Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups
The paper is concerned with the study of cohomology rings of finite-dimensional pointed Hopf algebras over an algebraically closed field \(k\) of characteristic zero. In particular, the authors are interested in answering the question when such Hopf algebras have finitely generated cohomology. The latter means that the cohomology ring \(H(H,k)\) is finitely generated and moreover, the ring \(H(H,M)\) is finitely generated \(H(H,k)\)-module for any \(H\)-module \(M\). The main results of the paper are summarized by the authors in the following theorems: \begin{itemize} \item[(a)] (Theorem 1.2.1) Let \(H\) be a finite-dimensional pointed Hopf algebra whose group of group-like elements is abelian. Then \(H\) has finitely generated cohomology. \item[(b)] (Theorem 1.2.2) Let \(H\) be a finite-dimensional basic Hopf algebra whose group of characters is abelian. Then \(H\) has finitely generated cohomology. \item[(c)] (Theorem 1.2.3) Let \(H\) be a finite-dimensional Hopf algebra that fits into an extension \(k\rightarrow K \rightarrow H \rightarrow L\rightarrow k\), where \(K\) is semisimple and \(L\) is either pointed with abelian group of group-like elements or else basic with abelian group of characters. Then \(H\) has fgc (finitely generated cohomology). \end{itemize}
scientific article; zbMATH DE number 7463879
scientific article