{
  "id": "2003.09457",
  "version": "v1",
  "published": "2020-03-20T18:52:02.000Z",
  "updated": "2020-03-20T18:52:02.000Z",
  "title": "Compactly supported $\\mathbb{A}^{1}$-Euler characteristic and the Hochschild complex",
  "authors": [
    "Niny Arcila-Maya",
    "Candace Bethea",
    "Morgan Opie",
    "Kirsten Wickelgren",
    "Inna Zakharevich"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We show the $\\mathbb{A}^{1}$-Euler characteristic of a smooth, projective scheme over a characteristic $0$ field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported $\\mathbb{A}^{1}$-Euler characteristic $\\chi^{c}_{\\mathbb{A}^{1}}: K_0(\\mathbf{Var}_{k}) \\to \\text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-20T18:52:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F42",
      "19E15",
      "13D03",
      "55M05"
    ],
    "keywords": [
      "euler characteristic",
      "hochschild complex",
      "grothendieck-witt group",
      "grothendieck group",
      "canonical bilinear form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}