Abelian quotients of categories of short exact sequences. (English) Zbl 1435.16003
The author considers abelian factor categories of the category of short exact sequences over an exact category \(\mathscr{C}\). The main theorem states that if \(\mathscr{C}\) has enough projectives, constituting a full subcategory \(\mathscr{P}\), the homotopy category \(\mbox{Ext}(\mathscr{C})\) of short exact sequence is equivalent to the category \(\mathbf{mod}(\mathscr{C}/[\mathscr{P}])\) of finitely presented \(\mathscr{C}/[\mathscr{P}]\)-modules. Dual versions of this fact, and some applications are given. For example, if \(\mathscr{C}\) has enough injectives, with corresponding full subcategory \(\mathscr{I}\), the existence of almost split sequences in \(\mathscr{C}\) is related to a Nakayama functor \(\mathscr{C}/[\mathscr{P}]\to\mathscr{C}/[\mathscr{I}]\).
Reviewer’s remark: The equivalence \(\mathrm{Ext}(\mathscr{C})\approx\mathbf{mod}(\mathscr{C}/[\mathscr{P}])\approx\mathbf{mod}(\mathscr{C}/[\mathscr{I}])\) plays an essential role in the concept of triad (see the formula (26) in the reviewer’s paper [W. Rump, J. Algebra 280, No. 2, 435–462 (2004; Zbl 1079.16008)] and [W. Rump, Commun. Algebra 33, No. 1, 73–95 (2005; Zbl 1081.16027)], proof of Theorem 3). For the existence of almost split sequences, see Zbl 1079.16008, Theorem 5.
Reviewer’s remark: The equivalence \(\mathrm{Ext}(\mathscr{C})\approx\mathbf{mod}(\mathscr{C}/[\mathscr{P}])\approx\mathbf{mod}(\mathscr{C}/[\mathscr{I}])\) plays an essential role in the concept of triad (see the formula (26) in the reviewer’s paper [W. Rump, J. Algebra 280, No. 2, 435–462 (2004; Zbl 1079.16008)] and [W. Rump, Commun. Algebra 33, No. 1, 73–95 (2005; Zbl 1081.16027)], proof of Theorem 3). For the existence of almost split sequences, see Zbl 1079.16008, Theorem 5.
Reviewer: Wolfgang Rump (Stuttgart)
MSC:
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 16G10 | Representations of associative Artinian rings |
| 18G05 | Projectives and injectives (category-theoretic aspects) |
| 18E10 | Abelian categories, Grothendieck categories |
| 18G80 | Derived categories, triangulated categories |
Keywords:
short exact sequence; quotient category; exact category; Auslander-Reiten sequence; Auslander-Reiten dualityReferences:
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