{
  "id": "2106.10630",
  "version": "v1",
  "published": "2021-06-20T05:41:32.000Z",
  "updated": "2021-06-20T05:41:32.000Z",
  "title": "A note on the hit problem for the Steenrod algebra in some generic degrees",
  "authors": [
    "Nguyen Khac Tin"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $\\mathcal P_n=\\oplus_{d\\geqslant 0} (\\mathcal P_n)_d \\cong \\mathbb Z_2[x_1,x_2,\\ldots ,x_n]$ be the graded polynomial algebra over the prime field of two elements $\\mathbb Z/2$, in $n$ generators $x_1, x_2, \\ldots , x_n$, each of degree 1. Being the mod-2 cohomology of the classifying space $B(\\mathbb Z/2)^n$, the algebra $\\mathcal P_n$ is a module over the mod-2 Steenrod algebra $\\mathcal A$. We study the {\\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\\mathcal{A}$. The purpose of this paper is to continue our study of the hit problem by developing a result in \\cite{ph31} for $\\mathcal P_n$ in the generic degree $d_t=r(2^t-1)+s.2^t$ with $r=n=5,\\ s=13,$ and $t$ an arbitrary positive integer. Note that in the case $t=1, d_1=5(2^1-1)+13.2^1=31,$ this problem has been studied by Phuc \\cite{ph31}. Moreover, base on these results, we get the dimension result for the graded polynomial algebra in the generic degree $m=(n-1).(2^{s+4}-1)+67.2^{s+4}$ with $s$ an arbitrary positive integer, and in the case $n=6.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-20T05:41:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55S10",
      "55S05",
      "55T15"
    ],
    "keywords": [
      "steenrod algebra",
      "generic degree",
      "hit problem",
      "graded polynomial algebra",
      "arbitrary positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}