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

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].

A simple note on the Yoneda (co)algebra of a monomial algebra

UDC 512.7 If is a monomial -algebra, it is well-known that is isomorphic to the space of (Anick) -chains for . The goal of this short note is to show that the next result follows directly from well-established theorems on -algebras, without computations: there is an -coalgebra model on satisfying that, for and , is a linear combination of , where , and . The proof follows essentially from noticing that the Merkulov procedure is compatible with an extra grading over a suitable category. By a simple argument based on a result by Keller we immediately deduce that some of these coefficients are .

On the Construction of d-Koszul algebras

An algebra has been constructed from a (D, A)-stacked algebra A, under the conditions that , A 1 and . It is shown that when the construction of algebra B is built from a (D, A)-stacked monomial algebra A then B is a d-Koszul monomial algebra.

The Gorenstein-projective modules over a monomial algebra

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.

Comparison morphisms between two projective resolutions of monomial algebras

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 $\operatorname{HH}^{\ast} (A)$.

Comparison morphisms between two projective resolutions of monomial algebras

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 $\operatorname{HH}^{\ast} (A)$.

Recognizing the Semiprimitivity of ℕ-graded Algebras via Gröbner Bases

Let K〈X〉=K〈X1,…,Xn〉 be the free K-algebra on X={X1,…,Xn} over a field K, which is equipped with a weight ℕ-gradation (i.e., each Xi is assigned a positive degree), and let [Formula: see text] be a finite homogeneous Gröbner basis for the ideal [Formula: see text] of K〈X〉 with respect to some monomial ordering ≺ on K〈X〉. It is shown that if the monomial algebra [Formula: see text] is semiprime, where [Formula: see text] is the set of leading monomials of [Formula: see text] with respect to ≺, then the ℕ-graded algebra A=K〈X〉 /I is semiprimitive in the sense of Jacobson. In the case that [Formula: see text] is a finite nonhomogeneous Gröbner basis with respect to a graded monomial ordering ≺ gr , and the ℕ-filtration FA of the algebra A=K〈X〉 /I induced by the ℕ-grading filtration FK〈X〉 of K〈X〉 is considered, if the monomial algebra [Formula: see text] is semiprime, then it is shown that the associated ℕ-graded algebra G(A) and the Rees algebra à of A determined by FA are all semiprimitive.