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
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
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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