A fibre product construction is used to give a description of quantum line bundles over the generic Podlés spheres obtained by gluing two quantum discs along their boundaries. Representatives of the corresponding K0-classes are given in terms of 1-dimensional projections belonging to the C*-algebra, and in terms of analogues of the classical Bott projections. The K0-classes of quantum line bundles derived from the generic Hopf fibration of quantum SU(2) are determined and the index pairing is computed. It is argued that taking the projections obtained from the fibre product construction yields a significant simplification of earlier index computations.

We describe a construction of an object associated to the fundamental group of ℙ1 − {0, 1, ∞} in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products in the motivic cohomology of the ground field.

We discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

Let K be a field. We study a class of left C-comodules over a basic left Euler coalgebra C by means of the Euler ℤ-bilinear form associated to C, the Euler characteristic χC(M,N) of left C-comodules M,N, and the defect ∂C(M,N) ∊ ℤ associated to any computable Euler pair (M,N) of left C-comodules. We show that (lgthM, lgthN) = χC(M,N) + ∂C(M,N), for any computable Euler pair (M,N) of comodules over a left Euler coalgebra C. One of the main results of the paper asserts that the defect ∂C(M,N) is zero and (lgthM,lgthN) = χC(M,N), if the comodules M,N are finite-dimensional.

In his 1973 paper [4] Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of Nenashev's paper [3], we are able to give an algebraic proof of Quillen's Resolution Theorem for K1 of an exact category. We view this as an advance towards the goal of giving an essentially algebraic subject an algebraic foundation.

Given a group G and a ring R, Loday [Lod, 1976] described an assembly map αG : hn(BG;L(R)) → Kn(RG) where L(R) is a spectrum with nth space K0(SnR) × BGL(SnR)+ for n ≥ 0 and Kn(RG) = πn(BGL(RG)+ × K0(RG)). Hambleton and Pederson, [HP, 2004], indicate a proof that this map is isomorphic to the map on homotopy groups from the assembly map . We will complete the proof of this.

Using cyclotomic specializations of equivariant K-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if , where the ni's are powers of a fixed prime number p, then the rank of an exceptional object on X is congruent to ±1 modulo p.