Corrigendum to "$m$-Periodic Gorenstein objects" [J. Algebra 621 (2023)]

Abstract

Let $(\mathcal{A,B})$ be a GP-admissible pair and $(\mathcal{Z,W})$ be a GI-admissible pair of classes of objects in an abelian category $\mathcal{C}$, and consider the class $\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}$ of $1$-periodic $(\omega,\mathcal{B})$-Gorenstein projective objects, where $\omega := \mathcal{A} \cap \mathcal{B}$ and $\nu := \mathcal{Z} \cap \mathcal{W}$. We claimed in \cite[Lem. 8.1]{HMP2023m} that the $(\mathcal{Z,W})$-Gorenstein injective dimension of $\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}$ is bounded by the $(\mathcal{Z,W})$-Gorenstein injective dimension of $\omega$, provided that: (1) $\omega$ is closed under direct summands, (2) $\mathrm{Ext}^1(\pi\mathcal{GP}_{(\omega,\mathcal{B},1)},\nu) = 0$, and (3) every object in $\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}$ admits a $\mathrm{Hom}(-,\nu)$-acyclic $\nu$-coresolution. These conditions are their duals are part of what we called ``Setup 1''. Moreover, if we replace $\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}$ by the class $\mathcal{GP}_{(\mathcal{A,B})}$ of $(\mathcal{A,B})$-Gorenstein projective objects, the resulting inequality is claimed to be true under a set of conditions named ``Setup 2''. The proof we gave for the claims $\mathrm{Gid}_{(\mathcal{Z,W})}(\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}) \leq \mathrm{Gid}_{(\mathcal{Z,W})}(\omega)$ and $\mathrm{Gid}_{(\mathcal{Z,W})}(\mathcal{GP}_{(\mathcal{A,B})}) \leq \mathrm{Gid}_{(\mathcal{Z,W})}(\omega)$ is incorrect, and the purpose of this note is to exhibit a corrected proof of the first inequality, under the additional assumption that every object in $\pi\mathcal{GP}_{(\omega,\mathcal{B},1)}$ has finite injective dimension relative to $\mathcal{Z}$. Setup 2 is no longer required, and as a result the second inequality was removed. We also fix those results in \S \ 8 of \cite{HMP2023m} affected by Lemma 8.1, and comment some applications and examples.