Gorenstein projective modules and Frobenius extensions

Gorenstein homological invariant properties under Frobenius extensions

“…Proof. The necessity holds by Lemma 2.2 in [25]. For the sake of completeness, we give the proof as follows.…”

“…However, π is not split as an A-homomorphism in general. The following lemma comes from [25], which is analogous to the results in [23] for separable algebras over commutative rings. (1) A/S is a separable extension.…”

“…In [25], Ren gave a partial answer for the above question. He proved that for a Frobenius extension, if a module over an extension ring is Gorenstein projective then it is also Gorenstein projective as a module over the base ring.…”

“…He proved that for a Frobenius extension, if a module over an extension ring is Gorenstein projective then it is also Gorenstein projective as a module over the base ring. Futhermore, the converse holds if the ring extension is a separable extension at the same time (see [25,Theorem A]). The first main result in this paper extends the result in [25] (2) A is CM-free if and only if so is S. It is well-known that determining the representation type of an algebra is fundamental and important in representation theory of Artin algebra.…”

“…Futhermore, the converse holds if the ring extension is a separable extension at the same time (see [25,Theorem A]). The first main result in this paper extends the result in [25] (2) A is CM-free if and only if so is S. It is well-known that determining the representation type of an algebra is fundamental and important in representation theory of Artin algebra. Auslander proved that there exists a 1-1 correspondence between the Morita equivalent classes of Artin algebras of finite representation type and that of Artin algebras with global dimension at most 2 and with dominant dimension at least 2 (see [1]).…”