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Shunhua Zhang

Publications and source records attributed to Shunhua Zhang.

At least 19 recordsLinked to original sources
arXiv2019

Algebras with finite relative dominant dimension and almost n-precluster tilting modules

In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce the almost n-precluster tilting module and establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules.

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arXiv2018

Gorenstein projective dimensions of modules over minimal Auslander-Gorenstein algebras

In this article we investigate the relations between the Gorenstein projective dimensions of $Λ$-modules and their socles for minimal n-Auslander-Gorenstein algebras $Λ$ in the sense of Iyama and Solberg \cite{IS}. First we give a description of projective-injective $Λ$-modules in terms of their socles. Then we prove that a $Λ$-module $N$ has Gorenstein projective dimension at most n iff its socle has Gorenstein projective dimension at most n iff $N$ is cogenerated by a projective $Λ$-module. Furthermore, we show that minimal n-Auslander-Gorenstein algebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.

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arXiv2016

Relative contravariantly finite subcategories and relative tilting modules

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Let $T$ be a tilting $A$-module and $B={\rm End}_A\ T$ be the endomorphism algebra of $T$. In this paper, we consider the correspondence between the tilting $A$-modules and the tilting $B$-modules, and we prove that there is a one-one correspondence between the basic $T$-tilting $A$-modules in $T^{\perp}$ and the basic tilting $B$-modules in $^{\perp}(D_BT)$. Moreover, we show that there is a one-one correspondence between the $T$-contravariantly finite $T$-resolving subcategories of $T^{\perp}$ and the basic $T$-tilting $A$-modules contained in $T^{\perp}$. As an application, we show that there is a one-one correspondence between the basic tilting $A$-modules in $T^{\perp}$ and the basic tilting $B$-modules in $^{\perp}(D_BT)$ if $A$ is a $1$-Gorenstein algebra or a $m$-replicated algebra over a finite dimensional hereditary algebra.

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arXiv2016

A new characterization of Auslander algebras

Let $Λ$ be a finite dimensional Auslander algebra. For a $Λ$-module $M$, we prove that the projective dimension of $M$ is at most one if and only if the projective dimension of its socle soc\,$M$ is at most one. As an application, we give a new characterization of Auslander algebra $Λ$, and prove that a finite dimensional algebra $Λ$ is an Auslander algebra provided its global dimension gl.d\,$Λ\leq2$ and an injective $Λ$-module is projective if and only if the projective dimension of its socle is at most one.

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arXiv2015

Some applications of $τ$-tilting theory

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, and $M$ be a partial tilting $A$-module. We prove that the Bongartz $τ$-tilting complement of $M$ coincides with its Bongartz complement, and then we give a new proof of that every almost complete tilting $A$-module has at most two complements. Let $A=kQ$ be a path algebra. We prove that the support $τ$-tilting quiver $\overrightarrow{Q}({\rm s}τ$-${\rm tilt} A)$ of $A$ is connected. As an application, we investigate the conjecture of Happel and Unger in [9] which claims that each connected component of the tilting quiver $\overrightarrow{Q}({\rm tilt} A)$ contains only finitely many non-saturated vertices. We prove that this conjecture is true for $Q$ being all Dynkin and Euclidean quivers and wild quivers with two or three vertices, and we also give an example to indicates that this conjecture is not true if $Q$ is a wild quiver with four vertices.

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arXiv2015

Endomorphism algebras of semi-tilting modules

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. We investigate the structure properties of the endomorphism algebras of semi-tilting $A$-modules, and prove that the endomorphism algebras arising from the mutations of semi-tilting $A$-modules can be realized as the endomorphism algebras of BB-tilting modules.

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arXiv2013

Repetitive cluster-tilted algebras

Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field $k$ and $\mathscr{C}_{F^m}$ be the repetitive cluster category of $H$ with $m\geq 1$. We investigate the properties of cluster tilting objects in $\mathscr{C}_{F^m}$ and the structure of repetitive cluster-tilted algebras. Moreover, we generalized Theorem 4.2 in \cite{bmrrt} (Buan A, Marsh R, Reiten I. Cluster-tilted algebra. Trans. Amer. Math. Soc., 359(1)(2007), 323-332.) to the situation of $\mathscr{C}_{F^m}$, and prove that the tilting graph $\mathscr{K}_{\mathscr{C}_{F^m}}$ of $\mathscr{C}_{F^m}$ is connected.

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arXiv2013

Mutation graphs of maximal rigid modules over finite dimensional preprojective algebras

Let $Q$ be a finite quiver of Dynkin type and $Λ=Λ_Q$ be the preprojective algebra of $Q$ over an algebraically closed field $k$. Let $\mathcal {T}_Λ$ be the mutation graph of maximal rigid $Λ$ modules. Geiss, Leclerc and Schr$\ddot{\rm o}$er conjectured that $\mathcal {T}_Λ$ is connected, see [C.Geiss, B.Leclerc, J.Schröer, Rigid modules over preprojective algebras, Invent.Math., 165(2006), 589-632]. In this paper, we prove that this conjecture is true when $Λ$ is of representation finite type or tame type. Moreover, we also prove that $\mathcal {T}_Λ$ is isomorphic to the tilting graph of ${\rm End}_ΛT$ for each maximal rigid $Λ$-module $T$ if $Λ$ is representation-finite.

