##
**Brauer and Galois groups of a Hopf algebra in a closed category.
(Grupos de Brauer y de Galois de un algebra de Hopf en una categoría cerrada.)**
*(Spanish)*
Zbl 0579.16004

Alxebra, 42. Santiago de Compostela: Universidad de Santiago de Compostela, Departamento de Algebra y Fundamentos. xxi, 305 p. (1985).

A Brauer group \(BM(R,H)\) whose elements are equivalence classes of R-Azumaya \(H\)-module algebras, for a cocommutative Hopf algebra \(H\) over a commutative ring \(R\) with unit, was introduced by F. W. Long [J. Algebra 30, 559–601 (1974; Zbl 0282.16007)]. Later, M. Beattie [J. Algebra 43, 686–693 (1976; Zbl 0342.16010)] showed that if the Hopf algebra \(H\) is also finitely generated projective and commutative, then there is a split exact sequence
\[
1\longrightarrow B(R)\longrightarrow BM(R,H)\longrightarrow\mathrm{Gal}(R,H)\longrightarrow 1,
\]
where \(\mathrm{Gal}(R,H)\) is the group of Galois \(H\)-objects over \(R\) defined by S. U. Chase and M. E. Sweedler [Hopf algebras and Galois theory. Berlin etc.: Springer (1969; Zbl 0197.01403)] and \(B(R)\) is the Brauer group of \(R\).

In the memoir under review, the author studies Brauer and Galois groups in the context of a symmetric closed category \({\mathcal C}\) with equalizers and coequalizers, using the language of 2-categories. In the two first chapters, Brauer groups for 1 and 2-Azumaya triples are defined following an idea of B. Pareigis [Lect. Notes Math. 549, 112–133 (1976; Zbl 0362.18011)]. These two groups coincide if the basic object of \({\mathcal C}\) is projective. Given a cocommutative (commutative) Hopf algebra \(\mathbb H\), the author defines the notions of Brauer groups of \(\mathbb H\)- module (comodule) 1 and 2-Azumaya triples. With this definition, the Brauer groups of a closed category are recovered, in both cases, if the Hopf algebra is trivial. In chapter 3, the author gives, by means of Morita theory, a direct description of the Brauer group of \(\mathbb H\)-module Azumaya triples. These groups are interpreted as the Brauer groups of a suitable closed category. These results indicate that closed categories yield a natural framework to develop the theory of these groups.

The rest of the memoir is dedicated to the study, at the same level of generality, of the Galois groups of (not necessarily commutative) \(\mathbb H\)-objects for a commutative and cocommutative finite Hopf algebra \(\mathbb H\). These groups are related to the above mentioned Brauer groups by means of a split exact sequence of abelian groups which generalizes the one given by Beattie.

In the memoir under review, the author studies Brauer and Galois groups in the context of a symmetric closed category \({\mathcal C}\) with equalizers and coequalizers, using the language of 2-categories. In the two first chapters, Brauer groups for 1 and 2-Azumaya triples are defined following an idea of B. Pareigis [Lect. Notes Math. 549, 112–133 (1976; Zbl 0362.18011)]. These two groups coincide if the basic object of \({\mathcal C}\) is projective. Given a cocommutative (commutative) Hopf algebra \(\mathbb H\), the author defines the notions of Brauer groups of \(\mathbb H\)- module (comodule) 1 and 2-Azumaya triples. With this definition, the Brauer groups of a closed category are recovered, in both cases, if the Hopf algebra is trivial. In chapter 3, the author gives, by means of Morita theory, a direct description of the Brauer group of \(\mathbb H\)-module Azumaya triples. These groups are interpreted as the Brauer groups of a suitable closed category. These results indicate that closed categories yield a natural framework to develop the theory of these groups.

The rest of the memoir is dedicated to the study, at the same level of generality, of the Galois groups of (not necessarily commutative) \(\mathbb H\)-objects for a commutative and cocommutative finite Hopf algebra \(\mathbb H\). These groups are related to the above mentioned Brauer groups by means of a split exact sequence of abelian groups which generalizes the one given by Beattie.

### MSC:

16T05 | Hopf algebras and their applications |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

14F22 | Brauer groups of schemes |

18D05 | Double categories, \(2\)-categories, bicategories and generalizations (MSC2010) |

16W20 | Automorphisms and endomorphisms |

16B50 | Category-theoretic methods and results in associative algebras (except as in 16D90) |