Let [Oscr ] be a complete discrete valuation ring with unique maximal ideal
J([Oscr ]), let K be its quotient field
of characteristic 0, and let k be its residue field [Oscr ]/J([Oscr ])
of prime characteristic p. We fix a finite group G,
and we assume that K is big enough for G, that is, K
contains all the [mid ]G[mid ]-th roots of unity, where [mid ]G[mid ] is the
order of G. In particular, K and k are both
splitting fields for all subgroups of G. Suppose that H is an
arbitrary subgroup of G. Consider blocks (block ideals) A and B
of the group algebras RG and RH,
respectively, where R∈{[Oscr ], k}. We consider the
following question: when are A and B Morita equivalent?
Actually, we deal with ‘naturally Morita equivalent blocks A and
B’, which means that A is isomorphic to
a full matrix algebra of B, as studied by B. Külshammer.
However, Külshammer assumes that H is normal
in G, and we do not make this assumption, so we get generalisations of the results of Külshammer.
Moreover, in the case H is normal in G, we get the same results as Külshammer; however, he uses the results
of E. C. Dade, and we do not.