1.4.
DIEUDONN´
E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 53
ker(p1) nilpotent. In other situations (such as gluing along closed subschemes,
which we need in the proof of Theorem 2.2.3) part (3) is used with surjective p2.
Proof. Part (1) is [40, Thm. 2.2(iv)] (upon noting that by the proof of [40,
Thm. 2.2(iii)], the kernel vanishes when M is R-flat). We prove part (2) by using
a limit argument suggested by D. Rydh. First assume that each Rj is finitely
generated over Rj. Consider the directed system {Rα} of finitely generated R-
subalgebras of R , and let Rj,α = ⊗R Rj, so lim

Rj,α = Rj for each j (since
R ⊗R Rj Rj is an isomorphism by part (1) applied to M = R ). Since each
Rj is finitely generated over Rj, it follows that for sufficiently large α0 the maps
Rj,α0 Rj are surjective for all j. In other words, for the cokernel R-module
M = coker(Rα0 R ), both M ⊗R R1 and M ⊗R R2 vanish. Hence, M = 0 by
[40, Thm. 2.2(ii)]. This says that the inclusion Rα0 R is an equality, so R is
finitely generated over R.
Now assume that each Rj is flat and finitely presented over Rj, so R is flat
and finitely generated over R. Form a presentation
0 I R[t1,...,tn] R 0.
By the R-flatness of R , I is R-flat and this sequence remains exact after applying
Rj ⊗R (·). Thus, the finite presentation of Rj over Rj implies that the ideal Ij =
Rj ⊗R I is finitely generated as a module over Pj = Rj[t1,...,tn] for each j. By part
(1) applied to the R-flat I, we have I = I1 ×I0 I2. Letting P = R[t1,...,tn], clearly
P = P1 ×P0 P2 for surjections P1,P2 P0, and Ij = Pj ⊗P I. Since I = I1 ×I0 I2,
a variation on the limit argument in the proof of (2) (now applied to modules over
R rather than algebras over R) shows that the P -module I = I1 ×I0 I2 is finitely
generated since each Ij is finitely generated over Pj.
To prove (3), first we prove the equivalences of categories. Assume p2 is sur-
jective. The key point in this case is that for closed immersions of schemes Z Y
and Z Y the associated pushout of ringed spaces Y
Z
Y (topological space
gluing and fiber product of structure sheaves) is again a scheme, identified in the
evident manner with the spectrum of a ring-theoretic fiber product when Y , Y ,
and Z are affine. The proof of this assertion is elementary and left to the reader;
it is made easier by first showing that the ringed space gluing has the expected
universal property among ringed spaces and is compatible with topological local-
ization, and then proving it is isomorphic in the expected way to the desired affine
scheme when Y , Y , and Z are affine. (See [21, §2] for further discussion of this
argument, and [40, §7], [62, Thm. 38], [81, §3], [101, Appendix A] for more general
existence results for pushouts.)
By part (1), the formation of the pushout Y
Z
Y is compatible with flat
base change over the pushout, so if X is a flat R-scheme then it is the pushout of
its closed subschemes XR1 and XR2 along their common closed subscheme XR0 =
(XR1 )R0 = (XR2 )R0 . The equivalence of categories in (3) therefore follows from
(1) and the existence of the general “gluing” of schemes along a closed subscheme
and its compatibility with Zariski-localization (so we may carry out computations
in the affine setting).
Suppose instead that the elements of ker(p1) are nilpotent. Now the relevant
pushout we must construct is relative to a closed immersion of schemes j : Z Y
that is topologically an equality (i.e., all sections of the defining quasi-coherent ideal
sheaf are locally nilpotent) and an affine map f : Z Y . For every affine open
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