Residual morphism

Since the morphism Xo Spec(Fp) is of finite type, the completions of the local rings of Xo at its closed points are Noetherian complete local rings with perfect residue fields. Over these rings we can find a Barsotti-Tate group H with H pk] Go restricted to these rings (see [IL] Theorem 4.4). Hence it follows from Proposition 3.4 that Mx(Go) is locally free over Ox/pkOx of rank h — height(G) at all the closed points of X. ... 

A morphism f X — S can always be considered as a family of schemes if we view it as the totality of its fibres, if f is flat we call it a fiat family of schemes parametrized bv 5 X is called the total space of the family f If 5 is connected and s e 5 is a k-rational point, f is also called a flat family of deformations of the fibre X(s). When S=Spec(A) with A a local k-algebra with residue field k, the flat morphism f is called a local family of deformations of %(q), the ribre over the closed point 5 if moreover the local k-algebra A is artinian, then f is called an infinitesimal family of deformations, or an infinitesimal deformation, of X(o). 

Let Y and S be schemes, with Y algebraic. A flat family of closed subschemes of Y parametrized bv 5 is a closed subscheme X c YxS such that the morphism X —+ S induced by the projection YxS —> 5 is flat. The family is called trivial if X - XxS for some closed subscheme X c Y. When 5 = 5pec(A) with A a noetherlan local it-algebra with residue field k we will call X a local family of closed subschemes of Y. If A is an art ini an local k-algebra with residue field k we will call X an... 

Theorem 2. Let O be a complete (noetherian) local ring, with maximal ideal m, residue field k. Let f be a morphism of finite type ... 

For any point x of "X. there is an i and a point u ofXii over x such that the corresponding morphism of residue fields is an isomorphism which maps to x widi die same residue field ... 

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