Auslander-Reiten sequences and global dimensions. (Q862064)

The authors prove the following result: Theorem. Let \(A\) be an Artin algebra, \(M\) a generator-cogenerator for \(A\)-mod. Then: (1) If \(\alpha_i\colon V_i\to M'_i\), \(i=1,\dots,s\), are left almost split morphisms in \(A\)-mod with all \(M'_i\) being in the additive closure of \(M\), then the global dimension of the endomorphism algebra of \(M\oplus\bigoplus_{i=1}^s V_i\) does not exceed the global dimension of the endomorphism algebra of \(M\). (2) If \(\beta_i\colon M'_i\to V_i\), \(i=1,\dots,s\), are right almost split morphisms in \(A\)-mod with all \(M'_i\) being in the additive closure of \(M\), then the global dimension of the endomorphism algebra of \(M\oplus\bigoplus_{i=1}^s V_i\) does not exceed the global dimension of the endomorphism algebra of \(M\).