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A-modules abelian algebraic K-theory assume bijective canonical Chern classes classifying space coefficient cohomology commutative diagram complex composition conductor congruence subgroup consider construction Corollary corresponding CW-complex defined definition denote Edited elements equivariant etale K-theory exact sequence extension fibration finite group functor G-map given group G H-space hence homology homomorphism homotopy equivalence induced injective integer invariant inverse irreducible isomorphism K-groups kernel Lecture Notes Lemma Let F Let G linear group matrix Milnor modules morphism natural map natural transformation notation Notes in Math obtain orthogonal pair polynomial prime projective projective modules Proof Proposition prove Quillen relations representations resp result ring satisfies semi-local semi-local ring simplicial set space spectral sequence Springer-Verlag St(R stability Steinberg group suffices Suppose surjective symbol Theorem theory topological trivial unimodular universal Vaserstein vector zero