arXiv subjects
Publications and source records attributed to Mindy Y. Huerta.
In this paper, we provide an interpretation of the existing reduction process for extriangulated categories in general. This process allows us to obtain a new category which, for well-known cases, admits a triangulated structure. We will show that in general this new category is extriangulated but not necessarily triangulated.
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.
In this work we introduce the notion of higher $\mathbb{E}$-extension groups for an extriangulated category $\mathcal{C}$ and study the quotients $\mathcal{X}_{n+1}^{\vee}/[\mathcal{X}]$ and $\mathcal{X}_{n+1}^{\wedge}/[\mathcal{X}]$ when $\mathcal{X}$ is an $(n+2)$-rigid subcategory of $\mathcal{C}$. We also prove (under mild conditions) that each one is equivalent to a suitable subcategory of the category of functors of the stable category of $\mathcal{X}_{n}^{\vee}$ and the co-stable category of $\mathcal{X}_{n}^{\wedge}$, respectively. Moreover, it can be induced an exact structure through these equivalences and we analyze when such quotients are weakly idempotent complete, Krull-Schmidt or abelian. The above discussion is also considered in the particular case of an $(n+2)$-cluster tilting subcategory of $\mathcal{C}$ since in this case we know that $\mathcal{X}_{n+1}^{\vee}=\mathcal{C}=\mathcal{X}_{n+1}^{\wedge}.$ Finally, by considering the category of conflations of a exact category, we show that it is possible to get an abelian category from these quotients.