THE LOCAL MODULI PROBLEM. APPLICATIONS TO ISOLATED HYPERSURFACE SINGULARITIES

) /

CONTENT

INTRODUCTION

§1. The prorepresenting substratum of the formal moduli.

§2. Automorphisms of the formal moduli.

§3. The Kodaira-Spencer map and its kernel.

§4. Applications

to

isolated hypersurface singula.rities.

§S. Plane curve singularities with k*-action.

§6. The generic component of the local-moduli suite.

§7. Appendix (by B. Martin & G. Pfister). An algorithm to compute the kernel of the Kodaira-Spencer· map for an irreducible plane curve .singularity

NOTATIONS

k field of characteristic o and algebraically closed k* the multiplicative group of k

the projective n-space

the category of local artinian k-algebras with residue field k

"

the category of local artinian H -algebras with residue field k the category of groups

aut5(X®kS) = {~EAut

5

^{(X®kS) 1~®}

5

^k=l},

^s

^{in 1}

autR(~"®H"R)

=

{~EAutR(X"®H"R) I~®Rk=l},

R in !H"

autR(~"~R)

=

{~EAutR(H"~R) I~<~ "~R) ~ ~ "~R}

H H

Introduction. ~e purpose of this paper is to contribute towards a better understanding of the local moduli problem in algebraic geome- try. Let k be field and let X be an algebraic object, say a pro-

jective k-scheme.

The local moduli problem may then be phrased as follows. Describe the set of isomorphism classes of objects X' occuring as "arbitrary

small deformations" of X.

In practice this means to define a natural filtration on the set of 'these isomorphism classes, such that each subset of the filtration may

be given an algebraic structure, say as a k-scheme or, more generally, as an algebraic space. We shall refer to any such, natural, filtra- tion

{M }

_,; as the lqcale moduli suite of X. This done, one would like 'to find the local structure of these new objects, their dimen- sions etc.

Our approach, which is rather general in scope, starts with a study of the infinitesimal automorphisms of the formal moduli of X, leaving the formal versal family invariant, see §1 and §2.

Coupled with a close look at the properties of the kernel V of the Kodaira-Spencer map of a formally versal family X+H

-

of X, whi~h we shall assume exists, this leads to a proof of the existence ·of a

"fine" local moduli suite in the category of algebraic spaces, see (3.18), provided tne objects X of our study, satisfY a set of rather strict conditions, see §3, (A

1 ^)~ (A

2) and (V' ). In particular these condit·ions imply the existence of an algebraization of the formal versal family, see (3.6) and (3.7).

One of the main results of this paper is the theorem (3.24) which asserts that the fine local moduli suite

f.t!,;}

is, generically the quotient of a canonical filtration {~,;} of the base space H by· the k-Lie-algebra

v.

In the process we obtain useful criteria for smoothability ·and non- smoothability of singularities, see (3.10).

We also show that for every ~ there is a flat family of Lie-algebras the fibers of which are the Lie-algebras of non liftable infinitesimal automorphisms L(X') of those X' representing the classes of M •

~

In §4 we specialize to the case of a hypersurface singularity in the algebroid sense ..

Finally in §S and §6 we treat the case of quasihomogenous plane curve singularities.

To illustrate the main ideas of this paper, let us consider a simple example, that of the cusp X'= Spec(k[x,y]/(x3+y2). As an affine scheme X admits an affine formally versal family, (see §4), the obvious algebraization of the formal versal family, given by

1t:X+H.

Refering to (4.2) we find that the Kodaira-Spencer map

maps

e

at0

1 -

g:Derk(H) +A (H,X,OX)

to 1 and to x, 1 and senting classes of

1 - aF aF

A (H,X;OX) = H(x,y]/(F,ax'ay).

X considered as repre-

The kernel V of g is easily seen to be generated as an H-module by the derivations

00 a 2 0

= to ~ + Jtl at 1 0

01 ~t2 a

+ to a

=.-

_at

0 et

9 1 1

3

Obviously, the,generic fiber of ~ is rigid, and by (3.26) the discriminant of ~ is therefore the determinant of V, i.e.

D.= (27t&+4t~)

as i t should. The closed points of the open subscheme ~

0 =

^{H D.}

correspond to the elliptic curves x³+y2+t 1x+t

0, D.:f:O which form a one dimensional family of isomorpnism-classes of affine curves, M

0 •

However the quotient

s

0

;v

is easily seen to be reduced to· a point.

This contradicts the assertion of (3.2~), and points to one of the main difficulties of the theory of local moduli. In fact the condi- tion (A

2) of §3 is not satisfied for affine schemes. Therefore the k- Lie-algebra V does not, in general, define the correct equivalence realtion on

_--r s.

If however, we choose to consider the cusp as a cone, or as a graded k-algebra, C

=

k[x,y,z]/(x3+y2z) we find that there is a formally versal family of graded k-algebras with, as before, H

=

^k[t₀^{, t}₁ ^{], and}

c =

H[x,y,z]/(x2+y2z+t

1xz2+t

0z3).

In this new category, the kernel V

Spencer map is generated by

&

₀ ^and

6

t~

Spec(k[t0, t1] O)

=

Spec(k[-xJ) where

of the corresponding Kodaira- th.e quotient

!:!Q

=

§..o/V

^{is now}

-i

t3 is a modular function for the elliptic curves in the family ~, in the classical sense. The point here is, of course, that for finite type graded k-algebras, the condition (A

2) is satisfied, so that we may apply (3.18) and (3.24).

However, we cannot, as the above example might lead us to believe, always reduce the local moduli problem of an affine scheme to the

corresponding problem for a graded k-algebra, as is easily seen in the example X= Spec(k[x,y]/(xn+yn)).

To treat the local moduli problem for affine schemes, i t seems there- fore that we have to find a r~placement for the Kodaira-Spencer kernel V -·as a means to defining the correct equivalence relation on the S •s. This may not be impossible, but we shall in this paper consen-

-~ /

trate on applications of the general theory of §3, to those categories of objects that satisfy (A

2). This, in particular, includes the category of isolated hypersurface singularities in the algebroid sense, see §4, §5 ·and §6.

The main results are summed up in the introductions to each paragraph.

This paper is the outgrowth of a colaboration between the two authors during the last 3 years. A first, very sketchy version appeared in 1983 [La-Pf].

