## Algèbres de type laura, algèbres de groupes gauches et groupes de (co)homologie

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##### Date de publication

2006##### Auteur(s)

Smith, David

##### Résumé

This thesis contains the main results of five of the papers signed or co-signed by the author, those are [127, 89, 128, 55, 40]. As a consequence, it deals with various topics in representation theory of algebras. The first one is about laura algebras. Those have been independently introduced by Assem and Coelho [10] and Reiten and Skowronski [109] at the early stage of this decade. The aim was to obtain a common treatment of both the caass of representation finite algebras and weakly shod algebras (see [49]). Since then, laura algebras have been heavily investigated [9, 109, 124, 10, 14, 56]. An important fact about laura algebras is their strong links with a particular type of Auslander-Reiten components, called quasidirected. In Chapters 2 and 3, we present the results of [127, 89] whose aim is to give numerous (unified) characterizations of laura algebras and quasi-directed components; many of them being expressed in terms of paths between indecomposable modules, see Theorems 2.2.1, 3.2.1 and 3.2.2. In Chapter 4, we present the main results of [128]. Those deal with a new family of algebras, called almost laura. We highlight some properties of those algebras and show that laura algebras stand as particular cases of almost laura algebras. In addition, we study more intensively the left or right supported almost laura algebras and conjecture that any such algebra is laura. Besides this, given an algebra A and a group G, one can define, following the works of de la Peäna [51] and Reiten and Riedtmann [107], the skew group algebra A [ G ]. It is known that A and A [ G ] often share many properties. In this document, we show that if G is a finite group whose order is invertible in A, then A is almost laura if and only if so is A [ G ] (see Section 4.6). Then, in Chapter 5, we prove that under some assumptions, the fact that A is piecewise hereditary (see [71, 78, 76]) implies that so is A [G ]. Finally, in Chapter v, we present the results of [40] concerning the Hochschild and simplicial (co)homology'groups of a pullback R of algebras A[subscript 1] and A[subscript 2] . We show that under proper assumptions there exist Mayer-Vietoris exact sequences relating the Hochschild, or simplicial, (co)homology groups of A[subscript 1], A[subscript 2] and R. We also establish some links allowing to compute the fundamental groups of a presentation of R in terms of fundamental groups of presentations of A[subscript 1] and A[subscript 2]. In order to help the reader, we always refer, when this is possible, to the paper in which a given result is presented.

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- Sciences – Thèses [589]

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