{
  "id": "math/0612448",
  "version": "v1",
  "published": "2006-12-15T15:26:10.000Z",
  "updated": "2006-12-15T15:26:10.000Z",
  "title": "Torsion classes of finite type and spectra",
  "authors": [
    "Grigory Garkusha",
    "Mike Prest"
  ],
  "categories": [
    "math.AG",
    "math.KT"
  ],
  "abstract": "Given a commutative ring R (respectively a positively graded commutative ring $A=\\ps_{j\\geq 0}A_j$ which is finitely generated as an A_0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y\\subset Spec R (respectively Y\\subset Proj A) of the form Y=\\cup_{i\\in\\Omega}Y_i, with Spec R\\Y_i (respectively Proj A\\Y_i) quasi-compact and open for all i\\in\\Omega, is established. Using these bijections, there are constructed isomorphisms of ringed spaces (Spec R,O_R)-->(Spec(Mod R),O_{Mod R}) and (Proj A,O_{Proj A})-->(Spec(QGr A),O_{QGr A}), where (Spec(Mod R),O_{Mod R}) and (Spec(QGr A),O_{QGr A}) are ringed spaces associated to the lattices L_{tor}(Mod R) and L_{tor}(QGr A) of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes perf(R) and the torsion classes of finite type in Mod R is established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-12-15T15:26:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "18E40",
      "18E30",
      "18F99"
    ],
    "keywords": [
      "finite type",
      "respectively tensor torsion classes",
      "ringed spaces",
      "perfect complexes perf",
      "thick subcategories"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12448G"
    }
  }
}