# 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.