{
  "id": "2402.15329",
  "version": "v2",
  "published": "2024-02-23T13:53:58.000Z",
  "updated": "2024-03-14T15:45:59.000Z",
  "title": "Iterations of the functor of naive $\\mathbb A^1$-connected components of varieties",
  "authors": [
    "Nidhi Gupta"
  ],
  "comment": "10 pages, comments are welcome, v2: added subsection 4.3",
  "categories": [
    "math.AG"
  ],
  "abstract": "For any sheaf of sets $\\mathcal F$ on $Sm/k$, it is well known that the universal $\\mathbb A^1$-invariant quotient of $\\mathcal F$ is given as the colimit of sheaves $\\mathcal S^n(\\mathcal F)$ where $\\mathcal S(F)$ is the sheaf of naive $\\mathbb A^1$-connected components of $\\mathcal F$. We show that these infinite iterations of naive $\\mathbb A^1$-connected components in the construction of universal $\\mathbb A^1$-invariant quotient for a scheme are certainly required. For every $n$, we construct an $\\mathbb A^1$-connected variety $X_n$ such that $\\mathcal S^n(X_n)\\neq \\mathcal S^{n+1}(X_n)$ and $\\mathcal S^{n+2}(X_n)=*$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-03-14T15:45:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F42"
    ],
    "keywords": [
      "connected components",
      "invariant quotient",
      "infinite iterations",
      "construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}