On the depth 2 condition for group algebra and Hopf algebra extensions. (Q961030)

Let \(R\) be a commutative ring. The authors prove that if \(H\) is a subgroup of finite index of a group \(G\), then the ring extension \(RH\subseteq RG\) has left (or right) depth 2 if and only if \(H\) is normal in \(G\), and they also give several other equivalent conditions in terms of the restriction and the induction functors. Then the depth 2 condition is considered for Hopf algebra extensions. If \(i\colon K\to H\) is a morphism of \(R\)-Hopf algebras, such that \(H\) is a faithfully flat finitely generated projective left \(K\)-module, then the extension \(i\colon K\to H\) is left normal if and only if it has right depth 2, and this is also equivalent to the extension being \(\overline H\)-Galois for a certain Hopf algebra \(\overline H\). It is also presented a result involving the concept of relative projectivity, and similar in spirit to the result about the depth 2 condition for extensions of group algebras.