Many authors have previously treated the same subject. In particular Zariski in his lecture notes [Z], published in 1973, laid the foun- dations to the study of hypersurface singularities in the algebroid sense. Obviously, his results together with those of Delorme [Del], Washburn [Wash], etc. have influenced upon our work, even though our methods are quite different, and our goals seemingly somewhat wider.

It should be explicitely mentioned that Palamodov in [Pal] studied the notion of prorepresenting substratum (§1) and that Saito in [s]

has the same calculation as we obtain in §4, of the kernel of the Kodaira-Spencer map. These ideas are, however, part of the folklore of the last 15 years, and were at the origin of one of the authors interest in this subject, see [La].

Finally, we are gratefull for the financial support by the Humboldt- Universitat of Berlin, DDR, by the University of Oslo, Norway· and by the Norwegian Res~arch Council, NAVF.

Given an algebraically closed field k of characteristic zero, we shall, troughout §l-§3 be concerned with an algebraic object X such as,

Example 1. X = £¹ a small category of k-schemes.

Put Ai i

=A (k,c,O ),

- c ^i;>O, see [La].

Example 2. X = Spec (A) , A any k-·algebra with isolated singula- rities. In particular, we shall be interested in the case where A= (k[[x

1, •.. ,Xn]]/(f)) is the local ring of an isolated hyper- surface singularity. I~ this case A⁰ = Derk(k[[x

1, .•. ,Xn]]/(f}), and

1 ()f ()f

A

=

(~}·k[[x

1

^{, ..•} ,xn]]/(x}•(ax-,···'ax-'f).

. l n

Example 3. X

=

^~' a small category of oy-Modules where Y is some k-

i i

scheme. Here A = Ext

0 (0 ,o ), i)Q are defined as in [La] with y ~ ~

Hom· replacing Der. See the concluding remark, loc.cit. p. 150, and see [La 2].

Example 4. X

=

E, a coherent 0 -Module. Ai

Pn Of

particular interest is the case where E is a vector bundle on Pn.

Assume now that

di~Ai<

^{... for} i = 1,2. Then, see e.g. [La], (4.2.4), there exist in all these cases a formal moduli HA (a prorepresenting hull for the deformation functor} of X, and a formal versal family

The first part of this paper is devoted to the study of. in this generality.

b

§1. THE PROREPRESENTING SUBSTRATUM OF THE FORMAL MODULI

Introduction. Let X be an object of the type we are considering, see the main Introduction above.

The basic notion in the study of local moduli of X is the notion of prorepresenting substratum of the formal moduli.

If HA is the formal moduli of X, then the prorepresenting sub- stratum HA

0 is the unique maximal quotient of HA for which the obvious composition

A A

Mor(H

0,-) + Mor(H ,-) + Defx is injectiv.

~This quotient exists in all gene~ality, and the object of this § is its construction, see (1 .3). Later, in §3, we shall want to extend this notion to the algebraization of the formal moduli. It turns out that this is facilitated by the introduction of the concept of the

th. . .

n equ~cohomo~og~cal substratum H(n) of HA and by proving that equicohomological substratum · H(O). A

HA 0 coincides with the 0-th

Let X be any algebraic object of the type discussed in the Introduc- tion, and consider the deformation functor

the corresponding cohomol9gy Ai

= Ai(k,X~OX),

i)O and the universal obstruction morphism

where T ⁱ

=

^{Sy~ (A i*} ) , (see [La ^1\ 1 ] ( 4. 2. 4) ) . HA

=

T 1 ® k

T2

Denote by

the formal moduli·of X, i.e. the prorepresentable hull of the defor- mation functor DefX , and put

" "

H

=

Spf (H ) .

In general there are lots of infinitesimal automorphisms of X, and obstructions for lifting these (see [~]). Therefore H" does not necessarily prorepresent Defx· However, as we shall see there is a universal prorepresenting substratum of H", corresponding to a quotient

of

H".

In fact, let us consider the category

!a

of all artinian local

"

H -algebras with residue field k.

Let

x"

be the formal versal family on H" defined by the identity

" "

1 "EMor(H ,H ) and consider the functor H

defined by:

Theorem (1.1). Assume

di~Ai

is countable i

=

0,1. Then there

( i ) (ii)

exists a morphism of complete local

0 :

a such that

oa (~~Td ^c= m2H~Tl a = (H~ ^{0 ) ® H"}

x"

k H~Tl k

"

H -algebra-s

is a prorepresenting hull for the functor aut

--x"

Proof. This follows from the proof of [Lal], ( 4. 2. 4) with aut

-x"

replacing Defx and i-1

A replacing Ai, i

=

^{1,2. Q.E.D.}

Recall that there is the usual automorphism functor of X,

Autx: sch/k + ~ defined by:

Assume AutX is represented by the k-scheme Aut(X) and let

"

^.

0 Aut (X), 1 1EAut(X) be the identity element. Thert the completion

of the local ring of Aut(X) at 1, represents the fiber-functor of AutX at 1EAutk(X), i.e. the functor

defined by

.Let aX be the prorepresentable hull of autX, such that with the assumption above

"

~ = 0Aut (X), 1 •

Notice that if AutX is smooth, then ax=Sy~(A o* " ) (see [Lal] Ch. 4).

Definition (1.2). Let the ideal

~~

H" be generated by the

coefficients of the.elements of

oa(~)·c ~ ~T

⁰

),

m being the maximal ideal of

H

^{~ Tl.} Then the prorepresenting substratum

is the formal sub-scheme defined by ^{OL •}

q

Put a0 ^/1.

=

a ^/1.

/M_,· •

Then

!!o

^/1.

=

Spf (a0 ) and we shall, mildly abusing ^/1.

the notations, also speak about the prorepresenting substratum By construction of

of a" for which

0 . i t is clear that a"

a 0 is the maximal quotient

is a^/1. ₀-smooth .•

Proposition (1.3). a~ is the maximal quotient of a" for which the canonical morphism of functors on 1,

is injective.

