48 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
generic point lands outside Z0 and whose closed point lands inside Z0. Making a
base change by the resulting map Spec(R) S0 yields a pair of R-homomorphisms
(A0)R (B0)R (one coming from h0, and the other from f0) that agree on the
closed fibers but are distinct on generic fibers, contradicting faithfulness of passage
to the special fiber for abelian schemes over a local ring.
By stability under generization, the closed subscheme Z0 in H0 is topologically
open, so the open subscheme U H with the underlying space of Z0 is a union of
connected components of H. The structure morphism U S is a homeomorphism,
so it is of finite type, not just locally of finite type. The map H S satisfies the
valuative criterion for properness since an abelian scheme over a discrete valuation
ring is the eron model of its generic fiber [10, 1.2/8], so the open and closed U
in H also satisfies the valuative criterion over S. This proves that the finite type
map U S is proper, yet it has ´ etale fibers of degree 1, so it is a closed immersion
(defined by a nilpotent ideal). The closed subscheme U S represents the lifting
condition on f0.
1.4.4.6. Remark. A globalization of the formal smoothness of the infinitesimal
deformation theory of an abelian variety is Grothendieck’s result that if R is a
ring containing an ideal J satisfying
J2
= 0 then every abelian scheme A0 over
R0 := R/J lifts to an abelian scheme over R. We sketch the proof, building on
the key case of an artinian local base that follows from the formal smoothness of
the infinitesimal deformation theory (and is a key step in the proof of the formal
smoothness in [88, Thm. 2.2.1]).
By direct limit arguments we may and do assume R is noetherian. The ob-
struction to lifting A0 to a smooth proper R-scheme A is a certain class ξ
H2(A0,
(ΩA0/R0
1 )∨
⊗R0 J). The formation of ξ is compatible with base change on R
(relative to base change morphisms for the cohomology of quasi-coherent sheaves),
so by Zariski localization and completion we see that the vanishing of ξ is reduced
to the case when R is a complete local noetherian ring. By the Theorem on Formal
Functions [34, III1, 4.2.1], the vanishing of ξ is reduced to the settled case when R
is an artinian local ring.
Now return to a general noetherian R, and fix a smooth proper R-scheme A
lifting A0. By smoothness we may choose a lift e A(R) of the identity section
e0 A0(R0). We claim that the subtraction morphism μ0 : A0 × A0 A0 defined
by (x, y) x y uniquely lifts to an R-morphism μ : A × A A carrying (e, e)
to e. Once such a μ exists, it is the subtraction for a unique group law due to
rigidity arguments explained in [83, Ch. 6, §3]. In particular, over an arbitrary
ring R (without noetherian hypotheses) μ is unique if it exists. Hence, by Zariski
localization it suffices to prove the existence of μ when R is a local noetherian ring,
and by fpqc descent with respect to R R we may assume R is complete. Formal
GAGA for morphisms [83, III1, 5.4.1] then reduces the existence problem to the
case of artinian local R, so by length induction we may assume J is killed by the
maximal ideal of R. This case is settled in [83, Ch. 6, §3, Prop. 6.15].
The deformation theory of p-divisible groups ends up with results similar to
the case of abelian varieties but proceeds by another path. To describe this, let κ
be a field of characteristic p 0, Λ a complete local noetherian ring with residue
field κ, and X0 a p-divisible group of height h 0 and dimension d 0 over κ (so
Previous Page Next Page