arXiv subjects

Jiaqun Wei

Publications and source records attributed to Jiaqun Wei.

At least 19 recordsLinked to original sources

arXiv2026

Reduction techniques for the derived delooping levels

The derived delooping level is a recently introduced homological invariant that provides an upper bound for the finitistic dimension of the opposite algebra. In this paper, we employ two reduction techniques-cleft extensions and recollements-to study the finiteness of the derived delooping level of finite-dimensional algebras over a field. By applying the theory of cleft extensions to bound quiver algebras, we establish arrow-removal operations that preserve the finiteness of the derived delooping level. In parallel, using recollement techniques, we develop vertex-removal operations with the same finiteness-preserving property. We conclude with several examples illustrating the applicability and effectiveness of these reduction methods.

math.RA↗

arXiv2026

Degree-shifted derived invariance of derived delooping levels

Gélinas introduced the delooping level of an Artin algebra as a homological invariant that bounds the big finitistic dimension of the opposite algebra. A natural question is whether such invariants are preserved under derived equivalences. Chen recently showed that the finiteness of the classical delooping level and its sub-derived variant is not derived-invariant, but the case of the finer derived delooping level of Guo and Igusa remained open. In this paper, we prove that the finiteness of the derived delooping level is \emph{degree-shift invariant} under derived equivalences: if two algebras are derived equivalent via a tilting complex of width $k_T$, then finiteness of the $(k+k_T)$-derived delooping level on one side implies finiteness of the $k$-derived delooping level on the other. Consequently, the finiteness of the derived $\infty$-delooping level is invariant under derived equivalences. Furthermore, we introduce the global derived delooping level and prove that, under a derived equivalence, its $\infty$-version changes by at most $k_T$. In particular, its finiteness is a derived invariant.

math.RA↗

arXiv2026

On simples in a cosilting heart and mutation of cosilting pairs

Let $A$ be a finite dimensional algebra. The lattice of torsion pairs in $\rm mod (A)$ is controlled by cosilting pairs, infinitely generated analogues of support $τ^-$-tilting pairs. Then, edges in the Hasse quiver (i.e. minimal inclusions of torsion-free classes) correspond to irreducible mutations of cosilting pairs. An important difference with classical $τ$-tilting theory is that not all indecomposable summands of a cosilting pair are mutable. So, it is very important to identify mutable indecomposable summands in a given cosilting pair. It is well-known that mutable summands correspond to injective envelopes of finitely presented simples in the HRS-tilted heart. Based on this correspondence, we first present a method for obtaining all simples in the HRS-tilted heart, and then give some necessary and sufficient conditions for an indecomposable summand of a given cosilting pair to be left mutable or right mutable.

math.RT↗

arXiv2026

Model structures arising from weak cotorsion pairs

Let $\mathcal{A}$ be an abelian category. Beligiannis and Reiten proved that there is a bijective correspondence between so-called projective model structures on $\mathcal{A}$ and hereditary cotorsion pairs in $\mathcal{A}$ with a contravariantly finite core. It is well-known that, tilting modules induce cotorsion pairs, so we may have a homotopicl interpretation of tilting modules. But a recent generalization of tilting modules, support $τ$-tilting modules, induce weak cotorsion pairs. In this paper, we define weak projective model structures and prove that there is a bijective correspondence between weak projective model structures and left weak cotorsion pairs satisfying some mild conditions. This is a generalization of Beligiannis-Reiten correspondence from the perspective and philosophy of $τ$-tilting theory. In particular, we prove that any support $τ$-tilting module induce a model structure, and there is bijective correspondence between support $τ$-tilting modules and a certain class of model structures.

math.RT↗

arXiv2026

$r$ICE-closed subcategories induced by the morphism category of projective modules

Let $Λ$ be an Artin $R$-algebra, and ${\rm proj}\mbox{-}Λ$ denotes the category of all finitely generated projective $Λ$-modules. Define $\CP(Λ) := {\rm Mor}({\rm proj}\mbox{-}Λ)$. Due to the favorable homological properties of $\CP(Λ)$, we initially examine several noteworthy objects and subcategories of $\CP(Λ)$, subsequently relating these findings to $\mmod Λ$. Following our examination of Image-Cokernel-Extension closed (hereafter referred to as ICE-closed) subcategories of $\CP(Λ)$, among other bijections, we demonstrate a bijection between rigid objects in $\CP(Λ)$ and ICE-closed subcategories of $\CP(Λ)$ with enough Ext-projectives. In order to translate the concept of ICE-closed subcategory from $\CP(Λ)$ to $\mmod Λ$, it is necessary to introduce the framework of rICE-closed subcategories of $\mmod Λ$. We then establish a bijection between $τ$-rigid modules in $\mmod Λ$ and rICE-closed subcategories of $\mmod Λ$ that possess an rExt-progenerator. This is a generalization of a bijection given by Enomoto for hereditary algebras. Our morphism approach improves a bijection given by Buan and Zhou by introducing r-cotorsion-torsion triples. We conclude our paper with further applications for $τ$-tilting theory.