Proof. Let

a"

1 be a quotient of are mapped onto the same element

a", and assume ~

1

^.~

2

EMor(a~,R)

~l

=

^~

₂

in Defx(R). This, of x"® R

~2

.course, means that there exists an R-isomorphism x"® R ~

~1

where at the left side R is considered as a"-module via ~l and at the right hand side R is considered as a"-module via ~

2

^•

We may assume, by induction, ~

1

^=~

2

^{(mod~) where n} is some ideal of R killed by the maximal ideal mR. Then ~ ® R/~ is an auto- morphism of X ^/1. ® R/n, corresponding to a morphism

a" -

^{a ®}

a"

₁ ^{+ R/!:!,·}

x"

^If

a ®a"

x" a"

^{1 is} formally a

1-smooth, then· obviously this morphism may be lifted to a morphism a ~ a

1 ⁺ R, proving that ~ ®R R/n is liftable

x"

as an automorphism to·some ~

1

^: x"® R

=

x"® R. But then

~2 ~2

is an isomorphism extending the identity of From this follows that

is injective.

10

Conversally assume

a"

₁ is a quotient of a" such that

"

p1: Mor(H

1,-) + Defx is injective. If H~-algebra; t~en any automorphism ^q; _n always be lifted to an automorphism of

of

R, an object of ~H' is any

" " "

X ® R/n = (X ® H

1 )® R/n may

a" - a"

H~ -

X"® R. It follows that a ® H~

H" x"a"

has to be formally smooth, which proves the proposition. Q.E.D.

Remark (1 .4). Recall that H/m2 represents the restriction of the deformation functor Defx to the subcategory _!2 = {RE!I~=O}

of l. Notice that, never the less, H/~2 is rarely a quotient of H~, see §4 for lots of examples~

Consider for any n>O the subfunctors Defx ⁿ of Defx defined by:

is a deformation of

Then one may prove th~t Defx has a prorepresentable hull H(n) n "

which is a quotient of a".

Definition (1.5). The formal subschema ^{Spf H(n)}

"

is called the n-th equicohomological substratum of a", and is denoted by ^!!(n).

"

Proposition (1.6). The prorepresenting substratum

a"

0 coincides with the 0-th equicohomological substratum

Proof. For any object R of !a' there exists a bijectiv map

exp:~er{A

⁰

(R,x":"R;OX"®"R)

⁺ AO(k,X;OX)} ⁺ autR(x":"R)

H.

the inverse of which is log.

In fact, any element a of ker{AO(R,x":"R;OX"®"R) + A⁰(k,X;OX)}

H

II

the module case) mapping any local section x of OXA® R into HA

~·OHA® R. Since for some HA

h , A

a omomorph~sm · v :H ( 0) +R,

n, ~H.OXA® n R

=

⁰

HA

exp a is defined.

then the map

Given

induced by some surjective morphism ~:R+S in

!r

is surjectiv. This implies that the corresponding

is surjectiv, thus by definition of there exists a unique morphism H₀^A +H(O)' such that the diagram ^A

HA A

+ H(O}

"' /1-v

HA 0 ^IJ.+ R conunutes.

On the other hand, given a homomorphism 1J.:H^A

0+R we may compose i t with the canonical homomorphism R+R(e:] to obtain a homomorphism

By definition of

the horizontal maps in

autR(e:J(XA:AR(e:J) oR[e:]

-t R .

A .

autR(X ® R}

HA

i t follows that for any surjective the following diagram are surjectiv, oR[e:]

s [

^e:]

+ aut₅ [e:] (xA:As [e:]

~

oS[e:].

"' s

~:R+S in .R.

I~

Since the vertical maps have sections, we find that AO(R,XA®AR;OHA® R)

=

^kero:[e] maps surjectively onto

H HA

AO(s,XA®AS;OXA®

8)

= kero~[eJ.

By definition of

H~O)

there exists a

H HA

unique morphism such that the diagram

HA A

+ H(O)

{-

/

^{{- v}

HA + R

0 1.1.

commutes. Consequently we find

a8 = a(

₀). Q.E.D.

Remark ( 1 • 7 ) • Let H(i) ^A be the i-th equicohomological substratum

of HA, and consider ~he subfamily A A A

1t ( i.) :X ( i ) +H ( i ) of 1t • Let·· !!.,., and let XA be the restric- be the intersection of the H(i) ^{A I} s, CD

i A A

A (H ,X ;OXA) _{CD CD} is HA-flat for all _CD tion of to

' _CD

i>O.

Suppose that is of finite type over H . ^A Then, in

""

is reflexive as an

A/

n

H m -module, for all

CD -

n;>O.

Now assuming we have a flat family ~:Y+Spec(S) such that Ai(S,Y;OY) is reflexive as an S-module for i = 1,2, there exist a morphism of complete S-algebras

such that the s-algebra

1"3

I 'I

'is the formal moduli of ^{Y(!3_) -} = T1 ^-1 (~) for all closed points

!3.. ESpec ( s) • The proof of this parallels the proof of [La/]~ ( 4. 4. 2) .

fl·

AUTOMORPHISMS OF THE EORMAL MODULI

Introducion. The prorepresentable substraum constructed in §1 is a closed_subproscheme ~8 ^{of In this} § we shall show that is the fixed proscheme of HA under the action of a subgroup functor

!.x

^{of aut} _{- A}

H

contained in the covering automorphisme group functor of the morphism p:Mor(H ,-) ^1\ ~ Defx·

Notice that this does not some natural subgroup of

imply that

a8

is the fixed proscheme of autk(H ), see 1\ (1.4).

As we shall see in §3, the group-functor iX does not, in general, extend to a group functor of automorphisms of an algebraization H of HA. To remedy this we consider the Lie-algebra-functor 1 ( 1t/\) of

ix·

The main result of this § is then (2.5), where we, in particular, prove that 1(1t/\) is an H/\-submodule and a sub k-Lie-algeqra of Derk(H ), such that A 1(1tA) ~ ^k ~ Ao(k,X;OX)/Ao/\ where

H 1t

is the Lie-ideal in A 0(k,X;OX) of those infinitesimal automorphisms of X

that can be lifted to X • Moreover ^A

defining a deformation of the Lie-algebra to

Consider any formal de~ormation of X, Spf (S ) . 1\

L(X)

=

Ao (k X•O )/AO

' I X 1\

1t

In particular we shall be interested in the formal versal family

1\ 1\ 1\ 1\

1t :X ~ ~

=

Spf ( H ) .

Let . aut be the subfunctor of Aut :1 ~ Sets such that for

--SA --SA -

IS

every object R of

!,

aut (R) is the subset of those

--s"

for which the following diagram commutes

s"

~ R

+

R

+

R •

Consider the subfunctors _--n: ^{I , i}

-Jt and l.

