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 = Rα ⊗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