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 N´ 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