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Transactions of the American Mathematical Society

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Some exact sequences associated with adjunctions in bicategories. Applications


Authors: J. Gómez-Torrecillas and B. Mesablishvili
Journal: Trans. Amer. Math. Soc. 371 (2019), 8255-8295
MSC (2010): Primary 16D20, 16T15, 18D05, 18D10, 18G30
DOI: https://doi.org/10.1090/tran/7625
Published electronically: February 14, 2019
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Abstract: We prove that the classical result asserting that the relative Picard group of a faithfully flat extension of commutative rings is isomorphic to the first Amitsur cohomology group still is valid in the realm of symmetric monoidal categories. To this end, we build some group exact sequences from an adjunction in a bicategory, which are of independent interest. As a particular byproduct of the evolving theory, we prove a version of Hilbert's theorem 90 for cocommutative coalgebra coextensions (=surjective homomorphisms).


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Additional Information

J. Gómez-Torrecillas
Affiliation: Departament of Algebra and CITIC, Universidad de Granada, E18071 Granada, Spain
Email: gomezj@ugr.es

B. Mesablishvili
Affiliation: I. Javakhishvili Tbilisi State University, Faculty of Exact and Natural Sciences, Department of Mathematics, 2 University St., 0186, Tbilisi, Georgia
Email: bachi@rmi.ge

DOI: https://doi.org/10.1090/tran/7625
Keywords: Bicategories, adjunctions, (firm) bimodules, coalgebras, Amitsur cohomology, descent data, Hilbert's theorem 90
Received by editor(s): October 4, 2016
Received by editor(s) in revised form: March 23, 2017
Published electronically: February 14, 2019
Additional Notes: This research partially supported by grant MTM2013-41992-P from the Ministerio de Economía y Competitividad of the Spanish Government and from FEDER
The second named author was supported by Shota Rustaveli National Science Foundation Grants DI/18/5- 113/13 and FR/189/5- 113/14
Article copyright: © Copyright 2019 American Mathematical Society