Lectures on
Deformations of Singularities
By
Michael Artin
Tata Institute of Fundamental Research
Bombay
1976
c
Tata
Institute of Fundamental Research, 1976
No part of this book may be reproduced in any
form by print, microfilm or any other means without written permission from the Tata Institute of
Fundamental Research, Colaba, Bombay 400 005
Printed in India
Anil D. Ved At Prabhat Printers. Bombay 400 004
And Published By
The Tata Institute of Fundamental Research.
Introduction
These notes are based on a series of lectures given at the Tata Institute
in January-February, 1973. The lectures are centered about the work of
M. Scahlessinger and R. Elkik on infinitesimal deformations. In general, let X be a flat scheme over a local Artin ring R with residue field
k. Then one may regard X as an infinitesimal deformation of the closed
fiber x0 = X × Spec(k). Schlessinger’s main result proven in part
Spec(R)
(for more information see his Harvard Ph.D. thesis) is the construction,
under certian hypotheses, of a “versal deforamtion space” for X0 . He
shows that ∃ a complete local k-algebra A = lim A/mnA and a sequence
of deformations Xn over Spec(A/mnA ) such that the formal A-prescheme
X = lim Xn is versal in this sense: For all Artin local rings R, every de−−→
formation S /S pec(R) of X0 may be obtained from some homomorphism
A → R by setting X = X × Spec(R).
Spec(A)
Note that by “versal deformation” we do not mean that there is in
fact a deformation of X0 over A. The versal deforamtion is given only
as a formal scheme. However, Elkik has proven (cf. “Algebrisation
du module formel d’une singularite isolée” Séminaire, E.N.S., 1971-72)
that such a deformation of X0 over A does exist when X0 is a affine and
has isolated singularities. We give a proof of this result in these lectures.
Finally, some of the work of M. Schaps, A. Iarrobino, and H. Pinkham is considered here. We prove schaps’s result that every CohenMacaulay affine scheme of pure codimension 2 is determinantal. Moreover, we outline the proof of her result that every unmixed CohenMacaulay scheme of codimension 2 in an affine space of dimension
< 6 has nonsingular deformations. For more details see her Harvard
iii
iv
Introduction
Ph. D. thesis, “Deforamtions of Cohen-Macaulay schemes of codimension 2 and non-singular deformation of space curves”. We also reproduce Iarrobino’s counterexample (cf. “Reducibility of the families of 0dimensional schemes on a variety”, University of Texas, 1970) that not
every 0-dimensional projective scheme (in P for n > 2) has a smooth deformation. Finally, we give some of Pinkham’s results on deformations
of cones over rational curves (cf. his Harvard Ph.D. thesis, “Deformations of algebraic varieties with Gm action).
There is of course much more literature in this subject. Two papers relevant to these notes are Mumford’s “Pathologies-IV” (Amer. J.
Math., Vol. XCVII, p. 847-849) in which expanding on Iarrobino’s
methods he proves that not every 1-dimensional scheme has nonsingular deformations, and M. Artin’s “Versal deformations and algebraic
stacks” (Inventions mathematicae, vol. 27, 1974), in which is shown
that formal versality is an open condition.
Thanks are due to the Institute for Advanced study for providing
excellent facilities in getting this manuscript typed.
Contents
Introduction
iii
1 Formal Theory and Computations
1
Definition of deformations . . . . . . . . . . . . . . .
2
Iarrobino’s example of a 0-dimensional scheme... . . .
3
Meaning of flatness in terms of relations . . . . . . . .
4
Deformations of complete intersections . . . . . . . .
5
The case of Cohen-Macaulay varieties of codim 2... . .
6
First order deformations of arbitrary X... . . . . . . . .
7
Versal deformations and Schlessinger’s theorem . . . .
8
Existence formally versal deformations . . . . . . . .
9
The case that X is normal . . . . . . . . . . . . . . . .
10 Deformation of a quotient by a finite group action . . .
11 Deformations of cones . . . . . . . . . . . . . . . . .
12 Theorems of Pinkham and Schlessinger on... . . . . . .
13 Pinkham’s computation for deformations of the cone...
.
.
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1
1
4
6
9
13
21
28
32
37
40
42
48
54
2 Elkik’s Theorems on Algebraization
1
Solutions of systems of equations . . . . . . . . . . .
2
Existence of solutions when A is t-adically complete
3
The case of a henselian pair (A, a) . . . . . . . . . .
4
Tougeron’s lemma . . . . . . . . . . . . . . . . . . .
5
Existence of algebraic deformations of isolated... . .
.
.
.
.
.
65
65
67
76
83
86
v
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.
.
Deformations of Singularities
Part 1
Formal Theory and
Computations
1 Definition of deformations
1
We work over an algebraically closed field k.
Let X be an affine scheme, X ֒→ An . Let A be finite (i.e., finitedimensional/k)
local algebra over k, so that A ≈ k[t1 , . . . , tr ]/a with
√
a = (t1 , . . . , tr ). Let T = Spec A.
Definition 1.1. An infinitesimal deformation XT (or XA ) of X is a scheme
∼
flat/T together with a k-isomorphism XT ×T Spec k −
→ X.
More generally, suppose we are given a commutative diagram
X

/ X̄
flat
/ Spec R
Spec k
where R is a ring of finite type over k and X is a scheme with closed
fiber isomorphic to X. We then say that X is a family of deformations or
a deformation of X over R.
Remark 1.1. XT is necessarily affine. In fact ∃ a closed immersion
XT ֒→ AnT (= Spec A[X1 , . . . , Xn ]) such that its base change with respect
1
2
1. Formal Theory and Computations
to the morphism Spec k → T (representing the closed point of Spec A is
the immersion X ֒→ An (= Ank ).
2
Let O = coordinate ring of X and O = k[X1 , . . . , Xn ]/I, with xi the
canonical images of Xi in O. To prove the remark, it suffices to prove
that if XA ֒→ AnA is an affine scheme (over A) imbedded in AnA , A′ is a
finite local algebra/k such that
0 → J → A′ → A → 0
is exact
with J 2 = 0 (J is an ideal of square 0) and XA′ is a scheme/A′ such that
XA′ ×Spec A′ Spec A ≃ XA , then the immersion XA ֒→ AnA can be lifted
to an immersion XA′ ֒→ AnA′ . (This reduction is immediate since the
ρ
ρ−1
maximal ideal is nilpotent. Say that mA′ = 0, and take J = mA′ , . . .).
We have an exact sequence
0 → I → OXA′ → OXA → 0,
where OXA′ , denotes the structure sheaf of XA′ . Now I 2 = 0 since J 2 = 0.
This implies that I, which is a priori a (sheaf of) OXA′ module(s), is
in fact a module over OXA′ /I, i.e., it acquires a canonical structure of
coherent OXA -module. Since XA is affine, it follows that H 1 (XA′ , I) = 0
(for it is = H 1 (XA , I)). Hence
0 → H o (XA′ , I) → H o (XA′ , XA′ ) → H o (X, XA ) → 0
3
is exact, i.e., in particular H o (XA′ , OXA′ ) → H o (X, OXA ) → 0 is exact.
Let xi be the coordinate functions on XA which define XA ֒→ AnA . The
xi can be lifted to ξi ∈ H o (XA′ , OXA′ ). Then the ξi define a morphism
ξ : XA′ → AnA′ . It follows at first that ξ is a local immersion; for this
it suffices to prove that ξ1 generate the local ring OXA′ ,x at every closed
point x of XA′ . Let Ix be the stalk of the ideal sheaf I at a closed point
x of X. We have 0 → Ix → OA′,x → OA,x → 0(OA′ ,x′ OA,x represent the
local rings at x of XA′ and XA respectively). We have Ix = J · OA′,x· . Now
j · θ1 = j · θ2 for j ∈ J and θ1 , θ2 in OA′,x such that their canonical images
in OA,x are the same. Let S be the subalgebra of OA′ ,x generated by ξi
over A′ . Then we see that I x = JS . Since J ⊂ A′ it follows that I x ⊂ S .
1. Definition of deformations
3
Since S maps onto OA,x′ given λ ∈ OA′ ,x· ∃s ∈ S such that λ − s ∈ I x .
This implies that λ ∈ S . This proves S = OA,x· . We conclude then that
ξ : XA′ ֒→ AnA′ is a local immersion. But ξ is a proper injective map
(since XA ֒→nA is a closed immersion). From this it follows that ξ is a
closed immersion. This proves the Remark.
Note that in the above proof we have not used the fact that XA′ is
flat/A′ .
Given the closed subscheme X ֒→ Ank let us define the following
two functors on the category of finite local algebras over k.
(Def. X) : (Finite local alg) → (Sets)
||
{Deformations of X/A}7→{isom. classes (over A) of schemes X flat/A
and suct that XA ⊗ k ≃ X}
(Emb. def. X):(Finite local alg)→(Sets)
{Embedded deformations/A 7→ closed subschemes XA of AnA flat/A,
such that XA ֒→ AnA by base change is the given XA ֒→ Ank }
These should be called respectively infinitesimal deformations of X 4
and infinitesimal embedded deformations of X. Then we have a canonical morphism of functors
f
(Emb.de f.X) −
→ (De f.X).
The above Remark says that f is formally smooth; that is, if A′ →
A → 0 is exact in (Finite local alg), we have
(Emb.de f.X)(A′ )
/ (De f.X)(A′ )
/ (De f.X)(A)
(Emb.de f.X)(A′ )
(by definition of a morphism of functors), and the canonial map
(Emb.de f.X)(A′ ) → (De f.X)(A′ ) ×(de f.X)(A) (Emb.de f.X)(A)
is surjective.
4
1. Formal Theory and Computations
2 Iarrobino’s example of a 0-dimensional scheme
which is not a specialization of d distinct points
5
Given X as above, we can ask whether it can be “deformed” into a nonsingular scheme. Here by a deformation we do not mean an infinitesimal
one, but a general family of deformations. Let us consider the simplest
case of Krull dimension 0. Then X = Spec O where O is a k-algebra of
finite dimension d. If d = 1, O ≈ k. If d = 2, O ≈ k × k or O ≈ k[t]/(t2 ).
If d = 3, we have



k3 or k × k[t]/t2
O =

or k[t]/t3 or k[X, Y]/(X, Y)2
In our particular case the question is whether X can be deformed
into d distinct points of An . Now An ֒→ Pn and we see easily that
this deformation implies a “deformation” of closed subschemes of Pn ,
i.e., if X can be deformed into d “distinct points” we see (without much
difficulty) that this can be done as an “embedded deformation” in An and
in fact as an embedded deformation in Pn . Let Hilbd denote the Hilbert
scheme of 0-dimensional subschemes Z ֒→ Pn such that if Z = Spec B
then B is of dim d over k. Then it is known that Hilbd is projective/k.
Let Ud denote the open sub-scheme of Hilbd corresponding to d distinct
points, i.e., those closed sub-schemes of Pn corresponding to points of
Hilbd which are smooth. We see that Ud is irreducible; in fact it is
d-fold symmetric product of Pn minus the “diagonals”. Now if every
0-dimensional subscheme can be deformed into a nonsingular one, then
Ud is dense in Hilbd and it follow that Hilbd is irreducible.
We shall now give the counterexample (due to Iarrobino) where
Hilbd is not irreducible. It follows therefore that a 0-dimensional scheme
cannot in general be deformed to a smooth one.
Theorem 2.1. Let Hilbd,n denote the Hilbert scheme of closed 0-dimensional subschemes of Pn of length d. Then Hilbd,n is irreducible for
n ≤ 2. For n > 2, Hilbd,n is not in general irreducible.
′
Proof. We give only the counter example for the case n > 2. Let O =
k[X1 , . . . , Xn ]/(X1 , . . . , Xn )r+1 . Let I be the ideal in (X)r /(X)r+1 (where
2. Iarrobino’s example of a 0-dimensional scheme...
5
(X) = (X1 , . . . , Xn )). Then I is a vector space over k and
Rank of
Rank of
I/k = Polynomial of degree
′
O /k = Polynomial of degree
(n − 1) in
n in
r
r.
6
Now I is an ideal in O ′ and in particular an O ′ module. Moreover, if λ ∈ max ideal of O ′ then λ · I = 0. Hence O ′ operated on I
through its residue field. In particular, it follows that any linear subspace (over k) of I is an ideal in O ′ and hence defines a closed subscheme of Spec O ′ . Take now θ = 12 Rank I or 21 Rank I − 12 according as Rank I is even or odd and d = Rank(O ′ /V), where V is a subspace of I of rank θ. Hence θ = Polynomial of degree(n − 1) in r and
d = polynomial of degree n in r. Let us now count the dimension of the
set Lθ of all linear subspaces of I of rank = θ. Then it is a Grassmannian
2 and dimLθ = 21 Rank I or 12 Rank I − 12 21 Rank I + 12 according as
Rank I is even or odd=(Polynomial of deg n − 1)2 in r = Polynomial of
degree (2n − 2) in r. Now if Ud is the subscheme of Hilbd of “d distinct
points” as above, then
dim Ud = d.n = n(Polynomial of degree
n in
r).
Now if r ≫ 0, we see that
dim Lθ > dim Ud .
Since Lθ can be identified as a subscheme of Hilbd , it follows now
that Ud is not dense in Hilbd . To see that Lθ is a subscheme of Hilbd ,
note that Spec O/I as a point set consists only of one point. The family
of subschemes of Pn parametrized by Lθ as a point given by (xo × Lθ ). It
suffices to check that on (x× Lθ ), we have a natural structure of a scheme
Γ such that
6
1. Formal Theory and Computations
Γ is a closed subscheme of Pn × Lθ and p−1 (x) is the subscheme of
Spec O/I defined by the linear subspace of I corresponding to x. Then 7
we see that p : Γ → Lθ is flat, for p is a finite morphism over Lθ such
that OΓ is a sheaf of OLθ -algebras; in particular, OΓ becomes a coherent
sheaf over OLθ . At every x ∈ Lθ , for OΓ ⊗ OLθ , x/maxideal(= OΓ ⊗ k),
the rank is the same. Lθ is reduced, this implies that OΓ is locally free
over OLθ . In particular, Γ is flat/Lθ .
3 Meaning of flatness in terms of relations
A module M over a ring A is said to be flat if the functor N 7→ M⊗A is
exact.
⇔ T orqA (M, N) = 0 ∀N/A.
⇔ T orlA (M, N) = 0 ∀finitely generatedN/A.
Let us now consider the case when A is a finite local k-algebra. Then
if N is an A-module of finite type, there is a composition series
N = N0 ⊃ N1 ⊃ . . . ⊃ Nℓ = 0,
such that
Ni /Ni + 1 ≈ k.
From this it follows immediately that
M f lat/A ⇔ T or1 (M, k) = 0
(if A finite local alg/k).
Let XA = Spec OA , A finite local algebra and OA an A-algebra of
Iinite type so that we have
0 → IA → P A → OA → 0
exact with PA A[X](X = (X1 , . . . , Xn )). Tensor this with k. Then we have
o → T or1 (OA , k) → IA ⊗ k → Pk → Ok → 0
8
3. Meaning of flatness in terms of relations
7
Let X = Spec Ok , X ≈ XA ⊗ k, I ideal of X in An . Then
XA is flat/A ⇔ T or1 (OA , k) = 0,
⇔ IA ⊗ k = I.
Take a presentation for the ideal IA in PA , i.e., an exact sequence
PℓA
/ Pm
A
(*)
0
??
??
??
??
>} IA
}}
}}
}
}}
/ PA
?






AA
AA
AA
AA
/ OA
/0
0
Then we have (because of the above facts): OA is A-flat ⇔ the above
presentation for IA tensored by k, gives a presentation for I, i.e., tensored by k gives again an exact sequence
Pℓk
/ Pm
k
(**)
0
>>
>>
>>
>>
~> I
~~
~
~~
~~
/ Pk
@
??
??
??
??
/O
/0
0
Suppose we are given:
O = K[X]/( f1 , . . . , fm ) and liftings ( fi′ ) of ( fi ) to elements in A[X].
Let I = ( f1 , . . . , fm ), IA = ( f1′ , . . . , fm′ ) and OA = A[X]( f1′ , . . . , fm′ ).
These data are equivalent to giving a lifting Pm
A → P A → OA → 0 of
the exact sequence Pm → P → O → 0, i.e., to giving an exact sequence
Pm
A → P A → OA → 0 such that its ⊗ A k is the given exact sequence 9
Pm → P → O → 0. Note that this need not imply that IA ⊗ k = I if
IA = Ker(PA → OA ) and I = Ker(P → O).
Suppose we are now given a “complete set of relations” (or a presentation for I) between fi ’s, i.e., an exact sequence
(*)
Pℓ → Pm → P → O → 0.
1. Formal Theory and Computations
8
′
Then giving a lifting of these relations to that of fi ’s (or IA ) is to
give
(**)
PℓA → Pm
A → PA → OA → 0
which extends the exact sequence Pm
A → P A → OA → 0, which is a
ℓ ⊂ Ker(Pm → P ) and such that (∗∗) lifts (∗).
complex at Pm
,
i.e.,
ImP
A
A
A
A
In this situation we have the following
Proposition 3.1. Suppose
(*)
Pℓ → Pm → P → O → 0
is exact and
(**)
PℓA → Pm
A → PA → OA → 0
is a complex such that the part
Pm
A → PA → OA → 0
is exact and (**) ⊗A k = (∗). Then OA is A-flat.
Proof. Suppose first that
(*)
10
PℓA → Pm
A → PA → OA → 0
11
is exact not merely a “complex at Pm
A . Then we claim that the flatness
of OA over A follows easily. For then (**) can be split up as:


PℓA →LA → 0, 0 → LA → Pm

A → IA → 0
exact.



0 → IA → P A → OA → 0
Therefore PℓA ⊗ k → LA ⊗ k → 0 and LA ⊗ k → Pm
A ⊗ k → 0 are
ℓ
m
exact. This implies that IA ⊗ k = coker(k ⊗ PA → k ⊗ PA )m i.e., cokernel
is pre-served by k⊗A . On the other hand, I = Coker(Pℓ → Pm ). Hence
IA ⊗ k = I. From this it follow that OA is flat/A as remarked before. Now
the hypotheses of our proposition amount to the fact all relations in I can
be lifted to IA . Given a relation in IA , i.e., an m-tuple (λ′1 , . . . , λ′m ) such
4. Deformations of complete intersections
9
P
that λ′i fi′ = 0, this descends to a relation in 1 by taking the canonical
′
images of λi in P. Take a complete set of relations for IA , i.e., an exact
sequence
′
PℓA → Pm
A → PA → OA → 0
(i)
(ℓ
′
need not be
ℓ),
then from our above argument this descends to a complete set of relations for I, i.e., tensoring (i) by k we get an exact sequence
′
Pℓ → Pm → P → O → 0.
Consequently, we have already shown in this situation that we must
have OA to be A-flat.
The criterion for flatness can given be formulated as follows:
′
′
Corollary . Let O = k[X]/( f1 , . . . , fm ) and OA = A[X]/( f1 , . . . , fm ) 11
′
where fi are liftings of fi . Then OA is A-flat ⇔ every relation among
′
′
( f1 , . . . , fm ) lifts to a relation among ( f1 , . . . , fm ).
Remark 3.1. It is seen immediately that
OA flat/A ⇒ IA flat/A.
For, the exact sequence 0 → IA → PA → OA → 0 by tensoring by k
gives
0
0
k
k
Tor2 (OA , k)
/ Tor1 (IA , k)
/ Tor1 (PA , k)
0
k
/ Tor1 (OA , k)
/1
/ Pk
/O
/0
This implies that Tor1 (IA , k) = 0, hence that IA is flat/A, by a previous
Remark. Repeating the procedure for OA , we see by succesive reasoning
that any resolution for O lifts to one for OA .
4 Deformations of complete intersections
Let X ֒→ An be a complete intersection. Let d = dimX (Krull dim)
and I = ( f1 , . . . , fn−d ) the ideal of X. Let O = P/I, where P = Pk =
1. Formal Theory and Computations
10
k[X1 , . . . , Xn ] Then it is well known that the “Koszul complex” gives a
resolution for O, i.e.,
2
1
∧ Pn−d → ∧ Pn−d (= Pn−d ) → P → O → 0
12
the homomorphisms being interior multiplication by the vector
( f1 , . . . , fn−d ) ∈ Pn−d ,
(e.g., Pn−d → P is the map (λ1 , . . . , λn−d ) 7→
2
n−d
Pn−d
P
i λi fi ).
The image of
in
gives the realtions in I, which shows that the relations
∧P
among the fi are (generated by) the obvious ones, i.e., fi z j − f j zi = 0.
Let A = k[t]/(t2 ) (which we write A = k + kt with t2 = 0). Then
deformations of X over A are called first order deformations. Lert IA be
the ideal in A[X1 , . . . , Xn ].
IA = (( f1 + g1 t), . . . , ( fn−d + gn−d t))
where gi ∈ Pk . We claim that for arbitrary choice of gi ∈ Pk , OA =
PA /IA is flat over A (of course we have seen that any deformation XA
of X can be defined by IA for sutiable choice of gi ). This is an immediate consequence of the fact that above explicit relations between fi can
obviously be lifted to relations between ( fi + gi t). This proves the claim.
Thus to classify embedded first order deformations it suffices to
′
write down conditions on (gi ), (gi ) in P so that in PA the ideals (( fi +gi t))
′
and (( fi + gi t)) are the same. We claim:
′
′
(( fi + gi t)) = (( fi + gi t)) ⇔ gi − gi ∈ I.
To prove this we proceed as follows:
′
(( fi + gi t)) ⊂ (( fi + gi t)) ⇔
′
fi + gi t =
(Set r = n − d.)
n−d
X
(αi j + βi j t)( f j + g j t)
j=1




r
r
r
X

X

X



=  αi j f j  + t  αi j g j +
βi j f j  ⇔
j=1
13
j=1
j=1
∃ (r × r) matrices (αi j ) and (βi j ) with coefficients in P such that
4. Deformations of complete intersections
11
   
f1
 f1   f1 
 .   . 
.
(a) (αi j )  ..  =  ..  , i.e.,(αi j − Id) (..) = (0), and
   
fr
fr
fr
 
   ′
g1 
 f1  g1 
 . 
   
(b) (αi j )  ..  + (βi j )  ...  =  ... .
 
