Finitistic dimension and endomorphism algebras of 𝒲-Gorenstein modules

2020 ◽  
pp. 2150046
Author(s):  
Kaili Wu ◽  
Jiaqun Wei
Keyword(s):  
Finitistic Dimension ◽  
Artin Algebra ◽  
Flat Dimension ◽  
Endomorphism Algebras ◽  
Gorenstein Modules

Let [Formula: see text] be an artin algebra, [Formula: see text] be a [Formula: see text]-Gorenstein [Formula: see text]-module and [Formula: see text], then [Formula: see text] is a [Formula: see text]-[Formula: see text]-bimodule. We use the restricted flat dimension of [Formula: see text] and the finitistic [Formula: see text]-dimension of [Formula: see text] to characterize the finitistic dimension of [Formula: see text], and obtain the following main result: if [Formula: see text] is [Formula: see text]-finite with [Formula: see text], then we have: (1) If [Formula: see text] or [Formula: see text], then [Formula: see text] (2) If [Formula: see text], then [Formula: see text]

THE FINITISTIC DIMENSION CONJECTURE AND RELATIVELY PROJECTIVE MODULES

2013 ◽  
Vol 15 (02) ◽  
pp. 1350004 ◽  
Author(s):  
CHANGCHANG XI ◽  
DENGMING XU
Keyword(s):  
Left Ideal ◽  
General Setting ◽  
Projective Dimension ◽  
Homological Dimension ◽  
Projective Modules ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
Finite Dimensional ◽  
New Formulation ◽  
Dimension Conjecture

The famous finitistic dimension conjecture says that every finite-dimensional 𝕂-algebra over a field 𝕂 should have finite finitistic dimension. This conjecture is equivalent to the following statement: If B is a subalgebra of a finite-dimensional 𝕂-algebra A such that the radical of B is a left ideal in A, and if A has finite finitistic dimension, then B has finite finitistic dimension. In the paper, we shall work with a more general setting of Artin algebras. Let B be a subalgebra of an Artin algebra A such that the radical of B is a left ideal in A. (1) If the category of all finitely generated (A, B)-projective A-modules is closed under taking A-syzygies, then fin.dim (B) ≤ fin.dim (A) + fin.dim (BA) + 3, where fin.dim (A) denotes the finitistic dimension of A, and where fin.dim (BA) stands for the supremum of the projective dimensions of those direct summands of BA that have finite projective dimension. (2) If the extension B ⊆ A is n-hereditary for a non-negative integer n, then gl.dim (A) ≤ gl.dim (B) + n. Moreover, we show that the finitistic dimension of the trivially twisted extension of two algebras of finite finitistic dimension is again finite. Also, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given. Our approach in this paper is completely different from the one in our earlier papers.

ENDOMORPHISM ALGEBRAS OF PROJECTIVE MODULES OVER LAURA ALGEBRAS

2004 ◽  
Vol 03 (01) ◽  
pp. 49-60 ◽  
Author(s):  
IBRAHIM ASSEM ◽  
FLÁVIO U. COELHO
Keyword(s):  
Projective Modules ◽  
Artin Algebra ◽  
Endomorphism Algebras ◽  
Tilted Algebras

Let A be a connected artin algebra, and e be an idempotent in A such that B=eAe is connected. We show here that if A is laura, left (or right) glued or weakly shod, so is B, respectively. Our proof yields also similar (and known) results for shod and quasi-tilted algebras.

On F-Gorenstein dimensions

2014 ◽  
Vol 13 (06) ◽  
pp. 1450022
Author(s):  
Xi Tang
Keyword(s):  
Equivalent Condition ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
Gorenstein Algebras ◽  
Artin Algebras ◽  
Dimension Conjecture

Over an artin algebra Λ, for an additive subbifunctor F of [Formula: see text] with enough projectives and injectives, we introduce F-Gorenstein dimensions in this paper. The new relative dimensions are useful to characterize F-Gorenstein algebras and F-self-injective algebras. In addition, with the aid of F-Gorenstein dimensions, we obtain an equivalent condition for the finitistic dimension conjecture to hold, that is, fin.dim Λ < ∞ for all artin algebras Λ if and only if rel.fin.Gdim F Λ < ∞ for all artin algebras Λ.

THE FINITISTIC DIMENSION AND CHAIN CONDITIONS ON IDEALS

2020 ◽  
pp. 1-8
Author(s):  
JUNLING ZHENG ◽  
ZHAOYONG HUANG
Keyword(s):  
Projective Dimension ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
Chain Conditions ◽  
Direct Sum ◽  
A Chain ◽  
Dimension Conjecture

Abstract Let Λ be an artin algebra and $0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$ a chain of ideals of Λ such that $(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$ for any $0\leq i\leq n-1$ and $\Lambda/I_{n}$ is semisimple. If either none or the direct sum of exactly two consecutive ideals has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. As a consequence, we have that if either none or the direct sum of exactly two consecutive terms in the radical series of Λ has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. Some known results are obtained as corollaries.

scholarly journals Excellent Extensions and Homological Conjectures

2018 ◽  
Vol 25 (04) ◽  
pp. 619-626
Author(s):  
Yingying Zhang
Keyword(s):  
Finite Group ◽  
Group Algebra ◽  
Finitistic Dimension ◽  
Artin Algebra ◽  
A Finite Group ◽  
Excellent Extension ◽  
Special Case ◽  
Skew Group Algebra ◽  
Dimension Conjecture ◽  
Homological Conjectures

In this paper, we introduce the notion of excellent extensions of rings. Let Γ be an excellent extension of an Artin algebra Λ, we prove that Λ satisfies the Gorenstein symmetry conjecture (resp., finitistic dimension conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a special case of excellent extensions, when G is a finite group whose order is invertible in Λ acting on Λ and Λ is G-stable, we prove that if the skew group algebra ΛG satisfies strong Nakayama conjecture (resp., generalized Nakayama conjecture), then so does Λ.

Endomorphism algebras of Gorenstein projective modules

2018 ◽  
Vol 17 (09) ◽  
pp. 1850177 ◽  
Author(s):  
Aiping Zhang
Keyword(s):  
Projective Modules ◽  
Text Representation ◽  
Artin Algebra ◽  
Gorenstein Algebra ◽  
Endomorphism Algebras ◽  
Gorenstein Projective Modules

Let [Formula: see text] be an Artin algebra, [Formula: see text] be a Gorenstein projective [Formula: see text]-module and [Formula: see text]. We give a characterization of modules on [Formula: see text] and show that if [Formula: see text] is [Formula: see text]-representation-finite, then [Formula: see text] is also [Formula: see text]-representation-finite. As an application, we prove if [Formula: see text] is a CM-finite [Formula: see text]-Gorenstein algebra, then [Formula: see text] is a [Formula: see text]-Igusa-Todorov algebra.

Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

2013 ◽  
Vol 20 (01) ◽  
pp. 123-140
Author(s):  
Teng Zou ◽  
Bin Zhu
Keyword(s):  
Simplicial Complex ◽  
Root System ◽  
Cluster Complex ◽  
Cluster Complexes ◽  
Endomorphism Algebras ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Hereditary Algebras ◽  
Cluster Tilting ◽  
Tilting Objects

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

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