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

Abstract

Let $(\mathcal{A},\mathcal{B})$ be a GP-admissible pair and $(\mathcal{Z},\mathcal{W})$ be a GI-admissible pair of classes of objects in an abelian category $\mathcal{C}$, and consider the class $\pi {\mathcal{G}\mathcal{P}}_{(\omega ,\mathcal{B},1)}$ 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 $(\mathcal{Z},\mathcal{W})$-Gorenstein injective dimension of $\pi {\mathcal{G}\mathcal{P}}_{(\omega ,\mathcal{B},1)}$ is bounded by the $(\mathcal{Z},\mathcal{W})$-Gorenstein injective dimension of $\omega$, provided that: (1) $\omega$ is closed under direct summands, (2) ${\mathrm{Ext}}^1(\pi {\mathcal{G}\mathcal{P}}_{(\omega ,\mathcal{B},1)},\nu )=0$, and (3) every object in $\pi {\mathcal{G}\mathcal{P}}_{(\omega ,\mathcal{B},1)}$ 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 $\pi {\mathcal{G}\mathcal{P}}_{(\omega ,\mathcal{B},1)}$ by the class ${\mathcal{G}\mathcal{P}}_{(\mathcal{A},\mathcal{B})}$ of $(\mathcal{A},\mathcal{B})$-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 ${\mathrm{Gid}}_{(\mathcal{Z},\mathcal{W})}(\pi {\mathcal{G}\mathcal{P}}_{(\omega ,\mathcal{B},1)})\le {\mathrm{Gid}}_{(\mathcal{Z},\mathcal{W})}(\omega )$ and ${\mathrm{Gid}}_{(\mathcal{Z},\mathcal{W})}({\mathcal{G}\mathcal{P}}_{(\mathcal{A},\mathcal{B})})\le {\mathrm{Gid}}_{(\mathcal{Z},\mathcal{W})}(\omega )$ 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 $\pi {\mathcal{G}\mathcal{P}}_{(\omega ,\mathcal{B},1)}$ 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 §\ 8 of \cite{HMP2023m} affected by Lemma 8.1, and comment some applications and examples.

10 pages. arXiv admin note: text overlap with arXiv:2207.00075