   ′ 
gr
fr
gr
Since the coordinates of the relation vectors are in I it follows from
(a) above that the element of (αi j − Id) are in I, i.e., (a) ⇒ (αi j ) ≡ (Id)
mod I). Then (b) implies that
′
   ′
g1  g1 
 ..   .. 
 .  ≡  .  (mod I).
   ′
gr
gr
′
Hence (( fi + gi )) ⊂ (( fi + gi t) ⇒ (gi − gi ) ∈ I. Hence (( fi + gi t)) =
′
′
′
(( fi + gi t)) ⇒ (gi − gi ) ∈ I. Conversely, suppose that (gi − gi ) ∈ I. Then
there is a matrix (βi j ) such that
   ′  
 f1  g1  g1 
     
(βi j  ...  =  ...  −  ... 
   ′   
gr
fr
gr
   ′
 
 f1   g1 
g1 
   
 .. 
Hence, (Id)  .  + (βi j )  ...  =  ... 
   ′ 
 
fr
gr
gr .
Taking (αi j ) = Id we see that the conditions (a), (b) are satisfied,
′
which implies that (( fi + gi t)) = (( fi + gi t)). This proves the claim and
thus we have classified all embedded (in An ) first order deformations of 14
X.
Now to classify first order deformations of X we have only to write
down the condition when two embedded deformations XA , XA′ are iso∼
′
morphic over A. Let θ be an isomorphism XA −
→ XA . By assumption,
′
X ⊗ k = XA ⊗ k = X. i.e., their fibres over the closed point of Spec A are
1. Formal Theory and Computations
12
X ⊂ An . We denote (of course) by Xν the canonical coordinate functions
′
of XA ֒→ AnA = Spec A[X1 , . . . , Xn ]. Let Xν = θ⋆ (Xν ). Then we have
′
Xν = Xν + ϕν (X)t
for some polynomials ϕν (X). Hence to identify two embedded deformations of X we have to consider the identification by the above change of
coordinates. Then
fi + gi t 7→ fi (Xν + ϕν (t)) + gi ((Xν + ϕν (t))t.
By Taylor expansion up to the first order, we get






X ∂ fi

= fi (X) + t 
ϕ
(X)
+ tgi (X)

ν




∂X
ν


n


X


∂ fi


ϕ
(X)
= fi (X) + t 
g
(X)
+

ν
i




∂X
ν=1
′
15
Hence ( fi + gi t) and ( fi + gi t) define the same deformation of the first
order up to change of coordinates above, which is equivalent to the fact
that there exists (ϕ1 , . . . , ϕn ) such that
   ′   ∂ f1
∂ f1  ϕ 
  1 
g1  g1   ∂X · · · ∂X
n
  . 
 .   .   1
  ..  .
 ..  −  ..  = 
···
(*)
   ′   ∂ fr
∂ fr   
gr
gr
∂X1 · · · ∂Xn ϕn
Recall that O = P/I and X = Spec O. Then embedded first order
deformatios are classified by O n−d = O ⊕ . . . ⊕ O (n − d times), which
is an O-module.
To classified all deformations, consider the homomorphism of P
modules
!
∂ fi
∂ fi
∂ fi
Jac
the images of
in P/I.
;
O → O n−d whose matrix is
∂X j ∂X j
∂X j
Let us call the euotient O n−d /ImO n by this map T . This is an Omodule and we see that its support is located at the singular points of
5. The case of Cohen-Macaulay varieties of codim 2...
13
X. In particular, if X has isolated singularities, T is a finite dimensional
vector space/k.
For example, consider the case that X is of codimension one, i.e.,
defined by one equation f . Then
!
∂f
∂f
T = k[X]/ f,
,...,
.
∂X1
∂Xn
The cone in 3-space has equation f = Z 2 − XY, and if chark , 2,
T = k[X, Y, Z]/( f, −Y, −X, 2Z) k.
Thus a universal first order deformation is given by
Z 2 − XY + t = 0.
5 The case of Cohen-Macaulay varieties of codim 2
in An (Hilbert, schaps)
16
The theorem that we shall prove now was essentially found by Hilbert.
This has been studied recently by Mary schaps.
Let P be as usual the polynomial ring P = k[X1 , . . . , Xn ]. Let (gi j )
be an
over P
#
" (r × r − 1) matrix
g11 · · · gr−1
. Let δi = (−1)i det((r − 1) × (r − 1) minor with ith row
gr,1 · · · gr,r−1
deleted).
#
#"
"
g1,1 · · · g1,r−1 g1,1 · · · g1,r−1 , g1,1
= 0 etc. This
Then (δ1 , . . . , δr )
gr,1 · · · gr,r−1 gr,1 · · · gr,r−1 , gr,1
implies that the sequence
Pr−1 −−−→ Pr −−−−−−→ P
(gi j )
(δ1 ,...,δr )
is a complex. Note that Pr−1 → Pr is injective if over the quotient field
of P. (gi j ) has rnak (r − 1), or equivalently, ∃ some xo ∈ An such that
(gi j (xo )) is of rank (r − 1).
1. Formal Theory and Computations
14
Theorem 5.1 (Hilbert, Schaps). (1) Let (gi j ) be an r×(r−1) matrix over
P and δi its minors as defined above. Let J be the ideal (δ1 , . . . , δr ).
Assume that V(J) = V(δ1 , . . . , δr ) is of codim ≥ 2 in An . Then
X = V(J) is Cohen-Macaulay, precisely of codim 2 in An . Further,
the sequence
(gi j )
(δ1 ,...,δr )
0 → Pr−1 −−−→ Pr −−−−−−→ P → P/J → 0
17
is exact, i.e., it gives resolution for P/J.
(2) Conversely, suppose given a Cohen-Macaulay closed subscheme
X ֒→ An of codim 2. Let X = V/(J), then P/J has a resolution
(gi j )
( f1 ,..., fr )
of length 3 which will be of the form 0 → Pr−1 −−−→ Pr −−−−−−→
P → P/J → 0, because hdP P/J = 2. (Depth P/J + hd p P/J = n,
depth P/J = n − 2, ⇒ hgP/J = 2). Note that fi need not be δi as
defined above. Then we claim that we have as isomorphism
0
0
/ Pr−1
/ Pr−1
(gi j )
(gi j )
/ Pr ( f1 ,..., fr )/ P
/ Pr
(δ1 ,...,δr )
≀
/P
i.e., ∃ a unit u ∈ P such that fi = uδi .
(3) The map of functors (Deformations of (gi j ))→ (Def X) is smooth,
i.e., -(i) deforming (gi j ) gives a deformation of X, (ii) any deformation of X can be obtained by deforming gi j , and (iii) given a deformtion XA of X defined by (gi j ) over A[X], A′ → A → 0 exact and
′
a deformation XA of X inducing XA , ∃(gi j ) over A′ [X] which defines
XA and the canonical image of (g′i j ) in A is (gi j ).
Proof. (1) Let (gi j ), δi , etc., be as in (l). We shall first prove that
(gi j )
(δi )
0 → Pr−1 −−−→ Pr −−→ P → P/J → 0
is exact, assuming codim X ≥ 2. This will complete the proof of (l).
For, it follows hdP P/J < 2. On the other hand, since codim X ≥ 2,
depth P/J ≤ (n − 2).
5. The case of Cohen-Macaulay varieties of codim 2...
18
15
But depth P/J + hdP P/J = n. This implies that dim P/J = depth
P/J = (n − 2) and hdP P/J = 2, which shows that X is CohenMacaulay of codim 2 in An .
Since V(J) , An , the δi ’s are not all identically 0, hence 0 →
Pr−1 → Pr is exact. Further we note that any xo < V(J), (gi j (xo ))
is of rank (r − 1) and in fact one of (δ1 , . . . , δr ) is nonzero at xo and
hence a unit locally at xo . This implies that Pr−1 → Pr → P split
exact locally at xo (i.e., if B is the local ring of An at xo , then tensoring by B gives a split exact sequence). Because of our hypothesis
that codim V(J) ≥ 2, if x is a point of An represented by a prime
ideal of height one and B its local ring. then tensoring by B makes
Pr−1 → Pr → P exact (i.e., the sequence is exact in codim 1). Let
K = Ker(Pr → P). We have K ⊂ Pr such that 0 → Pr−1 → K,
and 0 → K → Pr → P exact. Tensoring by B as above, it follows
that Pr−1 ⊗ B → K ⊗ B is an isomorphism (tensoring by B is a localization); i.e., the inclusion Pr−1 ⊂ K is in fact an isomorphism in
codim 1. Since Pr−1 is free sections of Pr−1 defined in codimension
1 exrtend to global sections. Moreover, K is torsion-free. Therefore,
the fact that Pr−1 ⊂ K is an isomorphism in codimension 1 implies
that it is an isomorphism everywhere. Therefore, Pr−1 → Pr → P is
exact, and this completes the proof of (1).
(2) Let I = ( f1 , . . . , fr ) be a Cohen-Macaulay codim 2 ideal in P. Since
hdP P/I = 2, there is a resolution
(*)
(gi j )
( fl ,..., fr )
0 → Pr−1 −−−→ Pr −−−−−−→ P → P/I → 0.
(Here we should take the warning about using free resolutions instead of using projective resolutions.) Let the complex (**) be defined by
(**)
(gi j )
(δ1 ...,δr )
0 → Pr−1 −−−→ Pr −−−−−−→ P → P/J → 0
δi being as before.
The sequence (*) is split exact at every point x < V(I). This implies
that some δi is a unit at x, and hence by direct calculation that (**)
19
1. Formal Theory and Computations
16
is also split exact, and P/J = 0, at x. This means
V(J) ⊆ V(I),
hence codim V(J) ≥ 2. Hence by (l) it follows that (**) is exact.
We shall now show that ∃ a unit u such that fi = uδi . Take the dual
of (*) and (**) dual: HomP (M, P) = M ∗), i.e.,
(Pr−1 )∗ ←−−−t (Pr )∗ ←−−−t P∗ ← 0
(∗)∗
(gi j )
(∗∗)∗
(P
(f)
) ←−−−t (P ) ←−−−t P∗ ← 0.
r−1 ∗
(gi j )
r ∗
(δi )
We claim that these sequences are exact. The required assertion
about the existence of u is an immediate consequence of this. For
( f )∗
(δi )∗
then P∗ −−−→ (P∗ )r and P∗ −−−→ (P∗ )r are injective maps into the
same submodule of (P∗ )r of rank 1 which implies that ( f ) and (δ)
(gi j )
( f1 ,..., fr )
differ by a unit. We note that Pr−1 −−−→ Pr −−−−−−→ P and Pr−1 →
(δ1 ,...δr )
Pr −−−−−−→ P are split exact in codimension 1. Consequently. it
follows from this and the fact that HomP (P/I, P) = HomP (P/J, P) =
0 that we have sequences
( fi )t
∗
(gi j )t
∗
0 → P∗ −−−→ Pr −−−→ Pr−1
(δi )∗
(gi j )t
∗
0 → P∗ −−−→ Prt −−−→ Pr−1
20
∗
and we must prove exactness at the Pr module. But we have
that Im fi∗ ⊂ Ker(gi j )t and Im(δi )∗ ⊂ Ker(gi j )t with equality at
the localization of each prime ideal of height 1. Consequently,
Im fi∗ = Ker(gi j )t , Im(δi )t = Ker(gi j )t (same argument as was used
above). This completes the proof of (2).
(3) Let A be an Artin local ring over k, and PA = A[X1 , . . . , Xn ]. Let
′
(gi j ) be an r × (r − 1) matrix over PA and (gi j ) the matrix over P
such that g′i j 7→ gi j by the canonical homomorphism A[X] → k[X].
′
′
Define δi analogous to δi . Suppose that codim V(δi ) in AnA is of
5. The case of Cohen-Macaulay varieties of codim 2...
17
codim ≥ 2 ⇔ codimV(δi ) in Ank is of codim ≥ 2 ⇔ condimV(δi ) =
2 because of (1). Consider
Pr−1
A
′
(gi j )
PrA
′
(δi )
(**)
0→
(*)
0 → Pr−1 −−−→ Pr −−→ P → P/I → 0.
−−−→
(gi j )
−−−→ PA → PA /IA → 0
(δi )
Now (**) is lifting of (*). Of course (*) is exact and PrA → PA →
PA /JA → 0 is exact. Besides, (**) is a complex. This implies by
proposition 3.1 that Pr−1
→ PrA → PA → PA /JA → 0 is exact
A
and PA /IA is A-flat and by Remark 3.1 (**) is exact/ (The exactness
of (**) can also be proved by a direct argument generalizing (1).)
′
This shows that any (infinitesimal) deformation (gi j ) of (gi j ) as in
(1) gives a (flat) deformation XA of X = Spec P/I and that XA is
“presented” in the same way as X.
Conversely, let XA be an infinitesimal deformation of X = V(δ1 , . . . , 21
δr ). Lift the generators for I to IA say ( fl , . . . , fr ). Then as we remarked before the exact sequence (*) can be lifted to an exact sequence
(∗∗)′
0→
Pr−1
A
′
(gi j )
( fi )
−−−→ PrA −−→ PA → PA /IA → 0
[Note that fi need not be (a priori) the determinants of minors of
′
′
′
′
(gi j ).] Let δi be the minors of (g′i j ), JA the ideal (δl , . . . δr ). Then as
we saw above
(**)
0→
Pr−1
A
′
(gi j )
−−−→
PrA
′
(δi )
−−−→ PA → PA /JA → 0
is exact and PA /JA is A-flat. Taking the duals of (**) and (∗∗)′ as
′
before, it follows that there is a unit u in PA such that fi = uδi . This
shows that IA = JA . Thus any deformation is obtained by a diagram
of type (**). The assertion of smoothness also follows from this
argument. This completes the proof of the theorem.
1. Formal Theory and Computations
18
Remark 5.1. Any Cohen-Macaulay X ֒→ An of codim 2 is defined by
the minors of an r × (r − 1) matrix (gi j ) (this is local, note the warning
about using free modules instead of projective ones). The gi j ’s define a
morphism
Φ
An −→ Ar(r−1) .
22
Let Q = k[Xi j ], 1 ≤ i ≤ r, 1 ≤ j ≤ r−1, so that Ar(r−1) = Spec k[Xi j ].
Then Φ∗ (Xi j ) = gi j . Consider (Xi j ) as an r × (r − 1) matrix over Q, and
let △i = (−1)i det(minor of Xi j with ith row deleted). Then the variety
V = V(△1 , . . . , △r ) → Ar(r−1) is called the generic determinantal variety
defined by r × (r − 1) matrices. It is Cohen-Macaulay and of codim 2 in
Ar(r−1) . Then Φ−1 (x) = X. This means that any Cohen-Macaulay codim
2 subscheme is obtained as the inverse image by An → Ar(r−1) of the
generic determinantal variety V of type r × (r − 1).
Remark 5.2. Other simple examples of codim 2, Cohen-Macaulay X
are
(a) any O-dimensional subscheme in A2 ,
(b) any 1-dimensional reduced X in A3 , and
(c) any normal 2-dimensional X in A4 .
More about the generic determinantal variety.
Now let X ⊂ Ar(r−1) denote the determinantal variety V(△i ) = V
defined above. Then any infinitesimal deformation XA of X is obtained
by the minors of a matrix (Xi j + mi j ) where mi j ∈ mA [Xi j ], where mA is
′
′
the maximal ideal of A. Set Xi j = Xi j + mi j . We see that Xi j 7→ Xi j is
′
just change of coordinates in AnA , i.e., A[Xi j ] = A[Xi j ]. This implies that
X is rigid, i.e., every deformation of X is trivial.
The singularity of X: X can be identified with the subset of Ar(r−1)
Homlinear (Ar , Ar−1 ) of linear maps of rank ≤ r − 2. Now G = GL(r) ×
GL(r−1) operates on Ar(r−1) in a natural manner. Lt Xk denote the subset
of Ar(r−1) of linear maps of rank equal to (r − k). We note that Xk is an
orbit under G and hence is a smooth, irreducible locally closed subset
5. The case of Cohen-Macaulay varieties of codim 2...
19
of Ar(r−1) . To compute its dimension we proceed as follows: If ϕ ∈
Hom(Ar , Ar−1 ) and ϕ ∈ Xk , then imϕ is a k-dimensional space. Hence 23
ϕ is determined by an arbitrary k-dimensional subspace of Ar (= ker ϕ)
an (r − k)-dimensional subspace of A(r−1) (= Imϕ) and an arbitrary linear
map of an (r − k)-dimensional linear space onto an (r − k)-dimensional
linear space.
Hence
dim Xk = (r − k)k + (r − k)[(r − 1) − (r − k)] + (r − k)2
= (r − k)k + (r − k)(k − 1) + (r − k)2 .
Suppose k = 2, i.e., consider linear maps of rank (r − 2). Then X2 is
obviously open in X and
dim X2 = (r − 2)2 + (r − 2) + (r − s)2
= (r − 2)2 + 1 + (r − s)
= (r − 2)(r + 1)
It follows that codim X2 = 2.
It is clear that X2 is dense in X (easily seen that every xo ∈ X is a specialization of some x ∈ X2 ). The implies that X is irreducible. Further
more, X2 is smooth, and in particular reduced. By the unmixedness theorem, since X is Cohen-Macaulay, it follows that X is reduced. Hence
X is a subvariety of Ar(r−1) .
′
Let X3 = X3 ∪ X4 . . . be the set of all linear maps of rank ≤ (r − 3).
′
Clearly X3 is a closed subset of X. To compute dim X3 , as we see easily
that x3 is a dense open set in X ′ 3 . Now
dim X3 = (r − 3)3 + (r − 3)2 + (r − 3)2 = (r − 3)(r + 2).
Hence codim X3 (in Ar(r−1) )= 6, for r ≥ 3. (If r = 2 it is a complete
intersection; X is the intersection of two linear spaces and hence X is
smooth.)
′
Hence codim X3 in X is 4. We claim also that every point of X3 is 24
singular on X. To prove this, suppose for example θ ∈ X3 . Since G acts
1. Formal Theory and Computations
20
transitively on X3 , we can assume it is the point

1

0
0
θ = 



0 0
1 0
0 1
0



0 




0 

∂△i = 0 ∀k, l. Thus θ ∈ X is smooth ⇔ θ ∈ X2 .
Then
∂Xkl θ
Thus the generic determinantal variety has an isolated singularity if
and only if r = 3, in which case dim X = 4, X ⊆ A6 (= Ar(r−1) ). We thus
get an example of a rigid isolated singularity (a 4-dimensional isolated
Cohen-Macaulay singularity).
Proposition 5.1. (Schaps). Let X0 ֒→ Ad (d ≤ 5) be of codim 2 and
Cohen-Macaulay. Then X0 can be deformed into a smooth variety.
Proof. We give only an outline. Since X0 is determinantal, it is obtained
as the inverse image of X in Ar(r−1) by some map
ϕ = (gi j ) : Ad → Ar(r−1)
′
′
Now codim X in Ar(r−1) is 6. “Perturbing” (gi j ) to (gi j ) = ϕ , the map
′
Ad → Ar(r−1) , defined by (gi j ) can be made “transversal” to X. This
′
implies that ϕ (Ad ) ∩ SingX = ∅[SingX is of condim 6] and in fact that
′−1
ϕ (X) is smooth.
25
Remark 5.3. It has been proved by Svanes that if X is the generic determinantal variety defined by determinants of (r × r) minors of an (m × n)
matrix, then X is rigid, except for the case m = n = r.
6. First order deformations of arbitrary X ...
21
6 First order deformations of arbitrary X and Schelessinger’s T 1
Let X = V(I) and A = k[t]/(t2 ). Let I = ( f1 , . . . , fm ). Fix a presentation
for I, i.e., an exact sequence
(ri j )
( fi )
Pℓ −−−→ Pm −−→ P → P/I → 0.
(*)
′
Let IA = ( f1′ , . . . , fm′ ) where fi = ( fi + tgi ), gi ∈ P. Then XA = V(IA ) is
A-flat if if (*) lifts to an exact sequence
PℓA
(**)
(ri′j )
( fi′ )
−−→ PA → PA /IA → 0.
−−−→ Pm
A −
In fact we have seen (cf. Proposition 3.1) that in order that (**) be exact
it suffices that (**) be a complex at Pm
A , i.e., X A = V(IA ) is A-flat iff there
is a matrix (ri′ j ) over PA extending (ri j ), such that
!
′ · · · r′
r11
′
′
11
( f1 , . . . , fm ) ′
=0
′
rm1 · · · rm1
′
Set ri j = ri j + tsi j , where si j ∈ P. Then XA is A-flat iff exists(si j over P
such that the matrix product
( f + gt)(r + st) = f r + t(gr + f s) = 0
where f = ( fi ), . . . , r = (ri j ). Now f r = 0 since (*) is exact. Thus 26
the flatness of XA over A is equivalent with the existence of a matrix is
s = (si j ) over P such that
(*)
(gr) + ( f s) = 0.
Consider the homomorphism
(g) : Pm → P
defined by (gi j ). Then condition (*) implies that (g) maps ImPℓ under
the homomorphism (ri j : Pℓ → Pm into the ideal I. Hence (g) induces a
homomorphism
(g) : Pm /Im(r) = I → P/I,
22
1. Formal Theory and Computations
i.e., g is an element of HomP (I, P/I) ≃ Hom p/I (I/I 2 , P/I) = HomOX
(I/I 2 , OX ) = the dual of the coherent OX module I/I 2 on X. The sheaf
HomOX (I/I 2 , OX ) = NX is called the normal sheaf to X in An . Thus
g is a global section of NX . Conversely, given a homomorphism g :
HomP (I, P/I) and a lifting of g to g = Pm → P, then (g) satisfies (*).
This shows that we have a surjective map of the set of first order deformations onto H o (X, NX ). Suppose we are given two liftings ( f + tg1 ) and
( f + g2 t) such that g1 , g2 define the same homomorphisms of I/I 2 into
P/I, i.e., g1 = g2 . We claim that if IA = (( fi +tg1 , i) and JA = (( fi +tg2,i )),
then IA = JA , i.e., the two liftings define the same sub-scheme of AnA .
This will prove that the canonical map
j : (First order def. ofX) → H o (X, NX )
27
is injective, which implies, since j is surjective, that j is needed bijective. The proof that IA = JA is immediate; from the computation in § 4
we see that
(( fi + tg2,i )) ⊂ (( fi + tg1,i )) ⇔ (JA ⊂ IA )
⇔ ∃(m × m) matrices (αi j ) and (βi j ) over P such that
 
 f1 
 
(a) (αi j − id)  ...  = 0
 
fm

  


 f1   g2,1 
 g1,1 

  


(b) (αi j )  ...  + (βi j )  ...  =  ... .

