Spectra of smash products
``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.''
scientific article; zbMATH DE number 4198245
scientific article