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In this paper we introduce, according to one of the main ideas of $\tau$-tilting theory, the $\tau$-tilting Hochschild cohomology in degree one of a finite dimensional $k$-algebra $\la$, where $k$ is a field. We define the excess of $\la$ as the difference between the dimensions of the $\tau$-tilting Hochschild cohomology in degree one and the dimension of the usual Hochschild cohomology in degree one. One of the main results is that for a zero excess bound quiver algebra $\la=kQ/I$, the Hochschild cohomology in degree two $HH^2(\la) $ is isomorphic to the space of morphisms $\Hom_{kQ-kQ}(I/I^2, \la).$ This may be useful to determine when $HH^2(\la)=0$ for these algebras. We compute the excess for hereditary, radical square zero and monomial triangular algebras. For a bound quiver algebra $\la$, a formula for the excess of $\la$ is obtained. We also give a criterion for $\la$ to be $\tau$-rigid., Comment: 14 pages

In this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising form a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT K-algebra with a path algebra of a finite quiver without oriented cycles is GLIT., Comment: 20 pages

In this survey, we review the fundamental properties of the Igusa-Todorov functions, the $\phi$-dimension, the $\psi$-dimension and their generalizations.

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 admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for it., Comment: Some small change in the Introduction. Theorem 2.9 was obtained in 2017 by N. Gao and C. Psaroudakis. The reference is added. To appear in J. of Algebra