  


g2,m
fm
g1,m
As was done in § 4, if (g1 ) = (g2 ), i.e.,
g1i − g2i ≡ 0(mod I) ∀i,
Then there exists (βi j ) such that (a) and (b) are satisfied with (αi j ) = Id.
This implies that JA ⊂ IA . In a similar manner IA ⊂ JA , which proves
that JA = IA . Thus we have proved
6. First order deformations of arbitrary X ...
23
Theorem 6.1. The set of The set first order embedded deformations of X
in An is canonically in one-one correspondence with NX (or H o (NX ))–
the normal bundle to X of X ⊂ An .
Remark 6.1. Suppose now we are given
0 →∈ A′ → A′ → A → 0 exact
where A′ , A are finite Artin local rings such that rkk (ǫA′ ) = 1 (in partic- 28
ular it follows easily that ǫ is an element of square 0). We see easily that
ǫA′ has a natural structure of an A-module and in fact ǫA′ ≃ A/mA as
A-module. (Given any surjective homomorphism A′ → A of Artin local
rings by successive steps this can be reduced to this situation.) Suppose
now that XA is a lifting of X defined by V(IA ), IA = ( f1′ , . . . , fm′ ) with fi′ s
as liftings of fi , X = V(I), I = ( f1 , . . . , fm ). We observe that even if A is
not of the form k[t]/t2 , in order that XA be flat/A and be a lifting of X it
is neccessary and sufficient that there exists a matrix (ri′ j ) such that
( f1′ , . . . , fm′ )(ri′ j ) = 0
( fi′ are liftings of fi ).
Suppose now are given two liftings XA1 , and XA2 , of XA , flat over A
defined by V(IA1 ′ ) and V(IA2 ′ ):
IA1 ′ = (g1 , . . . , gm ), IA2 , = (g′1 , . . . , g′m ).
Now XA is defined by an exact sequence
(*)
P1A
(ri′j )
−−−→ Pm
A → P A → P A /IA → 0
such that ⊗A K gives the presentation for X. An easy extension of the
argument given before (cf. Proposition 3.1) for characterization of flatness by lifting of generators for I shows that XA′ ′ , and XA2 ′ are flat/A.
They are presented respectively by
(si j )
(gi )
(s′i j )
(g′i )
(α)
1
P1A′ −−−→ Pm
A′ −−→ P A′ /IA′ , → 0 exact
(β)
−−→ PA′ → PA′ /IA2 ′ , → 0
P1A′ −−−→ Pm
A′ −
1. Formal Theory and Computations
24
such that (α⊗A , A) and (β⊗A , A) coincide with (*) and in fact it suffices 29
that (g)(s) and (g′ )(s′ ) are zero and s, s′ are liftings of r. We can write
g′i = gi + ǫhi ,
s′i j
= si j + ǫti j ,
with
with
hi ∈ P,
and
ti j ∈ P.
Then (g′ )(s′ ) = 0 iff (gi )(ti j ) + (hi )(si j ) = 0, and by the remark about ǫA′
as an A-module (or A′ -module) it follows that the right hand side above
is 0 if and only if
(†)
( fi )(ti j ) + (hi )(ri j ) = 0,
i.e., ∃(ti j ) and (hi ) over p such that this holds. Conversely, we see that
given XA1 , flat/A′ and (†), we can construct XA2 ′ , flat/A′ by defining g′i =
gi +ǫhi and lifting of relations by s′i j = si j +ǫti j . Now (†) gives rise to NX
as before. Thus fixing an XA1 , flat/A′ which is a lifting of XA , flat/A the
set of liftings XA′ of XA over A are in one-one correspondence with NX
(or H o (NX )) or in other words the set of flat liftings XA′ over XA is either
empty or is a principal homogeneous space under NX (or H o (NX )). In
the case A = k, A′ = k[t]/(t2 ) (or more generally A′ = A[ǫ]) we use
XA1 ′ = X ⊗k A′ (resp. XA1 ′ = XA ⊗A A′ ) as the canonical base point, and
using this base point the set of first order deformations gets identified
with NX or H o (NX ).
30
Remark 6.2. Compare the previous proof (cf. § 4) for the calculation
of first order (embedded) deformations of X − −a complete intersection
in An . In that proof we required the knowledge of the relations between
fi′ where X = V(I), I = ( f1 , . . . , fm ) whereas this id not required in the
present proof (we assume X arbitrary). This is because in the proof for
the complete intersection we obtained a little more, namely, (First order
embedded deformations of X)↔ {gi }, gi ∈ OX . The previous proof gives
also the fact that NX is free over OX of rank=codim X. Hence a better
proof for the case of the complete intersection would be to prove first the
general case and then show that I/I 2 is free of over OX of rank= codimX
(in An ) (this implies that NX = HomOX (I/I 2 , OX ) is free of rank=codim
X).
6. First order deformations of arbitrary X ...
25
Remark 6.3. The above proof also generalizes to the computation for
first order deformations of the Hilbert scheme or more generally the
Quotient scheme (in the sense of Grothendieck) as well as the consideration of Remark 6.1 above.
Schlessinger’s T 1 :
We have computed above the first order embedded deformations of
X. Now to compute the first order deformation of X (as we did in § 4)
we have to identify two first order embedded deformations XA , XA′ (⊂
AnA )A = k[t]/(t2 ) when there is an isomorphism θ : XA ≃ XA′ which
induces the identity auto-morphism X → X. As we saw before, the
isomorphism θ is induced by a change of coordinates in AnA , i.e., if Xν
are the canonical coordinates of An we started with and θ∗ (Xν ) = Xν′
then we have
Xν′ = Xν + ϕν (X)t, ν = 1, . . . , n(X ⊂ An ).
If a first order embedded deformation of X = V(I), I = ( fi ) is given by 31
( fi + gi t), then
fi + gi t 7→ fi ((Xν + tϕν ) + g((Xν + ϕν t))t
)
(X
∂ fi
ϕν (x) + tgi (X)
= fi (X) + t
∂Xi


n


X


∂ fi


= fi (X) + t 
gi (X) +
ϕν (X)
.




∂X
ν=1
′
Let λ be the canonical image of (gi ) in NX nad λ the canonical image
n
P
∂ fi
of gi +
∂Xν ϕν (X) in N X . Now ϕl , . . . , ϕn are arbitrary elements of the
ν=1
coordinate ring of An , and the canonicla image of
n
P
ν=1
∂ fi
∂Xν ϕν (X)
precisely the image under the canonical homomorphism
Θ n → N
A X
in NX is
X
where ΘAn reprsents the tangent bundle of An and X denotes its restriction to X. We have a natural exact sequence
I/I 2 → Ω1 → Ω1 → 0
An X
X
1. Formal Theory and Computations
26
where Ω1Z denotes the K ahler differentials of order one on a scheme
X/k. The dual of this exact sequence gives an exact sequence
0 → Θ → Θ1 → N ,
X
32
An X
X
where Θ1An X → NX is the homomorphism defined above. We define T X1
to be the cokernel of this homomorphism. So that we have
0 → Θ → Θ1 → N → T 1 → 0.
X
An X
X
X
Thus we have proved
Theorem 6.2. The first order deformation of X are in one-one correspondence with T X1 .
Remark 6.4. Suppose that A′ = A[ǫ] (i.e., A′ = A⊕ ∈ k). Then the
argument which is a combination of that of the theorem and Remark 6.1
above shows that
Def(A′ ) = Def(A) × Def(k[ǫ]),
i.e., the set of deformations of X over A′ which extend a given deformation over A are in one-one correspondence with the first order deformation of X. For the proof of this, we remark that a similar observation has
been proved above for embedded deformations. Now identifying two
embedded deformations XA′ and XA′ ′ which reduce to the same embedded deformation XA over A, the argument is the same as in the discussion
preceding Theorem 6.2, and the above remark then follows.
Remark 6.5. Deformations over A′ which extend a given deformation
over A form a prinipal homogeneous space under Def(k[ǫ]), or else form
the empty set.
33
Remark 6.6. Note that if X is smooth, then T X1 = 0. This implies that
any two embedded deformations XA , XA′ (A = k[ǫ]) of X are isomorphic
over A (in fact obtainable by a change of coordinates in AnA ).
Remark 6.7. If X has isolated singularities, T X1 as a vector space over k
has finite dimension. For, it is a finite OX -module with a finite set as
support.
6. First order deformations of arbitrary X ...
27
Proposition 6.1. Suppose that X = Xred . Then
T X1 ≃ Ext1OX (Ω1X , OX )(X ⊂ An ).
Proof. Let X = V(I). Then the exact sequence
I/I 2 → Ω1An X → Ω1X → 0
can be split into exact sequence as follows:
(i) 0 →∈→ I/I 2 → F → 0
(ii) 0 → F → Ω1An X → Ω1X → 0.
Since X = Xred , the set of smooth points of X is dense open in X, so
that ǫ, being concentrated at the nonsmooth points, is a torsion sheaf, in
particular, HomOX (ǫ, OX ) = 0. Writing the exact sequence Hom(·, OX )
for (i), we get that
0
/ HomO (F, OX )
X
/ Hom(I/I 2 , O )
X
/ 0 is exact .
NX
Writing the exact sequence Hom(·, OX ) for (ii), we get that
34
1
0 → ΘX → ΘAn X → F ∗ → E xtO
(Ω1X , OX ) → 0is exact
X
(since Ext1OX ΩA n X , OX = 0, ΩAn X being free ).
Now T X1 = coker ΘAn X → NX , and coker ΘAn X → F ∗ = Ext1OX
(Ω1X , OX ). Above F ∗ ≃ NX , and so
T X1 ≃ Ext1OX (Ω1X , OX ).
28
1. Formal Theory and Computations
7 Versal deformations and Schlessinger’s theorem
Let R be a complete local k-algebra with k as residue field (k-alg. closed
as before). We write Rn = R/mn+1
R where mR = m is the maximal ideal
of R. We are given a closed subscheme X of An (note that definitions
similar to the following could be given for more general X).
35
Definition 7.1. A formal deformation XR of X is: (i) a sequence {Xn },
Xn = XRn is a deformation of X over Rn , and (ii) iso-morphisms Xn ⊗Rn
Rn−1 ≃ Xn−1 for each n.
Suppose that A is a finite-dimensional local k-algebra. Then a kalgebra homomorphism ϕ : R → A is equivalent to giving a compatible
sequence of homomorphisma ϕn : Rn → A for n sufficiently large. This
is so because a (local) homomorphism ϕ : R → S of two complete local
rings R, S is equivalent to a sequence of compatible homomorphisms
ϕn : R/mnR → S /mnS , and in our case mnA = 0 for n ≫ 0. Given a formal
deformation XR of X and a homomorphism ϕ : R → A; Xn ⊗Rn A (via
ϕn : R → A as above) is up to isomorphism the same for ≫ 0. It is a
deformation of X over A. We define this to be XR ⊗ A (base change of
XR by Spec A → Spec R).
Definition 7.2. A formal deformation XR of X is said to be versal if the
following conditions hold: Given a deformation XA of X over a finite
dimensional local k-algebra A, there exists a homomorphism ϕ : R →
A and an isomorphism XR ⊗ A ≃ XA ; in fact, we demand a stronger
θ
− A of local
condition as follows: Given a surjective homomorphism A′ →
k-algebras, a deformation XA′ over A′ a homomorphism ϕ : R → A
and an isomorphism XA ⊗R A ≃ XA′ ⊗ A, there is a homomorphism
′
ϕ : R → A′ such that ϕ′ lifts ϕ and XR ⊗R A′ is isomorphic to XA′ . (Note
that it suffices to assume the lifting property in the case ker θ is of rank
1 over k.)
Let F : (Fin.loc.k − alg) → (Sets) be the functor defining deformations of X, i.e., F(A) = (isomorphism classes/A of deformations of
X/A). Given a formal deformation XR of X, let
G : (Fin.loc.k − alg.) → (Sets)
7. Versal deformations and Schlessinger’s theorem
29
be defined by G(A) = Homk−alg .(R, A). Then we have a morphism j :
G → F of functors defined by
ϕ ∈ Homk−alg .(R, A) → XR ⊗R A.
That a formal deformation is versal is equivalent to saying that the functor j is formally smooth.
36
Theorem 7.1. (Schlessinger). Let F :(Finite, local, k-alg.)→ (Sets) be a
(convaraint) functor. Then there is a formally smooth functor (as above)
i.e.,
HomK−alg .(R, ·) → F(·),
where R is acomplete local k-algebra with residue filed k, if
(1) F(k) = a single point.
(2) Given (ǫ) → A′ → A → 0 with A′ , A finite local k-algebras and
(ǫ) = Ker(A′ → A) of rank 1 over k and a homomorphism ϕ : B →
A let B′ = A′ ×A B{(α, β) ∈ A′ ×B such that their canonical images in
A are equal }. (Spec B′ is the “gluing” of Spec B and Spec A′ along
Spec A by the morphisms Spec A → Spec A′ . If ϕ is surjective, i.e.,
Spec A → Spec B is also a closed immersion, this is a true gluing.)
Then we demand that the canonical homomorphism
F(B′ ) → F(A′ ) ⊗F(A) F(B)
is surjective. (Note that B′ is also a finite-dimensional local kalgebra.)
(3) In (2) above, take the particular case A = k and A′ = k[ǫ] (ring of
dual numbers), then the canonical map defined as in (*) above
F(B′ )
/ F(k[ǫ]) ×F(k) F(B)
(F(B[ǫ])
/ F(k[ǫ]) × F(B))
is bijective, not merely surjective.
1. Formal Theory and Computations
30
(4) F(k[∈]) is a finite-dimensional vector space over k.
Remark 7.1. One first notes that F(k[ǫ]) has a natural structure of a
vector space over k without assuming the axiom (4) above: Given c ∈ k,
we have an isomorphism k[ǫ] → k[ǫ] defined by ǫ → c · ǫ (of course
1 → 1) which defines a bijective map
c∗ : F(k[ǫ]) → F(k[ǫ]).
By this we define “multiplication by c ∈ k” on F(k[ǫ]). By the third
axiom we have a bijection
F(k[ǫ1 , ǫ2 ]) ≃ F(k[ǫ1 ]) × pt F(k[ǫ2 ])
(k[ǫ1 , ǫ2 ] being the 4-dimmensional k-algebra with ǫ12 = ǫ22 = 0 and
basis 1, ǫ1 , ǫ2 , ǫ1 ǫ2 ). We have canonical homomorphism k[ǫ1 , ǫ2 ] → k[ǫ]
defined by ǫ1 → ǫ, ǫ2 → ǫ which gives a map F(k[ǫ1 , ǫ2 ]) → F(k[ǫ]), so
that we get a canonical map
F(k[ǫ]) × pt F(k[ǫ]) → F(k[ǫ]).
Here we use the fact that
F(k[ǫ] ⊗k k[ǫ]) = F(k[ǫ]) × pt F(k[ǫ])
This follows by axiom (3), and
k[ǫ] ⊗k k[ǫ] ≃ k[ǫ, ǫ ′ ].
38
We define addition in F(k[ǫ]) by this map, and then we check that this
map is bilinear. This gives a natural structure of a k-vector space on
F(k[ǫ]) if axioms (1), (3) hold.
Remark 7.2. The versal R can be constructed with mR /m2R ≃ F(k[ǫ]).
Then with this condition R is uniquely determined up to iso morphism.
(Note: We do not claim that given XA the homomorphism R → A is
unique.) The functor represented by R on (finite local k-alg.) is called
the hull of F which is therefore uniquely determined up to automorphism.
37
7. Versal deformations and Schlessinger’s theorem
31
Proof of Schlessinger’s Theorem 1. Let Cn denotes the full subcategory of (fin.loc.k-alg.) consisting of the rings A such that mn+1
= 0.
A
This category is closed under fibered products. We will show the existence of R by finding a versal Rn for F | Cn , for all n inductively.
Take the case n = 1. C1 is just the category of rings of the form A =
k ⊕ V where V is a finite-dimensional vector space. So, C1 is equivalent
to the category of finite-dimensional vector spaces. Let V be the dual
space to F(K[ǫ]), and put R1 = k ⊕ V. Then a map R1 → k[ǫ] is given by
a map V → ǫk. i.e., an element of F(k[ǫ]). So, Hom(R1 , k[ǫ]) = F(k[ǫ]).
Since, by axiom (3), F is compatiable with products of vector spaces,
R1 represents F | C1 .
Suppose now that Rn−1 is given, versal for F/Cn−1 , and let un−1 ∈
F(Rn−1 ) be the versal element. Let P be a power series ring mapping
onto Rn−1
0 → Jn−1 → P → Rn−1 → 0.
Choose an ideal Jn ,
Jn−1 ⊃ Jn ⊃ mJn−1 ,
which is minimal with respect to the property that un−1 lifts to un ∈ 39
F(P/Jn ), and put Rn = P/J. We test (Rn , un ) for versality. Let A′ → A
be a surjection with length 1, kerel Cn , and let a test situation be given:
a′ ∈ F(A′ )
A′
O
un
/ A,
Rn
O
O
/o /o /o /o /o o/ / a
Form R′ = Rn ×A A′ :
~
~
R
~>
~
P@
@@
@@
@@
′
Rn
/ A′
u′ /o /o o/ / a′
O
/A
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
un /o /o o/ / a
1. Formal Theory and Computations
32
Since P is smooth, a dotted arrow exists. By axiom (2), there is a u′ ∈
F(R′ ) mapping to un and a′ . Since Jn was minimal, R′ cannot be a
quotient of P. Hence im(P → R′ ) ≈ im(P → Rn ) = Rn , and so R′ → Rn
splits:
R′ ≈ Rn [ǫ] = Rn ×k k[ǫ].
Let v′ be the element of F(R′ ) induced from un by the splitting. The
versality will be checked if u′ = v′ , for then the map Rn → R′ → A′ is
the required one.
We can still change the splitting, and the permissible changes are by
elements of Hom(Rn , k[ǫ]) = Hom(R1 , k[ǫ]) = F(k[ǫ]). By axiom (3),
F(R′ ) = F(Rn ) × F(k[ǫ]). Both u′ and v′ have the same image un in
F(Rn ). So, we can make the required adjustment. This completes the
proof of Schlessinger’s theorem.
8 Existence formally versal deformations
40
Theorem 8.1. Let X ⊂ An be an affine scheme over k with isolated
singularities. Then for the functor F = Def.X = Def the conditions of
Schles-singer’s theorem are satisfied. In particular X admits a versal
deformation.
Proof. Since X has isolated singularities, we have rankk (Def.X)[k[ǫ]] =
rankk (T X1 ) < ∞. Hence it remains only to check the axioms (2) and (3).
Axiom (2): Given (ǫ) → A′ → A, ϕ′ : B → A a deformation XA of X
and two deformations XA′ , XB over XA
X A′
XB
!a a!
a! a!
!
XA
~
>~ ~>
>~ >~
Let B′ = A′ ×A B. We need to find a deformation XB′ over B′ inducing the given deformations XA′ and XB over A′ and B respectively. As
usual we write OXA′ = OA′ , . . ., etc. We now set OB′ = OA′ ×OA OB . We
have canonical homomorphisms B′ → B and B′ → A′ . It is easily seen
8. Existence formally versal deformations
33
that OB′ ⊗B′ B ≈ OB and OB′ ⊗B′ A′ ≈ OA′ . The only serious point to
check is that OB′ is flat/B′ . Since B′ = A′ ×A B, we have a diagram
0 → (ǫ) → B′ → B → 0
0 → (ǫ) → A′ → A → 0
with exact rows. (ǫ) is of rank 1 over k, so (ǫ) ≈ k as B′ modules.
Similarly,
0
/ ∈ Ok
0
/ ∈ Ok
/ O B′
/ OB
/0
/ O A′
/ OA
/ 0.
It follows easily that the exact sequence o →∈ Ok → OB′ → OB → 0 41
is obtained by tensorting 0 → (ǫ) → B′ → B → 0 wiht OB′ , and that
ǫ · Ok ≈ Ok . Now the faltness of OB′ is a consequence of
Lemma 8.1. Suppose that 0 → (ǫ) → B′ → B → 0 is exact with B′ , B
finite local k-algebras and rkk (ǫ) = 1 (so that (ǫ) ≈ k as B′ module).
Suppose that XB′ = Spec OB′ , is a scheme over B′ such that OB′ ⊗B′ B =
OB is flat over B,with XB = Spec OB a deformation of X = Spec Ok , and
that ker(OB′ → OB ) is isomorphic to Ok (as OB′ module). Then OB′ is
B′ flat; in particular, XB′ = Spec OB′ is a deformation of X over B′ .
Conversely, if OB′ is B′ flat giving a deformation of X = Spec Ok ,
OB′ has an exact sequence representation as in the lemma.
Proof. We have an exact sequence
0 → ǫOk → OB′ → OB → 0,
with ǫOk ≈ Ok . We claim that this is obtained by tensoring
(*)
0 → (ǫ) → B′ → B → 0
by OB′ and in fact that it remains exact because of our hypothesis that
Ker(OB′ → Ol ) is ≈ Ok as B′ -module. For, tensoring (*) by OB′ we have
ǫ ⊗ OB′ → OB′ → OB → 0 exact.
1. Formal Theory and Computations
34
But ǫ ⊗ OB′ ≈∈ ·Ok ≈ Ok . It follows then that the canonical - homomor- 42
phism ∈ ⊗OB′ → OB′ , is injective, i.e., (*) remains exact when tensorted
by OB′ . Consider
0
0
(R1)
/0
/ (ǫ)
/ m B′
/ mB
/0
(R2)
/0
/ (ǫ)
/ B′
/B
/0
⊗ B′ O B′ .
k
k
(C1)
(C2)
Tensoring (C1) by OB′ (over B′ ) and using the Tor sequence we
′
find that Tor1B (OB′ , k) = 0 iff OB′ flat iff the canonical homomorphism
′
OB′ ⊗B′ mB′ → OB′ ⊗B′ B′ = OB′ is injective. (We use the fact Tor1B
′
(OB′ , B ) = 0.) Now (C2) is an exact sequence of B-modules, and tensoring it by OB′ (over B′ ) amounts to tensoring it by OB (over B). Hence
(C2) ⊗B′ OB′ stays exact since OB is B-flat. Finally
OB′ ⊗B′ (ǫ)
(R1) ⊗B′ OB′
i
/
OB′ ⊗B′ m B′
a′
(R2) ⊗B′ OB′
43
0
/ OB′ ⊗B′ (ǫ)
/
OB′ ⊗B′ m B′ B′
/
/
OB′ ⊗B ′ m B
_
/
a
OB′ ⊗B′ B
/
0
exact
0
exact
(R2) ⊗B′ OB′ is exact as we observed above. To prove that OB′ is B′
flat, it is equivalent to proving that α′ is injective, i.e., Kerα′ = 0. Note
that OB′ ⊗B′ B ≈ OB and OB′ ⊗B′ mB ≈ OB ⊗B mB . Since OB is B-flat,
α is injective, and form the diagram it follows that α′ is also injective.
Further discussion of Axiom (2) and Axiom (3): Suppose we are given
0 → (ǫ) → A′ → A → 0 with rkk (ǫ) = 1 and a k-algebra homomorphism ϕ : B → A. Set B′ = B ×A A′ . Let us consider the canonical map
(*) in Axiom (2) of Schlessinger’s theorem for the functor Def.(X) in
more detail, (i.e., the map Def.(B′ ) → Def.(A′ ) ×Def.(A) Def.(B)). Suppose we are given deformations (not merely isomorphism classes) XB′
8. Existence formally versal deformations
35
over B′ , XA′ over A′ and XB over B and isomorphisms
ν1
ν2
XB′ ⊗B′ B −→ XB , ⊗B′ A′ −→ XA .
Now νi induce isomorphisms
ν1 ⊗ B 1 A
(XB′ ⊗B′ B) ⊗B A −−−−−→ XB ⊗B A,
(ν2 ⊗A′ 1A )
(XB′ ⊗B′ A′ ) ⊗A′ A −−−−−−−→ XA′ ⊗A′ A.
But now there is a canonical isomorphism
(XB′ ⊗B′ B) ⊗B A → (XB′ ⊗B′ A′ ) ⊗A′ A.
Hence the νi determine an isomorphism
(*)
θ
(XB ⊗B A) →
− (XA′ ⊗A′ A)
By the universal property of the “join”, we get a morphism
f : X B′ → Z B′
where ZB′ = Spec(OB ×OA OA′ )(XA = Spec OA is chosen to be one of the
objects in (*) and the homomorphisms OB → OA , OA′ → OA are then 44
defined uniquely but the fibre product OB ×OA OA′ is independent of the
choice for XA : It is fibre product of OB and OA′ by
OB OO
k O A′
OOO
kkkk
k
k
OOO
k
k
OOO
kkkk
O'
ukkkk
OB ⊗B A ∼θ OA′ ⊗A′ A
and thus well defined). We claim that f is is an isomorphism. From this
claim it follows as a consequence that given deformations XB and XA′
of X such that XB ⊗B A is isomorphic to XA′ ⊗A′ A i.e., given point ξ ∈
(def.X)(B) ×Def(X)(A) (Def.X)(A′ ), a deformation XB′ , which lifts XB and
XA′ depends (up to isomorphism) only on the chioce of the isomorphism
XB ⊗B A ≈ XA′ ⊗A′ A, so in particular, we have a surjective map
(*)
Isom(XB ⊗B A → XA,⊗A) → λ−1 (ξ)
1. Formal Theory and Computations
36
45
where λ : (Def.X)(B′ ) → (Def.X)(B)×(Def.X)(A) (Def.X)(A′ ) is the canonical map of Axiom (2). ((Def.X)(B) by definition=isomorphism classes
of deformations of X, i.e., all deformations of XB′ of X over B′ modulo
isomorphisms which induce the identity map on X.) Take the particular
∼
∼
case A = k, A′ = k[ǫ], ϕ : B → A; then XB ⊗B A −−→ X, XA′ ⊗A′ A −−→ X
can
can
and then the left hand side of (*) consists of a unique element, namely,
the one induced by the identity map X → X. This implies that λ−1 (ξ)
consists of a unique element, and completes the verification of Axiom
(3).Thus, finally it suffices to prove that the morphism f : XB′ → ZB′
is an isomorphism as claimed above. Note that ( f ⊗ k) is the identity
map X → X and that XB′ , ZB′ are flat over B. Hence our claim is a
consequence of
Lemma 8.2. Let XA1 = Spec OA1 and XA2 = Spec OA2 be two deformations
of X = Spec Ok and f ∗ : XA2 → XA1 ( f : OA1 → OA2 a homomorphism
of k-algebras) a morphism such that f ∗ ⊗ k is the identity ( f ⊗ k is the
identity). Then is an isomorphism. (In the proof, it would suffice to
assume OA2 flat/A).
Proof. The OAi can be realized as embedded deformations of X =
Spec Ok , so that we have a diagram
0
/ I1
A
/ PA
/ O1
A
/0
f
0
/ I2
A
/ PA
/ O2
A
/0
Let Xν′ = f (Xν ) where Xν are the variables in Pn . We have Xν′ = Xν + ϕν
where ϕν ∈ mA (X1 , . . . , Xn )(mA = max .ideal ofA). It follows easily
since A is finite over k (as we have seen before) that Xν 7→ Xν′ is just
a change of coordinates in AnA , so that f is induced by an isomorphism
∼
PA −
→ PA . We can assume without loss of generality that this is the
identity. Then it follows that f is induced by an inclusion IA1 ⊂ IA2 . This
implies that f is surjective. Let J = Kerf so that
0 → J → OA1 → OA2 → 0 is exact.
9. The case that X is normal
37
Since OA2 is A-flat, it follows that
f ⊗k
0 → J ⊗A k → OA1 ⊗ k −−−→ OA2 ⊗ k → 0
is exact.
Since f ⊗ k is the identity, it follows that (J ⊗ k) = 0 Since mA is in 46
the radical of OA1 , by Nakayama’s lemma, it follows that J = 0; hence f
is an isomorphism. This completes the proof of the theorem.
9 The case that X is normal
Let X ֒→ An be normal of dimension ≥ 2. Let U = X − SingX (Sing
X=Singular points of X). We use the well-known
Proposition 9.1. Let Z be any smooth (not necessarily affine) scheme
over k. Then the set of first order deformations of Z is in one-to-one
correspondence with H 1 (Z, ΘZ ), i.e.,
(Def.Z)(k[ǫ]) ≈ H l (Z, ΘZ )
(ΘZ
tangent bundle ofZ).
Proof. If Z is affine, we have seen that any first order deformation of
Z is trivial, i.e., it is isomorphic to base change by k[ǫ] (this was a con
sequence of the fact that when Z ֒→ An and Z is smooth, T Z1 = (0)).
Hence any first order deformation of Z is locally base change by k[ǫ].
Hence if {Ui } is an affine covering of Z, a first order deformation of Z is
given by {ϕi j }
ϕi j : (Ui ∩ U j ) ⊗ k[ǫ] → (Ui ∩ U j ) ⊗ k[ǫ],
{ϕi j } being automorphisms of (Ui ∩ U j ) ⊗ k[ǫ] satisfying the cocycle
condition. It is easy to see that first order deformations correspond to
cohomology classes. Now it is easy to see that for an affine scheme 47
W/k, (W = Spec B)
Autk[ǫ] (W ⊗ k[ǫ]) = Derivations of B/k(= H o (W, ΘW )).
The proposition now follows.
1. Formal Theory and Computations
38
Lemma 9.1. Let XA1 , XA2 be two deformations of X (not necessarily of the
first order, A =local, finite over k). Let U i = XAi U and let ϕ : U A1 → U A2
be an isomorphism over A. Then ϕ extends to an isomorphism (unique)
XA1 → XA2 .
Proof. If XA is a deformation of X as above, we shall prove
(*)
H o (X , O ) = H o (U , O )(U = X ).
A
A
A
UA
A
AU
The proposition is an immediate consequence of (*), for ϕ induces a
homomorphism
ϕ∗ : H o (U A2 OA )
/ H o (U 1 ϕA )
A
H o (XA2 , OX 2 )
/ H o (X 1 , OX 1 ).
A
A
A
This implies that ϕ is induced by a morphism ψ : XA1 → XA2 . It is
easily seen that (ψ ⊗ k) is the identity, and this implies easily that ψ is an
isomorphism.
To prove (*), we note first that if A = k, it is well known. In the
general case, we have a representation
0 → (ǫ) → A → Ao → 0 exact,
48
rkk (ǫ) = 1
By induction it suffices to prove (*) assuming its truth for XAo = XA ⊗A
Ao . Then we have
0 → ǫOXA → OXA → OXAo → 0 exact,
with ǫ · OXA ≈ Ok . This is because XA is flat over A. As a sheaf its
restriction to U is also exact. Then we get a diagram
0
/ H o (X, Ok )
≀
0
/ H o (U, Ok )
/ H o (XA , OA )
/ H o (U A , OA )
/ H o (XA , OA )
o
o
/0
≀
/ H o (U A , OA ).
o
o
The first and the last vertical arrows are isomorphisms, and it follows
then that the middle one is also an isomorphism. This proves the proposition.
9. The case that X is normal
39
Remark 9.1. The proposition is equivalent to saying that the morphism
of functors
(Def. X) → (Def. U)
obtained by restriction to U is a monomorphism.
Lemma 9.2. Suppose that X has depth ≥ 3 at all the points (not merely
closed points) of Sing X (e.g., dim X ≥ 3 and X is Cohen-Macaulay)
(X necessarily normal). Then every deformation U A of U extends to a
deformation XA of X.
Proof. Define XA by XA = Spec OXA with OA = OXA = H o (U A , OA ).
Suppose we have a presentation 0 → (ǫ) → A → Ao → 0 rkk (ǫ) = 1.
We prove this lemma again by an induction as in the previous proposi- 49
tion. This induces an exact sequence of sheaves by flatness
0
/ OU
A
/ OU
/ OU A
o
/0
·
(ǫ · OU )
Since depth of OX,x at x ∈ SingX ≥ 3, by local cohomology the maps


