 
Summary: A NOTE ON STRONGLY SEPARABLE ALGEBRAS
MARCELO AGUIAR
Abstract. Let A be an algebra over a eld k. If M is an A{bimodule, we let
M A and MA denote respectively the k{spaces of invariants and coinvariants of
M , and 'M : M A ! MA be the natural map. In this note we characterize those
algebras A for which 'M is a natural isomorphism as those separable algebras
for which the separability idempotent can be chosen to be symmetric, or as those
nite dimensional algebras for which the trace form is nondegenerate. These
algebras are called strongly separable. We also prove that the cotensor product of
C{bicomodules has a right adjoint if and only if the coalgebra C is cosemisimple,
and show that if C is strongly separable then the adjoint is given by maps of
C{bicomodules.
1. Introduction
Let k be a eld, G a group such that jGj 6= 0 in k and M a G{space. Then the
natural map between the spaces of G{invariants and G{coinvariants
M G = fm 2 M = gm = m 8 g 2 Gg ,! M !! MG = M=hgm m = g 2 G; m 2 Mi
is an isomorphism, since the map
MG !M G ;
m 7! 1
jGj
