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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

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

We review results on the first Hochschild cohomology vector space of a finite dimensional algebra, in particular for path algebras modulo a "pre-generated" ideal. In case of a monomial algebra whose quiver has no oriented cycles, a dimension formula is provided. The general monomial case is considered in a paper with M. Saorin., Comment: Published in 2000 and not widely available

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

Let $G$ be a group acting on a small category $\mathcal C$ over a field $k$, that is $\mathcal C$ is a $G$-$k$-category. We first obtain that $\mathcal C$ is resolvable by a category which is $G$-$k$-equivalent to it, on which $G$ acts freely on objects. This resolvent category enables to show that if the coinvariants and the invariants functors are exact, then the coinvariants and invariants of the Hochschild-Mitchell (co)homology of $\mathcal C$ are isomorphic to the trivial component of the Hochschild-Mitchell (co)ho\-mo\-logy of the skew category $\mathcal C[G]$. Otherwise the corresponding spectral sequence can be settled. If the action of $G$ is free on objects, there is a canonical decomposition of the Hochschild-Mitchell (co)homology of the quotient category $\mathcal C/G$ along the conjugacy classes of $G$. This way we provide a general frame for monomorphisms which have been described previously in low degrees., Comment: This version is improved after referee comments. This article differs from a previous article which has been withdrawn. The main point now is the resolving category. In addition, this work contains new results. It has been improved and reorganized. This article supersedes arXiv:1804.02223

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

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

Let $\mathcal C$ be category over a commutative ring $k$, its Hochschild-Mitchell homology and cohomology are denoted respectively $HH_*(\mathcal C)$ and $HH^*(\mathcal C).$ Let $G$ be a group acting on $\mathcal C$, and $\mathcal C[G]$ be the skew category. We provide decompositions of the (co)homology of $\mathcal C[G]$ along the conjugacy classes of $G$. For Hochschild homology of a $k$-algebra, this corresponds to the decomposition obtained by M. Lorenz. If the coinvariants and invariants functors are exact, we obtain isomorphisms $\left(HH_*(C)\right)_G\simeq HH^{\{1\}}_* (\mathcal C[G])$ and $\left(HH^*(\mathcal C)\right)^G\simeq HH^*_{\{1\}} (\mathcal C[G]), $ where $\{ {1}\}$ is the trivial conjugacy class of $G$. We first obtain these isomorphisms in case the action of $G$ is free on the objects of $\mathcal C$. Then we introduce an auxiliary category $M_G(\mathcal C)$ with an action of $G$ which is free on its objects, related to the infinite matrix algebra considered by J. Cornick. This category enables us to show that the isomorphisms hold in general, and in particular for the Hochschild (co)homology of a $k$-algebra with an action of $G$ by automorphisms. We infer that $\left(HH^*(\mathcal C)\right)^G$ is a canonical direct summand of $HH^*(\mathcal C[G])$. This provides a frame for monomorphisms obtained previously, and which have been described in low degrees., Comment: A new document is written and will be submitted to arXiv. It differs significantly from this one to be withdraw. It has a new main important viewpoint, namely the resolving category. In addition the new paper contains several new results. It has been improved and reorganized

Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra $\Lambda$ is triangular with respect to a system of non necessarily primitive idempotents, and if the algebras at the idempotents belong to $\mathcal H$, then $\Lambda$ is in $\mathcal H$. Secondly we consider a $2\times 2$ matrix algebra, with two algebras on the diagonal, two projective bimodules in the corners, and zero corner products. They are not triangular with respect to the system of the two diagonal idempotents. However, the analogous result holds, namely if both algebras on the diagonal belong to $\mathcal H$, then the algebra itself is in $\mathcal H$., Comment: To appear in Indiana University Mathematics Journal 28 pages

We consider an homogeneous action of a finite group on a free linear category over a field in order to prove that the subcategory of invariants is still free. Moreover we show that the representation type is preserved when considering invariants., Comment: 19 pages

We define the fundamental group of a Hopf algebra over a field. For this purpose we first consider gradings of Hopf algebras and Galois coverings. The latter are given by linear categories with new additional structure which we call Hopf linear categories over a finite group. We compare this invariant to the fundamental group of the underlying linear category, and we compute those groups for families of examples., Comment: Computations of the fundamental group of some Hopf algebras are added. The relation with the fundamental group of the underlying associative structure is now considered. We also analyse the situation when universal covers and/or gradings exist. Dedicated to Eduardo N. Marcos for his 60th birthday. 24 pages