{
  "id": "2501.02466",
  "version": "v1",
  "published": "2025-01-05T07:27:41.000Z",
  "updated": "2025-01-05T07:27:41.000Z",
  "title": "Comparing $τ$-tilting modules and $1$-tilting modules",
  "authors": [
    "Xiao-Wu Chen",
    "Zhi-Wei Li",
    "Xiaojin Zhang",
    "Zhibing Zhao"
  ],
  "comment": "10 Pages. Any comments are welcome",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "We characterize $\\tau$-tilting modules as $1$-tilting modules over quotient algebras satisfying a tensor-vanishing condition, and characterize $1$-tilting modules as $\\tau$-tilting modules satisfying a ${\\rm Tor}^1$-vanishing condition. We use delooping levels to study \\emph{Self-orthogonal $\\tau$-tilting Conjecture}: any self-orthogonal $\\tau$-tilting module is $1$-tilting. We confirm the conjecture when the endomorphism algebra of the module has finite global delooping level.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-05T07:27:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18G25",
      "18G65",
      "18G05",
      "18G20"
    ],
    "keywords": [
      "tilting module",
      "finite global delooping level",
      "endomorphism algebra",
      "tensor-vanishing condition",
      "tilting conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}