SELECTIONS OF FUNCTIONS AND THE PROBLEM OF LIFTING (DEFORMING) ALGEBRAIC STRUCTURES IV

(1)

The generalized 3-secant lemma

Introduction.. The following pages contain a proof of the conjec- ture formulated in Remark (5o2o14) of [La]o We shall use the notations of that paper, and we shall, in fact, typographi- cally make this paper fit into [La] and extend it in a natural way ..

This makes it necessary to erase Remark (5o2~11) renumber Lemma (5 .. 2o12) accordingly, and omit page 28o

The reader sh~uld note that the conclusion of the main theorem (5o2 .. 14) is proved only in characteristics greater than n ..

(2)

- 0 -

Suppose given a family of homogenous poly-

(*)

nomials of degree n

and a function

p : (P 1 , ... , P r} ^~k such that

Q. . (Pl)

~1, .... ,~s

=

(x. .. .. ox. ) 11 1s (P1 ) • p(P1 )

for all 1

=

1, ••• ,r, and all indices (i1 , ... ,is) .. Then there exists a homogenous polynomial Q of degree n-s such that

Proof. We know that f

coefficients.

Fix i 2 , ... ,is and find not contain the monomial tain the monomial ~.

(mod f)

contains and with non-zero

a,b E k such that Q . o, ~2^,... , J.s . - bf does

~ and Q1 . . - af does not con- '12, ••• ,~s

Then by (5.2.11) there exists a homogenous polynomial of degree n-1 , such that

In exactly the same way there are a' ,b' ,c,c'

'

^Q!^~2,^•••^,~s^. ^and

Q'.' .

~2' ⁰⁰⁰ '¹s such that

Q . . - b 'f = ^X

Qj_2'"""'is

o,~2, .... ,~s 0

Q2 . . ^- ^CIf = x2 Q! .

'~2'"""'~s ^~2,.... ^,~s and

(3)

Using (*) we find:

and since x is not a factor in f , this implies

0

and, exactly in tbe same way

thus

(mod f)

Now using (*) again, we find that

Q_ • )(Pl)

~2, ... ,J.s Thus

since

(4)

- 2 -

does not contain the monomial ~ where (i,j,k}

=

(0,1,2}.

Thus

Go on, obtain the result we want ..

Now, let n : R - S be a surjective homomorphism of local k-algebras with mR • kern ::;: 0 , S/B;s :::. k. Suppose given a commutative diagram of flat S-schemes

X ®

s

^cp^I~

y

^®

s <1.:. y'

k k

c

~ i~·h

Z'

and let

£a

denote the subdiagram consisting of cp' : X ® S - >

Y

^® S

k k

Suppose further that ^-~', ^'ll' and cp' are local complete intersections that may be lifted to R , then the obstruction

the diagram (or the category) .£ relative to

A~

(.£.,0c) ®kern

=

coker(l,m-n) -o - k

-o c

o(c,c ) - -o sits in

=

^[H⁰(X,Nx ® Oz) ^(f)H⁰(Y ,Ny ® Oy) /im(l ,m-n)} ®kern

Oy Oy k

and, in fact, is the class of the element

for lifting

(5)

- 3 -

where, having pi eked liftings n" of 'll' and ~ ¹¹ of ~ ' , o (a.) and

o(~)

are the obstructions in A1 (a' ;o2 ,)®kerTT =H⁰ (X,Nx®o2 )®kern

k ox k

respectively

A

¹

(~';o 2 ^,)®kern

⁼ ^H⁰^(Y,Ny®o²^)®kern for lifting

k Oy k

the morphisms a' and ~· with respect to T1¹¹ respectively ~"

and ^T]^{11 ..} We shall apply this observation to the situation above,

First, let's make the following,

Rt:mark

(5.2 .. 13)

Using the assumption that is a complete intersection, we know that ~ may be lifted everywhere.. Since Z = (P 1 , ... ,P r} is of dimension zero, Z

'-:r[

IP3 may also be

lifted everywhere. Using the observation above, we find (see

(2»

that

dim Iz (X, Y) ~ 2 Moreover, if ::lim Iz (X, Y)

=

2 is non singular at [Y} •

coker(l,m-n)

=

o thus I 2 (X,Y)

Next, let's compute the cup-product, ioeo the symmetric pairing

defined by the obstruction morphism ..

