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1-cocycle a e B(t acyclic Alan McConnell algebra map algebra with center Amitsur cohomology Amitsur complex B/radB Brauer group C-linear function C-linear Lie algebra C-space C+(U Cauchy sequence central simple algebra Chapter cl(t coboundary cocycle of H2(K/C cohomology group commutative A-algebra commutative ring complete noetherian defined Der(A derivation direct limit direct sum elements epimorphism exact sequence exists a C-linear finite separable forgetful functor free A-module free-finite functor given k>0 group cohomology hence Hochschild hypothesis isomorphism kernel l+U1 Lie algebra cohomology Lie algebra extension Lie algebra homomorphism limit is taken maximal commutative subfield maximal ideal monic monomorphism nilpotent noetherian local ring one-one p-semilinear function p-semilinear map Prop Proposition Bl radB regular restricted Lie residue class field restricted Lie algebra result ring with maximal S(tt separable extension split unit map suffices to show Theorem B3 vector space zero