On the automorphisms of the spectral completion of the algebraic numbers field

Let be the absolute Galois group of Q and let A=C(G,C) be the Banach algebra of all continuous functions defined on G with values in C. Let be the conjugation automorphism of C and let B be the R-Banach subalgebra of A consisting of continuous functions f such that for all σ∈G. Let ‖x‖=sup{|σ(x)|:σ∈G} be the spectral norm on and let be the spectral completion of . Using a canonical isometry between and B we study the structure of the group of R-algebras automorphisms of and the structure of its subgroup of all automorphisms of which when restricted to give rise to elements of G. We introduce a topology on and prove that this last one is homeomorphic and group isomorphic to G.     

Hidden constructions in abstract algebra (6): The theorem of Maroscia and Brewer & Costa

We give a constructive deciphering for a generalization of the Quillen-Suslin theorem due to Maroscia and Brewer & Costa stating that finitely generated projective modules over , where is a Prüfer domain with Krull dimension ≤1, are extended from .     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Finiteness of graded local cohomology modules

Let R=?_n∈N0R_n be a Noetherian homogeneous ring with local base ring (R₀,m₀) and irrelevant ideal R₊, let M be a finitely generated graded R-module. In this paper we show that is Artinian and is Artinian for each i in the case where R₊ is principal. Moreover, for the case where , we prove that, for each i∈N₀, is Artinian if and only if is Artinian. We also prove that is Artinian, where and c is the cohomological dimension of M with respect to R₊. Finally we present some examples which show that and need not be Artinian.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Fonction asymptotique de Samuel des sections hyperplanes et multiplicité

Let (A,m_A,k) be a local noetherian ring and I an m_A-primary ideal. The asymptotic Samuel function (with respect to I) : A?R∪{+∞} is defined by , ∀x∈A. Similarly, one defines, for another ideal J, as the minimum of as x varies in J. Of special interest is the rational number . We study the behavior of the asymptotic Samuel function (with respect to I) when passing to hyperplane sections of A as one does for the theory of mixed multiplicities.     

A uniform Artin-Rees property for syzygies in rings of dimension one and two

Let be a local Noetherian ring, let M be a finitely generated R-module and let I⊂R be an -primary ideal. Let be a free resolution of M. In this paper we study the question whether there exists an integer h such that IⁿF_i∩ker(∂_i)⊂I^n−hker(∂_i) holds for all i. We give a positive answer for rings of dimension at most two. We relate this property to the existence of an integer s such that I^s annihilates the modules for all i>0 and all integers n.     

A numerical characterization of the S2-ification of a Rees algebra

Let A be a local ring with maximal ideal . For an arbitrary ideal I of A, we define the generalized Hilbert coefficients . When the ideal I is -primary, j_k(I)=(0,…,0,(−1)^ke_k(I)), where e_k(I) is the classical kth Hilbert coefficient of I. Using these coefficients we give a numerical characterization of the homogeneous components of the S₂-ification of S=A[It,t⁻¹], extending previous results obtained by the author to not necessarily -primary ideals.     

Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let and be two atomic nests on complex Hilbert spaces H and K, respectively. Denote the nest algebra associated with and the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from onto and a thorough classification of numerical radius isometries from onto .     

Hilbert coefficients and Buchsbaumness of associated graded rings

Let A be a Noetherian local ring with the maximal ideal and I an -primary ideal. The purpose of this paper is to generalize Northcott's inequality on Hilbert coefficients of I given in Northcott (J. London Math. Soc. 35 (1960) 209), without assuming that A is a Cohen-Macaulay ring. We will investigate when our inequality turns into an equality. It is related to the Buchsbaumness of the associated graded ring of I.     

The zero-divisor graph of a reduced ring

In this paper the zero-divisor graph Γ(R) of a commutative reduced ring R is studied. We associate the ring properties of R, the graph properties of Γ(R) and the topological properties of . Cycles in Γ(R) are investigated and an algebraic and a topological characterization is given for the graph Γ(R) to be triangulated or hypertriangulated. We show that the clique number of Γ(R), the cellularity of and the Goldie dimension of R coincide. We prove that when R has the annihilator condition and ; Γ(R) is complemented if and only if is compact. In a semiprimitive Gelfand ring, it turns out that the dominating number of Γ(R) is between the density and the weight of . We show that Γ(R) is not triangulated and the set of centers of Γ(R) is a dominating set if and only if the set of isolated points of is dense in .