The Picard sequence of a fibration

Groups with cofinite Zariski topology and potential density

Exact sequences in cohomology and an application

In this paper we focus on the algebraic geometry of the variety of \(\ell \)-groups (i.e. lattice ordered abelian groups). In particular we study the role of the introduction of constants in functional spaces and \(\ell \)-polynomial spaces, which are themselves \(\ell \)-groups, evaluated over other \(\ell \)-groups. We use different tools and techniques, with an increasing level of abstraction, to describe properties of \(\ell \)-groups, topological spaces (with the Zariski topology) and a formal logic, all linked by the underlying theme of solutions of \(\ell \)-equations.

In this chapter, we study the topology of C ∞-manifolds. We define the de Rham cohomology of a manifold, which is the vector space of closed differential forms modulo exact forms. After sheafifying the construction, we see that the de Rham complex forms a so-called acyclic resolution of the constant sheaf ℝ. We prove a general result that sheaf cohomology can be computed using such resolutions, and deduce a version of de Rham’s theorem that de Rham cohomology is sheaf cohomology with coefficients in ℝ. It follows that de Rham cohomology depends only on the underlying topology. Using a different acyclic resolution that is dual to the de Rham complex, we prove Poincaré duality. This duality makes cohomology, which is normally contravariant, into a covariant theory. We devote a section to explaining these somewhat mysterious covariant maps, called Gysin maps. We end this chapter with the remarkable Lefschetz trace formula, which in principle, calculates the number of fixed points for a map of a manifold to itself.

Invariants for abelian groups and dual exact sequences

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is\nthe first step of Voevodsky's proof of this conjecture for arbitrary prime l)\nin a rather clear and elementary way. Assuming this conjecture, we construct a\n6-term exact sequence of Galois cohomology with cyclotomic coefficients for any\nfinite extension of fields whose Galois group has an exact quadruple of\npermutational representations over it. Examples include cyclic groups, dihedral\ngroups, the biquadratic group Z/2\\times Z/2, and the symmetric group S_4.\nSeveral exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn\nare proven in this way. In addition, we introduce a more sophisticated version\nof the classical argument known as "Bass-Tate lemma". Some results about\nannihilator ideals in Milnor rings are deduced as corollaries.\n

Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces, T=(S^1)^n, with free equivariant cohomology over the cohomology of BT. Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.

Classical algebraic K-theory: The functors K0, K1, K2

We show that any topologicalpair with normally em- bedded subspace has the strong shape ofa pair,such that theinclu- sion map of the subspace into the total space is a cofibration. Furthermore we prove that a strong shape morphism of pairsis a strong shape equivalence if and only ifitoperates as strong shape equivalence of the total spaces and of the subspaces considered seperately. Subject classification: Shape, strong shape, normal embeddings,

In this paper author has obtained the neutrix product of x+ −r and δ (α) (x), where α is a positive fractional number. Key Words: Neutrix limit, fractional-differentiation, sequence, function. Ams Subject Classification- 46F 1. INTRODUCTION Neutrix N is defined by J.G. vander Corput [2] as a commutative additive group of functions ν(ξ) defined on a domain N' with values in additive group N", where further if for some ν in N, ν(ξ) = γ for all ξ in N', then γ= 0. The functions in N are called negligible functions. Now let N' be a set contained in a topological space with a limit point b which does not belong to N'. If f(ξ) is a function defined on N' with values in N" and it is possible to find a constant β such that f(ξ) –β is negligible in N, then β is called the neutrix limit or N-limit of f as ξ tends to b and we write N− lim ξ→b f(ξ) = β, where β must be unique, if it exists. Introducing the neutrix limit, Fisher [3,4] defined the neutrix product of two distributions as –

Groupes d'Automorphismes des Boucles de Moufang Commutatives

Given a pair of short exact sequences in an abelian category A, extending results due to Pressman [7]. We construct nine bivariant long exact sequences that involve the groups extn(δ,β), extn(δ,α) and extn(γ,α) that interlock with the univariant hom-ext sequences determined by (i). Two applications of these sequences in the category of abelian groups are given. Firstly we describe an isomorphism that yields a “constructive” way to compute the adjusted cotorsion part of a reduced cotorsion group and secondly, given a torsion-free, reduced group G with no free summands, we indicate how to determine the p-ranks of the divisible group Ext(G,Z).

Let F be a number field, O the ring of algebraic integers in F, and let θ denote the inclusion O ↪ F. The localization exact sequence in algebraic K-theory splits into short exact sequences $$ 0 \to K_n O\xrightarrow{{\theta _\# }}K_n F \to \oplus _m K_{n - 1} \left( {O/m} \right) \to 0 $$ for all positive integers n, where θ # is the homomorphism induced by θ in K-theory and where m runs over the set of maximal ideals of O (see Section 5 of [Q1], Theorem 8 of [Q2] and Theoreme 1 of [S2]); in particular, θ # is always injective. On the other hand, G. Banaszak investigated the subgroup of divisible elements in K n F and explained the important role of these elements in relation with the Lichtenbaum-Quillen conjecture and etale K-theory (see [B1], [B2], [BG], [BZ]). If n is odd, K n F is a finitely generated abelian group and has therefore no non-trivial divisible elements. If n is even, K n F is a large torsion group but all its divisible elements belong to the image of θ # because ⊕ m K n-1 (O/m) is a direct sum of finite cyclic groups and hence contains no non-trivial divisible elements.

Grothendieck groups of subcategories

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments. Mathematics Subject Classification (2000): 11E57 (primary); 22E40 (secondary). 1.1. Let us recall some notation and basic notions. Following [1] we will assume that n is even and n ≥ 4. The group of orientation preserving isometries of hyperbolic n-space is isomorphic to SO(1, n)o, the connected component of the identity of the special orthogonal group of signature (1, n), which can be identified with SO0(1, n), the subgroup of SO(1, n) preserving the upper half space. This group is not Zariski closed in SLn+1 thus in order to construct arithmetically defined subgroups of SO(1, n)o we consider arithmetic subgroups of the orthogonal group SO(1, n) or, more precisely, of groups G = SO( f ) where f is an admissible quadratic form defined over a totally real number field k (see [1, Section 2.1]). We have an exact sequence of k-isogenies: 1 → C → G φ → G → 1, (1.1) where G(k) Spin( f ) is the simply connected cover of G and C μ2 is the center of G. This induces an exact sequence in Galois cohomology (see [5, Section 2.2.3]) G(k) φ → G(k) δ → H1(k, C) → H1(k, G). (1.2) The main idea of this note is that by using (1.2) certain questions about arithmetic subgroups of G can be reduced to questions about the Galois cohomology group H1(k, C). A coherent collection of parahoric subgroups P = (Pv)v∈V f of G (V f = V f (k) denotes the set of finite places of the field k) defines a principal arithmetic subgroup Received October 16, 2006; accepted in revised form March 12, 2007.