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2-cocycle abelian groups admissible B-comodule algebra algebra with structure Aut(A Azumaya algebra Azumaya F-comodule algebra Azumaya G-module algebra Azumaya H-comodule bijective bimultiplicative morphism Brauer group canonical homomorphism Cartier dual central extension central simple coalgebra cocommutative k-Hopf algebra commutative and cocommutative commutative diagram commutative k-group sheaves comodule structure defined e(Bc End(M End(V epimorphism Extcent(G,D Extcent(Sp extension of G F 8 H F 8 H-comodule F x Sp F-H-Azumaya algebras faithfully flat finite dimensional following commutative diagram G x G group of units group sheaf extensions grT(T H-comodule algebra Hence homomorphism of extensions homomorphism of group Hopf pairing induces a homomorphism invertible isomorphism classes k-algebra k-coalgebra k-finite projective k-functor k-group functor LEMMA Let G module monoid nilpotent notation phism Proof PROPOSITION prosmooth proves rad(A Skolem-Noether theorem smash product Sp H Sp H,u Sp*F Sp*H split exact sequence structure map surjective THEOREM torsor x e X(T y e Y(T