H 1 (X, OK ) → H 1 (U, OK )

are isomorphisms

o
o

H (X, OK ) → H (U, OK )
∼
The isomorphisms H i (X, Ok ) −
→ H i (U, Ok ) follow from Theorem ??,
p. 44, in Hartshorne’s Local cohomology. Indeed, we have the following exact sequence 0 → OX → OU F → 0. We must prove that
H o (F ) = H 1 (F ) = 0. Theorem ?? (loc. cit) states that for X a locally Noetherian prescheme, Y a closed pre-scheme and D a coherent
sheaf on X, the following conditions are equivalent: (i) Hyi (D) = 0,
i < n. (ii) depthY D ≥ n. Taking D = OX and Y = SingX, the
fact that H o (F ) = H 1 (F ) = 0 follows immediately. In particular,
H 1 (U, Ok ) = 0. Hence we get
0 → H o (U, OU ) → H o (U, OU A ) → H o (U, OU Ao ) → 0 is exact.
40
1. Formal Theory and Computations
Now H o (U, OU ) = H o (X, Ok ) = Ok and
H o (U, OU Ao ) = OAo flat/Ao
50
by induction hypothesis.
Hence
0 → Ok → OA → OAo → 0
is exact
and OAo flat/Ao . By proposition 3.1 it follows that OA is flat/A and so
XA represents a deformation of X.
Proposition 9.2. Let X ֒→ An be such that X is of depth ≥ 3 at ∀x ∈
Sing X. Then the morphism of functors
Def(X) → Def(U),
U = X − SingX
obtained by restriction is an isomorphism, i.e.,
(Def X)(A) → (Def U)(A) is an isomorphism ∀A
local finite over .k.
Proof. Immediate conseqence of the above two lemmas.
Remark 9.2. Suppose X ֒→ An is normal (with isolated singularities).
Then if ΘX is of depth ≥ 3 at ∀x ∈ Sing X is rigid. For, it suffices to
prove that first order deformations of X are trivial and by the preceding
to prove this for U. We have an isomorphism
H 1 (U, ΘU ) ← H 1 (X, ΘX )
because of our hypothesis. But H 1 (X, ΘX ) = 0 Since X is affine. This
implies the assertion.
10 Deformation of a quotient by a finite group action
51
Theorem 10.1. (Schlessinger–Inventiones’70). Let Y be a smooth affine
variety over k with chark = 0. Suppose we are given an action of a finite
group G on Y such that the isotropy group is trivial except at a finte
number of points of Y, so that the normal affine varirty X = Y/G has
10. Deformation of a quotient by a finite group action
41
only isolated singularities (at most the images of the points where the
isotropy is not trivial). Then if dimY = dim X ≥ 3, X is rigid. (It
suffices to assume that the set of points where isotropy is not trivial is
of codim≥ 3 ion Y. Of course, the singular points of X need not be
isolated, but even then X is rigid.)
Proof. Let U = X − SingX, and V = Y-(points at which isotropy is not
trivial). Then the canonical morphism V → U is an etale Galois covering with Galois group G. From the foregoing discussion, it suffices to
prove that H 1 (U, ΘU ) = 0 We have the Cartan-Leray spectral sequence
H p (G, H q (V, θV )) ⇒ H p+q (U, f∗G (θV ))
where f∗G (θV ) denotes the G-invariant subsheaf of the direct image of
the sheaf of tangent vectors on V. We have θU = f∗G (θV ). Hence the
above spetral sequence gives the spectral sequence
H p (G, H q (V, θV )) ⇒ H p+q (U, θU ).
Now H P (G, H q (V, θV )) = 0 for p ≥ 1 since G is finite and the characteristic is zero. Hence the spectal sequence degenerates and we have
isom.
H o (G, H q (V, θV )) −−−−→ H q (U, θU ).
In particular, H 1 (U, ΘU ) ≃ H o (G, H 1 (V, ΘV )). Now ΘY is a vector bun- 52
dle, and since dim Y ≥ 3 and (Y − V) is a finite number of points, ΘY is
of depth ≥ 3 at every point of (Y − V) (or because codim (Y − V) ≥ 3).
Hence H 1 (V, ΘV ) = H 1 (Y, ΘY ) = 0 It follows then that H 1 (U, ΘU ) = 0,
which proves the theorem.
Remark 10.1. Let Y be the (x, y)-plane, G = Z/2 operating by (x, y) 7→
(−x, −y). Then X = Y/G can be identified with the image of the (x, y)
plane in 3 space by the mapping (x, y) 7→ (x2 , xy, y2 ), so that X can
be identified with the cone v2 = uw in the 3 space (u, v, w). It has an
isolated singularity at the origin, but it is not rigid (cf., the computation
of T X1 which has been done for the case of a complete intersection, § 4
and § 6).
1. Formal Theory and Computations
42
11 Deformations of cones
Let X be a scheme and F a locally free OX -modlue of constant rank
r. The vector bundle V(F) associated to F is by definition Spec S (F),
i.e., “generalized spec” of S (F)–the sheaf of symmetric algebras of the
OX − module F. We get a canonical affine morphism p : V(F) → X.
The sections of V(F) over X (morphisms s = X → V(F) such that
p ◦ s = IdX can be identified with
HomOX − alg(S (F), OX ) ≈ HomOX − mod, (F, OX ) ≈ H o (X, F ∗ ),
53
where F ∗ is the dual of F. With this convention, sections of the vetor
bundle V(F) considered as a scheme over X are ≈ sections of F ∗ over
X.
π
Let PN+1 − (0, . . . , 0, 1) →
− PN be the “projection of PN+1 onto PN
from the point (0, . . . 0, 1)”, i.e., π is the morphism obtained by dropping
the last coordinate. Let L = PN+1 − (0, . . . , 0, 1). Then we see that L has
a natural structure of a line bundle over PN , and in fact

 ∞

M
OPN (−n) .
L ≃ Spec 
n=0
Proof that L ≃ Spec
n=0 OPN (−n) : We first remark that in general given F a locally free OX -module of constant rank r, the vector bundle V(F) = Spec S (F) associated to F (in the definition of Grothendieck)
is such that the geometric points of V(F) correspond to the dual of
the vector bundle F associated to F in the “usual” way, that is, F becomes theL
sheaf of sections
of F. Consequently, in order to prove that
∞
L ≃ Spec
n=0 OPN (−n) we must show that the invertible sheaf corresponding to L → PN is OPN (1). We do this by calculating the transition functions. If we denote the standard covering of PN by u0 , . . . , uN ,
π
then we see for PN+1 − (0, 0, . . . , 0, 1) →
− PN we have π−1 ((X0 , . . . , XN ))
is of form (X0 , . . . , XN , λ) and hence the patching data is of the form
L∞
Xi /X j
π−1 (ui ∩ u j ) ≃ C × (ui ∩ u j ) −−−−→ C × (ui ∩ u j ) ≃ π−1 (ui ∩ u j )
(where Xi are the standard coordinate functions on PN ). This is precisely the patching data for OPN (1). Consequently, OPN (1) is the invertπ
ible sheaf associated to PN+1 − (0, . . . , 0, 1) →
− PN . The set of points
11. Deformations of cones
43
PN ≈ S = (∗, . . . , ∗, 0) ⊂ PN+1 give a section of L over PN ; we can
identify this as the 0-section of the line bundle L. We have a canonical
∼
− (PN+1 − S ) sending (0, . . . , 0) ← (0, . . . , 0, 1),
isomorphism AN+1 ←
and hence L-(0-section)≈ AN+1
− (0, . . . , 0). Futher more, it is easily
L
∞
seen that L-(0-section)= Spec
−∞ OPN (−n) and it is the Gm bundle 54
associated to L.
Suppose now that Y is a closed subscheme of PN . Let LY = π−1 (Y).
Then LY is a line bundle over Y and from the preceding we have

 ∞

M
OY (−n) .
LY = Spec 
n=0
Let C be the cone over Y, i.e., if P is the canonical morphism P : (AN+1 −
(0, . . . , 0)) → PN induced by the isomorphism AN+1 ← (PN+1 − S )
defined above, we define C ′ = P−1 (Y) and then C ′ = C − (0, . . . , 0). The
point (0, . . . , 0) is the vertex of the cone C. So, C = Closure ofC ′ in
AN+1 . As before we have
 ∞

 M

LY − (0 − section) = Spec 
OY (n) .
n=−∞
Let C be the closure of C in PN+1 (C being identified iin PN+1 as above).
Then we see that
C = LY ∪ (0, . . . , 0, 1), C = LY − (0 − section).
We call (0, 0, . . . , 1) the vertex of the projective cone C, for (0, . . . 0, 1)
goes to the vertex of C under the canonical isomorphism (PN+1 − S ) →
AN+1 . If Y is smooth, C is smooth at every point except (possibly) at the
vertex.
Consider π : L → PN . Let T denote the line bundle on L consisting
of tangent vectors tangent to the fibers of L → PN . Then we have
0 → T → ΘL → π∗ ΘPN → 0 quad
Let us now suppose that Y is a smooth closed subscheme of PN . Let
πY : LY → Y be the canonical morphism. Then we have a similar 55
1. Formal Theory and Computations
44
exact sequence. Let T Y denote the bundle of tangents along the fibres of
LY → Y, so that we have a commutative diagram
(A)
0
0
0
0
/ TY
/ ΘL
Y
/ π∗ (ΘY )
Y
/0
0
/ T |L
Y
/ ΘL |L
Y
/ π∗ ((ΘPN )|Y )
/0
0
N LY
∼
/ π∗ (NY )
0
0
where NLY = normal bundle for the immersion LY ֒→ PN+1 , NY =
normal bundle for the immersion Y ֒→ PN . We have T Y ≃ T |LY from
which it follows that
∼
→ Coker[(π∗Y (ΘY )) → π∗ (OL )LY ],
Coker(ΘLY → ΘL|LY ) −
i.e.,
≃
N LY −
→ π∗ (NY ).
Let U = C-(vertex). Then if NU is the normal bundle for U ֒→
it is immediate that NU = j∗ (NLY ) where j : U → LY is the
canonical inclusion. If we denote by the same π the canonical morphism
π : U → Y, we deduce that NU = π∗ (NY ). Then restricting the bundles
AN+1 ,
11. Deformations of cones
45
in (A) to U we get
(B)
(B1)
0
/ T Y |U
(B2)
0
/ T Y |U
0
0
/ ΘU
/ π∗ (ΘY )
/0
/ ΘAN+1 −(0) |U
/ π∗ (Θ N |Y )
P
/0
NU
≃
π∗ (NY )
0
0
56
The exact sequence (B2) is obtained by restriction to U of the exact
sequence
(**)
0 → T N+1
→ ΘAN+1 −(0) → π∗ (ΘPN ) → 0
A
−(0)
where T AN+1 −(0) is the bundle of tangent vectors tangent to the fibres of
π : AN+1 − (0) → PN . We shall now show that we have
0
/ T |U
/ ΘAN+1 −(0) |U
0
/ π∗ (OPN )
/ π∗ (O N (1))N+1
P
(C)
≀≀
≀≀
/ π∗ (ΘPN |Y )
/0
/ π∗ (ΘPN )
/ 0,
≀≀
where the second row is the pull-back π∗ of the well-known sequence
0 → OPN → (OPN (1))N+1 → ΘPN → 0
on PN (the middle term of this exact sequence is the direct sum of OPN (1)
taken (N + 1) times). Let us examine the canonical homomorphism
h : T |U → ΘAN+1 −(0) |U . Let (z0 , . . . , zN ) be the coordinate of AN+1 .
Then H 0 (AN+1 , ΘAN+1 ) is a free module M over P = k[z0 , . . . , zN+1 ] 57
∂
∂
,...,
. We see easily that h is defined by restriction
with basis
∂z0
∂zN+1
to AN+1 − (0) of
ϕ:P→M
1. Formal Theory and Computations
46
∂
(1 is generator of P over P). Hence ϕ is a graded
∂zi
homomorphism, and is therefore defined by a homomorphism of sheaves
on PN . We see indeed ϕ = π∗ (ϕ0 ), where
where ϕ(1) = Σzi
ϕ0 : OPN → (OPN (1))N+1 .
(OPN (1) is defined by homogeneous elements of degree ≥ 1 in P considered as a module over P). From this the assertion (C) follows easily,
and we leave these details to the reader.
Let F be a coherent OPN module. Then we have
H p (L, π∗ F) ≃ H p (PN , π∗ π∗ F).
Now π∗ F is defined by the sheaf of OPN modules
∞
M
F(−n),
0
considered as a sheaf of modules over
π∗ π∗ F =
∞
L
∞
L
OPN (−n)), so that we find
0
F(−n) and hence
0
p
∗
H (L, π F) =
∞
M
H p (PN , F(−n)).
0
58
Similarly, if F is a coherent OY -module, we get
H p (LY , π∗ F) =
∞
M
H p (Y, F(−n)),
0
H p (U, π∗ F) =
∞
M
H p (Y, F(n))
−∞
H p (AN+1 − (0), π∗ F) =
(U = C − (0))
∞
M
−∞
H p (PN , F(n)).
and
11. Deformations of cones
47
Let us now suppose that Y is a smooth closed subvariety of PN (or
dimension ≥ 1) and that it is projectively normal, i.e., C is normal.
We have then
TC1 = Coker(ΘAN+1 |C → NC ).
But since C is normal (of dimension ≥ 2)
H 0 (ΘAN+1|C ) ≃ H 0 (U, ΘAN+1 −(0) |C ).
For, ΘAn+1 |C is a trivial vector bundle and hence this follows from the
fact H 0 (U, OU ) = H 0 (C, OC ). Now NC is a reflexive OC -module since
NC = HomOC (I/I 2 , OC ), where I is the defining ideal of C in AN+1 .
Because of this it follows that
H 0 (U, NU ) = H 0 (C, NC ).
Hence we have
TC1 = Coker(H 0 (U, ΘAN+1 −(0) |U ) → H 0 (U, NU )).
Now from (B) we get
59
p
H 0 (U, ΘAN+1 −(0) |U )
/ H 0 (U, π∗ (Θ N | ))
P Y
α
β
∼
q
H 0 (U, NU )
/ H 0 (U, π∗ N ).
Y
β
Hence TC1 = Coker α = Coker(Im p →
− H 0 (U, L
π∗ NY )). Now β
is a graded homomorphism of graded modules over
H 0 (Y, OY (n)),
n=0
namely, the gradings are
H 0 (U, π∗ (ΘPN |Y ))
H 0 (U, π∗ NY )
=
∞
L
−∞
=
H 0 (Y, (ΘPN |Y )(n))
∞
L
−∞
H 0 (Y, N
Y (n)).
1. Formal Theory and Computations
48
From (C), identifying ΘAN+1 −(0) |U ≈ π∗ (OPN (1))N+1 , we get
H 0 (U, ΘAN+1 −(0) |U ) =
∞
M
H 0 (PN , OPN (n + 1))
n=−∞
and then by (C), p is also a graded homomorphism. Hence (Im p) is a
graded submodule of H 0 (U, π∗ (ΘPN |Y )). From this it follows that TC1 has
a canonical structure of a graded module over k[z0 , . . . , zN+1 ] and in fact
that it is a quotient of the graded module
0
NC = H (C, NC ) =
∞
M
H 0 (Y, NY (n)).
n=−∞
Thus we get
60
Proposition 11.1. Let Y be a smooth projective subvariety of PN (of
dimension ≥ 1) such that the cone C over Y is normal (we have only
to suppose that C is normal at its vertex). Then TC1 has a canonical
structure of a graded module over k[z0 , . . . , zN+1 ], in fact it is a quotient
∞
L
H 0 (Y, NY (n)).
of the graded module
−∞
12 Theorems of Pinkham and Schlessinger on deformations of cones
Theorem 12.1. Let Y be as above, i.e., Y is smooth closed ֒→ PN and
dim Y ≥ 1. Then
(1) (Pinkham). Suppose that TC1 is negatively graded, i.e., TC1 (m) = 0,
m > 0. Then the functor
Hilb(C) → Def(C)
is formally smooth.
(2) (Schlessinger). Suppose that TC1 is concentrated in degree 0. Then
we have a canonical functor
Hilb(Y) → Def(C),
12. Theorems of Pinkham and Schlessinger on...
49
and it is formally smooth. In particular, every deformation of C is
a cone.
Recall that Hilb(C) is the functor such that for local finite k-algebras
A Hilb(C)(A) = {Z ⊂ Pn+1 × A, Z closed subscheme, Z is flat over A and
Z represents an embedded deformation of C over A}.
Proof. (1) Given 0 → (ǫ) → A′ → A → 0 exact with rkk (ǫ) = 1 and A′ ,
A finite local over k, we have to prove that the canonical map
(*)
Hilb(C)(A′ ) → Def(C)(A′ ) ×Def(C)(A) Hilb(C)(A)
is surjective. Take the case A′ = k[ǫ], A = k. In particular (∗) implies 61
that
(a) Hilb(C)(k[ǫ]) → Def(C)(k[ǫ]) is surjective, and
(b) given ξ ∈ Hilb(C)(A), let ξ be the canonical image of ξ
in Def(C)(A).
Then, if ξ can be extended to a deformation of C over A′ , then ∃η ∈
Hilb(C)(A′ ) such that η 7→ ξ (the difference between this and (∗) above
is that we do not insist that η 7→ ξ).
We claim now that (a) and (b) ⇒ (∗). (In particular, to prove (1)
it suffices to check (a) and (b).) Given ξ as above, the set of all η ∈
Hilb(C)(A′ ) such that η 7→ ξ (provided there exists an η0 such that η0 7→
ξ) has a structure of a principal homogeneous space under Hilb(C)(k[ǫ])
(see Remark 6.1). [This can be proved in a way similar to proving that
Hilb(C)(k[ǫ]) ≃ H 0 (C, Hom(I, OC )), where I is the ideal sheaf defining
C in PN+1 , cf. Remark 6.2.] Similarly, all deformations η which extend ξ
form a principal homogeneous space under Def(C)(k[ǫ]), provided there
exists one. Hence, if (b) is satisfied and η is a deformation extending ξ,
because of (a) there exists in fact η, η 7→ ξ and η 7→ η. This completes
the proof of the above claim.
Now Hilb(C)(k[ǫ]) = H 0 (C, NC ). We have C = LY ∪ (vertex) and C
is normal at its vertex. Hence
H 0 (C, NC ) = H 0 (LY , NLY ) =
∞
M
n=0
H 0 (Y, NY (−n)).
1. Formal Theory and Computations
50
If TC1 is negatively graded, it follows that the canonical map
∞
L
n=0
H 0 (Y, NY (−n)) ⊂
∞
L
H 0 (Y, NY (n))
n=−∞
TC1
62
is surjective. Hence we have checked that
Hilb(C)(k[ǫ]) → (Def C)(k[ǫ])
is surjective. It remains to check (b). Given ξ ∈ Hilb(C)(A), suppose
that ξ is locally extendable, i.e., given ξ : C A → PN+1
A , (i) ξ can be
′
extended locally to deformation over A at every point of C A , and (ii)
this extension can be embedded in PN+1
A′ , so as to extend ξ locally. We
observe that it is superfluous to assume (ii), for we have seen that in the
affine case (X ⊂ An )
(Embedded Def)(X) → Def(X)
is formally smooth. Then we see that to extend ξ to an η ∈ Hilb(C)(A′ ),
we get an obstruction in H 1 (C, NC ) (this is an immediate consequence of
the fact already observed, that extensions form a principal homogeneous
space, cf. Remark 6.5). We observe that for ξ the property (i) is satisfied.
Since there is an η ∈ Def(C)(A′ ) with η 7→ ξ (ξ ∈ Def(C)(A), ξ → ξ),
the condition (i) is satisfied for all x ∈ C. But now C is smooth at every
point of C − C. In this case we have already remarked before that (i) is
satisfied (cf., Proof of Proposition 9.1). Since C is normal, C is of depth
≥ 2 at its vertex, so that by local cohomology we get
H 1 (C, NC ) ֒→ H 1 (LY , NLY ).
We have
H 1 (LY , NLY ) =
∞
M
n=0
H 1 (Y, NY (−n)) ⊂
∞
M
n=−∞
H 1 (Y, NY (n)) = H 1 (U, NU ),
12. Theorems of Pinkham and Schlessinger on...
63
51
Now by hypothesis ξ ∈ Def(C)(A) can be extended to η ∈ Def(C)
(A′ ); in particular, η|U gives an extension of ξ|U . Hence the canonical image of this obstruction in H 1 (U, NU ) is zero. (We see that as
above, extending an embedded deformation of U gives an obstruction
in H 1 (U, NU ).) This implies that there is an η ∈ Hilb(C)(A′ ) such that
η 7→ ξ. This checks (b), and the proof of (1) is now complete.
(2) Given a deformation of Y, we get canonically a deformation of
U = C − (0). To get a canonical functor Hilb(Y) → Def(C), it suffices to
prove that a deformation of U can be extended to a deformation (which
is unique by an earlier consideration). If depth of C at its vertex is ≥ 3
this follows by an earlier result, but we shall prove this without using
it. Let 0 → (ǫ) → A′ → A → 0 be as usual, and let YA′ { YA be
deformations of Y in PN . Then we have an exact sequence of sheaves.
0 → OY → OYA′ → OYA → 0,
OY ≈ ǫ · OYA′ ,
OYA′ being A′ flat, etc. Similarly for Y = PN . Then we get a commutative diagram
0
0
/ H 0 (Y, O (n))
Y
/0
/ H 0 (YA′ , OY ′ (n))
A
/0
/ H 0 (YA , OY (n))
A
/0
H 0 (PN , OPN (n))
H 0 (PNA′ , OPN′ (n))
A
H 0 (PNA , OPN (n))
A
0
(C1)
0
(C2)
where n ≥ 0. It is immediate that (C1) and (C2) are exact. From this 64
1. Formal Theory and Computations
52
commutative diagram, by induction on rkk A′ , it follows that
H 0 (PNA′ , OPN′ (n)) → H 0 (YA′ , OYA′ (n)) → 0
A
is exact because (C1) and (C2) are exact and the first and third rows are
exact. Then as an easy exercise it follows that the second row is exact.
Then it follows that H 0 (YA′ , OYA′ , (n)) → H 0 (YA , OYA (n)) → 0 is exact.
∞
L
H 0 (YA′ , OYA′ (n)) (resp. OCA ). Then we get an exact
Define OCA′ =
sequence
n=0
0 → OC → OCA′ → OCA → 0,
OC ≃ ǫ · OCA′ .
Again by induction on rk A′ it follows that OCA′ is A′ -flat (for, OCA
is A-flat by induction hypothesis and by, an earlier lemma (Lemma 8.1)
this claim follows). Hence Spec OCA′ provides a deformation of C, i.e.,
it extends the deformation of U = C − (0) which is given by the deformation YA′ of Y in PN . This proves the required assertion. Since depth
of C at its vertex is ≥ 3 by an earlier result this is the case. Hence we
get a canonical functor
Hilb(Y) → Def(C).
From this stage the proof is similar to (1) above. As before it suffices
to check assertions similar to (a) and (b) above. We see that Hilb(Y)
(k[ǫ]) ≃ H 0 (Y, NY ). We have
H 0 (Y, NY ) ⊂
∞
L
n=−∞
H 0 (Y, NY (n))
TC1 .
65
The hypothesis that TC1 consists only of degree 0 elements, implies
that H 0 (Y, NY ) → TC1 is surjective, i.e.,
Hilb(Y)(k[ǫ]) → (Def C)(k[ǫ])
12. Theorems of Pinkham and Schlessinger on...
53
is surjective. This checks (a). To check (b), take 0 → (ǫ) → A′ →
A → 0 as usual, ξ ∈ Hilb(Y)(A), ξ 7→ η ∈ (Def C)(A). Suppose we are
given η′ ∈ (Def C)(A) extending η, then we have to find ξ ′ ∈ Hilb(Y)(A′ )
extending ξ. Since Y is smooth, the condition for local extension over
A is satisfied as above, and hence we get an obstruction element λ ∈
H 1 (Y, NY ). We have
H 1 (Y, NY ) ֒→
∞
M
H 1 (Y, NY (n)).
−∞
In a similar way, the element η|U defines an obstruction element
∞
L
µ ∈ H 1 (U, NU ) =
H 1 (Y, NY (n)). But by hypothesis this obstruction
−∞
is zero. By functorially and the fact that H 1 (Y, NY ) ֒→
∞
L
H 1 (Y, NY (n))
−∞
it follows that λ = 0. This proves the existence of ξ ′ and the theorem is
proved.
Examples where the hypothesis of the above theorem are satisfied.
Lemma 12.1. Let Y ֒→ PN be a smooth projective variety such that



