Preliminaries

DeMeyer, Frank; Ingraham, Edward

doi:10.1007/bfb0061227

Cited by 1 publication

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“…It is clear that is a finitely generated and projective -module. Using the characterization of Azumaya algebras (see for example [7, 16]), we need to check that is isomorphic as a -algebra to , which is proved as follows. Following the ideas of [9, Proposition 1.2], for any object in , we know that .…”

Section: Miyashita Action: the Azumaya Casementioning

confidence: 99%

Invertible Bimodules, Miyashita Action in Monoidal Categories and Azumaya Monoids

Ardizzoni¹,

Kaoutit²

2016

Nagoya Math. J.

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In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita's classical result and sheds light on other applications of geometric nature which can not be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

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