On the finite generation of the cohomology of bosonizations

Abstract

We use deformation sequences of (Hopf) algebras, extendingthe results of Negron and Pevtsova, to show that bosonizationsof some suitable braided Hopf algebras by some suitable finitedimensionalHopf algebras have finitely generated cohomology.In fact, our results are shown in more generality for smashproducts. As applications, we prove the bosonizations ofsome Nichols algebras (such as Nichols algebras of diagonaltype, the restricted Jordan plane, Nichols algebras of directsums of Jordan blocks plus points labeled with 1), bysome suitable finite-dimensional Hopf algebras, have finitely generated cohomology, recovering some known results as wellas providing new examples.