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We study homological properties of a family of algebras called toupie algebras. Our main objective is to obtain the Gerstenhaber structure of their Hochschild cohomology, with the purpose of describing the Lie algebra structure of the first Hochschild cohomology space, together with the Lie module structure of the whole Hochschild cohomology. [ABSTRACT FROM AUTHOR]

We develope the theory of ${\mathcal {E}}$ -relative Igusa-Todorov functions in an exact I T-context $({\mathcal {C}},{\mathcal {E}})$ (see Definition 2.1). In the case when ${\mathcal {C}}={\text {mod}}\, ({\Lambda })$ is the category of finitely generated left Λ-modules, for an artin algebra Λ, and ${\mathcal {E}}$ is the class of all exact sequences in ${\mathcal {C}},$ we recover the usual Igusa-Todorov functions, Igusa K. and Todorov G. (2005). We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov's results. Furthermore, we introduce the ${\mathcal {E}}$ -relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions. [ABSTRACT FROM AUTHOR]

Let Λ be an artin algebra and $\mathfrak {A}$ a two-sided idempotent ideal of Λ, that is, $\mathfrak {A}$ is the trace of a projective Λ-module P in Λ. We consider the categories of finitely generated modules over the associated rings ${\Lambda }/\mathfrak {A}, {\Lambda }$ and Γ = End( P) and study the relationship between their homological properties via the Igusa-Todorov functions. [ABSTRACT FROM AUTHOR]

In the first part we study nearly Frobenius algebras. The concept of nearly Frobenius algebras is a generalization of the concept of Frobenius algebras. Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove that the known constructions: direct sums, tensor, quotient of nearly Frobenius algebras admit natural nearly Frobenius structures. In the second part we study algebras associated to some families of quivers and the nearly Frobenius structures that they admit. As a main theorem, we prove that an indecomposable algebra associated to a bound quiver ( Q, I) with no monomial relations admits a non trivial nearly Frobenius structure if and only if the quiver Q is linearly oriented of type $\overrightarrow {\mathbb {A}_{n}}$ and I = 0. We also present an algorithm that determines the number of independent nearly Frobenius structures for gentle algebras without oriented cycles. [ABSTRACT FROM AUTHOR]

In Igusa and Todorov () introduced two functions ϕ and ψ, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin R-algebra A and the Igusa-Todorov function ϕ, we characterise the ϕ-dimension of A in terms of the bi-functors $\text{Ext}^{i}_{A}(-, -)$ and in terms of Tor's bi-functors $\text{Tor}^{A}_{i}(-,-).$ Furthermore, by using the first characterisation of the ϕ-dimension, we show that the finiteness of the ϕ-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz's result (Bongartz, Lect. Notes Math. 903, 26-38, (), Corollary 1) as follows: For an artin algebra A, a tilting A-module T and the endomorphism algebra B = End( T), we have that ϕ dim ( A) − pd T ≤ ϕ dim ( B) ≤ ϕ dim ( A) + pd T. [ABSTRACT FROM AUTHOR]

Let $\mathcal{C}$ be a length-category. Generalizing the Loewy length, we propose the layer length associated with a torsion theory, which is a new measure for objects of $\mathcal{C}$. As an application, we use the layer lengths and the Igusa-Todorov function to get a theorem (see Theorem 6.4) having as corollaries the main results of Huard et al. (Bull Lond Math Soc 41:367-376, ) and Wang (Commun Algebra 22(7):419-449, ). [ABSTRACT FROM AUTHOR]

We show that a right artinian ring R is right self-injective if and only if ψ( M) = 0 (or equivalently ϕ( M) = 0) for all finitely generated right R-modules M, where ψ, $\phi :\!\!\!\! \mod R \to \mathbb N$ are functions defined by Igusa and Todorov. In particular, an artin algebra Λ is self-injective if and only if ϕ( M) = 0 for all finitely generated right Λ-modules M. [ABSTRACT FROM AUTHOR]

We consider the algebras Λ which satisfy the property that for each indecomposable module X, either its projective dimension pdΛ X is at most one or its injective dimension idΛ X is at most one. This clearly generalizes the so-called quasitilted algebras introduced by Happel–Reiten–Smalø. We show that some of the niciest features for this latter class of algebras can be generalized to the case we are considering, in particular the existence of a trisection in its module category. [ABSTRACT FROM AUTHOR]