  In this paper we develop the theory of essential dimension of group schemes
over an integral base. Shortly we concentrate over a local base. As a
consequence of our theory we give a result of invariance of the essential
dimension over a field. The case of group schemes over a discrete valuation
ring is discussed. Moreover we propose a generalization of Ledet conjecture,
which predicts the essential dimension of cyclic $p$-groups in positive
characteristic, for finite commutative unipotent group schemes. And we show
some results and some consequences of this new conjecture.