math.RT↗

arXiv2026

Revising Auslander-Gruson-Jensen duality

For a ring $A$ there is a well-known duality between definable subcategories of right $A$-modules and definable subcategories of left $A$ modules. This is a consequence of Auslander-Gruson-Jensen duality $\rm mod\text{-}(mod\text{-}A)\rightarrow mod\text{-}(mod\text{-}A^{op})$. The existence of this duality arises from the fact that $\rm mod\text{-}(mod\text{-}A)$ is the free abelian category over the pre-additive category $A$ with a single object. In this note, first, we give a simple description of the free abelian category. This description clarifies Auslender-Gruson-Jensen duality and also the duality between definable subcategories of right $A$-modules and those of left $A$-modules.

math.RT↗

arXiv2026

AIR tilting subcategories of extended hearts

We introduce the notion of AIR tilting subcategories of extended hearts of $t$-structures on a triangulated category associated with silting subcategories. This notion generalizes $τ_{[d]}$-tilting pairs of extended finitely generated modules over finite-dimensional algebras to a more general framework, which includes both extended large modules over unitary rings and truncated subcategories of finite-dimensional derived categories of proper non-positive differential graded algebras. Within this setting, we establish a bijection between AIR tilting subcategories and silting subcategories. Furthermore, we define quasi-tilting and tilting subcategories of extended hearts, extending the corresponding notions from module categories, and investigate their fundamental properties along with the relationships among these tilting-related classes.

math.RT↗

arXiv2025

Monobricks in extriangulated length categories

In this paper, we introduce the notation of monobricks in an extriangulated length category as a generalization of the semibricks. We prove that there is a bijection between monobricks and left Schur subcategories. Then we show that this bijection restricts to bijection between cofinally closed monobricks and torsion-free classes. These extend the results of Enomoto for abelian length categories.

math.CT↗

arXiv2025

(projectively coresolved) Gorenstein flat modules over tensor rings

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize projectively coresolved Gorenstein flat modules over $T_R(M)$, showing that a $T_R(M)$ module $(X,u)$ is projectively coresolved Gorenstein flat if and only if $u$ is monomorphic and $coker(u)$ is a projectively coresolved Gorenstein flat $R$-module. A class of Gorenstein at modules over $T_R(M)$ are also explicitly described. We discuss applications to trivial ring extensions and Morita context rings.

math.RA↗

arXiv2025

The first Brauer-Thrall conjecture for extriangulated length categories

Let $(\mathcal{A},Θ)$ be a length category. We introduce the notation of Gabriel-Roiter measure with respect to $Θ$ and extend Gabriel's main property to this setting. Using this measure, when $(\mathcal{A},Θ)$ satisfies some technical conditions, we prove that $\mathcal{A}$ has an infinite number of pairwise nonisomorphic indecomposable objects if and only if it has indecomposable objects of arbitrarily large length. That is, the first Brauer-Thrall conjecture holds.

math.RT↗

arXiv2025

Distributivity in lattices of torsion classes over finite-dimensional algebras

Let $A$ be a basic finite-dimensional algebra and denote by $\operatorname{tors} A$ the collection of all all torsion classes of $A$. It has been proved in \cite{Demonet} that $\operatorname{tors} A$ is always a completely semidistributive lattice. In the present paper, we investigate the distributivity of this lattice and proved that the lattice is distributive if and only if the algebra is the finite direct product of finite-dimensional local algebras.

math.RT↗

arXiv2025

Extriangulated length categories: torsion classes and $τ$-tilting theory

This paper introduces the notion of extriangulated length categories, whose prototypical examples include abelian length categories and bounded derived categories of finite dimensional algebras with finite global dimension. We prove that an extriangulated category $\mathcal{A}$ is a length category if and only if $\mathcal{A}$ admits a simple-minded system. Subsequently, we study the partially ordered set ${\rm tor}_Θ(\mathcal{A})$ of torsion classes in an extriangulated length category $(\mathcal{A},Θ)$ from the perspective of lattice theory. It is shown that ${\rm tor}_Θ(\mathcal{A})$ forms a complete lattice, which is further proved to be completely semidistributive and algebraic. Moreover, we describe the arrows in the Hasse quiver of ${\rm tor}_Θ(\mathcal{A})$ using brick labeling. Finally, we introduce the concepts of support torsion classes and support $τ$-tilting subcategories in extriangulated length categories and establish a bijection between these two notions, thereby generalizing the Adachi-Iyama-Reiten bijection for functorially finite torsion classes.