·*

-'It

I (R) = {4~E~ "(R)IY"fR X

=

Y"~(S"f R)}

-1t

s

i (R) = { ~ E Ik < R >

1 x

^{"® k} = idx}

-'It

s

i*(R) = {~E~(R)

lx

^:1\k

=

^idX®R}

-1t

of _{- 1 \} aut

s

where we have written Y"~(S"fR)

by the morphism

for the pull-back of

41EAut (R)

s"

defined by

"

"

Y A xSpec (R)

In particular, corresponding to the formal versal family, we put

I(X) = IX(k)

=

_-n; ^{i 1\}

i(X)

=

ix(k).

Notice that i(X) = !x(k) ·= i~(k), ^by definition, consists of those auto-morphisms ⁴¹ of HA which leaves

fixed, i.e. s.t. the diagram

Mor*(H ,-) A ⁴¹

*

^A'

+ Mor (H ,-)

*

Jb

commutes. The group-functor ix thus measures the extent of non prorepresentability of Oefx .

Recall, from §1, that if AutX is smooth, then autx is prorepre- sented by To .

Now it is easy to show that autx, restricted to

!,

is smooth, in all generality. Let us prove it when X· is a k-algebra.

Consider surjective morphisms p:T+R and ~:R+S of 1 such that

where We may write

m

o = 2: r.o., i=l ~ ~

9R = ldXfR+O

riEker~, OiEOerk(X). Pick t.ET such that p(t.) = r. and

~ ~ ~

m

consider the derivation o•

= I

t.O.EDerk(X,XfT)• o• defines in i=l ~ ~

an obvious way a derivation o•eoerT(XfT)· Let

a•

= expo•

=

'do• 1 o•2 1 o•3 Th a• · 1 f ( T)

~ + ~1

·+J

₁ ^{+... . en ~sane ement o AutT}

Xf

^such

that e·~R = aR. An easy induction argument then shows that

aut:x(T)+~(R) is surjective, thus autX is smooth.

Theorem (2.1). There is a (non-canonical) morphism of the under- lying set-theoretical functors

such that

(i) <im~> = !x , as subfunctors, of Aut

--riA

(ii) H8 is the maximal quotient of HA trivializing i(X).

Proof. Consider the prorepresenting hull aX , of the group functor autXA' see §1. The identity .id: ax+ ax corresponds to the univer-

17

sal automorphism

By (1,1) aX= H ~To/~, where ~ is an ideal contained in the square of the maximal ideal of H ~ _{T .} a·

Consider now the trivial lifting x" _K: ^{~ T 0} of x" ~

a"

^ax

defined in terms of the quotient morphism q:

H~

^{To ++}

the lifting correspon~ing to the canonical homomorphism

. a"

~=

to H

~

T⁰

ax ^,

^i.e.

The automorphism

e

of

x"

~ a may be lifted· to an isomorphism

a" x

e

making the following diagram commutative

x" ~

^{To +}

x"f<_H~T

⁰

)

a

"' "'

x" _a"

~ _ax

_e

⁺

x"

_H

~"

^ax

where ~:

a".

⁺ H ~ T⁰ is some homomorphism such that ~oq = ioq.

k

By definition is the maximal quotient of for which

Consider the map which associates to every aEMork(T⁰ ,H~R) =

( o" " )

MorR T fR,HfR the composition

~(a): H ~ R k

Since 1 ® a maps H"

" . f

H ~ R, ~t ollows

H ~ R k

()V into the square of the maximal ideal of from ~oq = ioq that ~(a) reduces to the identity on the tangent level. Therefore ~(a.)EAutR(H ~-R), and we

obtain a map

+ Aut (R)

--HA

which, as one easily checks, is functorial. -·Furthermore, by construction,

x"® R k

x"

~ H ~ R

<t>(cx)

.

is an H ~ R-isomorphi~m, so we know· <t>(cx)Eix(R). We need only prove that ix(~) is generated by <t>R(autx(H~R))_. ^The ~est is clear.

Therefore the proof is reduced to proving the next ~reposition.

Q.E.D.

Proposition (2.2).

(i) Let 6El4or(H ,8) ^/\ correspond to .the deformation X

8EDefX(8).

Then for every ~Ei(X), the morphism ~ooEMor(H,8) corresponds to the same deformation x

8 .

(ii) If the surjections ++ 8, i=l,2, correspond to the same deformation x

8 , morphism cxnEautX(H~R)

k

then there exists a sequence of auto- such that p2 =lim <t>(cx )o<t>(cx _{n+a~ .} n _. n-1)o••

Proof. (i) is obvious. To prove (ii) consider the morphisms

pf: H~R ⁺⁺ 8/~2 = 82. By assumption, we have a commutative diagram (X"®kR)

, .j.

<x"~R)

.j.

X

® pl

®

PI

8 ⁺

-

't

82 ⁺

-

'"2

=

(X"®kR)

.j.

cx"~R)

.j.

X

® 8 p2

® 82 p2 2

where -. and -.

2 are isomorphisms.

I 'I

Since H

2 represents the deformation functor DefX restricted to the subcategory ~

2

^of

^!, pi =

p~ _and is therefore an auto- morphism. As such it corresponds to a morphism ~

2

^{: ~ +}

s

₂, which, composed with the canonical morphism

morphism ~

2

^{: H ~To +}

^s

^{2• Lift ~}

2

+ H ~ R, and consider the composition k

lj) (.a

2): H~R ⁺ H~TO ^{® R}

k ij>®lR k k

H ~ To ⁺ a gives us an HA-

k X '

to an HA-morphism a

2: H ~ To - k

By construction there is a commutative diagram

S®(H~Rl

XA®R ⁺ (XA®R) ® (H~R) -

k k

•<a

_{2 )}

Now, consider

Pi = •<a

₂^)op₁. Then, put

s

₃

=

^S/~3, and consider the commutative diagram

(XA®R) ®

s

⁺ (XA®R) ®

s

pI 1 -r:' k p2

"' "'

(XA®R) ® 53 ⁺

-

(XA®R) ®353

k p•3

'!:3 k P2-

1

"' "'

(XA®R) ® 52 ⁼ (XA®R) ® 52

pl2 p~

It follows from the commutativity of the lower square that p~ ^, therefore that ~

3

is an automorphism.