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arXiv2012

Endomorphism algebras arising from mutations

Let $A$ be a finite dimensional algebra over an algebraically closed field $k$, $\mathcal {D}^b(A)$ be the bounded derived category of $A$-mod and $A^{(m)}$ be the $m$-replicated algebra of $A$. In this paper, we investigate the structure properties of endomorphism algebras arising from silting mutation in $\mathcal {D}^b(A)$ and tilting mutation in $A^{(m)}$-mod.

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arXiv2011

Representation dimensions of triangular matrix algebras

Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $T_2(A)=(\begin{array}{cc}A&0 A&A\end{array})$ be the triangular matrix algebra and $A^{(1)}=(\begin{array}{cc}A&0 DA&A\end{array})$ be the duplicated algebra of $A$ respectively. We prove that ${\rm rep.dim}\ T_2(A)$ is at most three if $A$ is Dynkin type and ${\rm rep.dim}\ T_2(A)$ is at most four if $A$ is not Dynkin type. Let $T$ be a tilting A-$\module$ and $\ol{T}=T\oplus\ol{P}$ be a tilting $A^{(1)}$-$\module$. We show that $\End_{A^{(1)}} \ol{T}$ is representation finite if and only if the full subcategory $\{(X,Y,f)\ |\ X\in {\rm mod}\ A, Y\inτ^{-1}\mathscr{F}(T_A)\cup{\rm add}\ A\}$ of ${\rm mod \ T_2(A)}$ is of finite type, where $τ$ is the Auslander-Reiten translation and $\mathscr{F}(T_A)$ is the torsion-free class of ${\rm mod}\ A$ associated with $T$. Moreover, we also prove that ${\rm rep.dim\ End}_{A^{(1)}}\ {\ol T}$ is at most three if $A$ is Dynkin type.

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arXiv2011

Tilting modules over duplicated algebras

Let $A$ be a finite dimensional hereditary algebra over a field $k$ and $A^{(1)}$ the duplicated algebra of $A$. We first show that the global dimension of endomorphism ring of tilting modules of $A^{(1)}$ is at most 3. Then we investigate embedding tilting quiver $\mathscr{K}(A)$ of $A$ into tilting quiver $\mathscr{K}(A^{(1)})$ of $A^{(1)}$. As applications, we give new proofs for some results of D.Happel and L.Unger, and prove that every connected component in $\mathscr{K}({A})$ has finite non-saturated points if $A$ is tame type, which gives a partially positive answer to the conjecture of D.Happel and L.Unger in [10]. Finally, we also prove that the number of arrows in $\mathscr{K}({A})$ is a constant which does not depend on the orientation of $Q$ if $Q$ is Dynkin type.

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arXiv2010

Cluster-tilted algebras of type $D_n$

Let $H$ be a hereditary algebra of Dynkin type $D_n$ over a field $k$ and $\mathscr{C}_H$ be the cluster category of $H$. Assume that $n\geq 5$ and that $T$ and $T'$ are tilting objects in $\mathscr{C}_H$. We prove that the cluster-tilted algebra $Γ=\mathrm{End}_{\mathscr{C}_H}(T)^{\rm op}$ is isomorphic to $Γ'=\mathrm{End}_{\mathscr{C}_H}(T')^{\rm op}$ if and only if $T=τ^iT'$ or $T=στ^jT'$ for some integers $i$ and $j$, where $τ$ is the Auslander-Reiten translation and $σ$ is the automorphism of $\mathscr{C}_H$ defined in section 4.

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arXiv2010

Ringel-Hall Algebras of Duplicated Tame Hereditary Algebras

Let $A$ be a tame hereditary algebra over a finite field $k$ with $q$ elements, and ${\bar{A}}$ be the duplicated algebra of $A$. In this paper, we investigate the structure of Ringel-Hall algebra $\mathscr{H} (\bar{A})$ and of the corresponding composition algebra $\mathscr{C} (\bar{A})$. As an application, we prove the existence of Hall polynomials $g_{XY}^M$ for any $\bar{A}$-modules $M, X$ and $Y$ with $X$ and $Y$ indecomposable if $A$ is a tame quiver $k$-algebra, then we also obtain some Lie subalgebras induced by $\bar{A}$.

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arXiv2009

Partial tilting modules over $m$-replicated algebras

Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $δ(T)$ the number of non isomorphic indecomposable summands of $T$. In this paper, we prove that a partial tilting $A^{(m)}$-module $T$ is a tilting $A^{(m)}$-module if and only if $δ(T)=δ(A^{(m)})$, and that every partial tilting $A^{(m)}$-module has complements. As an application, we deduce that the tilting quiver $\mathscr{K}_{A^{(m)}}$ of $A^{(m)}$ is connected. Moreover, we investigate the number of complements to almost tilting modules over duplicated algebras.

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arXiv2009

Tilting mutation for $m$-replicated algebras

Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $A^{(m)}$ be the $m$-replicated algebra of $A$ and $\mathscr{C}_{m}(A)$ be the $m$-cluster category of $ A$. We investigate properties of complements to a faithful almost complete tilting $A^{(m)}$-module and prove that the $m$-cluster mutation in $\mathscr{C}_{m}(A)$ can be realized in ${\rm mod} A^{(m)}$, which generalizes corresponding results on duplicated algebras established in [Z1].

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