{
  "id": "1304.0754",
  "version": "v2",
  "published": "2013-04-02T19:59:36.000Z",
  "updated": "2014-04-19T02:13:52.000Z",
  "title": "The Phi-dimension: A new homological measure",
  "authors": [
    "Sonia Fernandes",
    "Marcelo Lanzilotta",
    "Octavio Mendoza"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "K. Igusa and G. Todorov introduced two functions $\\phi$ and $\\psi,$ which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become into a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin $R$-algebra $A$ and the Igusa-Todorov function $\\phi,$ we characterise the $\\phi$-dimension of $A$ in terms either of the bi-functors $\\mathrm{Ext}^{i}_{A}(-, -)$ or Tor's bi-functors $\\mathrm{Tor}^{A}_{i}(-,-).$ Furthermore, by using the first characterisation of the $\\phi$-dimension, we show that the finiteness of the $\\phi$-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result as follows: For an artin algebra $A,$ a tilting $A$-module $T$ and the endomorphism algebra $B=\\mathrm{End}_A(T)^{op},$ we have that $\\mathrm{Fidim}\\,(A)-\\mathrm{pd}\\,T\\leq \\mathrm{Fidim}\\,(B)\\leq \\mathrm{Fidim}\\,(A)+\\mathrm{pd}\\,T.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-19T02:13:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "artin algebra",
      "igusa-todorov function",
      "phi-dimension",
      "finitistic dimension conjecture",
      "endomorphism algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.0754F"
    }
  }
}