  We study divided power structures on finitely generated $k$-algebras, where
$k$ is a field of positive characteristic $p$. As an application we show
examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed
noetherian local ring of non-equal characteristic. We also show that Frobenius
neighbourhoods of a singular point of a general hypersurface of large dimension
have no liftings to mildly ramified rings of non-equal characteristic.
