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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., Published in 2000 and not widely available

23 pages; 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.

26 pages; The main objective of this paper is to present a theory for computing the Hochschild cohomology of algebras built on a specific data, namely multi-extension algebras. The computation relies on cohomological functors evaluated on the data, and on the combinatorics of an ad hoc quiver. One-point extensions are occurrences of this theory, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that we obtain via the study of trajectories of the quiver. We introduce cohomology along paths, and we compute it under suitable Tor vanishing hypotheses. The cup product on Hochschild cohomology enables us to describe the connecting homomorphism of the long exact sequence. Multi-extension algebras built on the round trip quiver provide square matrix algebras which have two algebras on the diagonal and two bimodules on the corners. If the bimodules are projective, we show that a five-term exact sequences arises. If the bimodules are free of rank one, we provide a complete computation of the Hochschild cohomology. On the other hand, if the corner bimodules are projective without producing new cycles in the data, Hochschild cohomology is that of the product of the algebras on the diagonal for large enough degrees. The word \emph{algebra} always means an associative $k$-algebra over a field $k$; an algebra is not necessarily finite dimensional.

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., 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 k be a commutative ring. We study the behavior of coverings of k-categories through fiber products and find a criterion for a covering to be Galois or universal. Fil: Cibils, Claude. Universite Montpellier II; Francia Fil: Redondo, Maria Julia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Bahía Blanca. Instituto de Matemática Bahía Blanca (i); Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentina Fil: Solotar, Andrea Leonor. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina

This paper has been withdrawn by the authors, due an error in Theorem 3.1 : the induced module is not in general a submodule.

We consider associative algebras λ over a field provided with a direct sum decomposition of a two-sided ideal M and a sub-algebra A - examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of λ. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes. Fil: Cibils, Claude. Université de Montpellier; Francia Fil: Marcos, Eduardo. Universidade de Sao Paulo; Brasil Fil: Redondo, Maria Julia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina Fil: Solotar, Andrea Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina

We describe the structure of the Grothendieck ring of projective modules of basic Hopf algebras using a positive integer determined by the composition series of the principal indecomposable projective module.

International audience; We compute the dimension at each degree of the Hochschild cohomology of a radical square zero finite dimensional algebra, in terms of the combinatorics of its quiver. The multiplicative structure is obtained as well, those algebras are finitely generated only for quivers without oriented cycles and for crowns.

En esta tesis estudiamos el problema de calcular resoluciones proyectivas de álgebrasasociativas. Nuestro punto de partida es la resolución de Bardzell para álgebrasmonomiales. Dada un álgebra asociativa, utilizamos el principio de sistemas de reducción de Bergman para asociarle álgebras monomiales. Mostramos que los diferencialesde la resolución de Bardzell de estas álgebras pueden modificarse para obtenerresoluciones proyectivas del álgebra de partida. Mas aún, damos un criterio paraque un complejo proveniente de una modificación de la resolución de Bardzell de unálgebra monomial asociada sea exacto. Aplicamos nuestro método a tres familias deálgebras: las intersecciones completas cuánticas, las álgebras de Weyl generalizadascuánticas y las álgebras down-up. En el caso de las álgebras down-up, utilizamos laresolución obtenida para calcular invariantes homológicos de estas álgebras. De estamanera probamos propiedades de regularidad y damos una solución al problema deisomorfismo para las álgebras down-up no noetherianas. Palabras clave: álgebras asociativas, cohomología de Hochschild, resoluciones proyectivas. This thesis is concerned with the problem of computing projective resolutions ofassociative algebras. Our starting point is Bardzell’s resolution for monomial algebras. Given an associatve algebra, we use Bergman’s principle of reduction systemsto associate monomial algebras to it. We prove that the differentials in Bardzell’s resolutionof these monomial algebras can be modified to obtain projective resolutionsof the original algebra. We also give sufficient conditions for a complex coming froma modification of Bardzell’s resolution of an associated monomial algebra to be exact. We apply our method to three families of algebras: Quantum complete intersections, Quantum generalized Weyl algebras and down-up algebras. In the case of down-upalgebras, we use the resolution obtained to compute homological invariants of thesealgebras. This way we prove regularity properties and we solve the isomorphismproblem for non-noetherian down-up algebras. Keywords: associative algebras, Hochschild cohomology, projective resolutions. Fil: Chouhy, Sergio Nicolás. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.