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Categories of exact sequences with projective middle terms

Research paper by KeYan Song, YueHui Zhang

Indexed on: 19 Dec '13Published on: 19 Dec '13Published in: Science China Mathematics



Abstract

Let A be a finite-dimensional algebra over an algebraically closed field k, \(\mathcal{E}\) the category of all exact sequences in A-mod, \(\mathcal{M}_P\) (respectively, \(\mathcal{M}_I\)) the full subcategory of \(\mathcal{E}\) consisting of those objects with projective (respectively, injective) middle terms. It is proved that \(\mathcal{M}_P\) (respectively, \(\mathcal{M}_I\)) is contravariantly finite (respectively, covariantly finite) in ɛ. As an application, it is shown that \(\mathcal{M}_P = \mathcal{M}_I\) is functorially finite and has Auslander-Reiten sequences provided A is selfinjective.