Now, copy the procedure above, get a

3: H ~ T^{0 +} H

f

^R such that

if p"

=

<!>(a: )op¹ then

. 1 3 1 p"4

=

p4

1 2 etc. See now that the correspon-

"

ding anEautX(H

f

R) n>2, have exactly the propert~es of (ii).

Q.E.D.

There is an obvious homomorphism of group functors

a: Autx

In fact, to each automorphism a:EAutX(R)

=

^AutR(X®R) there exists k

-an automorphism <!>(a:)EIX(R) such that the following diagram commutes x"®R

.r

X ® R k

+ a

(X"®R) ® (H ~ R) k q,(a:) k

.r -

X® R k

just as above. It is clear that the class of q,(a:) in Ix(R)I~~(R) is unique, and one checks easily that the map a: + a(a:)

=

^{class of}

$(a:) is a group homomorphism.

Let Aut~ (resp. aut~) be the subgroup functor of Autx (resp. autX) consisting for each R of those automorphisms of X

f

^R that lifts to

x"f

^R.

With these notations we have the following:

Corollary (2.3). (i) There is a canonical action of I(X)Ii(X) on (ii) AutX AutX

I

^{1 ::!} IX

I.*

~X as group functors.

Proof. By (2.1) H~ is the maximal quotient of H" such that for all R in ~ H8®R is a quotient of H"®RI{h-ihlhEH"®R, iE!x(R)}.

Let i ¹ Efx(R'). Since !x(R) is normal in Ix(R), i ¹ (h-ih) = i ¹h-i¹ih

=

(i¹h)-j(i ¹h) where iE!x(R) and where jEix(R) is defined by.i¹i

=

^{ji1 • Thus} I(X), which is contained in the group functor !x , ^operat~s on H

0. The rest is clear. Q.E.D.

:Z.l

Remark (2.4). (i) Notice that although

- - a

autx" is smooth on the group functor Autx" is not in general, smooth. An

0

example ·is furnished by any hyperelliptic curve X of genus )3. In this case we have a

=

^a₀ but the involution is never liftable to a"f~2, see [La-L¢].

(ii) I(X}/i(X) does not, in general, operate effectively on H 0• In fact if X= Spe.c(k[,!]/(f)) where f is quasihomogeneous, the torus action ~ E Autk(X) is not, in general, liftable to

~ operates trivially on

. (iii) The subgroup of consisting of those for which there exists an isomorphism

I(X)/i(X). In fact le~ ~ be any such automorphism and con- sider the restriction of x~ to the special fiber, $: X ; x.

Consider fuEther any representative a($-l)EI(X) of

a($-l)EI(X)/i(X). Let ~· be the restriction of a(~-1) to a~.

Then there is an isomorphism sition

x"

₀

1\ - 1\ 1\

Y~, :X

0 ⁺ x

0 ^~,a

0

and the compo-

reduces to the identity on the special fiber. It follows that

~o~ • EAu~ (a~) conserves the universal family x^1\

0, thus ~o~·

=

lao· But then ~ is induced by a($)EI(X)/i(X) ..

(iv) The action of I (X) on

a"

induces a linear action on the tangent space A1 (k,X;OX). Since i(X) acts trivially on the tangent space of a", we obtain a lin.ear action of I(X)/i(X) o~ Al(k,X;Ox>· This action is well understood and has been used in many instances, see e.g. [La-L¢]. It is, in view of

(2.3) (ii), given in terms of the action

defined by

=

^{~ *-1 0~ •}

*

~e above picture is better understood if we look at i t at the Lie- algebra level.

Given a functor F of groups on ~. Recall the definition of the Lie-algebra l i e ! , see [D-G],

. lie F(R) = ker {F(Rfk[e:]) + F(R)}.

Notice that

lie aut (k) --SA

= Der*(S ) A

,., and that by definition

where

As a~short-hand we shall write

~(~)=lie i (k).

-~

This notion has the advantage, that i t is readily relativized, and that it functions well with respect to functorality. In fact, con- sider together with the formal family

~= y + Spf(S ) = S ^A

a morphism of complete local k-algebras

Let

p:

s

A

~·: Y' + Spf(TA) = T

be the pull-back, and put

~(~,p) = {DEDer*(SA,TA)

I

^(Y'fk[e:] ~ (Y.~k[e:])~(TAfk[e:]) where ~ = p+e:•D and Xk[~]k

=

^idy}•

:2.3

Obviously ~(~,p) is the value at k of a ~unctor !(~,p) that the reader may·want to explicite. Now the restrictions of the natural morphisms

1\ 1\

Der* (S ,

s ) .

~

_1\

/

_1\.

Der*(S ,T )

defines maps,

~(n;) .l(~·)

1\ 1\

Der*(T ,T )

Before we state the main result of this §, let us put as another short-hand

and let us recall the canonical action

and the isomorphism Autx/Auti

=

fx/i;, see (2.3) (ii) and (2.4) (iv), above noticing that A¹ (k,X~OX)®kR

=

Al(R,X®R,OX®R). One checks that lie Auti(k)

=

^A0" is a Lie-ideal of lie Autx(k)

=

~

A⁰ (k,X~OX) and that we therefore obtain a morphism of Lie-algebras a:A⁰(k,X~OX) + End(Al(k,X~OX)) that- factors via

AO(k,X;OX)/AO/\ + lie(IX/i~)(k).

~-

Lemma (2.5). lie !x

=

^lie

i:.x·

2.'-/

Proof. Let oElie !x(R)I then oElie aut (R). Since lie aut (R)

=

---HA ---HA

A A

DerR(H~R) 1 o corresponds to a DEDer*R(H~R) such that

o

=

id+E•DEautR[E](H~R[e]). Moreover there exist isomorphisms w and x• such that

~·+ (XA~R[e])®

0

^(HfR[E])

t w +

commutes. Let x

=

(w-l®idk[e])x• and notice that the following

di~gram is commutative

~ I

This shows that oElie ~X(R) and we are through.

From this it follows that there exists an action

which we shall use extensively.

Now let us prove the main result of this

§.

Theorem ( 2. 6) • ( i) J. ( ^{1t 1} p ) = _{ DE De r ( S ^A 1 T· ) ^A

I

³

. *

for any local section y of OY and any E(sy)

=

p(s)•E(y)+D(s)•y}

sES A ¹

Q.E.D.

st.

