Examples of Projective Resolution of Lengths 3n/4n That Do Not Satisfy Homological Properties of Nakayama Algebra

In [1], Gyamfi et al. described homological properties in relation to Nakayama Algebras with projectives that satisfied condition Extn (M,N) = 0 for n ≫ 0 ⇐⇒ Extn (N,M) = 0 for n ≫ 0, [1]. The purpose of this paper is to give a similar characterization of Nakayama algebras. In particular, we present Ext-groups of the Nakayama algebras with projectives that do not satisfy the condition Extn (M,N) = 0 for n ≫ 0 ⇐⇒ Extn (N,M) = 0 for n ≫ 0. To do this, we consider the Ext-groups of Nakayama algebra with projectives of lengths 3n and 4n using combinations of modules of different lengths.