Gorenstein properties of split-by-nilpotent extension algebras

Let A be a finite-dimensional k-algebra over an algebraically closed field k.In this note, we study the Gorenstein homological properties of a split-by-nilpotent extension algebra.Let R be a split-by-nilpotent extension of A. We provide sufficient conditions to ensure when a Gorenstein-projective module over A induces a similar structure over R. We also study when a Gorenstein-projective R-module induces a Gorenstein-projective A-module.Moreover, we study the relationship between the Gorensteinness of A and R.