(ii} l(n:,p) is a T"-sub-module of Der

*

(S , T } .. ^{A A}

(ii') If p

=

^{id A'} the~ · l(n:} is a sub k-Lie-algebra of

s

(iii) There is a T"+k semi-linear map A.:l(n:,p) + A⁰(k X•O }/AO

I I X n;'

Der

* (

^{S ) .} A

which to any DEl(n:,p} associates the class

of

E®kEA ⁰ (k, x~ox).

T (iv) Suppose A

i.e.

s"

⁼

a"

^{and y A then A. induces}

1t = ^1t

,

= X ,

an isomorphism

l(n:,p}®k

=

AO(k,X70X}/A~,.

T

(v) A A

H0 is the maximal quotient of ·H trivializing l(n:}.

Moreover l(n:)®H~ is art H8-Lie algebra and a deformation of the Lie-algebra L(X}

=

A⁰ (k,X;OX}/A~.

Proof. Let DEl(n:,p} then ~

=

^{p+e:•D is} a morphism of complete local k-algebras. Put S

= s"

^{and T}

=

^{TA then}

is such that there exists an T®k[e:]-isomorphism

x

₀:Y®T[e:] ~ Y®T[e:]

s

~

lifting the identity on Y®T.

s

There are commutative diagrams of morphisms of k-schemes,

Xo

y ® T ^y ® T [e:] + y ® T [ e:

J

s s

~

I ^xo\. _-

^t _X ^t

y ® T [e:] + ^y ® T [ e:] X ® k [e:] + X ® k [e:]

s

~

'/

^k

\

\ I

I

p' ^{\ I ~I} X

e: ^{~ p} y

The difference El

=

^{cjllox -p~} corresponds to an element E of D e:

A⁰ (k, Y~Oy ~ T) • If y is a local section of ^~o y and s ES, then

I

Pe:* maps sy to p(s)y in

oy

~ T[e:], and ^4>1 maps sy to ( p ( s ) + e: • D ( s )) y in -· Oy

f

T [ e: ] • Since X D

x

0*( (p(s)+e:D(s)}y)

=

(p(s)+e:D(s)} (y+e:E(y}).

e:(p(s)E(yY+D(s)y} and consequently (*) E(sy)

=

p(s)E(y) + D(s)y.

is the identity on

*

Thus E ^{1 (} sy) =

*

If on the other hand EEA⁰(k,Y,Oy®T) is such.that (*) holds, for

s

some derivation DEDerk(S,T) then for 4>

=

^{p+e:D and} _{x 0* (y®l} >

=

Yflfr [e:] and y+e:E(y), we find that

x

₀^- is an isomorphism between

s

lifting the identity on Y®T, therefore DE~(~,p), and we

s

have proved (i).

If EiEA⁰(k,Y,Oy) one checks that

Since A⁰(k,Y:Oy®T)

s

.·

correspond to D.

].

is a T-module (ii) follows.

as in (i), f o r i

=

1,2, then

therefore ~(~) is a sub Lie-algebra of Derk(S,S) and (ii ^{1 )} fol- lows. (iii) is a consequenee of (i), as

x

₀^{®k =}

x

⁼ id+e:~(D), com-

T .

pare the right hand diagram above. Now, for the proof of (iv), we

- 0

first notice that ~ is surjective. In fact if EEA (k,X;OX), then x = idx+e:EEAutk[e:](X®k[e:]). As above (see (2.3) (ii}), there is a

~ :H+Hf[e:] such that x"fk [e:] and x"~H [e:] are isomorphic, the

4>

isomorphism lifting such that x"®1[e:]

H

idx+e:E. Tensorise with T and obtain cjl:H+T[e:]

and x"®T[e:] ar~ isomorphic, the isomorphism

4>

lifting idx+e:E. Since idx+e:EEker(Autk[e:](X®k[e:])+Autk(X)) we may arrange ~ such that ~

=

id+e:D, with DEDerk(H). In fact the

composition e:H1H[e:]+H is such that x"®H and x" are isomer-

a

'l-7

phic, th~ isomorphism lifting the identity on

x.

Then

a

is -1 -

necessarily an automorphism. Consider the composition

~

¹

:H~

+H$+H[€]

and see that has the required property. But then $

=

^p+ۥ0,

with DE~(ft,p), and we have proved that A is surjectiv.

To complete the proof of (iv) we shall prove that for any basis

{Ei}~=l

^of

Ao/A~.

^with

to E ;·eA 0 {k, x" 70x"®T)), the

E.=

^{A(D. ),} and

l. l. D.E~{~",p) corresponding

].

l. . H

D. ¹ s generate

]. ^~(~

"

^{,p) as a T-mod~le.}

Now pick any such basis and corresponding D. ¹ s, and

l.

E. Is.

l. Let hiET, then Ih ' D ' E ~ { ~A ^{I p )}

' l. l.

l.

corresponds to Ih.E ..

' l. l.

l.

Consequently we need only prove that for any DE~ {~"I p), the corres- pending E is a sum of the form Ih.E. , modulo A⁰(H,XA70XA®T).

i l. l. H

But this is easily achieved. In fact, the image of E in A⁰(k,X,OX) can be written as ^tN o- - - o

L' lh.E. + E(O), E(O)EA ^{I .}

l.= l. l. ~

Let E(O)EA⁰ (H,X";OXA~) be the preimage of E(O}, then D corres- ponds to E-E(O)

E-E(O)-E. N ₁h9E.

l.= l. l.

as well, and maps to zero in

- - tN

o-

E-E(O)

=

^L· _{l.= 1}^{h.E. ·} _l.·l. ^Therefore

.

"

A⁰(k,X 70X). Notice that this is a consequence of the fact that D and the D. ^1S map the maximal ideal m c

a"

into the maximal ideal

l.

n c T.

Consider the exact sequence

0 + A⁰ (k X"·O A® Tjn2) +

, I X " - H

Obviously:

A

⁰

(k,x";ox"®"~/~2) = Ao(k,x";ox)~~~~2

H

A ⁰ (k, x"; OXA® k)

=

^{A 0 (k., x" 7 OX) .}

. H"

Therefore the image of

(E-E(O)-L~=lh~Ei)

ⁱⁿ

A

⁰

(k,XA;OXA®T/~2)

Sits in AO(k XA·O A® n/n2) and in fact in the sub vector space

' I X A- - -

H

AO(k,X;OX) ®k~/£2· There must exist hiE~~ T. such that the image of {E~E(O)-Eh9E. )-E h~E. in AO(k XA·O A® Tjn2) sits in

l . l . l . l . ' ' X A -

H

A~, ~!!/~2.

