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 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
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