arXiv subjects
Publications and source records attributed to Rune Harder Bak.
This is a contribution to the theory of atoms in abelian categories recently developed in a series of papers by Kanda. We present a method that enables one to explicitly compute the atom spectrum of the module category over a wide range of non-commutative rings. We illustrate our method and results by several examples.
In this paper, we define what it means for an object in an abstract module category to be dualizable and we give a homological description of the direct limit closure of the dualizable objects. Our description recovers existing results of Govorov and Lazard, Oberst and R{ö}hrl, and Christensen and Holm. When applied to differential graded modules over a differential graded algebra, our description yields that a DG-module is semi-flat if and only if it can be obtained as a direct limit of finitely generated semi-free DG-modules. We obtain similar results for graded modules over graded rings and for quasi-coherent sheaves over nice schemes.
In 2004 and 2005 Enochs et al. characterized the flat and projective quiver-representations of left rooted quivers. The proofs can be understood as filtering the classes $Φ(\operatorname{Add}\mathscr X)$ and $Φ(\varinjlim\mathscr X)$ when $\mathscr X$ is the finitely generated projective modules over a ring. In this paper we generalize the above and show that $Φ(\mathscr X)$ can always be filtered for any class $\mathscr X$ in any AB5-abelian category. With an emphasis on $Φ(\varinjlim\mathscr X)$ we investigate the Gorenstein homological situation. Using an abstract version of Pontryagin duals in abelian categories we give a more general characterization of the flat representations and end up by describing the Gorenstein flat quiver representations over right coherent rings.