{
  "id": "1012.2176",
  "version": "v1",
  "published": "2010-12-10T04:00:47.000Z",
  "updated": "2010-12-10T04:00:47.000Z",
  "title": "Good tilting modules and recollements of derived module categories",
  "authors": [
    "Hongxing Chen",
    "Changchang Xi"
  ],
  "journal": "Proc. London Math. Soc. 104(2012) 959-996",
  "doi": "10.1112/plms/pdr056",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $T$ be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring $A$, and let $B$ be the endomorphism ring of $T$. In this paper, we prove that if $T$ is good then there exists a ring $C$, a homological ring epimorphism $B\\ra C$ and a recollement among the (unbounded) derived module categories $\\D{C}$ of $C$, $\\D{B}$ of $B$, and $\\D{A}$ of $A$. In particular, the kernel of the total left derived functor $T\\otimes_B^{\\mathbb L}-$ is triangle equivalent to the derived module category $\\D{C}$. Conversely, if the functor $T\\otimes_B^{\\mathbb L}-$ admits a fully faithful left adjoint functor, then $T$ is a good tilting module. We apply our result to tilting modules arising from ring epimorphisms, and can then describe the rings $C$ as coproducts of two relevant rings. Further, in case of commutative rings, we can weaken the condition of being tilting modules, strengthen the rings $C$ as tensor products of two commutative rings, and get similar recollements. Consequently, we can produce examples (from commutative algebra and $p$-adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence),or different lengths. This shows that the Jordan-H\\\"older theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-H\\\"ugel, K\\\"onig and Liu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-10T04:00:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "derived module category",
      "tilting module",
      "recollement",
      "fully faithful left adjoint functor",
      "total left derived functor"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.2176C"
    }
  }
}