Partial generalized crossed products and a seven-term exact sequence

Dokuchaev, Mikhailo; Paques, Antonio; Pinedo, Héctor; Rocha, Itailma

doi:10.48550/arxiv.1908.05820

Cited by 3 publications

(2 citation statements)

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“…This fits nicely the concept of a twisted partial group action in Exel's crossed products and the notion of the equivalence of twisted partial actions from [11]. The partial group cohomology of [13] found applications to partial projective representations [13], [22], to the construction of a Chase-Harrison-Rosenberg type seven terms exact sequence [20], [21], associated to a partial Galois extension of commutative rings [12], and to the study of ideals of (global) reduced C * -crossed products in [27]. It also inspired the treatment of partial cohomology from the point of view of Hopf algebras [4].…”

Section: Introductionsupporting

confidence: 57%

Globalization of group cohomology in the sense of Alvares-Alves-Redondo

2020

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Recently E. R. Alvares, M. M. Alves and M. J. Redondo introduced a cohomology for a group G with values in a module over the partial group algebra Kpar(G), which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action α of G on a (unital) algebra A we consider A as a Kpar(G)-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in A. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that A is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group H n par (G, A) is isomorphic to the usual cohomology group H n (G, M(B)), where M(B) is the multiplier algebra of B and B is the algebra under the enveloping action of α.

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Section: Introductionsupporting

confidence: 57%

Globalization of group cohomology in the sense of Alvares-Alves-Redondo

2020

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“…The cohomology from [6] is strongly related to the cohomology of inverse semigroups. It found applications to partial projective group representations in [6,12], to a generalization of the Chase-Harrison-Rosenberg seven term exact sequence for partial Galois extensions in [10,11], and to the study of ideals in reduced crossed products of C * -algebras by global actions in [14]. It also influenced a Hopf theoretic treatment of partial cohomology in [1] and a study of its affinity with extensions in [7,8].…”

Section: Introductionmentioning

confidence: 99%