{
  "id": "2307.11347",
  "version": "v1",
  "published": "2023-07-21T04:47:48.000Z",
  "updated": "2023-07-21T04:47:48.000Z",
  "title": "Classifying $t$-structures via ICE-closed subcategories and a lattice of torsion classes",
  "authors": [
    "Arashi Sakai"
  ],
  "comment": "16 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In a triangulated category equipped with a $t$-structure, we investigate a relation between ICE-closed (=Image-Cokernel-Extension-closed) subcategories of the heart of the $t$-structure and aisles in the triangulated categories. We introduce an ICE sequence, a sequence of ICE-closed subcategories satisfying a certain condition, and establish a bijection between ICE sequences and homology-determined preaisles. Moreover we give a sufficient condition that an ICE sequence induces a $t$-structure via the bijection. In the case of the bounded derived category $D^b({\\mathsf{mod}}\\Lambda)$ of a $\\tau$-tilting finite algebra $\\Lambda$, we give a description of ICE sequences in ${\\mathsf{mod}}\\Lambda$ which induce bounded $t$-structures on $D^b({\\mathsf{mod}}\\Lambda)$ from the viewpoint of a lattice consisting of torsion classes in ${\\mathsf{mod}}\\Lambda$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-21T04:47:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G10",
      "18E10",
      "18G80"
    ],
    "keywords": [
      "torsion classes",
      "ice-closed subcategories",
      "ice sequence induces",
      "triangulated category",
      "tilting finite algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}