54 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

U ⊂ Y , the preimage aﬃne open f

−1

(U ) in Z underlies an open subscheme of Y

that is also aﬃne (since aﬃneness is determined by the underlying reduced scheme

[24, A.2]). We define the pushout ringed space Y

Z

Y to have the same topolog-

ical space as Y and structure sheaf f∗(OY ) ×f∗(OZ

)

OY (via the identification of

OY as a sheaf on the topological space of Z). By working Zariski-locally on Y this

is easily checked to be a scheme, coinciding in the evident manner with the spec-

trum of the expected fiber product ring when Y , Y , and Z are aﬃne. Thus, the

equivalence of categories in (3) follows via (1) as in the case when p2 is surjective.

Finally, we analyze the behavior of various properties P of R-scheme morphisms

h : X → Y between two flat R-schemes of finite presentation when either p2

is surjective or ker(p1) consists of nilpotent elements. Let Yj = YRj and Xj =

XRj . Note that if ker(p1) consists of nilpotent elements then the map Spec(R) →

Spec(R2) is a closed immersion defined by an ideal of nilpotent elements, so the

same holds for X → X2 and Y → Y2. The two cases (surjective p2, or ker(p1)

consisting of nilpotent elements) now will be treated simultaneously.

Clearly the map induced by h between fibers over a point in Spec(R) is identified

with the analogous fiber map for the one of the pullbacks hj : Xj → Yj over Rj

with j ∈ {1, 2}. Hence, since X and Y are R-flat of finite presentation, the fibral

flatness criterion [34, IV3, 11.3.11] implies that h is flat if and only if h1 and h2 are

flat. By the fibral smoothness, ´ etaleness, and isomorphism criteria [34, IV4, 17.8.2,

17.9.5], the cases when P is “smooth”, “´ etale”, or “isomorphism” are also settled.

The cases of geometric fibers being connected or of pure dimension d are obvious.

Since h is finite if and only if it is quasi-finite and proper [34, IV3, 8.11.1],

and the finite type h is quasi-finite if and only if h1 and h2 are quasi-finite, the

case when P is “finite” is reduced the case when P is “proper”. When h1 and h2

are universally closed it is clear that h is universally closed, so it remains to treat

the case of separatedness; i.e., closedness of the diagonal ΔX/Y . Topologically

this map is visibly a gluing of ΔX1/Y1 and ΔX2/Y2 when p2 is surjective, and it

coincides topologically with ΔX2/Y2 when ker(p1) consists of nilpotent elements.

Thus, in both cases separatedness of h1 and h2 implies separatedness of h.

We have established enough compatibility for F = DefΛ(X0) with respect to

fiber products in CΛ to equip the tangent space tX0 with a natural κ-vector space

structure. Thus, to complete the verification of Schlessinger’s criteria we need to

address the finite-dimensionality of tX0 .

Equip the square-zero kernel of κ[ ] κ with trivial divided powers, so by

Grothendieck–Messing theory [75, Ch. IV, V] we can classify the deformations X

of X0 over κ[ ] in terms of the subbundle Lie(Xt)∨ of the Lie algebra of the universal

vector extension E(X) of X; here, E(X) is universal among extensions of X by the

vector group Lie(Xt)∨. (This application of Grothendieck–Messing theory does not

rest on the caveats as in Remark 1.4.4.10 since κ → κ[ ] has a section.) Via the

choice of divided powers, the Lie algebra Lie(E(X)) is canonically isomorphic to

the Lie algebra

Lie(E((X0)κ[

]

)) = Lie(E(X0)) ⊗κ κ[ ]

associated to the constant deformation (the “origin” of the tangent space), and

there are canonical exact sequences

0 →

Lie(Xt)∨

→ Lie(E(X)) → Lie(X) → 0