Restriction and Extension of Partial Actions

DOI10.1016/J.JPAA.2020.106391arXiv1909.12215MaRDI QIDQ6326057

Dirceu Bagio, H. Pinedo, Antonio Paques

Publication date: 26 September 2019

Abstract: Given a partial action

a l p h a = (A_{g}, a l p h a_{g})_{g i n m a t h c a l G}

of a connected groupoid

m a t h c a l G

on a ring

A

and an object

x

of

m a t h c a l G

, the isotropy group

m a t h c a l G (x)

acts partially on the ideal

A_{x}

of

A

by the restriction of

a l p h a

. In this paper we investigate the following reverse question: under what conditions a partial group action of

m a t h c a l G (x)

on an ideal of

A

can be extended to a partial groupoid action of

m a t h c a l G

on

A

? The globalization problem and some applications to the Morita and Galois theories are also considered, as extensions of similar results from the group actions case.

Mathematics Subject Classification ID

Actions of groups and semigroups; invariant theory (associative rings and algebras) (16W22) Sets with a single binary operation (groupoids) (20N02) Groupoids (i.e. small categories in which all morphisms are isomorphisms) (20L05) Groupoids, semigroupoids, semigroups, groups (viewed as categories) (18B40) Associative rings and algebras arising under various constructions (16S99)

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