{
  "abstract": "Let $S=\\kk[x_1,\\ldots,x_n]$ with its standard $\\mathbb N^n$-grading, and let $\\mathfrak m=(x_1,\\ldots,x_n)$. We study the essential images of the functors $M\\mapsto H^i_{\\mathfrak m}(M(-\\mathbf1))$ on categories of positively determined $S$-modules, where $\\mathbf1=(1,\\ldots,1)$. For a fixed determining bound $\\mathbf t\\in\\mathbb N^n$ and $0\\le i<n$, we identify the image in degree $i$ with the category of finite-dimensional representations of the subposet of $[\\mathbf0,\\mathbf t]$ consisting of multidegrees with at least $n-i$ positive coordinates. These images are Serre subcategories of the category of negatively $\\mathbf t$-determined modules. The top-degree image consists precisely of second cosyzygies in that category and is extension-closed, but is not abelian when $\\mathbf t\\ne\\mathbf0$. For the non-top image categories, we determine attached primes from projective covers and compute the Bass numbers at $\\mathfrak m$ and the injective dimensions over $S$ of projective and injective objects. In every non-top degree, we construct an exact functorial section selecting, for each nonzero target, a source of minimal projective dimension $n-i$ and maximal depth $i$ among all its sources. If $n\\ge2$ and $\\mathbf t\\ge\\mathbf1$, no fully faithful section exists in degrees $1$ and $n-1$, even without requiring exactness. In top degree, graded Matlis duality followed by $S$-duality gives a fully faithful functorial section. We also describe the global images obtained by varying the determining bound. The non-top global categories have enough injectives and no nonzero projective objects, while the top category corresponds to second cosyzygies in the category of finite-support representations of $\\mathbb N^n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11219",
  "author_total": 1,
  "authors": [
    {
      "name": "Dancheng Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4bsk1xhqqq3bb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11219",
  "fulltext_url": null,
  "id": "4bsk1xhqqq3bb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11219",
  "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": "W7212233318",
  "pdf_url": "https://arxiv.org/pdf/2609.11219",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4bsk1xhqqq3bb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2609.11219",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11219"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Multigraded Local Cohomology Categories: Homological Structure and Canonical Sources",
  "title_text": "Multigraded Local Cohomology Categories: Homological Structure and Canonical Sources",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/4bsk1xhqqq3bb",
  "venue": "arXiv",
  "versions": []
}