  We study a Hopf algebra $H$, which is finitely generated and projective over
a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions
in this setting, and provide a complete proof of Radford's formula for the
fourth power of the antipode, using Frobenius algebraic techniques. As further
applications, we extend Etingof and Gelaki's result that a separable and
coseparable Hopf algebra has antipode of order two, the result of Schneider
that Hopf subalgebras are twisted Frobenius extensions, and show that the
quantum double is always a Frobenius algebra.
