arXiv subjects

Yunxia Li

Publications and source records attributed to Yunxia Li.

3 recordsLinked to original sources

arXiv2025

Totally acyclic complexes and homological invariants over arbitrary rings

In this paper, we investigate equivalent characterizations of the condition that every acyclic complex of projective, injective, or flat modules is totally acyclic over a general ring R. We provide examples to illustrate relationships among these conditions and show that several are closely tied to the homological invariants silp(R), spli(R) and sfli(R). We also give sufficient conditions for the equality spli(R) = silp(R), thereby refining results due to Ballas-Chatzistavridis and Wang-Yang. Further, we extend a result of Christensen-Foxby-Holm on characterizations of Iwanaga-Gorenstein rings to the non-commutative setting. This generalizes a theorem of Estrada-Fu-Iacob, offering additional equivalent characterizations under a general assumption while also yielding characterizations of the Nakayama conjecture.

math.RA↗

arXiv2023

Auslander conditions and tilting-like cotorsion pairs

We study homological behavior of modules satisfying the Auslander condition. Assume that $\mathcal{AC}$ is the class of left $R$-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in $\mathcal{AC}$ belongs to $\mathcal{AC}$ for any ring $R$. As a consequence, we show that for any left Noetherian ring $R$, $\mathcal{AC}$ is a resolving subcategory of the category of left $R$-modules if and only if $_RR$ satisfies the Auslander condition if and only if each Gorenstein projective left $R$-module belongs to $\mathcal{AC}$. As an application, we prove that, for an Artinian algebra $R$ satisfying the Auslander condition, $R$ is Gorenstein if and only if $\mathcal{AC}$ coincides with the class of Gorenstein projective left $R$-modules if and only if $({\mathcal{AC}^{< \infty}},(\mathcal{AC}^{<\infty})^\bot)$ is a tilting-like cotorsion pair if and only if (${\mathcal{AC}^{< \infty}},\mathcal{I}$) is a tilting-like cotorsion pair, where $\mathcal{AC}^{<\infty}$ is the class of left $R$-modules with finite $\mathcal{AC}$-dimension and $\mathcal{I}$ is the class of injective left $R$-modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein.

math.RA↗

arXiv2019

When the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module

In this paper, we study when the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module. Applications go in three directions. The first is to characterize when the little finitistic dimension is finite. The second is to obtain equivalent formulations for a Wakamatsu tilting module to be a tilting module. The third is to give some new characterizations of Gorenstein rings.

math.RA↗