{
  "id": "1907.05114",
  "version": "v1",
  "published": "2019-07-11T11:22:38.000Z",
  "updated": "2019-07-11T11:22:38.000Z",
  "title": "Simple-minded reductions of triangulated categories",
  "authors": [
    "Haibo Jin"
  ],
  "comment": "23 pages",
  "categories": [
    "math.RT",
    "math.CT",
    "math.RA"
  ],
  "abstract": "We will introduce a new reduction process of triangulated category, which is analogue to the silting reduction and Calabi-Yau reduction. For a triangulated category $\\cal T$ with a pre-simple-minded collection (=pre-SMC) $\\cal R$, we construct a new triangulated category $\\cal U$ such that the SMCs in $\\cal U$ bijectively correspond to those in $\\cal T$ containing $\\cal R$. Secondly, we give an analogue of Buchweitz's theorem for the singularity category $\\cal T_{\\rm sg}$ of a SMC quadruple $(\\cal T,\\cal T^{\\rm p},\\mathbb S, \\cal S)$: the category $\\cal T_{\\rm sg}$ can be realized as the stable category of an extriangulated subcategory $\\cal F$ of $\\cal T$. Finally, we show the SMS (simple-minded system) reduction due to Coelho Sim\\~oes and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi-Yau reduction is the shadow of silting reduction due to Iyama and Yang.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-11T11:22:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "triangulated category",
      "simple-minded reductions",
      "calabi-yau reduction",
      "silting reduction",
      "reduction process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}