{
  "id": "1210.4525",
  "version": "v5",
  "published": "2012-10-16T18:50:38.000Z",
  "updated": "2015-04-24T07:25:37.000Z",
  "title": "Rationality problem for algebraic tori",
  "authors": [
    "Akinari Hoshi",
    "Aiichi Yamasaki"
  ],
  "comment": "To appear in Mem. Amer. Math. Soc., 146 pages, Section 4.3, Section 4.4 and Section 7 are added, minor typos are corrected",
  "categories": [
    "math.AG",
    "math.NT",
    "math.RA"
  ],
  "abstract": "We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\\leq {\\rm GL}(n,\\mathbb{Z})$ where $n\\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-12-29T15:22:18.000Z",
      "abstract": "We give a birational classification of algebraic tori of dimensions 4 and 5 over a field $k$. In particular, a birational classification of norm one tori whose Chevalley modules are of rank 4 and 5 is given. We show that there exist exactly 487 (resp. 7, resp. 216) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 4, and there exist exactly 3051 (resp. 25, resp. 3003) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 5. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to 6. Moreover, we show that they are stably permutation. We also verify that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\\leq 4$, and fails when the rank is 5. Indeed, there exist exactly 11 (resp. 131) $G$-lattices of rank 5 (resp. 6) which are decomposable into two different ranks. Moreover, when the rank is 6, there exist exactly 18 $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.",
      "comment": "125 pages, Section 4 is modified (thanks to S. Endo), minor typos corrected",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-04-24T07:25:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "12F20",
      "13A50",
      "14E08",
      "20C10",
      "20G15"
    ],
    "keywords": [
      "algebraic tori",
      "rationality problem",
      "retract rational",
      "computer algebra system gap",
      "birational classification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 146,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.4525H"
    }
  }
}