A Simple Note on the Yoneda (CO)Algebra of a Monomial Algebra

2021 ◽  
Author(s):  
E. Herscovich
Keyword(s):  
Monomial Algebra

Homological Aspects of Semigroup Gradings on Rings and Algebras

1999 ◽  
Vol 51 (3) ◽  
pp. 488-505 ◽  
Author(s):  
W. D. Burgess ◽  
Manuel Saorín
Keyword(s):  
Euler Characteristic ◽  
Division Ring ◽  
Artinian Ring ◽  
Global Dimension ◽  
Monomial Algebra ◽  
Finite Dimensional ◽  
Cartan Matrices ◽  
Torsion Free ◽  
Rings And Algebras ◽  
Finite Dimensional Algebras

AbstractThis article studies algebras R over a simple artinian ring A, presented by a quiver and relations and graded by a semigroup Σ. Suitable semigroups often arise from a presentation of R. Throughout, the algebras need not be finite dimensional. The graded K0, along with the Σ-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties.A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert Σ-series in the associated path incidence ring.The rationality of the Σ-Euler characteristic, the Hilbert Σ-series and the Poincaré-Betti Σ-series is studied when Σ is torsion-free commutative and A is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

2017 ◽  
pp. 1-31
Author(s):  
María Julia Redondo ◽  
Lucrecia Román
Keyword(s):  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Efficient Tool ◽  
Simple Description ◽  
Monomial Algebra ◽  
Cup Product ◽  
Lie Bracket ◽  
Projective Resolutions ◽  
Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

scholarly journals The Gorenstein-projective modules over a monomial algebra

2018 ◽  
Vol 148 (6) ◽  
pp. 1115-1134 ◽  
Author(s):  
Xiao-Wu Chen ◽  
Dawei Shen ◽  
Guodong Zhou
Keyword(s):  
Projective Modules ◽  
Stable Category ◽  
Monomial Algebra ◽  
Gorenstein Projective Modules ◽  
The Given

We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

scholarly journals THE FIRST COHOMOLOGY GROUP OF THE TRIVIAL EXTENSION OF A MONOMIAL ALGEBRA

2004 ◽  
Vol 03 (02) ◽  
pp. 143-159 ◽  
Author(s):  
CLAUDE CIBILS ◽  
MARÍA JULIA REDONDO ◽  
MANUEL SAORÍN
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Trivial Extension ◽  
Monomial Algebra ◽  
Finite Dimensional ◽  
Hochschild Cohomology Group ◽  
First Cohomology Group

Given a finite-dimensional monomial algebra A, we consider the trivial extension TA and provide formulae, depending on the characteristic of the field, for the dimensions of the summands HH1(A) and Alt (DA) of the first Hochschild cohomology group HH1(TA). From these a formula for the dimension of HH1(TA) can be derived.

scholarly journals Comparison morphisms between two projective resolutions of monomial algebras

2017 ◽  
pp. 1-31
Author(s):  
María Julia Redondo ◽  
Lucrecia Román
Keyword(s):  
Lie Algebra ◽  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Efficient Tool ◽  
Simple Description ◽  
Monomial Algebra ◽  
Cup Product ◽  
Lie Bracket ◽  
Projective Resolutions ◽  
Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Finite generation of the cohomology rings for the extension algebras of monomial algebras

2022 ◽  
Author(s):  
Hongbo Shi
Keyword(s):  
Cohomology Ring ◽  
Minimal Resolution ◽  
Finitely Generated ◽  
Finite Generation ◽  
Cohomology Rings ◽  
Monomial Algebra ◽  
Monomial Algebras ◽  
Resolution Graph ◽  
Dual Extension

We describe the cohomology ring of a monomial algebra in the language of dimension tree or minimal resolution graph and in this context we study the finite generation of the cohomology rings of the extension algebras, showing among others that the cohomology ring [Formula: see text] is finitely generated [Formula: see text] is [Formula: see text] is, where [Formula: see text] is the dual extension of a monomial algebra [Formula: see text] and [Formula: see text] is the opposite algebra of [Formula: see text].

The cohomology ring of a monomial algebra

1994 ◽  
Vol 85 (1) ◽  
pp. 11-23 ◽  
Author(s):  
E. L. Green ◽  
D. Zacharia
Keyword(s):  
Cohomology Ring ◽  
Monomial Algebra

On the Hochschild (co)homology of a monomial algebra given by a cyclic quiver and two zero-relations

2016 ◽  
Vol 45 (5) ◽  
pp. 2052-2073
Author(s):  
Tomohiro Itagaki
Keyword(s):  
Monomial Algebra ◽  
Cyclic Quiver

scholarly journals THE HOCHSCHILD COHOMOLOGY RING MODULO NILPOTENCE OF A MONOMIAL ALGEBRA

2006 ◽  
Vol 05 (02) ◽  
pp. 153-192 ◽  
Author(s):  
EDWARD L. GREEN ◽  
NICOLE SNASHALL ◽  
ØYVIND SOLBERG
Keyword(s):  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Dimensional Algebra ◽  
Monomial Algebra ◽  
Hochschild Cohomology Ring ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
The Ideal

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated K-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

scholarly journals An affine prime non-semiprimitive monomial algebra with quadratic growth

2006 ◽  
Vol 37 (4) ◽  
pp. 511-513 ◽  
Author(s):  
Agata Smoktunowicz ◽  
Uzi Vishne
Keyword(s):  
Quadratic Growth ◽  
Monomial Algebra
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