For certain product varieties, Murre's conjecture on Chow groups is investigated. More precisely, let k be an algebraically closed field, X be a smooth projective variety over k and C be a smooth projective irreducible curve over k with function field K. Then we prove that if X (resp. XK) satisfies Murre's conjectures (A) and (B) for a set of Chow-Künneth projectors {, 0 ≤ i ≤ 2dim X} of X (resp. for {()K} of XK) and if for any j, , then the product variety X × C also satisfies Murre's conjectures (A) and (B). As consequences, it is proved that if C is a curve and X is an elliptic modular threefold over k (an algebraically closed field of characteristic 0) or an abelian variety of dimension 3, then Murre's conjecture (B) is true for the fourfold X × C.

We introduce a refinement of the Bloch-Wigner complex of a field F. This refinement is complex of modules over the multiplicative group of the field. Instead of computing K2(F) and Kind3(F) - as the classical Bloch-Wigner complex does - it calculates the second and third integral homology of SL2(F). On passing to F× -coinvariants we recover the classical Bloch-Wigner complex. We include the case of finite fields throughout the article.

We summarise the construction of geometric cycles and their use in describing the Kasparov K-homology of a CW-complex X. When Kasparov K-homology is twisted by a degree three element of the Čech cohomology of X then there is a corresponding construction of twisted geometric cycles for the case where X is a smooth manifold however the method that was employed does not apply in the case of CW-complexes. In this article we propose a new approach to the construction of twisted geometric cycles for CW-complexes motivated by the study of D-branes in string theory.

Let be a generalized based category (see definition 1.2). In this paper, we construct a cohomology theory in the category of contravariant functors: where R is a commutative ring with identity, which generalizes Bredon cohomology involving finite, profinite or discrete groups.

We also study higher K-theory of the category of finitely generated projective objects in and the category of finitely generated objects in and obtain some finiteness and other results.

Given a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i (){p}.

Exact sequences in algebraic K-theory contain a lot of information. Here it is shown that by using K-theory exact sequences one can easily derive Bass’ description [1] of the SK1 of an ideal in a Dedekind domain in terms of relative reciprocities.

Assuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number field F with the orders of the tame kernels of some of its subfields. The details are given for fields F which are Galois over ℚ with Galois group the group ℤ/2 × ℤ/2, the dihedral group D2p; p an odd prime, or the alternating group A4. We include numerical results illustrating these formulas.

For an odd prime p we prove a Riemann-Hurwitz type formula for odd eigenspaces of the standard Iwasawa modules over F(μp∞), the field obtained from a totally real number field F by adjoining all p-power roots of unity. We use a new approach based on the relationship between eigenspaces and étale cohomology groups over the cyclotomic ℤp-extension F∞ of F. The systematic use of étale cohomology greatly simplifies the proof and allows to generalize the classical result about the minus-eigenspace to all odd eigenspaces.

Let X be a smooth, proper and geometrically irreducible curve X defined over a number field F and let χ be a regular and proper model of X over OF,Sl. In this paper we address the problem of detecting the linear dependence over ℤl of elements in the étale K-theory of χ. To be more specific, let P ∊ Ket2n(χ) and let ⋀̂ ⊂ Ket2n(χ) be a ℤl-submodule. Let rυ: Ket2n(χ) → Ket2n(χυ) be the reduction map for υ ∉ Sl. We prove, under some conditions on X, that if rυ() ∈ rυ (⋀̂) for almost all υ of then ∈ ⋀̂ + Ket2n(χ)tor.

As an application of our papers in hermitian K-theory, in favourable cases we prove the periodicity of hermitian K-groups with a shorter period than previously obtained. We also compute the homology and cohomology with field coeffcients of infinite orthogonal and symplectic groups of specific rings of integers in a number field.