On the coradical filtration of pointed coalgebras
The author studies some aspects of the coradical filtration of pointed Hopf algebras. In particular, he proves a useful stronger version of the Taft-Wilson theorem. He then considers field extensions. He proves: If \(C\) is a cocommutative \(K\)-coalgebra and \(K\subseteq L\) is a field extension, then \(E\otimes C\) is pointed if and only if, for each simple subcoalgebra \(D\subseteq C\), \(E\) contains the normal closure of the field \(D^*\) over \(K\). He considers moreover the coradical filtration of duals of inseparable field extensions. The problem is reduced to the case that the extension \(K\subseteq L\) is purely inseparable and \(E\) is a field containing the normal closure of \(L\) over \(K\). Then it is proved that \(E\otimes L^*=(E\otimes L^*)_1\), if and only if \(L/K\) is separable of \(\text{char}(K)=2=[L:L_s]\), where \(L_s\) is the separable closure of \(K\) in \(L\).
scientific article; zbMATH DE number 1867994