  If a nontrivial finite group coacts on a graded noetherian down-up algebra
$A$ inner faithfully and homogeneously, then the fixed subring is not
isomorphic to $A$. Therefore graded noetherian down-up algebras are rigid with
respect to finite group coactions, in the sense of Alev-Polo. An example is
given to show that this rigidity under group coactions does not have all the
same consequences as the rigidity under group actions.
