arXiv subjects

Juxiang Sun

Publications and source records attributed to Juxiang Sun.

9 recordsLinked to original sources

arXiv2026

Tilting pairs and Wakamatsu tilting pairs of subcategories over cleft extensions

Let $(\mathcal{B},\mathcal{A}, i, e, l)$ be a cleft extension of abelian categories. We prove that the functor $l$ preserves and reflects (Wakamatsu) tilting pairs of subcategories under certain conditions, unifying an abundance of known results. Then, we apply our results to the cleft extensions of module categories, and give characterizations of tilting pairs and Wakamatsu tilting pairs over $\theta$-extension of rings and tensor rings, which not only recover the earlier results in this direction, but also obtain some new conclusions.

math.RT↗

arXiv2026

Silting and tilting objects in cleft extensions of abelian categories

We establish connections between silting and tilting objects in an abelian category $\mathcal{B}$ and those in a cleft extension $\mathcal{A}$ of $\mathcal{B}$, which provides a method for constructing more silting and tilting objects. Then we apply our results to the cleft extensions of module categories, and characterize silting and tilting modules over $\theta$-extension of rings. Some known results over trivial extension of rings are extended and strengthened.

math.RT↗

arXiv2025

$\mathcal{X}$-Gorenstein projective and $\mathcal{Y}$-Gorenstein injective modules over tensor rings

Let $T_R(M)$ be a tensor ring and $\mathcal{X}$, $\mathcal{Y}$ be two classes of $R$-modules. Under certain conditions, we prove that a $T_R(M)$-module $(A, u)$ is $Ind(\mathcal{X})$-Gorenstein projective if and only if $u$ is monomorphic and $coker(u)$ is an $\mathcal{X}$-Gorenstein projective $R$-module. $\mathcal{Y}$-Gorenstein injective $T_R(M)$-modules are also explicitly described. As a consequence, the characterizations of Ding projective and Ding injective modules over $T_R(M)$ are obtained. Some applications to trivial ring extensions and Morita context rings are given.

math.RA↗

arXiv2025

Construction of Gorenstein projective modules over tensor rings

For a tensor ring $T_R(M)$, we obtain sufficient and necessary conditions to describe all complete projective resolutions and all Gorenstein projective modules. As a consequence, we provide a method for constructing Gorenstein projective modules over $T_R(M)$ from the ones of $R$. Some applications to trivial ring extensions, Morita context rings and triangular matrix rings are given.

math.AC↗

arXiv2025

Constructions of symmetric separable equivalences and their applications

Let $\Lambda$ and $\Gamma$ be symmetrically separably equivalent Artin algebras. We prove that there exist symmetrical separable equivalences between certain endomorphism algebras of modules. As applications, we provide several methods to construct symmetrical separable equivalences from given ones and discuss when the rigidity dimension is an invariant under symmetrical separable equivalences. Moreover, we show that a symmetrical separable equivalence preserves the Frobenius-finite type, Auslander-type condition, the (strong) Nakayama conjecture, the Auslander-Gorenstein conjecture and so on.

math.RT↗

arXiv2025

$\omega$-left approximation dimensions under Stable equivalence

In this paper, we investigate some transfer properties of $\omega$-left approximation dimensions of modules of stably equivalent Artin algebras having neither nodes nor semisimple direct summands. As applications, we give a one-to-one correspondence between basic (Wakamatsu) tilting modules, and prove that the Wakamatsu tilting conjecture is preserved under those equivalences.

math.RT↗

arXiv2025

Gorenstein categories and separable equivalences

Let $\mathscr{C}$ be an additive subcategory of left $\Lambda$-modules, we establish relations of the orthogonal classes of $\mathscr{C}$ and (co)res $\widetilde{\mathscr{C}}$ under separable equivalences. As applications, we obtain that the (one-sided) Gorenstein category and Wakamatsu tilting module are preserved under separable equivalences. Furthermore, we discuss when $G_{C}$-projective (injective) modules and Auslander (Bass) class with respect to $C$ are invariant under separable equivalences.

math.RT↗

arXiv2015

Gorenstein (semi)hereditary rings with respect to a semidualizing module

Let $C$ be a semidualizing module. We first investigate the properties of finitely generated $G_C$-projective modules. Then, relative to $C$, we introduce and study the rings of Gorenstein (weak) global dimensions at most 1, which we call $C$-Gorenstein (semi)hereditary rings, and prove that every $C$-Gorenstein hereditary ring is both coherent and $C$-Gorenstein semihereditary.

math.AC↗

arXiv2013

Global dimensions of rings with respect to a semidualizing module

In this paper, the notion of strongly G_C-projective and injective modules is introduced, where C is a semidualizing module. Using these modules we can obtain a new characterization of G_C-projective and injective modules, similar to the one of projective modules by the free modules. We then define and study the global dimensions of rings relative to a semidualizing module C, and prove that the global G_C-projective dimension of a ring R is equal to the global G_C-injective dimension of R.

math.RA↗