{
  "abstract": "We compute derived products and inverse limits in Grothendieck categories by exact coresolutions. For the product of $r$ doubled affine lines over $B[s]$, where $B$ is any nonzero commutative ring,the exact dimensions of products, epimorphic countable limits, and arbitrary countable limits are $r$, $r$, and $r+1$. We identify the top derived functors for every $B$-module of coefficients. Both obstruction modules contain the countable reduced product as a natural $B$-module direct summand, and a formal-series map relates them by an explicit cokernel sequence. A relative cofree coresolution gives degree shifts over an arbitrary Grothendieck category and computes the localization defect for arbitrary locally nilpotent diagrams. We also construct transfinite scalar extensions preserving a prescribed kernel and its derived limit;these give nonzero completion defects beyond any fixed ordinal.The general epimorphic top-degree vanishing question is not settled by these constructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34022",
  "author_total": 2,
  "authors": [
    {
      "name": "Rui Zeng"
    },
    {
      "name": "Guocheng Dai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7bnq6jd3k5r4x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34022",
  "fulltext_url": null,
  "id": "7bnq6jd3k5r4x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34022",
  "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.KT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7223810357",
  "pdf_url": "https://arxiv.org/pdf/2610.34022",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7bnq6jd3k5r4x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34022",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34022"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Derived products and completion defects of inverse limits",
  "title_text": "Derived products and completion defects of inverse limits",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/7bnq6jd3k5r4x",
  "venue": "arXiv",
  "versions": []
}