H 1 (Y, OY (n)) = 0, ∀n , 0, n ∈ Z


H 1 (Y, ΘY (n)) = 0, ∀n , 0, n ∈ Z,
and Y is projectively normal. Then TC1 consists only of degree 0 elements 66
(dim Y ≥ 2, follows from the hypothesis).
The hypothesis H 1 (Y, OY (n)) = 0, n , 0 implies that
TC1 = Coker(ΘAN+1 −(0) |U → NU ) = Coker(π∗ (ΘPN |Y ) → π∗ (NY ))
[as graded modules modulo elements of degree 0]. To prove this we
have only to write the cohomology exact sequence for (B2) of § 11 as
well as use (C) of § 11. To compute Coker(π∗ (ΘPN |Y ) → π∗ (NY )), use
the exact sequence
0 → π∗ (ΘY ) → π∗ (ΘPN |Y ) → π∗ (NY ) → 0.
1. Formal Theory and Computations
54
Now H 1 (π∗ (ΘY )) =
∞
L
H 1 (Y, ΘY (n)). Since by hypothesis H 1 (Y,
−∞
ΘY (n)) = 0 for n , 0, it follows by writing the cohomology exact sequence for the above, that Coker(π∗ (ΘPN |Y ) → π∗ (NY )) = TC1 has only
degree 0 elements. This proves the lemma.
Lemma 12.2. Let Y ⊂ PN be a smooth projective variety such that
H 1 (Y, OY (n)) = 0,
1
H (Y, ΘY (n)) = 0,
∀n > 0
∀n > 0.
Then TC1 has only elements in degree ≤ 0.
Proof. The proof is the same as for (2) above.
67
Remark 12.1. Given a smooth projective Y and an ample line bundle L
on Y the conditions in (1) (resp. (2)) above are satisfied for the projective
embeddings of Y defined by nL, n ≫ 0 if dim Y ≥ 1 (resp. dim Y ≥ 2).
Exercise 12.1. Let Y0 = 3 collinear points in P2 and Y1 = 3 noncollinear points in P2 . Take a deformation of Y0 to Y1 , which obviously
exists. Show that this deformation cannot be extended to a deformation
of the affine cone C0 over Y0 to the affine cone C1 over Y1 . [For, if such
a deformation exists, we see in fact that ∃ a deformation of C 0 to C 1 ,
closures respectively of C0 and C1 in P3 . We have a (finite) morphism
ϕ : C 1 → C 0 which is an isomorphism outside the vertex and not an
isomorphism at the vertex. We have
0 → OC 0 → ϕ∗ (OC 1 ) → k → 0.
This implies that the arithmetic genus of C 1 = (arithmetic genus of
C 0 + 1), but if there existed a deformation, they would be equal.]
13 Pinkham’s computation for deformations of the
cone over a rational curve in Pn
Lemma 13.1. Let Y ⊂ Pn be a connected curve of degree n, not contained in any hyperplane. The Y = Y1 ∪ . . . ∪ Yr where Yi are smooth
13. Pinkham’s computation for deformations of the cone...
55
rational curves of degree ni such that n = Σni , each Yi spans a linear
subspace of Pn of dimension ni and the intersections of Yi are transversal.
Proof. Let Y be an irreducible curve of degree < n in Pn . Then we
claim that there is a hyperplane H such that Y ⊂ H. Choose n distinct
points on Y. There exists a hyperplane H passing through these n points. 68
We claim that H contains Y for if H does not contain Y it follows that
deg(H · Y) > n, contradicting the fact deg Y < n. It then follows that if Y
is an irreducible curve of degree r, r < n, then there is a linear subspace
H ≃ P s in Pn such that Y ⊂ H ≃ P s .
Now let Y be a curve in Pn of degree n not contained in any hyperplane. Let Yi (1 ≤ i ≤ r) be the irreducible components of Y. Then if
ni = deg Yi , we have n = Σni . By the foregoing, if Hi is the linear subspace of Pn generated by Yi , then dim Hi ≤ ni . Since Yi ’s generate Pn ,
a fortiori the Hi ’s generate Pn . Since Y is connected, ∪Hi is also connected, i.e., we can write a sequence H1 , . . . , Hr such that H j ∩H j+1 , φ.
We find easily that if L j is the linear subspace generated by H1 , . . . , H j ,
r
P
then dim L j ≤ n1 + · · · + n j . Since n =
ni and dim Lr = n it follows
i=1
that dim L j = n1 + · · · + n j . It follows in particular that dim Hi = ni and
that L j ∩ H j = one point. It remains to prove that Yi is smooth and rational (the assertions about transversality are immediate by the foregoing),
and this is a consequence of the following
Lemma 13.2. Let Y be an irreducible curve in Pn of degree n not contained in any hyperplane. Then Y is a smooth rational curve, in fact
parametrized by t 7→ (1, t, t2 , . . . , tn ).
Proof. Let x1 be a singular point of Y. Choose n distinct points x1 , . . . ,
xn on Y. Then there is a hyperplane H such that xi ∈ H. Now x1 cannot
be smooth in H ∩ Y, for if it were so it would follow that x1 is also
smooth on Y. From this it follows easily
69
Deg(H · Y) > n
which is a contradiction. Hence every point of Y is smooth.
1. Formal Theory and Computations
56
To prove the assertion about parametric representation and ratiorality of Y, project Y from a point p in Y into Pn−1 (see § 11). We get
an irreducible curve Y ⊂ Pn−1 of degree (n − 1). Then it is smooth
and by induction hypothesis Y ′ can be parametrized as (1, t, . . . , tn−1 ).
The projection can be identified as the mapping obtained by dropping
the last coordinate. Hence the parametric form of Y can be taken as
(1, t, . . . , tn−1 , f (t)) where f (t) is a polynomial. Take the hyperplane H
as xn = 0; then by the hypothesis that Y is of degree n, it follows that f (t)
polynomial of degree n. Then by change of coordinates we see easily
that the parametric form of Y is t 7→ (1, t, t2 , . . . , tn ).
Lemma 13.3. Let Y be an irreducible curve in Pn of degree n, not contained in any hyperplane, or equivalently, a (smooth) curve parametrized by (1, t, . . . , tn ). Then Y is projectively normal.
Proof. Let L = OPn (1)|Y . It is well known that projective normality of
Y ⊆ Pn is equivalent to the fact that the canonical mapping
ϕν : H 0 (Pn , OPn (ν)) → H 0 (Y, Lν ) is surjective.
This follows from the following
70
Sublemma 13.1. Let X be a normal projective variety. Then X is projectively normal, i.e., X̂ (the cone over X) is normal if and only if H 0 (Pn ,
OPn (ν)) → H 0 (X, Lν ), L = OPn (1)|X , ν ≥ 0, is surjective.
L 0
Proof. In general we have A = {functions on X̂ − (0)} =
H (X, Lν )
ν∈Z
(see discussion in § 11). Now
L is ample we have H 0 (X, Lν ) = 0,
Lsince
0
H (X, Lν ). Now suppose X̂ is normal.
ν < 0, and therefore A =
ν≥0
Then {functions on X̂ − (0)} = {functions on X̂}. Also, X̂ ⊂ P̂n = An+1 ,
and
the same reasoning as before B = {functions on An+1 } =
L using
H 0 (Pn , OPn (ν)). But any function on X̂ can be extended to An+1
ν≥0
γ
which implies the natural map B →
− A is surjective. Therefore H 0 (Pn ,
0
ν
OPn (ν)) → H (X, L ) is surjective ∀ν ≥ 0.
13. Pinkham’s computation for deformations of the cone...
57
In general we have H 0 (Pn , OPn (ν)) → H 0 (X, Lν ) is
for
Lsurjective
ν large. Also ⊕H 0 (X, Lν ) is normal since X is. Thus
H 0 (X, Lν ) is
ν≥0
the integral closure of Im γ. Hence if we assume H 0 (Pn , OPn (ν)) →
H 0 (X, Lν ) is surjective ∀ν ≥ 0, we have that ⊕H 0 (X, Lν ) is normal. This
means X is normal.
Now returning to the proof of the lemma, Y ≃ P1 and L ≃ OP1 (n) so
that Lν ≃ OP1 (nν). It is clear that ϕ1 is injective if and only if Y is not
contained in any hyperplane. On the other hand, we have
dim H 0 (Pn , OPn (1)) = dim H 0 (P1 , OP1 (n))
= n + 1.
Hence ϕ1 is an isomorphism, i.e., Y ֒→ Pn is the immersion defined 71
by the complete linear system associated to OP1 (n). It is seen easily that
S ν (H 0 (P1 , OP1 (n)) → H 0 (P1 , OP1 (nν)) is surjective. This implies that ϕν
is surjective for all ν, and the lemma is proved.
Lemma 13.4. Let Y be as in the previous lemma. Then
H 1 (Y, Lν ) = 0,
1
H (Y, ΘY (ν)) = 0,
ν≥0
ν≥0
(the conditions in Lemma 12.1 are satisfied).
Proof. It is well known that ΘY ≃ OP1 (2) and H 1 (P1 , OP1 (ν)) = 0, ν ≥ 0.
This proves the lemma.
Proposition 13.1. Let Y be the nonsingular rational curve in Pn of degree n parametrized by t 7→ (1, t, . . . , tn ). Let C be the cone in An+1 over
Y and C its closure in Pn+1 . Then the function
Hilb(C) → Def(C)
is formally smooth. (Note by Hilb(C) we mean the restriction of the
usual Hilb to (local alg. fin./k) and defining deformations of C.)
Proof. This is an immediate consequence of Lemma 12.1.
58
72
1. Formal Theory and Computations
Let R be the complete local ring associated to the versal deformation of C (with tangent space = Def(C)(k[ǫ]) and R′ the completion of the local ring at the point of the scheme Hilb(C) corresponding
to C. Let ρ : R → R′ be the k-algebra homomorphism defined by
Hilb(C) → Def(C) to define this homomorphism we need not use the
representability of Hilb(C), it suffices to note that Hilb(C) also satisfies
Schlessinger’s axioms). From the above proposition it follows that the
homomorphism ρ is formally smooth. Hence we can conclude that R is
reduced (resp. integral, etc.) iff R′ is reduced (integral, etc.).
To study Hilb(C) we note that C is of degree n in Pn+1 and that it is
smooth outside its vertex.
Proposition 13.2. Let X ֒→ Pn+1 be a smooth projective surface of
degree n in Pn+1 , not contained in any hyperplane. Then X specializes
to C, i.e., there is a 1-parameter flat family connecting X and C.
Proof. Let H∞ be the hyperplane at ∞, with equation xn+1 = 0 (coordinates (x0 , . . . , xn+1 )). Let X be any closed subscheme of Pn+1 . Then
it is easy to see that X “specializes set theoretically” to the cone over
X · H0 (with vertex (0, . . . , 0, 1) and base X · H∞ ); in fact we define the
1-parameter family Xt , X1 = X, X0 = cone over X · H∞ (as point set) as
follows.
(x0 , . . . , xn+1 ) ∈ Xt ⇐⇒ (x0 , . . . , txn+1 ) ∈ X.
This family is obtained as follows: Consider the morphism
An+2 × A1 → An+2
defined by ((x0 , . . . , xn+1 ), t) 7→ (x0 , . . . , xn , txn+1 ). We see that it defines
rational morphism
: Pn+1 × A1 → Pn+1
73
and that ϕ is indeed a morphism outside the point x0 = ((0, . . . , 0, 1), 0).
Let us denote by ϕt the morphism (resp. rational for t = 0)
ϕt : Pn+1 → Pn+1 ; ϕt = ϕ|Pn+1 × {t}.
Then for all t , 0, ϕt is an isomorphism. Let Z ′ = ϕ−1 (X) (scheme
theoretic inverse image). We see that Z ′ is a closed subscheme of Pn+1 ×
13. Pinkham’s computation for deformations of the cone...
59
(A1 − x0 ). There exists a closed subscheme Z of Pn+1 ×A1 which extends
Z ′ and such that Z is the closure of Z ′ as point set. It is easy to see that
Z = Z ′ ∪ (0, . . . , 0, 1) (as sets). Let p : Z → A1 denote the canonical
morphism. Then ∀t ∈ A1 , t , 0, p−1 (t) ≃ Xt and for t = 0, p−1 (0) ≃
cone over X · H∞ (as sets).
The map (x0 , . . . , xn+1 ) → (x0 , . . . , xn , txn+1 ) (t , 0) defines an automorphism of Pn+1 , the image of X under this is denoted by Xt . The
scheme Xt specializes to a scheme X0 ֒→ Pn+1 (this follows for example by using the fact that Hilb is proper). From the above discussion it
follows that X0 = cone over H · X∞ (as point sets). It follows from this
argument that we can choose Z so that p : Z → A1 is flat (and then Z is
uniquely determined).
Suppose now that X is smooth, X · H∞ is smooth and that the cone
over H∞ · X (scheme theoretic intersection) is normal. Then we will
show that p−1 (0) = X0 = (cone over X · H∞ ) scheme theoretically. For
this, let I be the homogeneous ideal in k[x0 , . . . , xn+1 ] of all polynomials
vanishing on X. Let I = ( f1 , . . . , fq ) where fi = fi (x0 , . . . , xn+1 ) are homogeneous polynomials. Then Xt = V(It ) where It ( fi (x0 , . . . , xn , txn+1 ))
for t , 0. Let X0′ = V(I0′ ) where I0′ is the ideal I0′ = ( fi (x0 , . . . , xn , 0)). 74
Let I0 be the ideal of X0 . Then clearly I0′ ⊂ I0 . Also it is easy to see X0′ is
the scheme theoretic cone over X · H∞ . This means (X0′ ) red = (X0 ) red.
For, since I0′ ⊂ I0 we have X0′ ⊃ X0 . But by assumption X0′ is normal
hence in particular reduced. Therefore X0′ = X0 as required.
Let us now return to our particular case, where X is a smooth projective surface in Pn+1 of degree n not contained in any hyperplane. Then
we can choose a suitable hyperplane and call it H∞ such that H∞ · X is
smooth (Bertini) of degree n in Pn , and not contained in any hyperplane.
(The ideal defined by X ∩ H∞ is OX (−1). We have
0 → OX → OX (1) → OX∩H∞ (1) → 0
is exact. This gives
0
/ H 0 (X, OX )
O
surjection
0
/ H 0 (Pn+1 , O n+1 )
P
/ H 0 (OX (1))
O
/ H 0 (X ∩ H∞ , OX·H (1))
∞
O
exact
/ H 0 (OPn , OPn (1))
/ 0 exact.
injective
/ H 0 (OP (1))
n+1
60
75
1. Formal Theory and Computations
The second row is exact, and the first vertical arrow is a surjection.
The second vertical line is injective. This implies (via diagram chasing)
that the third arrow is an injection which implies X ∩H∞ is not contained
in any hyperplane. (We make this fuss because a priori degree does not
imply having the same Hilbert polynomial.))
Then by the preceding lemmas it follows that H∞ · X is a rational curve in Pn parametrized by (1, t, . . . , tn ) and then by the above discussion it follows that there is a 1-parameter flat family of closed subschemes deforming X to C (C is the closure in Pn+1 of the cone associated to X · H∞ ). This comples the proof of the proposition.
Remark 13.1. Let P be the Hilbert polynomial of C ֒→ Pn+1 . Let H
denote the Hilbert scheme of all closed subschemes of Pn+1 with Hilbert
polynomial P. Then H is known to be projective. The ring associated
with Hilb(C) above, is the completion of the local ring of H at the point
corresponding to C. Let H s be the open subscheme of H of points h ∈ H
such that (a) the associated subscheme Xh of Pn+1 is smooth and (b) Xh
is not contained in any hyperplane in Pn+1 . It is easy to see that (a) and
(b) define an open condition; that (a) defines an open condition is well
known, and that (b) also defines an open condition is easily checked.
The foregoing shows that as a point set H s corresponds to the set of
smooth surfaces in Pn+1 of degree n not contained in any hyperplane and
further H s contains points associated to projective cones over smooth
irreducible rational curves of degree n in Pn .
The varieties in Pn+1 corresponding to points of H s have been classified by the following
76
Theorem 13.1 (Nagata). Let X ֒→ Pn+1 be a closed smooth (irreducible) surface of degree n, not contained in any hyperplane. Then
either (a) X is a rational scroll, i.e., it is a ruled surface where the rulings are lines in Pn+1 and it is P1 bundle over P1 ; in fact X ≈ Fn−2 ;
where Fr we denote a P1 bundle over P1 with a section B having selfintersection (B)2 = −r, and then it is embedded by the linear system
|B + (n − 1)L|, where L denotes the line bundle corresponding to the ruling, or (b) n = 4 and X = P2 given by the Veronese embedding of P2 in
13. Pinkham’s computation for deformations of the cone...
61
P5 , i.e. the embedding defined by OP2 (2) :
(x0 , x1 , x2 ) 7→ (x20 , x21 , x22 , x1 x2 , x2 x0 , x0 x1 ),
or
(c) n = 1, X = P2 .
Remark 13.2. Let X = P2 , and take the Veronese embedding, with
U0 = x20 , U1 = x21 , U2 = x22
V0 = x1 x2 , V1 = x2 x0 , V2 = x0 x1 .
Then X is a determinantal variety,
matrix