math.RT↗

arXiv2025

Universal enveloping H-pseudoalgebras of DGP pseudoalgebras

The notions of Poisson $H$-pseudoalgebras are generalizations of Poisson algebras in a pseudotensor category $\mathcal{M}^{\ast}(H)$. This paper introduces an analogue of Poisson-Ore extension in Poisson $H$-pseudoalgebras. Poisson $H$-pseudoalgebras with the differential graded setting induces the notions of differential graded Poisson $H$-pseudoalgebras (DGP pseudoalgebras, for short). The DGP pseudoalgebra with some compatibility conditions is proved to be closed under tensor product. Furthermore, the universal enveloping $H$-pseudoalgebras of DGP pseudoalgebras are constructed by a $\mathcal{P}$-triple. A unique differential graded pseudoalgebra homomorphism between a universal enveloping $H$-pseudoalgebra of a DGP pseudoalgebra and a $\mathcal{P}$-triple of a DGP pseudoalgebra is obtained.

math.AC↗

arXiv2025

Wakamatsu-tilting subcategories in extriangulated categories

Let $\mathscr{C}$ be an extriangulated category with enough projectives and injectives. We give the definitions of Wakamatsu-tilting subcategories and Wakamatsu-cotilting subcategories of $\mathscr{C}$ and show that they coincide with each other. Moreover, the definitions of $\infty$-tilting subcategories and $\infty$-cotilting subcategories given by Zhang, Wei and Wang also coincide with them. As a result, Wakamatsu-tilting subcategories success all properties of $\infty$-tilting subcategories and $\infty$-cotilting subcategories. On the other hand, we glue the Wakamatsu-tilting subcategories in a special recollement and show that the converse of the gluing holds under certain conditions.

math.RT↗

arXiv2024

Thick subcategories and silting subcategories in recollement

Let $(\mathcal{A}, \mathcal{B}, \mathcal{C}, i^{*}, i_{\ast}, i^{!},j_!, j^\ast, j_\ast)$ be a recollement of extriangulated categories. We show that there is a bijection between thick subcategories in $\mathcal{C}$ and thick subcategories in $\mathcal{B}$ containing $i_{\ast}\mathcal{A}$. Futhermore, the thick subcategories $\mathcal{V}$ in $\mathcal{B}$ containing $i_{\ast}\mathcal{A}$ can induce a new recollement relative to $\mathcal{A}$ and $j^{\ast}\mathcal{V}$. We also prove that silting subcategories in $\mathcal{A}$ and $\mathcal{C}$ can be glued to get silting subcategories in $\mathcal{B}$ and the converse holds under certain conditions.

math.RT↗

arXiv2024

A bijection between support $τ$-tilting subcategories and $τ$-cotorsion pairs in extriangulated categories

Let $\mathscr{C}$ be an extriangulated category with enough projectives and injectives. We give a new definition of tilting subcategories of $\mathscr{C}$ and prove it coincides with the definition given in [19]. As applications, we introduce the notions of support $τ$-tilting subcategories and $τ$-cotorsion pairs of $\mathscr{C}$. We build a bijection between support $τ$-tilting subcategories and certain $τ$-cotorsion pairs. Moreover, this bijection induces a bijection between tilting subcategories and certain cotorsion pairs.

math.RT↗

arXiv2024

Quotients of extriangulated categories induced by selforthogonal subcategories

Let C be an extriangulated category. We prove that two quotient categories of extriangu?lated categories induced by selforthogonal subcategories are equivalent to module categories by restriction of two functors E and Hom, respectively. Moreover, if the selforthogonal sub?category is contravariantly finite, then one of the two quotient categories is abelian. This result can be regarded as a generalization of Demonet-Liu and Zhou-Zhu.

math.RA↗

arXiv2024

Some properties of congruence lattices of path semigroups

Each quiver corresponds to a path semigroup, and such a path semigroup also corresponds to an associative K-algebra over an algebraically closed field K. Let Q be a quiver and S_Q, KQ be its path semigroup, path algebra, respectively. In this paper, we study some properties of the congruence lattice of S_Q. First, we show that there is a one-to-one correspondence between congruences on S and certain algebraic ideals of KQ. Based on such a description, we consider acyclic quivers and show that the congruence latices of such path semigroups are strong upper semimodular but not necessarily lower semimodular. Moreover, we provide some equivalent conditions for the congruence lattices to be modular and distributive.

math.GR↗