Nous construisons une suite exacte longue, exprimant les modules d'extension d'une algèbre triangulaire en fonctions de ceux associés aux algèbres de sa diagonale, dans deux cas : soit son bimodule est un module-à-gauche projectif soit c'est un module-à-droite plat. Ceci complète un résultat de Palmér et Roos sur la dimension globale des algèbres triangulaires.
We construct a long exact sequence where the extensions modules of a triangular algebra are involved with extension modules of its diagonal algebras. This holds in two cases: either its bimodule is right-projective or it is left-flat. This completes a result of Palmér and Roos about the global dimension of triangular algebras.
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Belkacem Bendiffalah 1
@article{CRMATH_2004__339_6_387_0, author = {Belkacem Bendiffalah}, title = {Modules d'extensions des alg\`ebres triangulaires}, journal = {Comptes Rendus. Math\'ematique}, pages = {387--390}, publisher = {Elsevier}, volume = {339}, number = {6}, year = {2004}, doi = {10.1016/j.crma.2004.07.007}, language = {fr}, }
Belkacem Bendiffalah. Modules d'extensions des algèbres triangulaires. Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 387-390. doi : 10.1016/j.crma.2004.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.007/
[1] B. Bendiffalah, D. Guin, Cohomologie de l'algèbre triangulaire et applications, Algebra Montpellier Announcements 01 (2003)
[2] A generalization of the ring of triangular matrix, Nagoya Math. J., Volume 18 (1961), pp. 13-25
[3] Tensor Hochshild Homology and Cohomology, Lectures in Pure and Applied Math., vol. 210, Dekker, New York, 2000
[4] Hochschild cohomology and fundamental groups of incidence algebras, Commun. Algebra, Volume 29 (2001) no. 5, pp. 2269-2283
[5] On the deformation of algebra morphisms and diagrams, Trans. Amer. Math. Soc., Volume 279 (1983) no. 1, pp. 1-50
[6] Hochschild cohomology rings and triangular rings, Proc. of the 9th International Conference, Beijing 2000, vol. II, Beijing Normal University Press, 2002, pp. 192-200
[7] B. Keller, Derived invariance of higher structures on the Hochschild complex, Preprint, 2003
[8] Hochschild cohomology of triangular matrix algebras, J. Algebra, Volume 233 (2000) no. 2, pp. 502-525
[9] Formules explicites pour la dimension homologique des anneaux de matrices généralisées, C. R. Acad. Sci. Paris, Ser. I, Volume 273 (1971), pp. 1026-1029
[10] Explicit formulæ for the global homological dimensions of trivial extensions of rings, J. Algebra, Volume 27 (1973), pp. 380-413
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