Thus there exists E (1)

E~•A

⁰ (HA ,XA

;OX~T)

such that ,

E-(E(O)+E(l))-L~ 1 ^(h9+h~)E.

maps to zero in AO(k,XA;OXA® T/n2).

l.= . l. l. l. HA -

Continue, considering the exact sequences induced by 0 +

~P~~~+l

⁺

T/~p+l

^{+ T/np +} 0, we obtain· E =

I~=lhiEi+Ij= 0 ^E(j)

^{where hiET,}

E(j)EAO(HA,XA;OxA:T) and Ij=

0E(j) converges to an element of

0( A A 0 )

A H ,X ~ XA®T •

H

...

Now, to prove (v), notice that it follows from (iv), with ^{p :H +HA} ₀^{A ,} is the maximal quotient of HA trivial- and'from (1.6) that H~

izing .t(1tA). Therefore if A A A A

H = H

/ot

any &E.t ( 1t ) maps H into 4'l..

0 Moreover, if kEH, &₁,&

2E.t(1tA) then [&

1,ko

2 ] = k[&₁,&₂]+&₁(k)&_{2 •} Consequently ~·.t(1tA) is a Lie-ideal of .t(1tA) and i(1tA)® HA₀ is

HA

an H~-Lie-algebra. ^Q.E.D.

Remark ( 2 • 7 ) • In particular we have proved that if

then .t(1t) is of finite type as HA-module and vice versa.

Moreover the rank of .t ( 1t) is bounded by

di~

^{(A10 /A o A) .}

1t

For every DE.t(1t), D is a derivation of HA mapping the maximal ideal m into itself. Therefore D induces an endomorphism of· the tangent space of HA, D*: Al(k,X;OX) + Al(k~X;OX). Because of (2.6)

(iv) the representation D + D*,

factors via A0 (k,X;OX)/A 0A. In particular, we find a repre-

1t

sentation, of Lie-algebras

which we shall make explicite. Put for EEA (k,X;OX)' E* ^{- 0 -} = p(E). ^- Notice first that any element may be considered as a morphism of complete local k-algebras

Consider

i(~A,~)

=

{DEDer*(HA,k[~]} IXA~k[~]fk[E]

<jl

=

\+E•D, X ® k

=

id } •

k [ € ] XA fk [ ~ ^{] ..._}

By (2.2) (ii) there exists an element ~Eix(k[E]) such that

(~®idk[E]oa

=

^{~+E•D. Since ~}

=

^id+E•E EElie i(X) we obtain D

=

^~oE. On the other hand if EElie i(X) then certainly

~oEEi(~A,~). Since ~·Der*(-HA,k[~]) = 0, (2.6) (iv} implies .the- following

Corollary (2.8). The orbit of ~ under 1(~) is equal to .

where is the subspace of that lift to the family X~

A⁰ (~,X;OX}

on k [ ~].

More explicitely, we have the following.

Proposition (2.9). Let EEA (k,X;OX} ^{- 0} then in the following way:

of those elements

is defined

Given ~EAl (k, X;OX), let X~ be the corresponding lifting of X to k[~], and consider the lifting situation; g2=~2=o

'30

Spec ( k [ e ] [ Tl ] ) ⁺ XI; ®k [e] - - - + Xl;®k [e]

t t t

.

Spec(kk[e]) ⁺ X®k (e] ⁺ X®k [e]

t ^t id+e•E _t

<.

Spec(k) ⁺ X = X

Let o(id+e:E) be the obstruction. for lifting the automor- phism id+e:E as an automorphism of ~l;®k[e], then

o(id+e:E)EAl(k[e], X®k[e:]rOx®k[e]) = Al(k,XrOx)~k[e:], has the form e•E*(I;).

Proof. Let

DE1(~)

correspond to

~EAO(k,XArOxA)

as in (2.6) (i) and consider EEA⁰ (k,X~Ox), E = E®Hk. D operates on· A¹(k,Xr0x) via the automorphism o = id+e:•DEAutk[e:](HAfk(e:]). By construction, puting X = id+e•E

0 the following diagram commutes ·

XA®k (e:]

xo

A

+ (X ®k (e:]) ® (HA®k [e:])

0

t t

X®k(e:] ⁺ X®k(e:]

id+e:•E

Reduce the diagram modulo ~A and obtain

t t

X®k(e:] + X®k(e:]

