Some exact sequences associated with adjunctions in bicategories. Applications

TL;DR

This paper extends classical algebraic results about Picard groups and cohomology to symmetric monoidal categories, using bicategory adjunctions to construct exact sequences and proving a version of Hilbert's theorem 90 for coalgebra coextensions.

Contribution

It introduces a new framework using bicategory adjunctions to generalize exact sequences and cohomological results in a categorical setting.

Findings

Classical Picard group isomorphism extended to symmetric monoidal categories.

Constructed exact sequences from bicategory adjunctions.

Proved a version of Hilbert's theorem 90 for cocommutative coalgebra coextensions.

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 stills valid in the realm of symmetric monoidal categories. To this end, we built 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 cocommutatvie coalgebra coextensions (=surjective homomorphisms).