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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

Let $B\subset A$ 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., Comment: Updated version. We have replaced the misleading "smooth algebra" by the standard "algebra of finite global dimension". The Appendix is now incorporated in a more direct and shorter form at Section 5

For an extension of associative algebras $B\subset A$ over a field and an $A$-bimodule $X$, we obtain a Jacobi-Zariski long nearly exact sequence relating the Hochschild homologies of $A$ and $B$, and the relative Hochschild homology, all of them with coefficients in $X$. This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness., Comment: A typo in the degree of the torsion vector spaces of the first page of the spectral sequence is corrected. The degree where the exact long sequence ends at Theorem 6.5 is therefore updated. To appear in Bulletin of the London Mathematical Society

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., Comment: 18 pages, submitted to a peer-reviewed journal

A main purpose of this paper is to prove that the class of finite dimensional algebras which verify Han's conjecture is closed under split bounded extensions., Comment: An extra hypothesis is added for Proposition 3.3, and the proof is modified accordingly

We provide a formula for the change of the dimension of the first Hoch\-schild 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., Comment: 17 pages - small changes made

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra., Comment: Typos corrected To appear in Proc. Amer. Math. Soc

For an abelian category $\mathcal{A}$, we define the category PEx($\mathcal{A}$) of pullback diagrams of short exact sequences in $\mathcal{A}$, as a subcategory of the functor category Fun($\Delta, \mathcal{A}$) for a fixed diagram category $\Delta$. For any object $M$ in ${\rm PEx}(\mathcal{A}),$ we prove the existence of a short exact sequence $0 {\to} K {\to} P {\to} M {\to} 0$ of functors, where the objects are in PEx($\mathcal{A}$) and $P(i) \in {\rm Proj(\mathcal{A})}$ for any $i \in \Delta$. As an application, we prove that if $(\mathcal{C}, \mathcal{D}, \mathcal{E})$ is a triple of syzygy finite classes of objects in $\mathrm{mod}\,\Lambda$ satisfying some special conditions, then $\Lambda$ is an Igusa-Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa-Todorov.

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.

We develope the theory of the $\mathcal{E}$-relative Igusa-Todorov functions in an exact $ IT$-context $(\mathcal{C},\mathcal{E}).$ In the case when $\mathcal{C}=$mod$\, (\Lambda)$ is the category of finitely generated left $\Lambda$-modules, for an artin algebra $\Lambda,$ and $\mathcal{E}$ is the class of all exact sequences in $\mathcal{C},$ we recover the usual Igusa-Todorov functions. 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.

In this paper we study the behaviour of the Igusa-Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the $\phi$-dimension and $\psi$-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite $\phi$-dimension and finite $\psi$-dimension.

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

Let $A$ be an artinian algebra, and let $\mathcal{C}$ be a subcategory of mod$A$ that is closed under extensions. When $\mathcal{C}$ is closed under kernels of epimorphisms (or closed under cokernels of monomorphisms), we describe the smallest class of modules that filters $\mathcal{C}$. As a consequence, we obtain sufficient conditions for the finitistic dimension of an algebra over a field to be finite. We also apply our results to the torsion pairs. In particular, when a torsion pair is induced by a tilting module, we show that the smallest classes of modules that filter the torsion and torsion-free classes are completely compatible with the quasi-equivalences of Brenner and Butler., Comment: 22 pages