R$ and an arbitrary $R$-module $M$, this applies to the idealization $S=R\\ltimes M$ without any projectivity, flatness, finite generation, or finite flat-dimension assumption on $M$. We obtain a list of equivalent characterizations of the condition that $R\\ltimes M$ is $n$-perfect, together with \\[ \\cotD(R\\ltimes M)=\\cotD(R). \\] We further derive exact preservation of the flat-projective-dimension spectrum, strict $n$-perfectness, pointwise localization, arbitrary iterated idealizations, $n$-trivial extensions, truncated polynomial rings, and upper triangular matrix rings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22853",
  "author_total": 2,
  "authors": [
    {
      "name": "Wei Qi"
    },
    {
      "name": "Xiaolei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/10n2hhrdqsnav.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22853",
  "fulltext_url": null,
  "id": "10n2hhrdqsnav",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22853",
  "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",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214055873",
  "pdf_url": "https://arxiv.org/pdf/2609.22853",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/10n2hhrdqsnav.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22853",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22853"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the n-perfectness under split nilpotent extensions",
  "title_text": "On the n-perfectness under split nilpotent extensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/10n2hhrdqsnav",
  "venue": "arXiv",
  "versions": []
}