id+e:E where

o

2=id + e•D

2 and

n

₂ is the linearization of D, i.e. the derivation on

a

2 = HA /r~_2 induced by D. Notice that· sin.ce

~~~2

=

Al(k,XrOx)*, 1; is a linear form on ~~~2 inducing a .k-algebra homomorphism I;:H

2 ⁺ k[TJ}· Consider· the homomorphism

31

id+e:•E

where (X~®k[e:]) ® (k[n]®k[e:])

=

^X~®k[e:].

!;®lk [e:] .

The composition o

2

o(~®lk[~J):H

2

^{®k[e:J +} k(n)®k[e:] is by definition of

E*

¹ ~®~[e:] + e:{E*(~)®lk[e:]) where E*(~) E Al(k~X~OX) is '- considered as an alg_ebra homomorphism E* ( ~) : H₂ ⁺ k [ n ] ¹ as

explained above.

But then (x"~k[e:]) ® (H₂®k[e:]) ® (k[n]®k[e:]) is the lifting 62 ~®lk[e:]

of X®k[e:] corresponding to ~·n+E*(~)·n•e: E A¹ (k[e]~X~k[e:]~(n)).

From the existence of the commutative diagram (*) we deduce

(id+e:•E>*-l(id+e:E)*(~®lk[e:J> = !;®lk(eJ+e:(E*(~;)®lk[€]>

where (id+e:E)*: Al (k [e:] ^1X®k [e:] ^I Ox®k [e:]) ⁺ Al (k [e:] ^1X®k [e: ], Ox®k [e:]) is the automorphism induced by id+e:E: X®k [e:] ⁺ X®k[e:] and (id+e:E)*

the one induced by id+e:E: ox®k [e:] ⁺ Ox®k [e:]. But then .e:•E*(~;) the obstruction for lifting id+e:E to X~® k [e:].

Q.E.D.

Corollary (2.10). Let E E AO(k1X;OX) then E* E End(Al) is defined by

e:•E*

=

(id+e:E)*(id+e:•E)*-¹-id Thus p

=

^a.

is

Proof. This is exactly the contention of the proof above. Q.E.D.

k [ x

1, • • • 1 X ]

Corollary (2.11). Let X= .Spec( R IJ) and let

DEDerk(k[~]IJ).

Then the action of p(D) on

al {k₁ X, oX) ;;; Hom( j ¹ k

[~]I

^{J) IDer} is defined as follows. Let

~EHl(k1^X;OX) be represented bY. a homomorphism ~:J+k[~]IJ and lift D to a derivation E:k[]!.]+k[~] then p(D)(~) is

represented by the homomorphism

, Proof. Use f2.10). Q.E.D.

·.In particular we have:

Corollary (2.12). Let X= Spec(k[.!]l(f) and let

EEDerk(k[~]l(f))

· IN

^{ef -}

be defined by .

1 ^~E. (x.)

=

q(x)f(x). Then if ^~ is the

~= vX, ~ ~ - "-

~

class of the polynomial ~ (_!) in al (kl k [.!]I (f) ¹ k [.!]1 (f)) = k[x]

-

l(f~a- 1••·1~ ef ef ) 1 the action of E* on ~ is given by

Xl vXn

E*(~)

=

^{class of} < n

l:

i=l

E(x. )-q(x)~(x))

~

- -

Corollary (2.13). the rank of the linearized action of ~(~) on HA is equal to the dimension of the maximal orbit of

Al(k~XrOX) under the action of A⁰(k1X;ox>·

Proof. By definition the linearized action of ~(~) on is the one given in terms of the action of -~1~> on the tangent space

(~1~2)* = Al(k1X;OX). The rank of ~(~) is then almost by defini- tion the dimension of the maximal orbit of Al(X10X) under ~(~).

Q.E.D.

33

§3. THE KODAIRA-SPENCER MAP AND ITS KERNEL

Introduction. In this § we shall _apply the results of §1 and §2 to study the local properties of formally versal families of objects of the type we are concerned with, see (3.6) for the definition of formal versality, and notice that the families we are talking about are algebraic families, not formal ones.

-

We start by defining the Kodaira-Spencer map g:Derk(S)+A¹ (S,Y~OY) associated to a flat family n:Y+S.

The first part of the § is a study of the properties of the kernel Vn of g. As we shall see Vn is a sub k-Lie-algebra o~ Derk(S) with nice functorial properties.

We shall not venture into the difficult problem of when formally versal algebraic families exists. At this ·point, we take the easy way out; assuming that the objects X we are handeling are such that

(A

-

1) there exists an algebraization ~:X+H

=

^Spec(H) of the formal versal family ~A:XA+HA

and such that

(V) ~ is formally versal.

We then formulate a set of conditions (v•) .akin to the conditions studied by M. Artin, see [Ar], which imply formal versality if ~

exists. Assuming (v•) we also prove that the infinitesimal notions of §1 and §2, such as the prorepresenting stratum and the Lie-

1 b ^o(~A)

a ge ra ^~ ,. are formalizations of local notions, ^~ and V respectively, see (3.5).

Since the main objectiv of this § is the construction of a local moduli space, we shall have to impose the following condition on our objects X:

11 :Y+S is a flat family with fiber _X

=

11 ^-1 (~) for some closed point

e:E+S of s isomorphism

~E~, then there exists an etale neighbourhood and a morphism p:~+~ such that there is an

* *

e (n}~p (~)between the pull-backs of 11 and· ~.

We know, see [E], that (A

1) holds for affine schemes with isolated singularities. In [Grl], Theoreme- (5.4.5), Grothendick gives

conditions for when (A

1) holds for projective schemes. Moreover, (A2) holds when X is-projective, when X is a finitely generated graded k-algebra -and we restrict to deformations within the cate- gory of graded k-algebras, or when X is a complete local k-alge- bra of finite type, restricting this time the deformations to the category of such formal, or algebroid, schemes. This last example is going to be treated in some detail in §4-§6.

The main results of this § can now be summed up as follow-s. Assume (A1), (A

2) and (V'). The flattening filtration {s~}~ of H corresponding to the H-module V, is stable under the operations defined by

v.

There exist reasonably good quotiens M

-~ of

s ,

-~

in the category of algebraic spaces, carrying families the pull- back of which are the restrictions ~ of ~ to S , see (3.18),

~ -~

such that the collection {~~} deserve the name, local moduli suite of

x.

Perhaps of more interest for the applications we have in mind, we prove, see (3.24), that for every component S of

s

-~ there is . -~,c

an open dense subschema of the reduced normalization S' of

-~,c

on which the action of the kernel of the corresponding Kodaira- Spencer map has a strict quotient in the category of schemes.

-~,

s

c

Notice that we do not know wether V acts rationally on H, there- fore we cannot invoke the general results of Rosenlicht, see [R],- or Dixmier and Raynaud, see [D-R}.

Assuming (V') we prove, see (3.12), that V®H

0 is a deformation

H

of the Lie-algebra L(X)

=

^A0 (k,X;Ox)/A~. In fact, see (3.17) we obtain for every a flat family of Lie-algebras defined on the fibers of which are the k-Lie-algebras L(X(t)), tES .

- -

^-,;

Thus we are lead .to the study of deformations of Lie-algebras.

s

^I

- , ;

Now, we might have included Lie-algebras among our objects of study from the beginning of. For reasons to be explained elsewhere, we shall treat it apart. In particular we have to modify the defini- tion of cohomology of Lie-algebras to obtain a natural setting.

In the above process we also find a necessary conditiori and a sufficient condition for an object satisfying (A

1 ), (A

2) and (V) to have a rigidification, see (3.11). ·Moreover, we prove a result which was conjectured by Wahl, see [W], and recently also proved by Greuel and Loojinga [G-L] on the-dimension of a smoothing component of ~, see (3.10).

Let S be any k-algebra .and consider a flat family

rp Y ⁺ Spec ( s) •

Corresponding to the simplicial k-algebra v' 1

s :::.

+, v2

+ +

s ~s

t

s ~s ~s o o o ,

+

v' ₁

= =

1®id

one may define a series of obstructions for descent of Y to k.

3b