U0 V2

 V2 U1
V1 V0
defined by (2 × 2) minors of the

V1 

V0 

U2
The cone C over the rational quartic curve has a determinantial representation defined, for instance, by letting U2 specialize to V2 , i.e., substituting V2 for U2 in the above matrix.
Remark 13.3. The general scroll can be checked to be the determinantal
variety defined by (2 × 2) minors of the (2 × n) matrix
#
"
x0 . . . (xn−1 + xn+1 )
,
x1 . . .
xn
and the cone C is obtained by setting xn+1 = 0 in this matrix.
Let us now take the case n = 4. Any two scrolls (resp. Veronese
surfaces) in P5 are equivalent under the projective group. Hence H s
split up into two orbits under the projective group: say, H s = K1 ∪ K2 .
We note also that there exists no flat family of closed subschemes of
Pn+1 connecting a scroll and a Veronese. For if there existed one, the
Veronese would be topologically isomorphic to a scroll (say, we are
over C). This is not the case: One can see this, for example, by the
fact that H 2 (scroll, Z) = Z2 , H 2 (P2 , Z) = Z. Since Ki are orbits under
PGL(5), they are locally closed in H s , and the nonexistence of a flat
77
62
1. Formal Theory and Computations
family connecting a member of K1 to K2 shows that Ki (i = 1, 2) are in
fact open in H s , so that H s is the union of disjoint open sets K1 and K2 ; in
particular, H s is reducible. Let ξ be the point of H determined by C [we
fix a particular C, namely, the cone in Pn+1 with vertex (0, . . . , 0, 1) and
base the rational curve in Pn parametrized by (1, t, t2 , . . . , tn )]. We saw
above that the closure of H s contains ξ. This implies that H is reducible
at ξ.
78
We shall now show that any “generalization” of C is again a projective cone over a smooth rational curve in Pn not contained in any
hyperplane. In other words, consider a 1-parameter flat family of closed
subschemes whose special member is C and whose generic member is
W. Then W has only isolated (normal) singularities, is of degree n and
not contained in any hyperplane. We shall now prove more generally
that a surface W in Pn+1 (which is not smooth) having these properties
is a cone of the type C. To prove this let θ be a singular point of W.
Then we can find a hyperplane L throught θ such that (L · W) is smooth
outside θ (Bertini’s theorem). Now L ≈ Pn , L · W is connected, (L · W)
is not contained in any hyperplane (same argument as for X ∩ H∞ in the
proof of Proposition 13.2), and it is of degree n. Since it has only one
singular point, it follows by Lemmas 13.1 and 13.2 that (L · W) consists of n lines meeting at θ. This happens for almost all hyperplanes L
passing through θ. We take coordinates in Pn+1 so that θ = (0, . . . , 0, 1)
and take projection on the hyperplane M = {xn+1 = 0}. We can suppose
that the choice of L is so made that L · W is smooth and not contained
in any hyperplane, so that L · W is the rational curve parametrized by
(1, t, . . . , tn ) with respect to suitable coordinates in L ≈ Pn . Let B be the
projected variety in M. It follows that almost all the lines joining θ to
points of B are in W from which it follows immediately that in fact all
these lines are in W. In particular, we have B = W · M and W is the cone
over B. This proves the required assertion.
Thus any “generalization” of C is either a cone of the same type as
C, a scroll, or a Veronese. It follows then that in the neighborhood of
ξ, H consists of only (parts) of three orbits under PGL(5), namely, K1 ,
K2 and ∆, where ∆ is the orbit under PGL(5) determined by C. In a
neighborhood of ξ, K 1 and K 2 are patched along ∆.
13. Pinkham’s computation for deformations of the cone...
63
It follows that Spec R where R is the complete local ring defined by
Def(C) is again reducible and has only two irreducible components.
Remark 13.4. It has been shown by Pinkham that R is reduced, that 79
the irreducible component corresponding to scrolls is of dimension 2
and the one corresponding to Veronese is of dimension 1, that they are
smooth, and that they intersect at the unique point corresponding to C
having normal crossings at this point.
Remark 13.5. The above argument can be extended to show that Spec R
for n ≥ 4 is irreducible. Pinkham shows Spec R has an embedded component at the point corresponding to C and outside this it corresponds to
scrolls.
Remark 13.6. The reason that for n = 4 we have got two components
is that in this case the cone over the rational quartic is a determinantal
variety in two ways, namely, it can be defined by (2 × 2) minors of
"
x0 . . . x3
x1 . . . x4
#
or of

 x0
 x2

x4
Remark 13.7. Case n ≤ 4. Exercise: Discuss.
x2
x1
x3

x4 

x3 

x2
Part 2
Elkik’s Theorems on
Algebraization
1 Solutions of systems of equations
80
Let A be a commutative noetherian ring with 1 and B a commutative
finitely generated A-algebra, i.e., B = A[X1 , . . . , XN ]/( f1 , . . . , fq ), F =
( f1 , . . . , fq ). Then finding a solution in A to F(x) = 0, i.e., finding a
vector x = (a1 , . . . , aN ) ∈ AN such that
fi (x) = 0
(1 ≤ i ≤ q),
is equivalent to finding a sectin for Spec B over Spec A.
Let J be the Jacobian matrix of the fi ’s defined by
 ∂f
 1
 ∂X
 1
.
J =  ..

 ∂ fq
∂XN
...
∂ f1 

∂XN 


∂ fq 

∂XN
(q × N) matrix.
We recall that at a point z ∈ Spec B ֒→ AN represented by a prime
ideal P in A[X1 , . . . , XN ], Spec B is smooth over Spec A at z if and only
if: There is a subset (a) = (a1 , . . . , a p ) of (1, 2, . . . , q) such that
65
2. Elkik’s Theorems on Algebraization
66
(i) there exists a (p × p) minor M of the (p × N) matrix
∂ fai
such that
∂X j
M . 0(mod P), and
(ii) ( f1 , . . . , fq ) and ( fa1 , . . . , fa p ) generate the same ideal at z.
81
If the conditions (i) and (ii) are satisfied at z, then Spec B is of relative codimension p in AN (i.e., relative to Spec A).
Let F(α) be the ideal ( fa1 , . . . , fa p ). The condition (ii) above is equivalent to the following: There is a g ∈ A[X1 , . . . , Xn ] such that z < V(g)
and
(F(a) )g = (F)g .
(The subscript g means localization with respect to the multiplicative set
generated by g. This implies that gr (F) ⊂ (F(a) ) for some r. Conversely
suppose given a g ∈ A[X1 , . . . , XN ], such that
g(F) ⊂ F(a)
(i.e., g ∈ conductor of F in Fa , (F(a) : F)).
Then at all points z ∈ Spec B such that g(z) , 0, (F) and F(a) generate the same ideal (since F(a) ⊂ F). Hence the condition (ii) above can
be expressed as:
(ii) There is an element g ∈ K(a) = conductor of F in F(a) (i.e., the
set of elements g such that gF ⊂ F(a) ) such that g(z) , 0.
Let ∆(a) = ideal generated
! by the determinants of the (p × p) minors
∂ fai
. Let H be the ideal in A[X1 , . . . , XN ] defined
of the (p× N) matrix
∂X j
by
X
H=
K(a) ∆(a)
(a)
i.e., the ideal generated by the ideals {K(a) ∆(a) } where (a) ranges over all
subsets of (1, . . . , q). Then we see that at a point z ∈ Spec B ֒→ AN ,
Spec B is smooth over Spec A ⇔ H generates the unit ideal at z ⇔ z <
V(H). Hence we conclude:
z ∈ Spec B is smooth over Spec A if and only if z < V(H) ∩ Spec B.
2. Existence of solutions when A is t-adically complete
67
2 Existence of solutions when A is t-adically complete
82
With the notations as in the above § 80, we have
Theorem 2.1. Suppose further that A is complete with respect to the
(t)-adic topology, (t) = principal ideal generated by t ∈ A (i.e., A =
lim A/(t)n ). Let I be an ideal in A. Then there is a positive integer q0
←−−
such that whenever
F(a) ≡ 0(mod tn I), n ≥ n0 , n > r
(i.e., fi (a) = 0(mod tn I), ∀1 ≤ i ≤ q, n ≥ n0 , n > r) and
tr ∈ H(a)
(H(a) is the ideal in A generated by H evaluated at a, or equivalently,
the closed subscheme of Spec A obtained as the inverse image of V(H)
s
by the section Spec A →
− AN defined by (a1 , . . . , aN )), then we can find
(a′1 , . . . , a′N ) ∈ AN such that
F(a′ ) = 0 and a′ ≡ a(mod tn−r I).
Remark 2.1. The condition tr ∈ H(a) implies that V((t)) ⊃ V(H(a)).
So the section s : Spec A ֒→ AN defined by (a1 , . . . , aN ) does not pass
through V(H) except for the points t = 0. Roughly speaking, the above
theorem says that a section s of ANA over Spec A which is an approximate
section of Spec B over Spec A and not passing through V(H) except over
t = 0 can be approximated by a true section of Spec B over Spec A.
Proof of Theorem 2.1. (1) We claim that it suffices to prove the following: ∃h ∈ AN (represented as a column vector) such that
(*)



F(a) ≡ J(a)h(mod t2n−r I),


h ≡ 0(tn−r I).
and
83
2. Elkik’s Theorems on Algebraization
68
To prove this claim we use the Taylor expansion
F(a − h) = F(a) − J(a)h + O(h2 ) (error terms quadratic in h).
Now h2 ∈ t2(n−r) I. We note further that
t2n−r I = tr · t2n−2r I ⊂ t(2n−2r) I.
We have F(a) − J(a)h ∈ t2n−r I and hence
F(a) − J(a)h ∈ t2n−2r I,
i.e., (∗) implies



F(a − h) ∈ t2(n−r) I,


h ∈ t(n−r) I.
and
Set a1 = a − h, a0 = a. Then this gives
F(a1 ) ∈ t2(n−r) I,
a1 − a0 ∈ t(n−r) I.
Hence by iteration we can find ai ∈ A such that
i
(i) F(ai ) ∈ t2 (n−r) I, i ≥ 0, and
i−1 (n−r)
(ii) (ai − ai−1 ) ∈ t2
I, i ≥ 1.
Now by (ii), a′ = lim ai exists, and a′ ≡ a(mod tn−r I). Further (i)
←−−
implies that F(a′ ) = 0. This completes the proof of the claim.
(2) We claim that it suffices to prove that there is a z ∈ AN such that
tr F(a) ≡ J(a)z(mod t2n I),
and
n
z ≡ 0(t I).
84
For, we see easily that there is an h ∈ AN such that tr h = z and
h ∈ tn−r I. Then, if J = JF denotes the Jacobian for F, we have
tr (F(a) − JF (a)h) ≡ 0(mod t2n I),
2. Existence of solutions when A is t-adically complete
69
h ≡ 0(mod tn−r I).
Set tr F = G; then these relations can be expressed as



G(a) − JG (a)h ≡ 0(mod t2n I), and


h ≡ 0(mod tn−r I).
In particular we certainly have



G(a) − JG (a)h ≡ 0(mod t2n−r I),
(**)


h ≡ 0(mod tn−r I).
and
These are just the same as the relations (∗) as in Step (1) above, with
F replaced by G. Hence we conclude (as for F) that there is an a′ such
that



G(a′ ) = 0, and


a′ ≡ a(mod tn−r I).
Thus we conclude that there is an a′ such that



tr F(a′ ) = 0, and


a′ ≡ a(mod tn−r I).
Note that F(a′ ) ∈ tn−r I since
F(a′ ) = F(a) + JF (a)(a′ − a)(mod t2(n−r) I)
and F(a) ∈ tn I. Thus we have
tr F(a′ ) = 0,
and
F(a′ ) ∈ tn−r I.
85
We would like to conclude that F(a′ ) = 0; this may not be true;
however, we have
Lemma 2.1. Let T q = {a ∈ A|tq a = 0} and let q0 be an integer such that
T q0 = T q0 +1 = T q0 +2 = . . ., etc., (note that T q ⊂ T q′ for q′ ≥ q and A
being noetherian, the sequence T 1 ⊂ T 2 . . . terminates). Then
T s ∩ (tm )A = (0)
for
m ≥ q0 .
2. Elkik’s Theorems on Algebraization
70
Proof. Let a ∈ T q0 ∩ tm A. Then a = tm a′ = tq0 (tm−q0 a′ ). Since tq0 a = 0,
we have tn+q0 a′ = 0. But by the choice of q0 , we have in fact tq0 a′ = 0.
Hence a = tq0 tm−q0 a′ = tm−q0 tq0 a′ = 0. This proves the lemma
By the lemma if (n − r) ≥ q0 , then tr F(a′ ) = 0, and F(a′ ) ∈ tn−r I
implies that F(a′ ) = 0. Thus the claim (2) is proved.
(3) Let (β) = (β1 , . . . , β p ) denote a subset of p elements of (1, . . . , N).!
∂ fαi
Given (α) = (α1 , . . . , α p ) ⊂ (1, . . . , q), let δα,β , be the p× p minor
∂Xα j
of the Jacobian matrix J. Then we claim that it suffices to prove the following:
Given k ∈ K(α) and δαβ ∈ ∆α (δαβ defined as above) then there exists
a z ≡ 0(mod tn I) such that
k(a)δαβ (a)F(a) ≡ G(a)z(mod t2n I).
P
For, we have tr = λαβ kα (a)δαβ (a). Then by hypothesis, given kα
α,β
and δαβ , we have zαβ such that zαβ = 0(mod tn I) and
86
kα (a)δαβ (a)F(a) ≡ J(a)zαβ (mod t2n I);
P
then we see that if we set z = λαβ zαβ , we have
αβ
tr · F(a) ≡ J(a)z(mod t2n I),
and
n
z ≡ 0(mod t I),
which is the claim (2).
(4) We can assume without loss of generality that (α)!= (1, . . . , p)
∂ fi 1 ≤ i ≤ p
.
,
and (β) = (1, . . . , p) so that δα,β = det M where M =
∂X j 1 ≤ j ≤ p
Then we have
"
#
M ∗
J=
.
∗ ∗
Let N be the matrix formed by the determinants of the (p−1)×(p−1)
minors of M so that we have
MN = δ · Id, δ = δαβ , α, β as above.
2. Existence of solutions when A is t-adically complete
71
h i
We denote by N0 the (N × p) matrix by adding zeros to N.
Let k ∈ K(α) , (α) as above and δ be as above. Then we claim that if
z is defined by
 
" #  f1 
N  . 
 .  , Z is an (N × 1) matrix;
Z=k
0  . 
fp
then if z = Z(a), we have
k(a)δ(a)F(a) ≡ J(a)z(mod t2n I),
and
n
z ≡ 0(mod t I).
By the foregoing, if we prove this claim, then the proof of the theorem would be completed.
To prove this claim, we observe that the relation z ≡ 0(mod tn I) is 87
immediate. Since k ∈ K(α) , (α) as above, we have
(I)
k fj =
p
X
i=1
λi j fi , 1 ≤ j ≤ q.
p
∂ fj
P
∂ fi
= λi j
(mod F), 1 ≤ j ≤ q.
∂X1 i=1 ∂X1
Substituting (a), we get
This gives k
p
k(a)
X
∂ fj
∂ fi
(a) ≡
(a)(mod F(a))
λi j
∂X1
∂X
1
i=1
≡
p
X
λi j (a)
i=1
∂ fi
(a)(mod tn I).
∂X1
Let u ∈ AN be an element such that u ≡ 0(mod V), where V is some
ideal of A. Let v = J(a) · u ∈ AN . Then
k(a)v j = k(a)
N
X
∂ fj
uℓ
∂Xℓ
ℓ=1
2. Elkik’s Theorems on Algebraization
72

p
N 
X
X
∂ fi 

(a) uℓ (mod tn IV)
≡
 λi j (a)
∂X
1
ℓ=1 i=1
 N

p
X
X ∂xi


≡
λi j (a) · 
(a)uℓ  (mod tn IV)
∂X1
i=1
ℓ=1
≡
p
X
λi j (a)vi (mod tn IV).
i=1
In other words,
(II)
k(a)v j =
p
X
λi j (a)vi (mod tn IV),
i=1
88
for
1 ≤ j ≤ q,
where u ∈ AN , v = J(a)u, and u ≡ 0(mod V).
Now take for u ∈ AN and v the elements


 
"
#  f1 (a) 
" #  f1 



N(a)  . 
N  . 
 . .
 ..  (a), v = J(a)
u=
0  . 
0  
f p (a)
fp
Then u ≡ 0(mod tn I) and we have
k(a)v j =
p
X
λi j (a)vi (mod t2n I).
i=1
Moreover,




#  f1 (a) 
#"
#  f1 (a)  "
M(a) ∗ N(a)  . 
N(a)  . 
 . 
 . =
v = J(a)
0  . 
∗
∗
0  . 
f p (a)
f p (a)


 δ(a) f1 (a) 


..


.

 

"
#  f1 (a) δ(a) f p (a)

 

M N
..

=
(a)  ...  = 
.

 

∗ N
 w p+1 
f p (a)


..


.


wN
"
2. Existence of solutions when A is t-adically complete
73
Hence vi = δ(a) fi (a), 1 ≤ i ≤ p. By (II),
k(a)vi ≡
p
X
λi j (a)vi (mod t2n I), so that
k(a)vi ≡
p
X
λi j (a) fi (a)δ(a)(mod t2n I).
i=1
i=1
By (I),
p
X
λi j (a) fi (a)δ(a) = k(a)δ(a) f j (a).
i=1
Hence it follows that
" #" #
"
#
N fi
f1 (a)
k(a)J(a)
≡ k(a)δ(a)
0 fp
fq (a)
(mod t2n I)
which is precisely the claim in Step (4) above. This completes the proof 89
of the theorem as remarked before.
Remark 2.2. A better proof of the theorem is along the following lines:
Introduce a set of relations for F so that we have an exact sequence
R
F
→ P → P/F → 0.
→ Pq −
Pℓ −
 f1 
 
