# Galois H-objects with a normal basis in closed categories. A cohomological interpretation.

José N. Alonso Alvarez; José Manuel Fernández Vilaboa

Publicacions Matemàtiques (1993)

- Volume: 37, Issue: 2, page 271-284
- ISSN: 0214-1493

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topAlonso Alvarez, José N., and Fernández Vilaboa, José Manuel. "Galois H-objects with a normal basis in closed categories. A cohomological interpretation.." Publicacions Matemàtiques 37.2 (1993): 271-284. <http://eudml.org/doc/41519>.

@article{AlonsoAlvarez1993,

abstract = {In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:BMinn(C,H) ≅ B(C) ⊕ H2(H,K)In particular, if C is the symmetric closed category of C-modules with K a field, H2(H,K) is the second cohomology group introduced by Sweedler in [21]. Moreover, if H is a finitely generated projective, commutative and cocommutative Hopf algebra over a commutative ring with unit K, then the above decomposition theorem is the one obtained by Beattie [5] for the Brauer group of H-module algebras.},

author = {Alonso Alvarez, José N., Fernández Vilaboa, José Manuel},

journal = {Publicacions Matemàtiques},

keywords = {Algebra de Hopf; Grupo de Galois; Grupo de Brauer; Categorías cerradas; Cohomología; symmetric monoidal category; bifunctor; equalizers; co-equalizers; Azumaya monoid; finite Hopf algebra; -module monoid; -comodule monoid; Galois -object; category of modules; cocommutative Hopf algebra; split exact sequence; Brauer group; Azumaya -algebras; symmetric closed categories; Sweedler cohomology group},

language = {eng},

number = {2},

pages = {271-284},

title = {Galois H-objects with a normal basis in closed categories. A cohomological interpretation.},

url = {http://eudml.org/doc/41519},

volume = {37},

year = {1993},

}

TY - JOUR

AU - Alonso Alvarez, José N.

AU - Fernández Vilaboa, José Manuel

TI - Galois H-objects with a normal basis in closed categories. A cohomological interpretation.

JO - Publicacions Matemàtiques

PY - 1993

VL - 37

IS - 2

SP - 271

EP - 284

AB - In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:BMinn(C,H) ≅ B(C) ⊕ H2(H,K)In particular, if C is the symmetric closed category of C-modules with K a field, H2(H,K) is the second cohomology group introduced by Sweedler in [21]. Moreover, if H is a finitely generated projective, commutative and cocommutative Hopf algebra over a commutative ring with unit K, then the above decomposition theorem is the one obtained by Beattie [5] for the Brauer group of H-module algebras.

LA - eng

KW - Algebra de Hopf; Grupo de Galois; Grupo de Brauer; Categorías cerradas; Cohomología; symmetric monoidal category; bifunctor; equalizers; co-equalizers; Azumaya monoid; finite Hopf algebra; -module monoid; -comodule monoid; Galois -object; category of modules; cocommutative Hopf algebra; split exact sequence; Brauer group; Azumaya -algebras; symmetric closed categories; Sweedler cohomology group

UR - http://eudml.org/doc/41519

ER -

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