{
  "abstract": "In this paper, we further investigate the role of the graded Grothendieck group $K_0^{\\gr}$ and its positive cone (the talented monoid) as an effective tool for distinguishing structural types of algebras associated to higher-rank $k$-graphs. We study the prime spectrum (the space of all prime $Γ$-order ideals equipped with a Zariski-like topology) of a general commutative $Γ$-monoid and establish that this space is spectral in the sense of Hochster, provided the monoid has the refinement property and every $Γ$-order ideal is finitely generated. As a result we are able to show that the prime spectrum of the talented monoid of a row-finite $k$-graph without sources and with a finite set of vertices is spectral. For any row-finite $k$-graph $Λ$ without sources, one of our main results says that the space of all graded prime ideals of the Kumjian--Pask algebra $\\KP(Λ)$ is homeomorphic to both the space of all prime $\\mathbb{Z}^k$-order ideals and the space of all prime $\\mathbb{Z}^k$-filters of the talented monoid $T_Λ$. Another main result of this paper provides a complete topological description of regular $Γ$-order ideals of a refinement $Γ$-monoid: a $Γ$-order ideal $J$ is regular if and only if the corresponding closed (resp., open) set $V(J)$ (resp., $D(J)$) is regular closed (resp., regular open) in the prime spectrum. As an application of these results, we establish a lattice isomorphism between the lattice of all regular $\\mathbb{Z}^k$-order ideals of the talented monoid and the lattice of all regular graded ideals of the Kumjian--Pask algebra via a spectrum-theoretic approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16632",
  "author_total": 2,
  "authors": [
    {
      "name": "Roozbeh Hazrat"
    },
    {
      "name": "Promit Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a91v633nse4x3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16632",
  "fulltext_url": null,
  "id": "a91v633nse4x3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16632",
  "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.GN"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7213435268",
  "pdf_url": "https://arxiv.org/pdf/2609.16632",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a91v633nse4x3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16632",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16632"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The prime spectrum of the talented monoid of a higher-rank graph and applications",
  "title_text": "The prime spectrum of the talented monoid of a higher-rank graph and applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/a91v633nse4x3",
  "venue": "arXiv",
  "versions": []
}