Gorenstein projective and injective dimensions over Frobenius extensions

Abstract

Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, ${}_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module ${}_{R}M$ is Gorenstein projective (injective). (2) if $\mathrm{G}\text{-}{proj.dim}_{A}M<\infty$, then $\mathrm{G}\text{-}{proj.dim}_{A}M=\mathrm{G}\text{-}{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\mathrm{G}\text{-}\mathrm{gldim}(A)=\mathrm{G}\text{-}\mathrm{gldim}(R)$.

A corrigendum version of Comm. Algebra,46(12):5348-5354, 2018. A typo in Proposition 3.2 is fixed, and the assumption that the extension is split is added for Theorem 3.3, 3.4, and Corollary 3.5. arXiv admin note: text overlap with arXiv:1707.05885

Cited by in corpus (1)

• Frobenius functors and Gorenstein homological properties