Homogeneous SK1 of simple graded algebras

For a simple graded algebra S = Mn(E) over a graded division algebra E, a short exact sequence is established relating the reduced Whitehead group of the homogeneous part of S to that of E. In particular it is shown that the homogeneous SK1 is not in general Morita invariant. Along the way we prove the existence and multiplicativity of a Dieudonné determinant for homogeneous elements of S.