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AN EXACT SEQUENCE INVOLVING THE PICARD AND BRAUER
The Functor PGLM
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abelian groups Alg(R algebra Amitsur cohomology automorphism Brauer group c.s. R-algebra category and F Clearly cocycle cofinal cohomology sets colimit commutative R-algebra commutative ring composition considered Corollary Cp(A D_(R defined Definition denote diagram is commutative direct sums directed category easily seen End Q endomorphism evaluation map F and G faithfully flat full subcategory functor F Galois cohomology group cohomology group functor group homomorphism group transformation Hence identity map inclusion map integer isomorphism classes Kernal Kf(R Lemma lim F lim G map f maximal ideal Mf(R module monic monoid Moreover morphism mult mult(u multiplicatively exact natural transformation need only show non-abelian notation object of Mp(R one-one orbit Pic(R pointed sets Proof Proposition quasi-abelian R-algebra map rank one projective S-isomorphism satisfied sequence of functors Simp simplicial summable category Suppose F switch map Theorem 3.1 torsion element unique map vertical maps X K(R