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One key aim of the author [Realization of Voevodsky's motives, J. Algebraic Geom. 9 (2000), no. 4, 755-799] was to construct a realization functor from Voevodsky's triangulated category of geometrical motives to her own triangulated category of mixed realizations. This note corrects a mistake in this construction. The new argument consists of a rearrangement of the original construction together with a careful analysis of hypercovers of complexes of varieties.

Using the 'slice filtration', defined by effectivity conditions on Voevodsky's triangulated motives, we define spectral sequences converging to their motivic cohomology andétale motivic cohomology. These spectral sequences are particularly interesting in the case of mixed Tate motives as their E 2 -terms then have a simple description. In particular this yields spectral sequences converging to the motivic cohomology of a split connected reductive group. We also describe in detail the multiplicative structure of the motive of a split torus.

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The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the "special values" of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters.We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.