This is interesting only in case the imbedding dimension of I 2 (X;Y) is 3 ' as we have just seeno

The cup-product is the obstruction for lifting the universal diagram

to T1;m3, lifting cp® 1 trivially.

(6)

- 4 ~

We have already seen that the imbedding

w'

is defined by the ideal

Cx

3 ,f- Qx3 ) where

Moreover,

2

x3 ⁼ ^x3 ⁺ ^'E ^{x .}^y~

j=O J J

1 - 1 2 ( } 1 2

11 : Z ® T

/m =

_{P 1 ,}^{o ....},P

r

^®T

/m

^~

I k

is defined by an ideal of the form

Let the imbedding X Y 1P3 be defined by the ideal

~ ~ k[x₀ ,x1 ,x3

J.

Lift

w'

and 11' to T1/m3 by lifting the corresponding ideals, trivially, to

Notice that set theoretically

Given these liftings 1jJ" and 11" , the obstructions for lifting a' and

~·

to T1;m3 are the elements:

o(a') E H⁰(X,Nx® Oz) ® m²/m3 k

o(l3') E H⁰ (X,Nx® Oz) ® m²/m3

k

defined by:

(7)

""' 5 -

where for any h E OC ,

. . 2 .

o(a' )(h)

=

{h( x ,a,x ,a':'"'l ""l ₂x , ._. a .x " "'l y. *) }r . , ,

0 ' 0 0 . J 0 J l = '

"-'i

a = 1 , Vi

0

and where

o(i3')(x3)

=

o o(l3')(f-Q;x3 )

=

J=O

. . . . 2 .

[ _{f x}( ""l "'l ) ( "'l "'l "'l * } ^r

0 ,a,x ,a.2x - Q x ,a.,x ,a₂x )( E a .x y .) . ,

' 0 0 0 ' 0 0 . J 0 J l=t

J=O The cup-product is therefore defined by the element

Suppose this cup-product is zero, then we may, in exactly the same way compute the ~d Massey-product. etc.,

We shall not bother to make these products explicite.. However, we shall deduce from their existence the following theorem.

Theorem (5.2.14) Let X, Y and Z be given o.s above, and assume char k > n. Then ciim{Y} Iz(X;Y)

=

2 unless there exists a hypersurface of the form:

containing X. In that case Iz(X;Y) is non-singular of dimension 3 •

Proof. We know that the imbedding dimension of Iz (X; Y) is 2 or 3.,

In the first case is non-singular of dimension 2 at [Y}

(see (5.2 ..

13)),

at least when _r

= e

ⁿ⁺²²⁾^•

Suppose that the dimension of Iz(X;Y) at [Y} is greater than 2 , thus 3. Then, obviously, Iz(X;Y) must be noll-singular at [Y}., Consequently all Massey products involved must be zero.

In

particular

(8)

- 6 -

the cup-product described above must be zero.

Moreover, the tangent t ~(P.) of X at P. is normal to the

-x ¹ ¹

direction x . (P. )nf (P. ) - Q . (P. )n = x. (P. ) (nf (P. ) - Q(P. )n ) •

J 1 - 1 J 1 -x3 ^J ¹ ⁷ ¹ ¹ -x3

Since nf(P.)- Q(P. )n is the normal at P. of the surface

- 1 1 -x3 ¹

at P. is contained V(f-Qx3 ) the tangent space of ^X l

in the tangent space of V(f-Qx3 ) tence of surjective homomorphisms

at P. ⁴ This implies the exis-

l

where ideal

mP./ 2 ~P-; 2 ( ~ )

mp

²

1 mP. ~ 1 ~- + f-I<I!;X3 ^P. ^~ ^i/mp.⁺^Otp.

l l l l l

mP.

l is the maximal ideal of 0 3 , (f-Qx3 )P. is the

IP ,P. 1

OLP.

l

is ¹ the ideal 0{. .. 0 3 IP ,P.

l

0

Thus and since, by assumption, X cuts V(x3 ) regularly at Pi , the ideal ~- therefore being generated by

en

and x3 ' we find l

f - Qx3 E otP. + (x3)P. ' 2

l l

and an element R2 . E 0 3 such that

, l IP ,Pi

(4)

Since the cup product is zero, there exist elements u E H⁰ (Z,Nz) ® m²;m3

v E H⁰(Y,Ny) ® m2 /m3, such that

o(a')=l(u)

v = L: (1 .. 'Q . . )y>ty*.

i.::

^j ^lJ ^lJ ^l ^J

o(l3')

=

m(u)- n(v) •

(9)

Since

we find:

a.D.d

- 7 -

f - Qx3

=

h 2 . (mod :£!p ) 2

,~ i

=

o(a')(h2 .)

