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abelian affine algebraic K-theory Berlin bijective central extension Chern classes coefficients cohomology commutative diagram compute conductor congruence subgroup consider construction Corollary corresponding CW-complex defined definition denote element etale K-theory exact category exact sequence functor G-central given group G H-space handle additions hence homology homomorphism homotopy equivalence induces injective integers isomorphism K-groups kernel Lecture Notes Lemma Lie algebras linear group matrix maximal ideal Mennicke Milnor modules morphism Morse point MQ(A natural map natural transformation noetherian notation Notes in Math number field obtain pairing polynomial ring projective projective modules proof Proposition prove Quillen quotient R-module resp result satisfies semi-local ring Serre simplex simplicial complex simplicial set space spectral sequence Springer-Verlag St(R stability Steinberg group subbundle Suppose surjective Suslin symbols Theorem Topology trivial unimodular Vaserstein vector bundle X-ring y-pair zero