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Résumé: Nous étudions les propriétés locales des espaces de Berkovich sur  Z. À l’aide de théorèmes de Weierstraß, nous montrons que les anneaux locaux de ces espaces sont noethériens, réguliers dans le cas des espaces affines et excellents. Nous prouvons également que le faisceau structural est cohérent. Nos méthodes s’adaptent à d’autres anneaux de base (corps valués, anneaux de valuation discrète, anneaux d’entiers de corps de nombres, etc.) et traitent de façon unifiée espaces complexes et p-adiques. [ABSTRACT FROM AUTHOR]

Abstract: We show that the Hecke algebra for modular forms mod 2 of level 1 is isomorphic to the power series ring , where and . [Copyright &y& Elsevier]

In this article we define and study a Harder-Narasimhan filtration of finite flat group schemes over an unequal characteristic valuation ring. We also study that type of filtrations for families of finite flat group schemes parametrised by formal schemes. [ABSTRACT FROM AUTHOR]

Abstract: The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. After its description, we prove a Mayer–Vietoris exact sequence in this framework. In the case of a Galois extension of a number field with rings of integers A, B respectively, this K-theory of the “norm functor” is an extension of a subgroup of the ideal class group of F by the Tate cohomology group . The Mayer–Vietoris exact sequence enables us to describe in a quite explicit way a quotient of the subgroup of the ideal class group , where N is the norm. We also prove a short exact sequence where is the group of semi-local units of F. Finally, we conclude this paper by applications of our methods to Number Theory. [Copyright &y& Elsevier]

Abstract: We prove quantifier elimination in Witt vectors over an algebraically closed fields (or in power series), considered as a valued module over the Ore ring of Frobenius polynomials. We get that these structures do not have the independence property. To cite this article: L. Bélair, F. Point, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: We describe the additive structure of the graded ring of quasimodular forms over any discrete and cocompact group . We show that this ring is not finitely generated. We calculate the exact number of new generators of weight k (even). This number is constant for k sufficiently large and equals , where I and are the ideals of modular forms and quasimodular forms, respectively, over Γ of strictly positive weight. To cite this article: N. Ouled Azaiez, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: We obtain a complete description of closed ideals of the algebra , , where is the Dirichlet space and is the algebra of analytic functions satisfying the Lipschitz condition of order α. To cite this article: B. Bouya, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: We give a bound on the Castelnuovo–Mumford regularity of a homogeneous ideal I, in a polynomial ring A, in terms of the number of variables and the degrees of generators, when the dimension of is at most two. This bound improves the one obtained by Caviglia and Sbarra in [G. Caviglia, E. Sbarra, Characteristic-free bounds for the Castelnuovo–Mumford regularity, Prépublication, math.AC/0310122]. In the continuation of the examples constructed in Chardin and D''Cruz [M. Chardin, C. D''Cruz, Castelnuovo–Mumford regularity: examples of curves and surface, J. Algebra 270 (2003) 347–360], we use families of monomial curves to construct homogeneous ideals showing that these bounds are quite sharp. To cite this article: M. Chardin, A.L. Fall, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

Abstract: 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. To cite this article: B. Bendiffalah, C. R. Acad. Sci. Paris, Ser. I 339 (2004). [Copyright &y& Elsevier]

A classical result of Kronecker, stated at the end of the Section 10 of Kronecker (J. Reine Angew. Math. 92 (1882) 1–123), is that any radical of a finitely generated ideal in a polynomial ring of n variables is the radical of an ideal generated by n+1 elements. We give a constructive and elementary proof of a generalisation presented in (Michigan Math. J. 31 (1984) (2) 167–180): in a ring of Krull dimension ⩽ n a radical of a finitely generated ideal is the radical of an ideal generated by n+1 elements. To cite this article: T. Coquand, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

Let V be a mixed characteristic complete discrete valuation ring, P a smooth formal scheme over V, P its special fiber, X a smooth subscheme of P, T a divisor in P such that TX=T∩X is a divisor in X and D†P the weak completion of the sheaf of differential operators on P. We prove that the unit-root F-isocrystals on X&z.drule;TX overconvergent along TX are coherent over D†P,Q. To cite this article: D. Caro, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

We define an involution which reduces to a block triangular form the Gram matrices of the algebra of planar binary trees. This leads us to conjecture the existence of a tower of algebras admitting the latter as its Grothendieck ring. To cite this article: F. Hivert et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]

Abstract: A finitely generated structure is said to be QFA (for quasi-finitely axiomatizable, see [A. Nies, Separating classes of groups by first order sentences, Internat. J. Algebra Comput. 13 (2003) 287–302]) if there exists a first order sentence satisfied by S such that every finitely generated structure satisfying it is isomorphic to S. We prove that every structure which is bi-interprétable with the ring of integers is QFA and prime. We apply this result on the one hand to some metabelian groups and on the other, to commutative rings. To cite this article: A. Khelif, C. R. Acad. Sci. Paris, Ser. I 345 (2007). [Copyright &y& Elsevier]