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.
@article{urn:eudml:doc:41519,
title = {Galois H-objects with a normal basis in closed categories. A cohomological interpretation.},
journal = {Publicacions Matem\`atiques},
volume = {37},
year = {1993},
pages = {271-284},
mrnumber = {MR1249231},
zbl = {0808.16041},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41519}
}
Alonso Alvarez, José N.; Fernández Vilaboa, José Manuel. Galois H-objects with a normal basis in closed categories. A cohomological interpretation.. Publicacions Matemàtiques, Tome 37 (1993) pp. 271-284. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41519/