{
  "id": "1510.07098",
  "version": "v1",
  "published": "2015-10-24T01:27:57.000Z",
  "updated": "2015-10-24T01:27:57.000Z",
  "title": "Applications of Exact Structures in Abelian Categories",
  "authors": [
    "Junfu Wang",
    "Zhaoyong Huang"
  ],
  "comment": "15 pages, accepted for publication in Publicationes Mathematicae Debrecen",
  "categories": [
    "math.RT",
    "math.CT",
    "math.RA"
  ],
  "abstract": "In an abelian category $\\mathscr{A}$ with small ${\\rm Ext}$ groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors $\\mathcal{F}$ of ${\\rm Ext}^{1}_{\\mathscr{A}}(-,-)$ such that $\\mathscr{A}$ has enough $\\mathcal{F}$-projectives and enough $\\mathcal{F}$-injectives and Quillen exact structures $\\mathcal{E}$ with enough $\\mathcal{E}$-projectives and enough $\\mathcal{E}$-injectives. In this case, we get a strengthened version of the translation of the Wakamatsu lemma to the exact context, and also prove that subcategories which are $\\mathcal{E}$-resolving and epimorphic precovering with kernels in their right $\\mathcal{E}$-orthogonal class and subcategories which are $\\mathcal{E}$-coresolving and monomorphic preenveloping with cokernels in their left $\\mathcal{E}$-orthogonal class are determined by each other. Then we apply these results to construct some (pre)enveloping and (pre)covering classes and complete hereditary $\\mathcal{E}$-cotorsion pairs in the module category.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-24T01:27:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian category",
      "applications",
      "orthogonal class",
      "quillen exact structures",
      "one-to-one correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}