{
  "id": "2402.13142",
  "version": "v2",
  "published": "2024-02-20T16:56:20.000Z",
  "updated": "2024-03-18T18:09:58.000Z",
  "title": "Prüfer modules over wild hereditary algebras",
  "authors": [
    "Frank Lukas"
  ],
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let $A$ be a hereditary finite dimensional algebra over an algebraically closed field $k$. A brick is defined as a finitely generated module with a division ring as an endomorphism ring. Two non-isomorphic bricks $X,Y$ are said to be orthogonal if $Hom(X,Y)=Hom(Y,X)=0$ and a class of pairwise orthogonal bricks is called a semi-brick. We show that a semi-brick $\\mathcal{X}$ allows the construction of Pr\\\"ufer modules. We consider the category $Filt(\\mathcal{X})$ of modules with a filtration in $\\mathcal{X}$ and show how to construct injective objects in this category. We call the infinite dimensional, indecomposable injective objects in $Filt(\\mathcal{X})$, Pr\\\"ufer modules and show that they share many properties with the Pr\\\"ufer modules over tame hereditary algebras as defined by C. M. Ringel in \\cite{ringel1979infinite}. The results of this paper can also be applied to tame hereditary algebras. The construction gives a strict filtration of the generic module $Q$. We give an alternative proof for the classification of torsion-free divisible modules in order to show how useful this filtration is. In two appendices we show that the module category of a wild hereditary algebra has large classes of semi-bricks. The elements $X$ of these semi-bricks have the property $dim_k Ext(X,X)\\geq 2$. Bricks $X$ with $dim_k Ext(X,X)=1$ arise from full and exact embeddings of module categories of tame hereditary algebras.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-03-18T18:09:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G60"
    ],
    "keywords": [
      "wild hereditary algebra",
      "tame hereditary algebras",
      "prüfer modules",
      "module category",
      "hereditary finite dimensional algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}