Spectra of smash products
type: Item
instance of: scholarly article
title: Spectra of smash products
zbMATH Open document ID: 0726.16026
DOI: 10.1007/BF02764612
author: William Chin
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: ``Let \(T=R\#H\) be a smash product where H is a finite dimensional Hopf algebra. We show that the ideals of T invariant under the dual \(H^*\) of H are extended from H-invariant ideals of R. This allows us to transport the study of ideals in T to invariant ideals. When the Hopf algebra is pointed the relationship between an ideal and its invariant ideal is shown to be manageable. Restricting to prime ideals, this yields results on the prime spectra of R and T. We obtain Krull relations for \(R\subseteq T\) for some H, including Incomparability whenever H is commutative (or more generally when \(H^*\) is pointed after base extension). The results generalize and unify a number of results known in the context of group and restricted Lie actions.'' ``Unlike common approaches to ideals in smash products, the approach taken here yields a simultaneous description of the prime spectrum of R{\#}H in terms of the space of H-primes of R, not merely those primes of R{\#}H having trivial intersection with the coefficient ring. Furthermore our approach is less dependent on having a `nice' basis for H and does not require dealing with X-inner actions as for smash products of group algebras and enveloping algebras. However, we must assume that either the Hopf algebra H or its dual \(H^*\) are pointed (after a finite base field extension) in order to get the most useful results on primes. Because of this we are not able to obtain proofs for all smash products of nonabelian finite groups and noncommutative restricted enveloping algebras.''
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16S40
zbMATH DE Number: 4198245
zbMATH Keywords: smash product
zbMATH Keywords: finite dimensional Hopf algebra
zbMATH Keywords: H-invariant ideals
zbMATH Keywords: prime ideals
zbMATH Keywords: prime spectra
zbMATH Keywords: Incomparability
zbMATH Keywords: restricted enveloping algebras
reviewed by: D. S. Passman
MaRDI profile type: Publication
cites work: Q3933122
cites work: Crossed Products and Inner Actions of Hopf Algebras
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cites work: Hopf algebra actions
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cites work: Actions of divided power Hopf algebras
cites work: Prime ideals in crossed products of finite groups
cites work: Q4145843
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cites work: Q5590089
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description: scientific article; zbMATH DE number 4198245