scholarly journals The finiteness of the Gorenstein dimension for Artin algebras

2019 ◽  
Vol 18 (06) ◽  
pp. 1950112
Author(s):  
René Marczinzik
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Artin Algebra ◽  
Artinian Rings ◽  
Gorenstein Dimension ◽  
Finite Global Dimension ◽  
Artin Algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

scholarly journals TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS

2013 ◽  
Vol 94 (1) ◽  
pp. 133-144
Author(s):  
ZHAOYONG HUANG ◽  
XIAOJIN ZHANG
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Tilted Algebra ◽  
Gorenstein Algebras

AbstractLet $\Lambda $ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda $ admits a trivial maximal 1-orthogonal subcategory of $\text{mod } \Lambda $, then, for any indecomposable module $M\in \text{mod } \Lambda $, the projective dimension of $M$ is equal to one if and only if its injective dimension is also equal to one, and $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $\Lambda $ is a tilted algebra.

scholarly journals Some properties of non-commutative regular graded rings

1992 ◽  
Vol 34 (3) ◽  
pp. 277-300 ◽  
Author(s):  
Thierry Levasseur
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Supplementary Condition ◽  
Graded Rings ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Commutative Case ◽  
Finite Global Dimension ◽  
Image Position ◽  
Dimension One

Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a supplementary condition. Before stating it, we recall that the grade of a finitely generated (left or right) module is defined by

scholarly journals Gabriel–Roiter Measure, Representation Dimension and Rejective Chains

2020 ◽  
Vol 71 (2) ◽  
pp. 619-635
Author(s):  
Teresa Conde
Keyword(s):  
Global Dimension ◽  
Alternative Proof ◽  
Representation Dimension ◽  
Finite Global Dimension ◽  
Measure Representation ◽  
Artin Algebras

Abstract The Gabriel–Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel–Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel–Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel–Roiter measure $\mu$ in an abelian length category $\mathcal{A}$, there exists an object $X^{\prime}$ that depends on $X$ and $\mu$, such that $\Gamma =\operatorname{End}_{\mathcal{A}}(X\oplus X^{\prime})$ has finite global dimension. Analogously to Iyama’s original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

scholarly journals The Cartan matrix as an indicator of finite global dimension for Artinian rings

1985 ◽  
Vol 95 (2) ◽  
pp. 157-157 ◽  
Author(s):  
W. D. Burgess ◽  
K. R. Fuller ◽  
E. R. Voss ◽  
B. Zimmermann-Huisgen
Keyword(s):  
Global Dimension ◽  
Cartan Matrix ◽  
Artinian Rings ◽  
Finite Global Dimension

scholarly journals The Cartan determinant and generalizations of quasihereditary rings

1998 ◽  
Vol 41 (1) ◽  
pp. 23-32 ◽  
Author(s):  
W. D. Burgess ◽  
K. R. Fuller
Keyword(s):  
Global Dimension ◽  
Primitive Idempotent ◽  
Artinian Rings ◽  
Matrix Reduction ◽  
Finite Global Dimension ◽  
Compare And Contrast

The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(R/I), where I is a protective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the “projective” condition on heredity ideals. These rings, called left k-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

Elementary Superalgebras

2012 ◽  
Vol 19 (04) ◽  
pp. 673-682
Author(s):  
Bo Hou ◽  
Shilin Yang
Keyword(s):  
Weight Function ◽  
Global Dimension ◽  
Hochschild Homology ◽  
Trace Function ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Complete Set

Let Λ be a finite-dimensional superalgebra over a field K. A characterization of an elementary superalgebra Λ is given by a quiver and a weight function. It is shown that Λ is elementary if and only if its Hochschild extension is elementary. Furthermore, if Λ is elementary of finite global dimension and {e1, …, en} is a complete set of gr-primitive orthogonal idempotents of Λ, then the following equalities hold: [Formula: see text] where ΦΛ is the Coxeter matrix of Λ, tr is the trace function of a matrix, HHi(Λ) and HHi(Λ) are the i-th Hochschild homology and cohomology, respectively.

scholarly journals A new characterization of Auslander algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750219
Author(s):  
Shen Li ◽  
Shunhua Zhang
Keyword(s):  
Projective Dimension ◽  
Global Dimension ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Auslander Algebra

Let [Formula: see text] be a finite dimensional Auslander algebra. For a [Formula: see text]-module [Formula: see text], we prove that the projective dimension of [Formula: see text] is at most one if and only if the projective dimension of its socle soc[Formula: see text][Formula: see text] is at most one. As an application, we give a new characterization of Auslander algebras [Formula: see text] and prove that a finite dimensional algebra [Formula: see text] is an Auslander algebra provided its global dimension gl.d[Formula: see text][Formula: see text] and an injective [Formula: see text]-module is projective if and only if the projective dimension of its socle is at most one.

Relative Gorenstein global dimension

2016 ◽  
Vol 26 (08) ◽  
pp. 1597-1615 ◽  
Author(s):  
Driss Bennis ◽  
J. R. García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Upper Bound ◽  
Projective Dimension ◽  
Global Dimension ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Tilting Module ◽  
Injective Dimension ◽  
Wakamatsu Tilting Module ◽  
Bass Class ◽  
Gorenstein Injective

We study the relative Gorenstein projective global dimension of a ring with respect to a weakly Wakamatsu tilting module [Formula: see text]. We prove that this relative global dimension is finite if and only if the injective dimension of every module in Add[Formula: see text] and the [Formula: see text]-projective dimension of every injective module are both finite (indeed these three dimensions have a common upper bound). When RC satisfies some extra conditions we prove that the relative Gorenstein projective global dimension of [Formula: see text] is always bounded above by the [Formula: see text]-projective global dimension of [Formula: see text], these two dimensions being equal when the class of all [Formula: see text]-Gorenstein projective [Formula: see text]-modules is contained in the Bass class of [Formula: see text] relative to [Formula: see text]. Of course we also give the dual results concerning the relative Gorenstein injective global dimension.

scholarly journals Injective homogeneity and the Auslander–Gorenstein property

1995 ◽  
Vol 37 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Zhong Yi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Global Dimension ◽  
Noetherian Rings ◽  
Connected Component ◽  
Injective Dimension ◽  
The Right

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

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