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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 describe how the Hochschild (co)homology of a bound quiver algebra changes when deleting or adding arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun, and a length one relative projective resolution of tensor algebras. [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]

We use the characteristic polynomial of the Coxeter matrix of an algebra to complete the combinatorial classification of piecewise hereditary algebras which Happel gave in terms of the trace of the Coxeter matrix. We also give a cohomological interpretation of the coefficients (other than the trace) of the characteristic polynomial of the Coxeter matrix of any finite dimensional algebra with finite global dimension. [ABSTRACT FROM AUTHOR]

Abstract: We show that the main results of Happel–Rickard–Schofield (1988) and Happel–Reiten–Smalø (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and show that if is a finite group acting on a piecewise hereditary algebra over an algebraically closed field whose characteristic does not divide the order of , then the resulting skew group algebra is also piecewise hereditary. [Copyright &y& Elsevier]

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree. or if and only if its first Hochschild cohomology group H²(A) with coefficients in AAA vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹ (A) = 0 and it contains no full convex subcategory which is hereditary of type A. or if and only lilt is separated and contains no full convex subcategory which is hereditary of type A. [ABSTRACT FROM AUTHOR]