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Duality for groupoid (co)actions. (English) Zbl 1322.16027

From the introduction: Partial ordered groupoid actions on rings were considered by D. Bagio and the authors [J. Algebra Appl. 9, No. 3, 501-517 (2010; Zbl 1227.16026)] as a generalization of partial group actions, as introduced by M. Dokuchaev and R. Exel [in Trans. Am. Math. Soc. 357, No. 5, 1931-1952 (2005; Zbl 1072.16025)]. And in [D. Bagio and A. Paques, Commun. Algebra 40, No. 10, 3658-3678 (2012; Zbl 1266.16045)] this notion was extended to the general context of groupoids.
Our purpose is to present a generalization of Cohen-Montgomery duality Theorem for group actions [M. Cohen and S. Montgomery, Trans. Am. Math. Soc. 282, 237-258 (1984; Zbl 0533.16001), Theorem 3.2] (resp., group coactions or, equivalently, group grading [ibid., Theorem 3.5]) to the setting of groupoid actions (resp., groupoid coactions or groupoid grading) (see Theorems 3.7 and 4.5).
This paper is organized as follows. In the next section we give preliminaries about groupoids, groupoid actions, skew groupoid rings, weak bialgebras, groupoid gradings, groupoid coactions and weak smash products, these later as introduced by S. Caenepeel and E. De Groot [in Contemp. Math. 267, 31-54 (2000; Zbl 0978.16033)]. In that section we will be concerned only with the results strictly necessary to construct the appropriate environment to prove our main theorems, whose proofs we set in the sections 3 (actions) and 4 (coactions).

MSC:

16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
16S40 Smash products of general Hopf actions
16W50 Graded rings and modules (associative rings and algebras)
16T05 Hopf algebras and their applications
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
16S35 Twisted and skew group rings, crossed products
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References:

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