Exact sequences for the Galois group. (English) Zbl 0884.18010

Fernández Vilaboa, J. M.; Villanueva Novoa, E.; González Rodríguez, R.

Commun. Algebra 24, No. 11, 3413-3435 (1996).

Let \({\mathcal C}\) be a symmetric closed category with equalizers and co-equalizers and let \({\mathbf H}\) be a finite commutative and co-commutative Hopf algebra in \({\mathcal C}\). To these data, one may associate a Picard group \(\text{Pic} ({\mathcal C}, {\mathbf H})\) and a Brauer group \(BM({\mathcal C}, {\mathbf H})\) of left \({\mathbf H}\)-modules, as well as a Galois group \(\text{Gal}_{\mathcal C} ({\mathbf H})\).
Let \(F:{\mathcal C} \to {\mathcal C}'\) be a monoidal functor between categories of the above type. Then the authors construct, through \(K\)-theoretic methods, exact sequences involving the above groups and generalizing similar ones due to Caenepeel, Beattie and the authors.

Reviewer: A.Verschoren (Antwerp)

MSC:

18D15	Closed categories (closed monoidal and Cartesian closed categories, etc.)
19D23	Symmetric monoidal categories
16W30	Hopf algebras (associative rings and algebras) (MSC2000)
18D10	Monoidal, symmetric monoidal and braided categories (MSC2010)

Keywords:

symmetric closed category; Hopf algebra; Picard group; Brauer group; Galois group

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