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We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ. [ABSTRACT FROM AUTHOR]

Let B ⊂ A be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that B satisfies Han's conjecture if and only if A does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we give a structure result for extensions of an algebra given by a quiver and admissible relations, and examples of non split left or right bounded extensions. [ABSTRACT FROM AUTHOR]

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper. [ABSTRACT FROM AUTHOR]

We provide a formula for the change of the dimension of the first Hochschild cohomology vector space of bound quiver algebras when adding new arrows. For this purpose we show that there exists a short exact sequence which relates the first cohomology vector spaces of the algebras to the first relative cohomology. Moreover, we show that the first Hochschild homologies are isomorphic when adding new arrows. [ABSTRACT FROM AUTHOR]

In this paper we study the behaviour of the Igusa–Todorov functions for radical square zero algebras. We show that the left and the right ϕ -dimensions coincide, in this case. Some general results are given, but we concentrate more in the radical square zero algebras. Our study is based on two notions of hearth and member of a quiver Q . We give some bounds for the ϕ and the ψ -dimensions and we describe the algebras for which the bound of ψ is obtained. We also exhibit modules for which the ϕ -dimension is realised. [ABSTRACT FROM AUTHOR]

Abstract: Let R be a finite dimensional k-algebra over an algebraically closed field k and modR be the category of all finitely generated left R-modules. For a given full subcategory of modR, we denote by the projective finitistic dimension of . That is, . It was conjectured by H. Bass in the 60''s that the projective finitistic dimension has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and G. Todorov defined in [K. Igusa, G. Todorov, On the finitistic global dimension conjecture for artin algebras, in: Representations of Algebras and Related Topics, in: Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 201–204] a function , which turned out to be useful to prove that is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of modR instead of a class of algebras. That is, we suggest to take the class of categories , of θ-filtered R-modules, for all stratifying systems in modR. We prove that the Finitistic Dimension Conjecture holds for the categories of filtered modules for stratifying systems with one or two (and some cases of three) modules of infinite projective dimension. [Copyright &y& Elsevier]

We study the class of algebras having a path of nonisomorphisms from an indecomposable injective module to an indecomposable projective module and any such path is bounded by a fixed number. We show that such an algebra is an iteration of one-point extensions starting at a product of tilted algebras. This allows us to describe, for instance, its Auslander–Reiten quiver. [Copyright &y& Elsevier]