Seja S(gl_n) a álgebra simétrica da álgebra de Lie das matrizes de tamanho nxn sobre o corpo C dos números complexos. Para \\xi em gl_n*=gl_n, seja F_{\\xi}(gl_n) a asubálgebra de Mishchenko-Fomenko de S(gl_n) construída pelo método de deslocamento de argumento associada ao parâmetro \\xi. É conhecido que se \\xi é um elemento semisimples regular ou nilpotente regular então a subálgebra F_{\\xi}(gl_n) é gerada por uma sequência regular em S(gl_n). Nesta tese é provado que em gl_3 o resultado estende para todo \\xi em gl_3, isto é, as subálgebras de Mishchenco-Fomenko F_{\\xi}(gl_3) são geradas por uma sequência regular em S(gl_3), uma consequência deste fato é que os módulo irredutíveis sobre certas subálgebras comutativas da álgebra envolvente universal U(gl_3) podem ser levantados a módulos irredutiveis sobre U(gl_3). Além disso, é provado que em gl_4 esse resultado é válido para todo elemento nilpotente \\xi em gl_4. O caso geral, que é determinar quando as subálgebras de Mishchenko-Fomenko F_{\\xi}(gl_n) , com \\xi em gl_n, são geradas por uma sequência regular em S(gl_n), é ainda um problema aberto. / Let S(gl_n) be the symmetric algebra of the Lie algebra of the matrices of size nxn over the field C of complex numbers. For \\xi in gl_n*=gl_n, let F_{\\xi}(gl_n) be the Mishchenko-Fomenko subalgebra of S(gl_n) constructed by the argument shift method associated with the parameter \\xi. It is known that if \\xi is a semisimple regular element or nilpotent regular element then the subalgebra F_(g_ln) is generated by a regular sequence in S(gl_n). In this thesis we prove that in gl_3 the result is extended to all \\xi in gl_3, this is, the Mishchenco-Fomenko subalgebras F_{\\xi}(gl3) are generated by a regular sequence in S(gl_3), A consequence of this fact is that the irreducible modules over certain commutative subalgebras of the universal enveloping algebra U(gl_3) can it be lifted to irreducible modules over U(gl_3). Furthermore, is proved that this result is true for all elements nilpotente \\xi in gl_4. The general case, which is determined when the Mishchenko-Fomenko subalgebras F_{\\xi}(gl_n), with \\xi in gl_n, are generated by a regular sequence in S(gl_n), it is still an open problem.

Cette thèse d'algèbre commutative porte principalement sur la théorie de la profondeur. Nous nous efforçons d'en fournir une approche épurée d'hypothèse noethérienne dans l'espoir d'échapper aux idéaux premiers et ceci afin de manier des objets élémentaires et explicites. Parmi ces objets, figurent les complexes algébriques de Koszul et de Cech dont nous étudions les propriétés cohomologiques grâce à des résultats simples portant sur la cohomologie du totalisé d'un bicomplexe. Dans le cadre de la cohomologie de Cech, nous avons établi la longue suite exacte de Mayer-Vietoris avec un traitement reposant uniquement sur le maniement des éléments. Une autre notion importante est celle de dimension de Krull. Sa caractérisation en termes de monoïdes bords permet de montrer de manière expéditive le théorème d'annulation de Grothendieck en cohomologie de Cech. Nous fournissons également un algorithme permettant de compléter un polynôme homogène en un h.s.o.p.. La profondeur est intimement liée à la théorie des résolutions libres/projectives finies, en témoigne le théorème de Ferrand-Vasconcelos dont nous rapportons une généralisation due à Jouanolou. Par ailleurs, nous revenons sur des résultats faisant intervenir la profondeur des idéaux caractéristiques d'une résolution libre finie. Nous revisitons, dans un cas particulier, une construction due à Tate permettant d'expliciter une résolution projective totalement effective de l'idéal d'un point lisse d'une hypersurface. Enfin, nous abordons la théorie de la régularité en dimension 1 via l'étude des idéaux inversibles et fournissons un algorithme implémenté en Magma calculant l'anneau des entiers d'un corps de nombres. / This Commutative Algebra thesis focuses mainly on the depth theory. We try to provide an approach without noetherian hypothesis in order to escape prime ideals and to handle only basic and explicit concepts. We study the algebraic complexes of Koszul and Cech and their cohomological properties by using simple results on the cohomology of the totalization of a bicomplex. In the Cech cohomology context we established the long exact sequence of Mayer-Vietoris only with a treatment based on the elements. Another important concept is that of Krull dimension. Its characterization in terms of monoids allows us to show expeditiously the vanishing Grothendieck theorem in Cech cohomology.We also provide an algorithm to complete a omogeneous polynomial in a h.s.o.p.. The depth is closely related to the theory of finite free/projective resolutions. We report a generalization of the Ferrand-Vasconcelos theorem due to Jouanolou. In addition, we review some results involving the depth of the ideals of expected ranks in a finite free resolution.We revisit, in a particular case, a construction due to Tate. This allows us to give an effective projective resolution of the ideal of a point of a smooth hypersurface. Finally, we discuss the regularity theory in dimension 1 by studying invertible ideals and provide an algorithm implemented in Magma computing the ring of integers of a number field.