arXiv subjects
Publications and source records attributed to Junling Zheng.
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.
The extension dimensions of an Artin algebra give a reasonable way of measuring how far an algebra is from being representation-finite. In this paper we mainly study extension dimensions linked by recollements of derived module categories and singular equivalences of Morita type with level, and establish a series of new inequalities and relationships among their extension dimensions.
In this paper, we investigate the behavior of Igusa-Todorov distances, extension and Rouquier dimensions under cleft extensions of abelian categories. We apply our results to Morita context rings, trivial extension rings, tensor rings and arrow removals.
We introduced the notion of Orlov spectra of Abelian categories, and study its some properties. In particular, we give precise result of Orlov spectra of algebras with type $\mathbb{A}_{n}$.
There is a well-known class of algebras called Igusa-Todorov algebras which were introduced in relation to finitistic dimension conjecture. As a generalization of Igusa-Todorov algebras, the new notion of $(m,n)$-Igusa-Todorov algebras provides a wider framework for studying derived dimensions. In this paper, we give methods for constructing $(m,n)$-Igusa-Todorov algebras. As an application, we present for general artin algebras a relationship between the derived dimension and the representation distance. Moreover, we end this paper to show that the main result can be used to give a better upper bound for the derived dimension for some classes of algebras.
A new homological dimension, called the Igusa-Todorov distance, is introduced to measure how far an Artin algebra is from being an Igusa-Todorov algebra. An upper bound for the dimension is established in terms of the Loewy length, leading to the conclusion that every Artin algebra has a finite Igusa-Todorov distance.Using this dimension, we derive an upper bound for the dimension of the singularity category. Furthermore, we investigate how the Igusa-Todorov distance behaves under various relationships between algebras. Specifically, we demonstrate that stable equivalences preserve the Igusa-Todorov distances for algebras without nodes, prove that it is an invariant under singular equivalence of Morita type with level, and establish bounds for the distances of algebras involved in a recollement of derived module categories. Consequently, the Igusa-Todorov distance is an invariant under derived equivalences of algebras.
We introduce Igsua-Todorov distances of Artin algebra, prove its invariance under derived equivalence, present its application to exterior algebra, and establish the link between the dimension of the singularity category and this distance.
In this paper, our primary focus is on investigating the extension dimensions of syzygy module categories associated with Artin algebras, particularly under various equivalences. We demonstrate that, for sufficiently large $i$, the $i$-th syzygy module categories of derived equivalent algebras exhibit identical extension dimensions. Furthermore, we establish that the extension dimension of the $i$-th syzygy module category is an invariant under both stable equivalence and separable equivalence for each nonnegative integer $i$.
We show that the difference of the extension dimensions of two derived equivalent algebras is bounded above by the minimal length of a tilting complex associated with a derived equivalence, and that the extension dimension is an invariant under the stable equivalence. In addition, we provide two sufficient conditions such that the extension dimension is an invariant under particular derived equivalences.
For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $Λ$, we give the upper bounds of the extension dimension of $Λ$ in terms of the radical layer length of $Λ$ and certain relative projective (or injective) dimension of some simple $Λ$-modules, from which some new upper bounds of the derived dimension of $Λ$ are induced.
We give an upper bound for the dimension of the bounded derived categories of $(m,n)$-Igusa-Todorov algebras which is a generalization of $n$-Igusa-Todorov algebras, where $m,n$ are two nonnegative integers. As an applications, we get a new upper bound for the dimension of bounded derived categories in terms of the projective dimensions of certain of simple modules as well as radical layer length of artin algebra $Λ$.
Let $Λ$ be an artin algebra, and $\mathcal{V}$ a subset of all simple modules in $\modΛ$. Suppose that $Λ/\rad Λ$ has finite syzygy type, then the derived dimension of $Λ$ is at most $\ell\ell^{t_{\mathcal{V}}}(Λ_Λ)+\pd\mathcal{V}.$ In particular, if the global dimension of $Λ$ is finite, then the derived dimension of $Λ$ is at most $\ell\ell^{t_{\mathcal{V}}}(Λ_Λ)+\pd\mathcal{V}.$ This generalized the famous result which state that the derived dimension of $Λ$ is less than or equal to the global dimension of $Λ$.
Let $Λ$ be an artin algebra. We obtain that $Λ$ is syzygy-finite when the radical layer length of $Λ$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the category $\mod Λ$ of finitely generated right $Λ$-modules in terms of the projective of certain class of simple right $Λ$-modules and also get the left big finitistic dimension conjecture holds.
Let $Λ$ be an artin algebra. We give an upper bound for the dimension of the bounded derived category of the category $\mod Λ$ of finitely generated right $Λ$-modules in terms of the projective and injective dimensions of certain class of simple right $Λ$-modules as well as the radical layer length of $Λ$. In addition, we give an upper bound for the dimension of the singularity category of $\mod Λ$ in terms of the radical layer length of $Λ$.
Let $\A$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\A$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\A$ are identical, and they are at most the representation dimension of $\A$ minus two. By using it, for a right Morita ring $\La$, we establish the relation between the extension dimension of the category $\mod \La$ of finitely generated right $Λ$-modules and the representation dimension as well as the right global dimension of $Λ$. In particular, we give an upper bound for the extension dimension of $\mod Λ$ in terms of the projective dimension of certain class of simple right $Λ$-modules and the radical layer length of $Λ$. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.