(Here F denotes  ...  as well as the ideal generated by fi and P =
fq
A[X1 , . . . , XN ], and R is the matrix of relations of F; it has entries in
P.) In matrix notation, we have F · R = 0. Differentiating with respect
to X1 , we obtain
JR ≡ 0(mod F),
!
∂ fi
where J is the Jacobian matrix
(it is an N × q matrix and is there∂X j
fore the transpose of the matrix J introduced in the theorem above). Let
P = P/F and define J, R similarly. Then we obtain a complex
(*)
ℓ R
q J
→P −
→P
P −
N
2. Elkik’s Theorems on Algebraization
74
R
J
→ PN is a complex mod F). It is not difficult to see that
→ Pq −
(or: Pℓ −
at a point x ∈ Spec(P/F), the complex (∗) is homotopic to the identity
(in particular, is exact), i.e., denoting by a subscript x the localization at
x, there is a diagram
r
j
w
x
1
Px
90
R
/ Pq
x
J
/ PN ,
x
with Rr + jJ = Id
if and only if Spec(P/F) is smooth at x. For example, suppose that
Spec(P/F) is smooth
! at x. Then we can assume without loss of gener∂ fi 1≤i≤p
, is a unit in P x . From this it follows easily
,
ality that det
∂Xi 1≤ j≤p
q
p
that there is a direct summand Q1 ֒→ Px , Q1 ≈ P x such that Im(J) =
Im(J|Q1 ) and J|Q1 : Q1 → Im(J) is an isomorphism (J|O1 denotes the
restriction to Q1 ). Indeed, we can take Q1 to be the submodule generN ∂ fi
P
ated by the first p-coordinates. (Note we have J(ei ) =
ξ1 , with
ℓ=1 ∂X1
q
N
(ei ) a basis of P , and (ξi ) a basis of P ). We see also that Im(R) is of
q
rank (q − p) and is a direct summand in P . In fact the relations
fj =
p
X
λi j fi ,
j≥ p+1
p
X
λi j ei ,
j ≥ p + 1,
i=1
give elements of the form
ej =
i=1
in Im R. Suppose now
p
P
µi fi is a relation; then as we have seen before
i=1
(on relations for a complete intersection), µi ∈ ( f1 , . . . , fi ) so that µi = 0.
Hence we conclude that (Im R) is precisely the submodule generated by
p
P
(e j −
λi j ei ), j ≥ p + 1, which shows that Im R is of rank (q − p)
j=1
q
and is a direct summand in P . Now we see easily that the complex (∗)
2. Existence of solutions when A is t-adically complete
75
is homotopic to the identity at x if and only if, (i) (∗) is exact and (ii)
Im R is a direct summand which admits a complement Q1 such that J|Q1
is an isomorphism and Im J ≃ Im(J|Q1 ). Now we have checked these
conditions when Spec(P) is smooth at x. Conversely if (i) and (ii) are 91
satisfied (at x), it can be checked that Spec(P) is smooth at x.
Suppose more generally that we are given a complex
1 R
q J
→R →
− P
P −
N
(we keep the same notation). Then we can define an ideal H in P
which measures the nonsplitting of the complex as follows: H is the
ideal generated by elements h such that there are maps r, j:
r
j
x
P
1
x
R
/
P
q
J
/
N
P , such that Rr + jJ = Id.h.
It can be shown that H is the following ideal: Take a (p × p) minor
M in J and a “complementary” (q − p) × (q − p) minor K in R (involving “complementary indices”); then H is the ideal generated by the
elements (det M)(det K).
In our case, Spec P−V(H ) is the open subscheme of smooth points.
It follows then that H and the ideal H (H is the ideal defined before and
H denotes the image of H in B = P/F) have the same radical. In our
case we have then
Rr + jJ ≡ e
h(mod F),
for some e
h∈P
with image
h in H .
Multiplying by F (where F is a vector now), we have
FRr + F jJ ≡ Fe
h(mod F 2 ).
Since FR = 0, this gives
F jJ ≡ Fe
h(mod F 2 ).
We have used the transposes of the original J; setting z = jt F t and
taking “evaluation at a”, we get
92
2. Elkik’s Theorems on Algebraization
76
h(a)F(a) ≡ J(a)z(mod t2n I).
Now a power of h is contained in H since H and H have the same
radical. This is the crucial step in the proof of the theorem above. Hence
from this the proof of the theorem follows easily.
Before going to the next theorem, let us recall the following facts
about a-adic rings (cf. Serre, Algebre Locale, Chap. II, A). Let (A, a)
be a Zariski ring, i.e., A is a noetherian ring, is an ideal contained in the
Jacobson radical Rad A of A, and A is endowed with the a-adic topology,
i.e., a fundamental system of neighborhoods of 0 is formed by an . Since
T n
a = (0), it follows that this topology is Hausdorff. Let A denote
n
93
the a-adic completion. Then A is noetherian. If M is an A-module of
finite type we can consider the a-adic topology on M and similarly it
is Hausdorff and if M denotes the a-adic completion of M, we have
M = M ⊗A A. In fact the functor M 7→ M is exact. Suppose now that
M = M. Then we note that any submodule N of M is closed with respect
to the a-adic topology (for, the quotient topology in M/N is the a-adic
topology and since M/N is of finite type and a ⊂ Rad A, this topology is
Hausdorff and hence N is closed). If A = A, i.e., A is complete, then any
module of finite type is complete for the adic topology so that we don’t
have to assume further that M = M.
Let A = A, M be as usual and t ∈ a. Then M/tM is complete for the
a/ta-adic topology, for this is simply the a-adic topology on M/tM.
Let A = A and V be an ideal in A. Given an r, suppose that the
relation ar ⊂ V + am , m ≫ 0, holds. Then ar ⊂ V, for our relation
T
implies that ar (A/V) ⊂ am (A/V). Since A/V is Hausdorff for the
m
a-adic topology, it follows that ar A/V = (0), which implies ar ⊂ V.
3 The case of a henselian pair (A, a)
Theorem 3.1. With the same notations for A, B as in the pages preceding Theorem 2.1, suppose further that (A, a) is a henselian pair1 (in
1
Definition. By a henselian pair we mean a ring A and an ideal a ⊂ Rad A (=
Jacobson radical of A) such that given F = ( f1 , . . . , fN ), N elements of A[X1 , . . . , XN ]
3. The case of a henselian pair (A, a)
77
particular a ⊂ Rad A), and that A is the a-adic completion of A. SupN
pose we are given a ∈ A such that F(a) = 0 (i.e., a formal solution)
and ar · A (or briefly ar ) ⊂ H(a) for some r. (This means that the section of Spec B over Spec A where B = A|X|/( fi ) represented by a passes
through smooth points of the morphism Spec B → Spec A except over
V(a).) Then for all n ≥ 1 (or equivalently for all n sufficiently large)
there is an a ∈ AN such that F(a) = 0, and a ≡ a(mod an ).
Proof. (1) Reduction to the case a principal.
Let a = (t1 , . . . , tk ). Let us try to prove the theorem by induction on
k. If k = 0, then the theorem is trivial. So assume the theorem proved
for (k − 1). We observe that the couple (A/tkℓ , (t1 , . . . , tk−1 )) is again a
henselian pair ∀ℓ ≥ 1 (here t1 , . . . , tk−1 denote the canonical images of
ti in A/tkℓ ). Set t = tk and A1 = A/(tℓ ). We note also that the a-adic 94
topology on A1 is the same as the (t1 , . . . , tk−1 ) = a1 -adic topology. If A,
A1 denote the corresponding completions, we get a canonical surjective
homomorphism A → A1 whose kernel is tℓ ·A. Let b be the canonical image of a in A1 . Then we have F(b) = 0. Besides, we see that a1s ⊂ H(b)
for some s (this follows from the fact that V(H) ∩ Spec B = locus of
nonsmooth points for Spec B → Spec A and the set of smooth points
behaves well by base change and for us the base change is A → A1 .
We canonot say that s = r, for the ideal H (or rather H(mod B)) which
we have defined using the base ring A does not behave well with respect to base change. The ideal H does behave well with respect to
base change, and if we had used this ideal we could have got the same
integers). Hence by induction hypothesis, for all m ≥ t, there exists a
b ∈ A1N such that F(b) = 0, and b ≡ b(mod a1M ).
Lift b to an element a1 ∈ AN and choose ℓ so that ℓ ≥ m. Then we
see that
a1 ≡ a(mod am ),
and
F(a1 ) ≡ 0(mod (tℓ )).
and x0 ∈ An , x0 = (x01 , . . . , x0N ) such that F(x0 ) ≡ 0mod (a) and such that det
invertible mod a, then ∃x ∈ AN , x ≡ x0 mod (a) with F(x) = 0.
!
∂ fi
| 0 is
∂X j x
2. Elkik’s Theorems on Algebraization
78
m
(The fact that b ≡ b(mod am
1 ) implies (a1 − a)+ x ∈ a with x ∈ Ker(A →
ℓ
m
A1 ) = (t ). Now t ∈ a, and if ℓ ≥ m, (t) ∈ a .) We claim that if m ≫ 0
(and consequently ℓ ≫ 0), we have ar ⊂ H(a1 ). By hypothesis we have
ar ⊂ H(a), and from the relation a1 ≡ a(mod am ), we get
ar ⊂ H(a1 ) + am
95
(as ideals in A; to deduce this we use the Taylor expansion). Then, as we
remarked before the theorem for m ≫ 0, this implies that ar ⊂ H(a1 ).
Since t ∈ a it follows that tr ∈ H(a1 ).
Let At denote the t-adic completion. Then the following relations in
A,
for all ℓ ≥ 0, F(a1 ) ≡ 0(mod (t I ))
there exists a1 ∈ AN such that tr ∈ H(a1 )
hold a fortiori in At , and hence by Theorem 2.1 we can find a′ ∈ AtN
such that
F(a′ ) = 0,
and
a′ ≡ a1 (mod (tℓ−r )).
Note that the pair (A, (t)) is also henselian. Hence if the theorem
were true for k = 1, we would have
∀n, ∃ a ∈ AN such that F(a) = 0, and
a ≡ a′ (mod tn A).
But we have
a′ ≡ a1 (mod (t s )At ), for s sufficiently large, and
a1 ≡ a(am · A), m sufficiently large.
These imply a ≡ a(mod an ) (since t ∈ a and At ⊂ A), which implies
the theorem. Hence we have only to prove the theorem in the case k = 1,
i.e., a principal.
3. The case of a henselian pair (A, a)
79
(2) A general lemma:
Lemma 3.2. Let A, B be as in the pages preceding the theorem, i.e., B =
A[X1 , . . . , XN ]/( f1 , . . . , fq ), F = ( f1 , . . . , f0 ). Let C be the symmetric
algebra on F/F 2 over B, so that Spec C is the conormal bundle over 96
Spec B (F/F 2 as a B-module is the conormal sheaf over Spec B). Let f ,
g, h denote the canonical morphisms
Spec C −
→ Spec B →
− Spec A, g = h ◦ f.
f
h
Let V be the open subschemes of Spec B where Spec B → Spec A is
smooth and V ′ = f −1 (V). Then we have the following:
(a) g : Spec C → Spec A is smooth and of relative dimension N (over
A) on V ′ and
2N+q
(b) ∃ an imbedding Spec C ֒→ AA
(A-morphism) such that the
restriction of the normal sheaf (for this imbedding) to every affine
open subset U ֒→ V ′ is trivial.
Proof of Lemma 3.2. We set
C = B[Y1 , . . . , Yq ]/I
A[X, Y]/(F, I), K = (F, I).
On V we have the exact sequence
(i)
0 → F/F 2 → ΩA[X]/A ⊗A[X] B → ΩB/A → 0.
(This is an abuse of notation; strictly speaking we have to write
(F/F 2 )|V . . ., etc.) On V we have the exact sequence
(ii)
0 → f ∗ (ΩB/A ) → ΩC/A → ΩC/B → 0,
∗
and
2
ΩC/B ≈ f (F/F ).
Taking f ∗ of the first sequence, we get sequences
0 → f ∗ (F/F 2 ) → (Free) → f ∗ (ΩB/A ) → 0
97
2. Elkik’s Theorems on Algebraization
80
0 → f ∗ (ΩB/A ) → ΩC/A → f ∗ (F/F 2 ) → 0
which are exact on any affine open set U in V ′ . These exact sequences
are split on U, so that we conclude
ΩC/A
is free on
U.
On V ′ we have the exact sequence
0 → K/K 2 → ΩA[X,Y]/A ⊗A[X,Y] C → ΩC/A → 0.
Thus it follows that on U
K/K 2 ⊕ (Free) = (Free)
|{z}
of rank N
(From the exact sequence (ii) it follows that ΩC/A is of rank N over C and
this implies the assertion (a).) If we introduce N more indeterminates
Z1 , . . . , ZN , then
(*)
C = A[X, Y, Z1 , . . . , Zn ]/(K, Z1 , . . . , Zn ).
Let K ′ = (K, Z1 , . . . , ZN ). Then we see easily that
K ′ /K ′ 2 = K/K 2 ⊕ (Free of rk N).
98
It follows that K ′ /K ′ 2 is free on U. Thus for the embedding
2N+q
Spec C ֒→ AA , the restriction of the normal bundle K ′ /K ′ 2 to U
is trivial. This completes the proof of the lemma.
(3) We saw in (1) above, that for the theorem it suffices to prove it
in the case a = (t). The condition ar A ⊂ H(a) becomes tr ∈ H(a). (Note
that H(a) is the ideal in A generated by evaluating at a elements of H
and H is an ideal in A[X1 , . . . , XN ] not in A[X1 , . . . , XN ].)
We claim now that there is an h ∈ H such that
h(a) = (unit) · tr .
A priori it is clear there is an h ∈ H · A[X] such that h(a) = tr . Since
A is the t-adic completion of A, we can find an h ∈ H such that
h ≡ h(mod (tr+1 ))
3. The case of a henselian pair (A, a)
81
(i.e., the coefficients of h and h differ respectively by an element of
(tr+1 )). This implies that
h(a) − h(a) = ∗tr+1 ,
hence
r
h(a) = t (1 + ∗t).
Now (1 + ∗t) is a unit in A. This proves the required claim.
(4) The final step.
Since Spec C is a vector bundle over Spec B, we have the 0-section
x
N
Spec C → Spec B. We are given a ∈ A such that F(a) = 0. Now a
determines a section of Spec(B⊗A A) over A, or equivalently a morphism
s : Spec A → Spec B forming a commutative diagram
s
Spec AI
II
II
II
II
$
/ Spec B
u
u
uu
u
u
uu
uz u
Spec A
Using the 0-section, s can be lifted to a morphism s1 : Spec A → 99
Spec C:
Spec AI
s1
II
II
II
II
$
/ Spec C
u
u
u
uuu
u
u
uz u
Spec A
Now by (3), s1 carries (Spec A − V(t)) to Spec C[1/h], where h is
as in (3) (here h denotes the canonical image in C of the h in (3)). We
observe that
Spec C[1/h] ⊂ V ′ ,
(V ′ = f −1 (V), V = locus of smooth points of Spec B → Spec A) and V ′
is contained in the locus of smooth points of the map Spec C → Spec A.
We have C = A[X, Y, Z]/K ′ where K ′ = (F, I, Z1 , . . . , ZN ). The section
2N+q
(a′ extends the
s1 defines a solution K ′ (a′ ) = 0, where a′ ∈ A
section a). We have tk ∈ H(a′ ) for a suitable k. Hence the conditions,
similar to those of Spec B → Spec A, are now satisfied for the map
82
2. Elkik’s Theorems on Algebraization
Spec C → Spec A. It is immediately seen that it suffices to solve for the
case Spec C → Spec A, in fact if we get a solution a′ ∈ A2N+q , we have
to take for a the first N coordinates of a′ .
Let U = Spec C[1/h]. Then the restriction of the normal bundle of
2N+q
the imbedding Spec C ֒→ AA
is trivial on U and, U being smooth
over A, it follows easily that U is open in a global complete intersection. By this we mean there exist g1 , . . . , gN+q ∈ A[X, Y, Z] such that
V(g1 , . . . , gN+q ) has dimension N and we have an open immersion
(Spec C[1/t]) = Spec C[1/h] ֒→ Spec A[X, Y, Z]/(g1 , . . . , gn+q ).
Let G = (g1 , . . . , gN+q ). Then we have
Kt′ = Gt .
100
(We denote the localization with respect to t by a subscript. Note that
localization with respect to t is the same as localization with respect to
2N+q
b.) The given solution a′ in A
is such that K ′ (a′ ) gives rise to a
′
solution G(a ) = 0 by changing the gi , multiplying them by suitable
powers of t. Conversely, suppose we have solved the problem for G,
i.e., we have found a′ ∈ A2N+q such that G(a) = 0 and a′ ≡ a(mod tn ).
Then we see easily that there is a θ such that
tθ · F(a′ ) = 0.
Since F(a) = 0 by Taylor expansion, it follows that F(a′ ) ≡ 0
(mod tn ). Now if n ≫ 0, by Lemma 2.1, it follows that F(a′ ) = 0.
Thus it suffices to solve the problem for G, i.e., for the morphism
Spec C ′ → Spec A,
C ′ = A[X, Y, Z]/G.
We have seen that a′ defines a section of this having the required
properties. Further, Spec C ′ is a global complete intersection and
smooth over A in A2N+q , i.e., we have reduced the theorem to the following lemma.
4. Tougeron’s lemma
83
4 Tougeron’s lemma
Lemma 4.1. Let (A, a)
! be a henselian pair and fi ∈ A[Y1 , . . . , YN ], 1 ≤
∂ fi
be the Jacobian matrix, 1 ≤ i ≤ m, 1 ≤ j ≤ N.
i ≤ m. Let J =
∂Y j
Suppose we are given y0 = (y01 , . . . , y0N ) ∈ AN such that
f (y0 ) ≡ 0(mod ∆2 V)
where V(V) = V(a) (or ⇔ V is also a defining ideal for (A, a)) and
∆ is the annihilator of the A-module C presented by the relation matrix 101
(i.e., C is the cokernel of the homomorphism AN → Am whose matrix is
J(y0 )). Then there is a y ∈ AN such that
f (y) = 0 and
y ≡ y0 (mod ∆V).
Proof. The henselian property of (A, a) is used in the following manner: Let F = (F1 , . . . , F N ) be N elements of A[Y1 , . . . , YN ] and y0 =
(y01 , . . . , y0N ) ∈ AN such that
(i) F(y0 ) ≡ 0(mod a)
!
∂Fi
(ii) det
is a unit
∂Y j y=y0
(P)
(mod a).
Then there is a y ∈ AN such that F(y) = 0 and y ≡ y0 (mod a).
Let δ1 , . . . , δr generate the annihilator of ∆. This implies that there
exist N × m matrices such that
JNi = δi I,
J = J(y0 ),
I = Id(m×m) .
Write
f (y0 ) =
X
δi δ j ǫi j ,
i, j
ǫi j = (ǫi j1 , . . . , ǫi jν , . . . , ǫi jm ), ǫi j ∈ V
ց
m components.
2. Elkik’s Theorems on Algebraization
84
We try to solve the equations


r

 0 X
δi Ui  = 0
f y +
i=1
102
for elements Ui = (Uil , . . . , UiN ) ∈ AN (we consider vectors in An to
be column matrices). Expansion by Taylor’s formula in vector notation
gives
X
X
0 = f (y0 ) + J · ( δi Ui ) +
δi δ j Qi j ,
i, j
where
J is an (m × N) matrix,
f (y0 ) is an (m × 1) matrix,
Ui is an (N × 1) matrix (not (1 × N) matrix as it is written),
and
Qi j , ǫi j
are
(m × 1) matrices.
Expanding, we get
r
X
( δi Ui )
0=J·
+
i, j
|i=1{z }
(m×N)(N×1) matrix
=
=
r
X
i=1
r
X
i=1
δi (JUi ) +
X
(δi J) · Ui +
X
i, j
(m×1) matrix
δi · JN j · (Qi j + ǫi j ) (δ j Id = JN j )
X
i
δi δ j (Qi j + ǫi j )
| {z }
X
δi J(
N j (Qi j + ǫi j ))
Thus it suffices to solve the r equations
X
N j (Qi j + ǫi j ),
(*)
0 = Ui +
j
(δi are scalars).
J
1 ≤ i ≤ r.
This is an equation for an (N×1) matrix. Thus (∗) gives Nr equations
in the Nr unknowns Uiν , 1 ≤ i ≤ r, 1 ≤ ν ≤ N.
4. Tougeron’s lemma
85
We note that Qi j are vectors of polynomials in Uiν all of whose terms
are of degree ≥ 2. Let F = F1 , . . . , F NR ∈ A[Uiν ] represent the right
hand side of (∗). Write Z1 , . . . , ZNr for the indeterminates Uiν . Then
!
∂Fk
= Id + M, M = (mαβ ), (Nr × Nr) matrix
∂Zℓ
where M is an Nr × Nr matrix of polynomials in Zℓ and every mαβ has 103
no constant term.
Let x0 ∈ ANr represent the vector (0, . . . , 0); then we have
F(z0 ) ≡ 0(mod V)
ǫi jν ∈ V.
since
Without loss of generality we can suppose V = a since V is also a
defining ideal for (A, a)
F(Z 0 ) ≡ 0(mod a).
!
∂Fk
Further, we have
is a unit in A/a. Hence by the henselian
∂Z1 Z=z0
property of (A, a), we have a solution z of (∗) in A such that z ≡ z0
(mod a), i.e., z ≡ 0(mod a) since z0 ≡ (0). Set
y = y0 + z.
Then we have
y ≡ y0 (mod ∆a)
and
f (y) = 0,
which proves the lemma.
Remark 4.1. It is possible to take V such that V ⊂ ar , for if (A, a) is
a henselian pair, the henselian property is true for V, V ⊂ a. Then the
above proof also goes through for this case.
Corollary. Let (A, a) be a henselian pair and
f1 , . . . , fm ∈ A[y1 , . . . , yN ]
2. Elkik’s Theorems on Algebraization
86
and ϕ the canonical morphism
ϕ : Spec A[y1 , . . . , yN /( f1 , . . . , fm ) → Spec A
B
104
Suppose that ϕ is smooth and a relative complete intersection. Then
given y ∈ Â (a-adic completion of A) such that
f (y) = 0,
there is a y ∈ A such that f (y) = 0 and
y ≡ y(mod ac )
for any given c ≥ 1.
Proof. We can take c = 1 for ac is also a defining ideal for (A, a). We
can a fortiori find y0 ∈ AN such that
f (y0 ) ≡ 0(mod a)(⇐ f (y) = 0).
y0 then defines B a section of Spec(B) over Spec A/a. The morphism is a
complete intersection and smooth at points of this section. This implies
that the Ideal generated by canonical images in A/a of the determinants
of the (m × m) minors of J(y0 ) is the unit ideal in A/a, i.e., the canonical
image of ∆ in A/a is the unit ideal. Since a is in the Jacobson radical,
it follows that ∆ is itself the unit ideal. Indeed ∆ being the unit ideal in
A/a implies there exists a u ∈ ∆ such that u ≡ 1(mod a), hence u − 1 ∈ a,
hence u = 1 + r, r ∈ a. Since a ⊂ Rad A, u is a unit in A.
5 Existence of algebraic deformations of isolated
singularities
π
Definition 5.1. A family of isolated singularities is a scheme X →
− S
over S = Spec A, A a k-algebra such that (i) π is flat, of finite presentation and X is affine, X = Spec O and (ii) if Γ is the closed subset of X
where π is not smooth, then Γ → S is a finite morphism (for this let us
say that Γ is endowed with the canonical structure of a reduced scheme).
5. Existence of algebraic deformations of isolated...
87
Definition 5.2. We say that two families X → S and X ′ → S of isolated 105
singularities are equivalent or isomorphic if there is a family of isolated
singularities X ′′ → S (note that S is the same) and étale morphisms
X ′′ → X ′ , X ′′ → X such that
(i) the following diagram is commutative
′′
X
} BBB
BBétale
étale }}}
BB
}}
B!
}
~}
XA
X′
AA
|
|
AA
|
AA
||
A }|||
S
and
(ii) these maps induce isomorphisms
Γ′′ B
Γ
~
∼ ~~~
~
~
~~
~
BB
BB∼
BB
B!
Γ′ .
An equivalence class of isolated singularities represented by X → S
is therefore the henselization of X along Γ.
Consider in particular a one point “family” X0 → Spec k with an
isolated singularity. (We could also take a finite number of isolated singularities.) We see easily that the formal deformation space of X0 (in the
sense of Schlessinger defined before) depends only on the equivalence
class of X0 . Let A be the formal versal deformation space associated
to X0 (we can speak of the versal deformation space of X0 by taking
Zariski tangent space of A = dim T X1 0 ), i.e., A is a complete local ring,
and we are given a sequence {Xn } of deformations over An
Xn = Spec On ,
An = A/mn+1
A ,
OA ⊗ An−1 On−1
satisfying the versal property mentioned before. Note that by a versal
deformation it is not meant that there is a deformation of X0 over A. The 106
2. Elkik’s Theorems on Algebraization
88
following theorem, proved by Elkik, asserts that in fact a deformation of
X0 over A does exist (i.e., in the case of isolated singularities). It is the
crucial step for the existence of “algebraic” versal deformations for X0 .
Theorem 5.1. Let X0 = Spec O0 , O0 = k[X1 , . . . , Xm ]/( f ), ( f ) = ( f1 ,
. . . , fr ). Let Xn = Spec On be as above, but we suppose moreover that X0
is equidimensional of dimension d. Then there is a deformation X ′ over
A such that X ′ ⊗ An ≈ On , and if X ′ = Spec O ′ then O ′ is an A-algebra
of finite type. (We do not claim that O ′ has the same presentation as
O0 .)
Proof. Let O = lim On , On = An [X1 , . . . , Xm ]/( f (n) ), where fi(n) ∈
←−−
An [X] is a lifting of fi ∈ k[X]. Let A[X]∧ denote the m-adic (m = mA )
completion of A[X]. We see that A[X] is the set of formal power series
P
ai X (i) such that ai → 0 in the m-adic topology of A. Then f i = lim fi(n)
is in A[X] and we see easily that
O = A[X]∧ /( f ).
(For, we see that we have a canonical homomorphism
α : A[X]∧ /( f ) → O
obtained from the canonical homomorphism A[X]∧ /( f ) → An [X]/( f (n) ).
It is easy to see that α is an isomorphism.)
The proof of the theorem is divided into the following steps:
107
(1) It is enough to find a (flat) deformation X ′′ over A of X0 such that
O ′′ ⊗A A1 ≈ O1 , where X ′′ = Spec O ′′ (recall that A1 = A/m2 ).
For, given a (flat) deformation X ′′ over A we get deformations
= X ′′ ⊗ An over An . For each n, we get then by the versal property
of A, a homomorphism
αn : A → A n .
{Xn′′ }
Note that αn is defined by
(αn )m : Am → An ,
m ≫ 0,
5. Existence of algebraic deformations of isolated...
89
so that Xm ⊗Am An ≈ Xn′′ . These {αn } are consistent and hence {an } patch
up to define a homomorphism of rings
α : A → A.
The hypothesis that O ′′ ⊗A A1 ≈ O1 implies that
α ≡ Id(mod m2A ).
This condition on α implies that α is an isomorphism; for it follows
that α induces an isomorphism on the Zariski tangent spaces, so that
Im α contains a set of generators of mA , hence α is surjective; further,
this condition implies that the induced homomorphisms αn : A/mnA →
A/mnA are surjective, and these vector spaces being finite-dimensional,
it follows that αn is an isomorphism for all n (in particular injective). It
T
follows easily that Ker α ⊂ mnA = (0), i.e., α is injective. Hence α is
n
an isomorphism.
Now define the deformation X ′ over A as the pull back of X ′′ over A
by the isomorphism α − 1. It is easily checked that X ′ ⊗A An ≈ Xn , and
this proves (1).


