arXiv subjects

Xiu-Hua Luo

Publications and source records attributed to Xiu-Hua Luo.

7 recordsLinked to original sources

arXiv2026

Gorenstein-Projective Modules over the Ring of Dual Integers

The ring of dual integers is the bounded polynomial ring $\mathbb Z[\eps]=\mathbb Z[T]/(T^2)$ with integer coefficients. We describe the (finitely generated) Gorenstein-projective $\mathbb Z[\eps]$-modules as the torsionless $\mathbb Z[\eps]$-modules, while the stable category of $\Gproj\mathbb Z[\eps]$ modulo projectives is shown to be equivalent to the orbit category $\mathcal D^b(\mathbb Z)/[1]$ of the derived category of the integers. It follows that the latter carries the structure of a triangulated category. \smallskip The category $\Gproj\mathbb Z[\eps]$ is related to the embeddings of a subgroup in a free abelian group and has a quotient which is equivalent to the category of finite abelian groups. In fact, we present a cube which has as vertices eight related categories and as edges in each of the three directions functors which are related to push-down functors modulo the shift; canonical functors to stable categories; and homology functors, respectively. We note that in $\Gproj\mathbb Z[\eps]$ uniqueness of direct sum decomposition fails.

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arXiv2025

Auslander-Reiten translations in the monomorphism categories of exact categories

Let $Λ$ be a finite dimensional algebra. Let $\mathcal C$ be a functorially finite exact subcategory of $Λ$-mod with enough projective and injective objects and $\mathcal S (\mathcal C)$ be its monomorphism category. It turns out that the category $\mathcal S (\mathcal C)$ has almost split sequences. We show an explicit formula for the Auslander-Reiten translation in $\mathcal S (\mathcal C)$. Furthermore, if $\mathcal C$ is a stably $d$-Calabi-Yau Frobenius category, we calculate objects under powers of Auslander-Reiten translation in the triangulated category $\overline{\mathcal S(\mathcal C)}$.

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arXiv2022

Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules

Given a finite dimensional algebra $A$ over a field $k$, and a finite acyclic quiver $Q$, let $Λ= A\otimes_k kQ/I$, where $kQ$ is the path algebra of $Q$ over $k$ and $I$ is a monomial ideal. We show that $(\mathcal X,\mathcal Y)$ is a (complete) hereditary cotorsion pair in $A$-mod if and only if $({\rm smon}(Q,I,\mathcal X), {\rm rep}(Q,I,\mathcal Y))$ is a (complete) hereditary cotorsion pair in $Λ$-mod. We also show that $A$ is left weakly Gorenstein if and only if so is $Λ$. Provided that $kQ/I$ is non-semisimple, the category $^{\perp}Λ$ of semi-Gorenstein-projective $Λ$-modules coincides with the category of separated monic representations ${\rm smon}(Q,I,^{\perp}A)$ if and only if $A$ is left weakly Gorenstein.

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arXiv2022

A reflection equivalence for Gorenstein-projective quiver representations

For $Λ$ a selfinjective algebra, and $Q$ a finite quiver without oriented cycles, the algebra $ΛQ$ is a Gorenstein algebra and the category ${\rm Gproj}ΛQ$ of Gorenstein-projective $ΛQ$-modules is a Frobenius category. For a sink $v$ of $Q$, we define a functor $F(v) : \underline{\rm Gproj}ΛQ\to \underline{\rm Gproj}ΛQ(v)$ between the stable categories modulo projectives, where $Q(v)$ is obtained from $Q$ by changing the direction of each arrow ending in $v$. The functor is given by an explicit construction on the level of objects and homomorphisms. Our main result states that $F(v)$ is an equivalence of categories. In the case where the underlying graph of $Q$ is a tree, we deduce that the stable category $\underline{\rm Gproj}ΛQ$ does not depend on the orientation of $Q$. Moreover, if $Q$ is a quiver of type $\mathbb A_3$ and $Λ=k[T]/(T^n)$ the bounded polynomial algebra, we use the symmetry of the octahedron in the octahedral axiom to verify that the composition of twelve reflections yields the identity on objects.

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arXiv2018

Gorenstein projective bimodules via monomorphism categories and filtration categories

We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras $A$ and $B$, we use the special monomorphism category Mon(B, A-Gproj) to describe some Gorenstein projective bimodules over the tensor product of $A$ and $B$. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for Mon(B, A-Gproj) being the category of all Gorenstein projective bimodules. In addition, If both $A$ and $B$ are Gorenstein, we can describe the category of all Gorenstein projective bimodules via filtration categories. Similarly, in this case, we get the same result for infinitely generated Gorenstein projective bimodules.

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arXiv2016

Monic monomial representations I Gorenstein-projective modules

For a $k$-algebra $A$, a quiver $Q$, and an ideal $I$ of $kQ$ generated by monomial relations, let $Λ: = A\otimes_k kQ/I$. We introduce the monic representations of $(Q, I)$ over $A$. We give properties of the structural maps of monic representations, and prove that the category ${\rm mon}(Q, I, A)$ of the monic representations of $(Q, I)$ over $A$ is a resolving subcategory of ${\rm rep}(Q, I, A)$. We introduce the condition ${\rm(G)}$. The main result claims that a $\m$-module is Gorenstein-projective if and only if it is a monic module satisfying ${\rm(G)}$. As consequences, the monic $\m$-modules are exactly the projective $\m$-modules if and only if $A$ is semisimple; and they are exactly the Gorenstein-projective $\m$-modules if and only if $A$ is selfinjective, and if and only if ${\rm mon}(Q, I, A)$ is Frobenius.

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arXiv2011

Monic representations and Gorenstein-projective modules

Let $Λ$ be the path algebra of a finite quiver $Q$ over a finite-dimensional algebra $A$. Then $Λ$-modules are identified with representations of $Q$ over $A$. This yields the notion of monic representations of $Q$ over $A$. If $Q$ is acyclic, then the Gorenstein-projective $\m$-modules can be explicitly determined via the monic representations. As an application, $A$ is self-injective if and only if the Gorenstein-projective $\m$-modules are exactly the monic representations of $Q$ over $A$.

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