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In this paper we introduce the notion of strong Galois $H$-progenerator object for a finite cocommutative Hopf quasigroup $H$ in a symmetric monoidal category ${\sf C}$. We prove that the set of isomorphism classes of strong Galois $H$-progenerator objects is a subgroup of the group of strong Galois $H$-objects introduced in \cite{JKMS}. Moreover, we show that strong Galois $H$-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if $H$ is finite, we find exact sequences of Picard groups related with invertible left $H$-(quasi)modules and an isomorphism $Pic(_{{\sf H}}{\sf Mod})\cong Pic({\sf C})\oplus G(H^{\ast})$ where $Pic(_{{\sf H}}{\sf Mod})$ is the Picard group of the category of left $H$-modules, $Pic({\sf C})$ the Picard group of ${\sf C}$, and $G(H^{\ast})$ the group of group-like morphisms of the dual of $H$.
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참고문헌 (18)
참고문헌 신청
J.N. Alonso Álvarez / 2015 / Projections and Yetter–Drinfel'd modules over Hopf (co)quasigroups / Journal of Algebra 443:153~199
J. N. Alonso Alv´arez / 1996 / A Picard-Brauer five term exact sequence for braided categories, in Rings, Hopf algebras, and Brauer groups / Dekker :11~41
Jose N. Alonso ´Alvarez / 2017 / THE GROUP OF STRONG GALOIS OBJECTS ASSOCIATED TO A COCOMMUTATIVE HOPF QUASIGROUP / 대한수학회지 54(2):517~543
J. M. Barja P´erez / 1977 / Morita theorems for triples in closed categories
H. Bass / 1968 / Algebraic K-Theory / W. A. Benjamin, Inc
J. N. Alonso Alv´arez / 1996 / A Picard-Brauer five term exact sequence for braided categories, in Rings, Hopf algebras, and Brauer groups / Dekker :11~41
Jose N. Alonso ´Alvarez / 2017 / THE GROUP OF STRONG GALOIS OBJECTS ASSOCIATED TO A COCOMMUTATIVE HOPF QUASIGROUP / 대한수학회지 54(2):517~543
J. M. Barja P´erez / 1977 / Morita theorems for triples in closed categories
H. Bass / 1968 / Algebraic K-Theory / W. A. Benjamin, Inc
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