=

l(u)(h2 .)

=

_{u.(h2 .)}

,~ ,~ ~ ,~

(f-Qx 3 ^)(x 0 ^,a~x 0 ^,~x

_{0 j=O}^, ?

^{~ ~x y~)}

_J ₀ _J

=

u. _~(f-Qx3)-. < . ^I: ^{Q ••}"'iJ ^(P.~ ~ ^)y~y~J ^•

~-J

Here we have used the fact that

n(u) = [n(u). }:' _{1 ..}

~ ~=

Subtracting the .first formula from the second, we obtain using ( 4) ,

or,

- ~j(P1⁾ ⁼ xixj • 2 • R2 , 1 (P1 ) - Qii (Pl)

=

xi .. 2 R2,l (Pl)

l

=

1, .... ,r

for i /: j , 1 = 1, ... , r for

Since char k > n we may assume char k /: 2 , therefore (5o2.12) garanties the existence of an (n-2) form Q_{2 E k[x}₀ ,x₁,x2 J(n-2 )

(10)

~ 8 -

such that for and Q ..

]_]_

Thus

Put Q1

= -

Q , then by constru.ction the ideal

cx

₃^,f-Qx-..+_{? .} ^L:_{< .}^{Q . .}_{lJ ]_ J}^;y~y~)

l _ J defines the imbedding

Now

for 1=1, ... ,r therefore

and, since ~ 1 + Q2 E !£p ⁼ ^Otp +

Cx

₃^{)p ,}

' 1 1 1

the existence of elements h3 1 E ^0(.p and

' 1

we deduce fron this R3 ,1 E 0 3 such that

JP ,Pl

Now computing the Massey-products one after the other, knowing they are vanishing, we obtain a series of forms ~ E k[x₀ ,x1 ,x2 J(n-i) such that the ideal

defines the imbedding

for all q _:: 1 ..

y ~ T1/mn+q ~ 1P3~ T1/mn+q

T

Consider the universal family

(11)

- 9 -

and let 0 be the local ring of Iz (X; Y) at (Y} .. Take the base change Spec(O) ~ Iz(X;Y) and consider the corresponding family

Spec(O)

We know that 0 ^"

=

T • ¹ Let

17

be the ideal of O[x₀ ,x₁,x₂,x3

J

defining the imbedding ^Y ~ ]?3 ^®^{0 •} What we have proved above

0 k

amounts to the following

This, however, implies that the hypersurface V(f

+.~ ~x~)

^{of JP3®}⁰

~=1 k

-

containing a component of the closed contains a component of ^y

0

fiber, i .. eo containing a component of our original Y ..

In fact, dehomogenizing at xj , ^j

=

o, .... , 3 we find, using [Se]

(II, 11, Cor 4) that there exist elements m. E m and

J -o

x₀ x₁ x₂ X;z.

0 [ - - - -"'-] x.'X.'X.'X. such that·.

J J J J

1 n n i

( - ) ( f _xj + _i=1~L: Q. x3) ( ¹⁺^{m .}^{g . )}

J J

denotes the ideal

tJ

dehomogenized at X. o

where

h

J xj

Let

1D

be any prime containing

41

such that a component of contained in V(1) , then obviously ^{( 1}+ m .g.) %. rf)

J J

rx.

J

thus

Y is

(12)

- 10 --

n .

f +I: _. _,.,Q.x~~

3

^E^r"P·}

~=I

n . 3

Consequently the surface V(f +i:1 Qix'3) of JP contains a 0omponent of im cp containing a component of Y.

Since this component of i.m cp must necessarily contain infinite

n .

many points of X , we have proved that X c V(f + I: Q x~)

- i=1 i 3 ^o Q.EoD ..

Bibliograp;ro::

[La] Laudal, OoAo: Sections of functors and the problem of lifting (deforming)algebraic structureso

Preprint Series no~ 24(1975), Institute of Mathematics, University of Osloo