 ∂ f i 


Let us set X = Spec O. Consider the Jacobian matrix J = 
∂X j
1 ≤ i ≤ r, 1 ≤ j ≤ m (( f ) = ( f 1 , . . . , f r )). Let Γ denote the locus 108
of points in X = Spec O where rk J < (m − d). Then Γ is a closed
∂fi
subscheme in X. (We note that f i ∈ A[X]∧ and
∈ A[X]∧ so that
∂X j
the Jacobian matrix J is a matrix of elements is A[X]∧ .) Hence if x ∈
Spec A[X]∧ (in particular if x ∈ Spec O = X ֒→ A[X]), we can talk of
the rank of J at x, i.e., the matrix J(x) whose elements are the canonical
images in k(x) (residue field at x) of the elements of J. It follows then
easily that the locus of points x of X where J(x) is of rank < (m − d) is
closed in X; in fact we see that Γ = V(I) where I is the ideal generated
by the determinants of all the (m − d) × (m − d) minors of J ′ where
J ′ is J with elements replaced by their canonical images in O. It is
clear that Γ ∩ X0 is precisely the set of singular points of X0 , which
is by our hypothesis a finite subset of X0 . It can then be seen without
90
2. Elkik’s Theorems on Algebraization
much difficulty that Γ is finite over A (Γ is endowed with the canonical
structure of a reduced scheme or a scheme structure from the ideal I
introduced above). The proof of this is similar to the fact: quasifinite
implies finite in the “formal case”, i.e., in the situation A → B where A,
B are complete local rings and B is the completion of a local ring of an
A-algebra of finite type.
Let Γ = Spec O/∆ and let ∆0 be the ideal in O0 defined by ∆ (i.e., the
canonical image of ∆ ⊗ k in O0 ). By the Noether Normalization lemma
we can find y01 , . . . , y0d in ∆0 such that O0 is a finite k[y01 , . . . , y0 d] module
such that the set of common zeros of y0i is precisely the set of singular
points of X0 . Lift y0i to elements y1 , . . . , yd in ∆ so that y1 , . . . , yd vanish
on Γ. Then we have
109
(2) O is a finite A[y]∧ module (and Γ is precisely the locus of yi = 0).
This is again obtained by an argument generalizing “quasifinite implies finite in the “formal” cases.”
(3) The open subscheme X − Γ of X is regular (over A) (i.e., X − Γ is
flat over A and the fibres are regular).
This is a generalization of the Jacobian criterion of regularity to the
adic and formal case.
(4) Outside the set {y = 0} in Spec A[y]∧ (this is a section of Spec
A[y]∧ over Spec A and Γ lies over this set), O is locally free
over A[y]∧ say of rank r, i.e., p∗ (OX−Γ ) is a locally free sheaf
of OSpec A[y]∧ −{yi =0} modules (p : X → Spec A[y]∧ canonical morphism).
For, p∗ (OX ) is finite over Spec A[y]∧ . Now Spec A[y]∧ is regular
over Spec A. We have seen that X − Γ is regular over Spec A, so that it
is in particular Cohen-Macaulay over Spec A. Now a Cohen-Macaulay
module M (of finite type) over a regular local ring B is free (cf. Serre’s
“Algèbre locale”) and from this (4) follows. (We can use this property
only for the corresponding fibres, but then the required property is an
easy consequence of this.)
5. Existence of algebraic deformations of isolated...
91
(5) Set P̂ = A[y]∧ . Then for O considered as a P̂ module we have a
representation of the form
(ai j )
P̂m −−−→ P̂n → O → 0
(exact as P̂ modules)
with rk(ai j ) ≤ (n−r) (i.e., determinants of all minors of rk(n−r+1)
of (ai j ) are zero).
More generally, let us try to describe an R-algebra B having a rep- 110
resentation of the form

(ai j )



Rm −−−→ Rn → B → 0 exact sequence
(*)



rk(ai j ) ≤ (n − r).
of R-modules
The we have
Lemma 5.1. ∃ an affine scheme V (of finite type) over Spec Z such that
every R-algebra of the form (∗) is induced by an R-valued point of V.
Proof of Lemma. To each R we consider the functor F(R)
(
)
set of all commutative R-algebras B
F(R) =
.
with a representation of the form (∗)
One would like to represent the functor F by an affine scheme, etc.
We don’t succeed in doing this, but we will represent a functor G such
that we have a surjective morphism G → F.
An algebra structure on B is given by an R-homomorphism
B ⊗R B → B.
Then in the diagram
Rn ⊗ RRn
∃
Rn
/ B ⊗R B
/B
2. Elkik’s Theorems on Algebraization
92
the homomorphism Rn ⊗R Rn → Rn factors via a homomorphism Rm ⊗
Rn → Rn (of course not uniquely determined). Now the following sequence
ψ
(Rm ⊗R Rn ) ⊕ (Rn ⊗ Rm ) −
→ Rn ⊗ Rn → B ⊗ B → 0
111
is exact where ψ = (ai j ) ⊗ Id ⊕ Id ⊗ (ai j ). [We have
can hom
(Ker(Rn → B) ⊗ Rn ) ⊕ (Rn ⊗ (Ker Rn → B)) −−−−−−→ Ker(Rn ⊗ Rn → B ⊗ B) → 0.
This implies exactness of the given diagram.] Again there exists a lifting of the above commutative diagram
(I0 )
(Rm ⊗ Rn ) ⊕ (Rn ⊗ Rm )
∃ c
ψ
(ai j )=a
Rm
/ Rn ⊗ Rn
/ B ⊗R B
/0
/B
/0
∃ b
/ Rn
On the other hand, giving a commutative diagram
ψ
(Rm ⊗ Rn ) ⊕ (Rn ⊗ Rm )
(I)
c
/ Rn ⊗ Rn
b
Rm
(ai j )=(a)
/ Rn
where ψ = (ai j ) ⊗ Id ⊕ Id ⊗ (ai j ) determines the commutative diagram
(I0 ).
The algebra structure on B induced by (I0 ) is associative if the diagram
Rn ⊗ Rn ⊗L Rn
b·(Id⊗b)
LLL
LLL
Id⊗b LLL&
b·(b⊗Id)
/ Rn
/ Rn
Rn ⊗ Rn ⊗L Rn
>
L
|
|>
L
|
|
L
LLL
,
,
||
||
||
||
b⊗Id LLL&
|| b
|| b
Rn ⊗ Rn
Rn ⊗ Rn
and the map Rn ⊗ Rn ⊗ Rnb·(Id⊗b)−b·(b⊗Id) → Rn factorizes through Rm →
5. Existence of algebraic deformations of isolated...
93
Rn , i.e.,
(II)
Rn ⊗ Rn ⊗LRn
b·(Id⊗b)−b·(b⊗Id)
LLL
LLL
L
∃α LLL
&
Rm
/ Rn
=
|
||
|
||
|| (ai j )=a
or, b · (Id ⊗b) − b · (b ⊗ Id) is zero in B.
Let e
b : Rn × Rn → Rn be the homomorphism obtained by changing 112
n
b : R × Rn → Rn by the involution defined by x ⊗ y 7→ y ⊗ x in R ⊗ R.
Then the algebra structure is commutative if there is a homomorphism
δ : Rn ⊗R Rn → Rm such that
(III)
b−b
Rn ⊗R RIn
II
II
I
δ III
$
Rm
/ Rn
|>
|
||
||(ai j )=a
|
|
commutes.
Finally the identity element 1 ∈ B can be lifted to an element e ∈ Rn
(determines a homomorphism R → Rn ) and there is a map ǫ : Rn → Rm
such
Rne ⊗ b −L Id
LL
LLL
ǫ LLL
L%
(IV)
Rm
/ Rn
|=
|
||
||a=(ai j )
|
|
commutes, i.e., e ⊗ b − Id = 0 in B.
Let us then define the functor G : (Rings) → (Sets) as follows:
G(R) =(i) Set of homomorphisms a : Rm → Rn such that rk a ≤ (n − r)
(i.e., determinants of all minors of a of rk(n − r + 1) vanish),
together with (0)
(ii) Set of homomorphisms b, c
Rn ⊗ Rn
,
(Rm ⊗ Rn ) ⊕ (Rn ⊗ Rm )
c
b
Rn
Rm
2. Elkik’s Theorems on Algebraization
94
113
such that the following diagram is commutative:
(I)
a⊗Id ⊕ Id ⊗a
(Rm ⊗ Rn ) ⊕ (Rn ⊗ Rm )
/ Rn ⊗ Rn
c
b
a
Rm
/ Rn
together with
(iii) an R-homomorphism α : Rn ⊗Rn ⊗Rn → Rm such that the following
is commutative
(II)
Rn ⊗ Rn ⊗LRn
bo(Id ⊗b)−bo(b⊗Id)
LLL
LLL
LLL
L&
Rm
/ Rn
|=
|
||
|| a
|
|
,
and
(iv) an R-homomorphism δ : Rn ⊗ Rn → Rm such that
(III)
b−b̃
Rn ⊗ RHn
HH
HH
HH
δ HH$
Rm
/ Rn
|=
|
||
|| a
|
|
commutes, and
(v) an R-homomorphism e : R → Rn and ǫ : Rn → Rm such that
(IV)
Rn B
e⊗b−Id
BB
BB
ǫ BBB
!
Rm
/ Rn
|=
|
||
|| a
|
|
commutes.
It is now easily seen that G(R) can be identified with a subset S ֒→
RP such that there exist polynomials Fi (X1 , . . . , X p ) over Z such that
5. Existence of algebraic deformations of isolated...
114
95
s = (s1 , . . . , s p ) ∈ S iff Fi (s1 , . . . , s p ) = 0, and the set {Fi } and p are
independent of R. From this it is clear that G(R) is represented by a
scheme V of finite type over Z and since G(R) → F(R) is surjective,
Lemma 5.1 follows immediately.
It follows in particular that O is represented by a homomorphism
ϕ : Spec P̂ → V(P̂ = A[y1 , . . . , ym ]∧ ).
(6) The image of Spec P̂ − {y = 0} in V lies in the smooth locus of V
over Spec Z.
We have a representation
(*)
P̂[z1 , . . . , zs ] → O → 0
(homomorphisms of rings and homomorphisms as P modules). Now
P̂[z1 , . . . , zs ] is regular over A and X − Γ is regular over Spec A − {y =
0}. Hence the immersion X ֒→ Spec P̂[z1 , . . . , zs ] ≃ A s , being an AP̂
morphisms, is a local complete intersection at every point of X − Γ (we
use the fact that a regular local ring which is the quotient of another
regular local ring is a complete intersection in the latter; we use this
fact for the corresponding local rings of the fibres and then lifting the
m-sequence, etc.). Now codim X in A s is s. Now take a closed point
P̂
x0 ∈ Spec P̂ − {y = 0}. Then tensoring (∗) by k(x0 ) we get
k(x0 )[z1 , . . . , zs ] → O ⊗P̂ k(x0 ) → 0
exact.
Now Spec Ô ⊗P̂ k(x0 ) is precisely the fibre of X over x0 for the morphism X ֒→ Spec P̂. We claim that Spec O ⊗P̂ k(x0 ) is also a local complete intersection in k(x0 )[z1 , . . . , zs ] wherever x0 ∈ Spec P − {y = 0}
s
) 115
(it is a 0-dimensional subscheme of Spec k(x0 )[z1 , . . . , zs ] ֒→ Ak(x
0)
and in fact that Spec O ⊗P̂ R0 ֒→ Spec R0 [z1 , . . . , zs ] is a morphism
of local complete intersection over R0 for any base change Spec R →
Spec P̂−{y = 0} (i.e., flat and the fibres of the morphism over Spec R is a
local complete intersection). To prove this we note that O⊗P̂ R0 is locally
free (of rank r) (Spec R0 → Spec P̂ − {y = 0}) and Spec R0 [z1 , . . . , zs ] →
Spec R0 is a regular morphism. In particular a Cohen-Macaulay morphism. Now the claim is an immediate consequence of
2. Elkik’s Theorems on Algebraization
96
Lemma 5.2. Let B, C, R be local rings such that B, C are R-algebras
flat over R and B is Cohen-Macaulay over R. Let f : B → C be a surjective (local) homomorphism of R-algebras such that C is a complete
intersection in B. Then for all R → R0 → 0, the surjective homomorphism
B ⊗ R0 → C ⊗ R0 → 0
(the morphism Spec(C⊗R0 ) ֒→ Spec(B⊗R0 )) is a morphism of complete
intersection over Spec R0 .
Proof. Since the flatness hypothesis is satisfied, it suffices to prove that
C ⊗ k is a complete intersection in B ⊗ k(k = R/mR ). Now we have
0→I →B→C→0
116
exact,
I = ker B → C, and I = ( f1 , . . . , fs ), fi is an m-sequence in B. Now the
codimension of C, Spec(C ⊗ k) in Spec(B ⊗ k), is equal to the codimension of Spec C in Spec B, which is s. (This follows by flatness of B, C
over R and the fact that flat implies equidimensional.) Let f i denote the
canonical images of fi in B⊗k. We have then (B⊗k)/( f 1 , . . . , f s ) = C⊗k.
Now (B ⊗ k) is Cohen-Macaulay and the codimension of Spec(C ⊗ k) in
Spec(B ⊗ k) is s. It follows by Macaulay’s theorem that f 1 , . . . , f s is an
m-sequence in B ⊗ k. This implies that C ⊗ k is a complete intersection
in B ⊗ k, and proves Lemma 5.2.
The complete intersection trick. We go back to the proof of (5). Let
λ : Spec R0 → Spec P̂ be a morphism such that R0 is Artin local and
λ(Spec R0 ) ⊂ Spec P̂ − {y = 0}. Consider the morphism (ϕ ◦ λ) :
Spec R0 → V. Then (ϕ ◦ λ) defines an R0 algebra B which is a free
R0 -module of rank r (in particular flat over R0 ), and B is a morphism
of local complete intersection over R0 (B is of relative dimension 0 over
R0 ). Let R → R0 → 0 be such that Spec R is an infinitesimal neighborhood of Spec R0 . Then the assertion (5) follows if we show that
(ϕ ◦ λ) : Spec R0 → V factors through Spec R → V.
Now B is defined by
a0
−→ Rn0 → B → 0,
Rm
0 −
5. Existence of algebraic deformations of isolated...
97
where a0 = (a0i j ). Now Spec B → Spec R0 is a morphism of local
complete intersection for a suitable imbedding and Spec B. It is clear
that Spec B → Spec R0 is in fact a morphism of global complete intersection for the corresponding imbedding, for it is easily seen that a
0-dimensional closed subscheme of Ank which is a local complete intersection, is in fact a global intersection. We have seen in § 4, Part 1, That
the functor of global deformations of a complete intersection is unobstructed i.e., formally smooth. Hence there is a flat R-algebra B′ such 117
a0
that B′ ⊗R R0 ≈ B. Hence the sequence Rm
−→ Rn0 → 0 can be lifted
0 −
a
− Rn → B′ → 0; the proof of this is in spirit
to an exact sequence Rm →
analogous to imbedding a deformation (infinitesimal) of X ֒→ An in the
same affine space. A homomorphism (Rn0 → B) over R0 is given by n
elements in B. Hence this homomorphism can be lifted to Rn → B′ and
n
it becomes surjective. Now it is seen easily that Rm
0 → R0 can be lifted
m
n
to R → R . It follows that determinants of all minors of a of rank
(n − r + 1) vanish. Thus the relations (0) above can be lifted to R.
Consider the relations (I). We are ginve b0 , c0 such that the following
diagram is commutative:
n
n
m
(Rm
0 ⊗ R0 ) ⊕ (R0 ⊗ R0 )
a0 ⊗Id ⊕ Id ⊗a0 =ψ0
/ Rn ⊗ Rn
0
c0
0
b0
Rm
0
a0
With the above lifting of (a0 ) to a we get
/ Rn .
0
98
(Rm ⊗ Rn ) ⊕ (Rn ⊗ Rm )
a⊗Id ⊕ Id ⊗a /
Rn ⊗ Rn
can
(*)
∃ c
(B)
a
Rm
b∃
(A)
/ B ′ ⊗R B ′ → 0
/0
can
/ Rn
/B
/ 0.
2. Elkik’s Theorems on Algebraization
5. Existence of algebraic deformations of isolated...
99
The two rows are exact. We claim that there is a b which lifts b0 , and
such that the square (A) is commutative. In fact this is quite easy, for it
is clear that we can find b′ : Rn ⊗ Rn → Rn such that (A) is commutative.
Let b′0 be the reduction mod R0 of b′ . Then we see that the composition
118
of the canonical map Rn0 → B with b0 − b′0 is zero. Hence
b0 (z) ≡ b′0 (z)
(mod Im a0 , or Ker Rn0 → B).
Now b0 (z) − b′0 (z) can be lifted to elements in Im a, so that taking for
{za } a canonical basis in Rn → Rn we define b : Rn ⊗ Rn → Rn
b(za ) = b′ (za ) + θa ,
where θa ∈ Im a lifts b0 (a) − b′0 (z). It is now clear that b lifts b0 and the
square (A) is commutative. This proves the claim.
By a similar argument as above there is a c such that c reduces to c0
and the square (B) is commutative. (If necessary we can prolong to the
left the exact sequence of the second row in the diagram (*).) Thus it
follows that the relations in I can be lifted to R.
A similar argument shows that a0 , δ0 , e0 and ǫ0 which are given
representing the point Spec R0 → V can be lifted to R so that the diagrams (II), (III) and (IV) are still commutative. This means that the
Spec R0 → V can be lifted to an R-valued point of V. As we remarked
before, the assertion (5) now follows.
We go back to the usual notations in the theorem. Then: (7) Let
a = mA (y1 , . . . , yd ) ideal in A[y]. Then P̂ = A[y]∧ is also the a-adic
completion of A[y] (of course P̂ is also the mA · (A[y])-adic completion
of A[y]).
P
The proof of this assertion is immediate for a convergent series fi ,
i
fi ∈ A[y] in the a-adic topology is precisely one such that the coefficients of fi tend to zero in the mA -adic topology and the degree of the
monomials → ∞. This implies that a convergent series is precisely a 119
formal power series in {yi } such that the coefficients (in A) tend to 0 in
the mA -adic topology. This is the description of P̂ we had and (7) is
proved. (8) Now for the morphism ϕ : Spec P̂ → V we have that the
image of Spec P̂ − {y = 0} is in the smooth locus of V over Z. We note
2. Elkik’s Theorems on Algebraization
100
that V(a) = (Spec k[y]) ∪ {y = 0} (k residue field of A), i.e., V(a) contains {y = 0} so that the image of Spec P̂ − V(a) (by ϕ) is also in the
smooth locus of V over Z. Now the a-adic completion of A[y] is P̂. Let
e denote the henselization of (P, a), P = A[y]. Now apply Theorem 2
P
proved above. Hence we can find an étale map Spec R → Spec P which
is trivial over V(a) and morphism ϕ′ : Spec R → V (note that P̂ is also
e such that
the a-adic completion of P)
ϕ′ ≡ ϕ(mod N ),
for any given N.
(Note that the a-adic (i.e., aR-adic, a is not an ideal in R) completion of
R is also A|y|∧ ). Let O′ be the R-algebra defined by ϕ′ . Then we have
O ≡ O(mod aN ) (i.e., O ′ /aN = O/aN ). Now O ′ becomes an A-algebra
and then we have
O ′ ≡ O(mod mNA )
for (mA R)N ⊃ aN . Take in particular N = 2. Thus we can find an Ralgebra O ′ of finite type and consequently of finite type over A such
that
O ′ ≡ O(mod m2A ).
Thus to conclude the proof of the theore, it suffices to prove that O ′
is flat over A.
120
(9) Choice of O ′ such that O ′ is flat/A.
We had a presentation of O as follows:
a
− P̂n → O → 0.
P̂m →
We claim that we have a presentation such that
(*)
a
θ



− P̂n → O → 0,
− P̂m →
P̂ℓ →



a

− P̂n → O → 0 is exact, and
where a · θ = 0 and P̂m →





θ⊗kA
a⊗kA

P̂ℓ ⊗A kA −−
−−→ P̂m ⊗ kA −−−−→ P̂n ⊗ kA → O ⊗ kA → 0 is exact,
where kA = A/mA .
5. Existence of algebraic deformations of isolated...
101
This follows if we prove that O is flat over A (cf., § 3, Part 1).
However, (*) can be established directly as follows: We can find an
exact sequence of the form
θ0
a0
P̂ℓ ⊗ kA −→ P̂m ⊗ kA −−→ P̂n ⊗ kA → O ⊗ kA → 0.
(Note that Spec O ⊗ kA = X0 = Spec O0 is the scheme whose deformation we are considering and that Xn = O ⊗ A/mn+1
= Spec On are
A
infinitesimal deformations of X0 .) The above exact sequence can be
lifted to an exact sequence
θn
an
→ P̂m ⊗ A/mn+1
−→ P̂n ⊗ A/mn+1
P̂ℓ ⊗ A/mn+1
A −
A → On → 0
A −
since On is flat over A/mn+1
A . Passing to the limit, we have an exact
sequence
θn
an
P̂ℓ ⊗ A/mn+1
→ P̂m ⊗ A/mn+1
−→ P̂n ⊗ A/mn+1
A −
A −
A → On → 0
since On is flat over A/mn+1
A . Passing to the limit, we have an exact
sequence
θ
a
− Pm →
− Pn → O → 0
P̂ℓ →
a
− Pn → O → 0 is exact. This proves the 121
such that a · θ = 0, and P̂m →
existence of (*).
Now define a functor G′ which is a modification of G as follows:
G′ (R) = Set of {θ, a, b, c, α, δ, e, ǫ where a, b, c, α, δ, e, ǫ are as in definiθ
a
− Rn with a · θ = 0}.
− Rm →
tion of G(R); and θ is defined by Rℓ →
Then as in the case of G(R), we see that G′ is represented by a
scheme V ′ of finite type over Z. The given representation for O as in
(∗) above gives rise to a morphism ψ : Spec P → V ′ . We claim that as
in the case of V, the image of Spec P − {y = 0} lies in the smooth locus
of V ′ . With the same notations for R, R0 as in the proof of the statement
for the case V, it suffices to prove the following: Given
θ0
a0
Rℓ0 −→ Rm
−→ Rn0 → B → 0
0 −
2. Elkik’s Theorems on Algebraization
102
a0
such that a0 · θ0 = 0 and Rm
−→ Rn0 → B → 0 is exact, and a flat lifting
0 −
′
B over R (hence free over R), we have to lift this sequence to R (the
proof of the lifting of the quantities involved is the same as for G(R)).
As we have seen before, for G(R) we have a lifting
a
Rm →
− Rn → B′ → 0.
Now Im θ0 are a set of relations. Since B′ is flat over R these relations can also be lifted, i.e., we have a lifting
θ
a
− Rn → B′ → 0
− Rm →
Rℓ →
a
122
− Rn → B′ → is exact. This proves
such that a · θ = 0 and Rm →
the required claim and hence it follows ψ(Spec P − {y = 0}) lies in the
smooth locus (over Z) of V ′ .
Applying Theorem 2, we can find an étale Spec R → Spec P, P =
A[y] and a morphism ψ′ : Spec R → V ′ such that
ψ′ ≡ ψ(mod a2 ).
Let O ′ be the R-algebra defined by ψ′ . Then as we have seen before,
we have
O ′ ≡ O(mod m2A ).
We claim that O ′ is flat over A. For this we observe that we have a
fortiori
O ′ ≡ O(mod mA ).
This implies that O ′ /mA · O ′ = O/mA O ≈ O0 . Let
a′
θ′
(I′ )
P̂ℓ −→ P̂m −→ P̂n → O ′ → 0
a′ ◦ θ′ = 0, P̂m → P̂n → P̂n → O ′ → 0 exact
be a representation of O ′ . Recall we have the representation for O
(I)
θ
a
P̂ℓ →
− |
P̂m →
− P̂n{z
→ O → }0,
exact
and
a ◦ θ = 0.
5. Existence of algebraic deformations of isolated...
103
Now tensoring (I′ ) with A/mnA yields
a′ ⊗A/mnA
θ′ ⊗A/mnA
→ P̂m ⊗ A/mnA −−−−−−−→ P̂n ⊗ A/mnA → O ′ ⊗ AmnA → 0,
P̂ℓ ⊗A/mnA −−−−−−−
|
{z
}
exact
and
(a′ ⊗ A/mnA ) · (θ′ ⊗ A/mnA ) = 0.
We have (I ′ ) ⊗ A/mA = (I) ⊗ A/mA , as a consequence of the fact that 123
≡ O(mod mA ). By (I) I⊗A/mA is exact. This implies that (I ′ )⊗A/mA
is exact. So we have that (I ′ ) ⊗ A/mA is exact, and (I ′ ) ⊗ A/mnA has the
property, (a′ ⊗ A/mnA ) · (θ′ ⊗ A/mnA) = 0 and
O′
a′ ⊗A/mnA
P̂m ⊗ A/mnA −−−−−−−→ P̂n ⊗ A/mnA → O ′ ⊗ A/mnA → 0,
for all n. This implies, as we saw in the first few lectures, that O ′ ⊗ A/mnA
is flat over A/mnA for every n.
Now O ′ is an A-algebra of finite type and so O ′ ⊗ A/mnA flat over
A/mnA for all n implies that O ′ is flat over A (cf. SGA exposes on flatness).
The proof of the theorem is now complete.
Remark 5.1. The fact that O ′ is flat over A can also be shown in a
different manner. This can be done using only the functor G (i.e., V), but
a better approximation (i.e., better than N = 2) for O ′ may be needed.
This uses the following result of Hironaka: Let B be a complete local
ring, b ⊂ m an ideal in B and M a finite B-module locally free (of rank
r) outside V(b). Then there is an N such that whenever M ′ is a finite Bmodule locally free of rank r outside V(b) and M ′ = M(mod bN ), then
M ′ ≈ M. Take in our present case B = A[[y]] so that we have

A=
==
==
==
==
==
==
==
==
=
/ A[y]∧ 
<
yy
yy
y
yy
- yy
=R
||
|
|
||
. ||
A[y]
/ A[[y]]
104
124
2. Elkik’s Theorems on Algebraization
R étale over A[y] such that A[y]/a ≈ R/a and that all the extensions are
faithfully flat. Take a coherent sheaf on Spec R; to verify that it is flat
over A it suffices to verify that its lifting to Spec A[[y]] is flat over A.
Take b to be a · A[[y]]. Then by taking a suitable approximation for O ′
it follows that the liftings to A[[y]] of O ′ and O are isomorphic. This
implies that O ⊗ A/mn ≈ O ′ ⊗ A/mn , hence that O ′ is flat over A, since
Spec(O ⊗ A/mn ) = Xn is flat over A.