This volume contains previously unpublished papers on algebraic $K$-theory written by Leningrad mathematicians over the last few years. The main topic of the first part is the computation of $K$-theory and $K$-cohomology for special varieties, such as group varieties and their principal homogeneous spaces, flag fiber bundles and their twisted forms, $\lambda$-operations in higher $K$-theory, and Chow groups of nonsingular quadrics. The second part deals with Milnor $K$-theory: Gersten's conjecture for $K^M_3$ of a discrete valuation ring, the absence of $p$-torsion in $K^M_*$ for fields of characteristic $p$, Milnor $K$-theory and class field theory for multidimensional local fields, and the triviality of higher Chern classes for the $K$-theory of global fields.
Part I: Computations in $K$-theory.: Chow groups of quadrics and the stabilization conjecture by N. A. Karpenko Simplicial definition of $\lambda$-operations in higher $K$-theory by A. Nenashev On algebraic $K$-theory of generalized flag fiber bundles and some of their twisted forms by I. A. Panin On algebraic $K$-theory of some principal homogeneous spaces by I. A. Panin $K$-theory and $\scr K$-cohomology of certain group varieties by A. A. Suslin $SK 1$ of division algebras and Galois cohomology by A. A. Suslin Part II: Milnor $K$-theory.: On class field theory of multidimensional local fields of positive characteristic by I. Fesenko On $p$-torsion in $K^M *$ for fields of characteristic $p$ by O. Izhboldin Triviality of the higher Chern classes in the $K$-theory of global fields by A. Musikhin and A. A. Suslin Milnor's $K 3$ of a discrete valuation ring by A. A. Suslin and V. A. Yarosh.