Corrigendum to "$m$-Periodic Gorenstein objects" [J. Algebra 621 (2023)]
Mindy Y. Huerta, Octavio Mendoza, M. A. Pérez
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 $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ of $1$-periodic $(ω,\mathcal{B})$-Gorenstein projective objects, where $ω:= \mathcal{A} \cap \mathcal{B}$ and $ν:= \mathcal{Z} \cap \mathcal{W}$. We claimed in \cite[Lem. 8.1]{HMP2023m} that the $(\mathcal{Z,W})$-Gorenstein injective dimension of $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ is bounded by the $(\mathcal{Z,W})$-Gorenstein injective dimension of $ω$, provided that: (1) $ω$ is closed under direct summands, (2) $\mathrm{Ext}^1(π\mathcal{GP}_{(ω,\mathcal{B},1)},ν) = 0$, and (3) every object in $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ admits a $\mathrm{Hom}(-,ν)$-acyclic $ν$-coresolution. These conditions are their duals are part of what we called ``Setup 1''. Moreover, if we replace $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ 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 $\mathrm{Gid}_{(\mathcal{Z,W})}(π\mathcal{GP}_{(ω,\mathcal{B},1)}) \leq \mathrm{Gid}_{(\mathcal{Z,W})}(ω)$ and $\mathrm{Gid}_{(\mathcal{Z,W})}(\mathcal{GP}_{(\mathcal{A,B})}) \leq \mathrm{Gid}_{(\mathcal{Z,W})}(ω)$ 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 $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ 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 §\ 8 of \cite{HMP2023m} affected by Lemma 8.1, and comment some applications and examples.