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Abstract: Given a subroup G of a stable (in the model-theoric sense) group Γ, in particular when Γ is a group of finite Morley rank, the traces on G of the definable subsets of Γ have a remarkable property: if the definable closure of G is connected, they are either supergeneric, or supergenerically complemented, in the sense of the definition given at the very beginning of this paper. An example of this situation is provided by the linear groups: for some n, G is a subgroup of , where K is a field that we may take algebraically closed; the definable sets in the sense of are its constructible subsets, i.e. the boolean combinations of a finite number of its Zariski closed subsets. For any group G, the supergeneric subsets of G form a filter of large sets, which, to my best knowledge, is defined here for the first time. This paper undertakes the study of supergenericity in a general context, with no hypotheses of a model-theoric nature, but with a special attention given to the very specific properties of genericity possessed by the definable subsets of a stable group. It can be read without any knowledge of Logic, provided that one is ready to skip the proofs of the theorems showing precisely that these definable sets have these properties. [Copyright &y& Elsevier]

Abstract: Let G be the identity component of an orthogonal or a symplectic p-adic group (not necessarily split) or an inner form of a general linear p-adic group. In a previous paper, it was shown that the Bernstein components of the category of smooth representations of G are equivalent to the category of right modules over some Hecke algebra with parameters, or more general over the semi-direct product of such an algebra with a finite group algebra. The aim of the present paper is to show that this equivalence preserves the tempered spectrum and the discrete series representations. [Copyright &y& Elsevier]

Abstract: In this paper, we first characterize the subsets of the Bruhat–Tits tree of , K a complete valued field, that are the sets of fixed points of a subgroup G of . When G is irreducible, the are the “strips” in the tree. We then evaluate the form of the strip , in particular in terms of algebraic invariants of the group G. We give two applications to representation theory, one about a new generalization of Ribet's lemma on extensions, the other about the trace-convergence of a sequence of representations. [Copyright &y& Elsevier]

Abstract: In this paper we give a complete answer to the isotropy of an Albert bilinear form over the function field of a quadric in characteristic 2. As a consequence, we complete the classification of nongood bilinear forms of height and degree 2 given in Laghribi and Rehmann (2009) . [Copyright &y& Elsevier]

Abstract: In this paper we define the notion of “graded algebra with symmetries”. This notion is a generalization of the extended differential forms. We prove that for a graded algebra with symmetries T, we associate a subalgebra which generalizes the noncommutative differential forms. Using this algebra , we can define the Hochschild and cyclic homologies, cup i-products and the Steenrod squares. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate some basic classical results on the Krull dimension of -graded commutative rings by using the concept of iterated boundary ideals and iterated boundary monoids. After having pointed out the link between the Krull dimension (Kdim) and the boundary ideals and monoids, we characterize in the same way the graded dimension (dimgr). The main goal is to compare , and , where is a graded ring (noetherian or not). To do this, we establish various properties using the boundary ideals, in particular some typical boundary ideals inclusions. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to give an effective version of the Strong Artin Approximation Theorem for binomial equations. First we give an effective version of the Greenberg Approximation Theorem for polynomial equations, then using the Weierstrass Preparation Theorem, we apply this effective result to binomial equations. We prove that the Artin function of a system of binomial equations is bounded by a doubly exponential function in general and that it is bounded by an affine function if the order of the approximated solutions is bounded. [Copyright &y& Elsevier]

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 consider groups endowed with root data associated with non-necessarily finite root systems. We generalise to these groups the twisting methods of Chevalley groups initiated by Steinberg and Ree. The resulting theorem (proved in 1988) can be applied to Kac–Moody groups: see for instance two papers published by J. Ramagge in 1995 [J. Ramagge, On certain fixed point subgroups of affine Kac–Moody groups, J. Algebra 171 (2) (1995) 473–514; J. Ramagge, A realization of certain affine Kac–Moody groups of types II and III, J. Algebra 171 (3) (1995) 713–806]. [Copyright &y& Elsevier]

Abstract: In this paper, we define in the framework of noncommutative differential forms the notion of cup i-product introduced by Steenrod in [N.E. Steenrod, Product of cocycles and extensions of mappings, Ann. Math. 48 (2) (1947)]. This cup i-product permits to define an operator who will be a generalisation of the operator b defined in Hochschild and cyclic homology. With the operator we define in an explicitly way the Steenrod and Thomas–Pontrjagin squares. [Copyright &y& Elsevier]

Abstract: This paper shows a mean (using multilinear alternating forms) of getting unimodular vectors in a module over a commutative ring, without noetheriannity hypothesis. We show an elementary approach of Serre''s splitting off theorem, Bass''s stable range and cancellation theorems and Forster–Swan''s theorem. [Copyright &y& Elsevier]

Abstract: Let n be a positive integer. We say that a group G satisfies the condition , if every set of elements of G contains a pair such that , for some positive integer k. In this paper, we study finite groups G satisfying this condition. In particular, if G is a finitely generated soluble group, then , where is the hypercentre of G. [Copyright &y& Elsevier]

This paper presents an axiomatic description in a base free way of infinite root systems as those found in the study of Kac–Moody–Borcherds' algebras. In 1979, R. Moody [Adv. Math. 33 (1979) 144] posed the problem of finding such a description. In [Comm. Algebra 23 (1995) 4791], J.G. Bliss has already presented two answers to this question with his notions of “geometric root systems” and “rational root systems.” This new answer to Moody's question fits into the framework of the earlier axiomatic theory of the « syste`mes ge´ne´rateurs de racines » developed in [N. Bardy, Me´m. Soc. Math. Fr. (N.S.) 65 (1996)]. [Copyright &y& Elsevier]

Given any commutative field k, denote R=Oq(Mn(k)) the coordinate ring of quantum n×n matrices over k and assume q is a nonzero element in k which is not a root of unity. Recall that R is generated by n2 variables Xi,α ((i,α)∈⟦1,n〉2) subject (only) to the following relations:If xyzt is any 2×2 sub-matrix of X=(Xi,α), then: (a) yx=q−1xy, zx=q−1xz, tz=q−1zt, ty=q−1yt, zy=yz; (b) tx=xt−(q−q−1)yz.Denote R the k-algebra generated by the same variables Xi,α subject to the same relations, except relations (b) which are replaced by: (c) tx=xt; so that R is just the algebra of regular functions on some quantum affine space of dimension n2 over k.The theory of “derivative elimination” defines a natural embedding ϕ :Spec(R)→Spec(R) and asserts that: