Asymmetry of Ext-groups

In this paper, we will use techniques of noncommutative projective geometry to construct examples of algebras R over a field k not satisfying the following two types of symmetric behaviors of Ext-groups: (EE) For any pair of finitely generated R-modules (M,N), for all i∈N if and only if for all i∈N. (ee) For any pair of finitely generated R-modules (M,N), for all i≫0 if and only if for all i≫0. In particular, and contrary to the commutative case, we give a simple example of a noncommutative noetherian Gorenstein (Frobenius) local algebra satisfying (uac) (uniform Auslander condition) but not satisfying (ee).