LIFTINGS(DEFORMATIONS) OF GRADED ALGEBRAS

Introduction

Chapter 1 • Cohomology groups of graded algebras Chapter 2. Deformation functors and formal moduli.

Chapter 3. Relations to projective geometry.

Chapter 4. Positive and negative grading.

Chapter 5. The existence of a k-algebra

which is unliftable to characteristic zero.

- 1 - INTRODUCTION

In this paper we study formal deformations of graded algebras and corresponding problems in projective geometry. Given a

graded algebra A, we may forget the graded structure and deform (lift) A as an algebra. Clearly we also have a deformation theory respecting the given graded structure of A • This deform- ation theory is closely related to the corresponding theory of X = Proj(A). One objective of this paper is to compare these three theories of deformation.

A basic tool is the cohomology groups of Andr~ and Quillen.

Let S -+ A be a graded ringhomomorphl.SIB and let M be a graded A-module.. ·we shall see that the groups

Hi(S,A,M)

are graded A-modules whenever S is noetherian and S -+ A is finitely generated. In fact, if we let

vHi(S,A,M)

correspond to S-derivations of degree v, we shall prove that there are canonical isomorphisms

co .

ll

vH¹ (S,A,M) ~ Hi(S,A,M)

'Y=- co

for every i > 0.

Deformations of A (forgetting the graded structure) are classified by the groups Hi(s,A,A) for i=1,2. Restricting to graded deformations, we shall see that they are class.i£ie4 by the subgroups

0Hi(s,A,A)

for i=1,2 • These generalities are proved or at least stated in chapter 1_.

Let ^TT R ^-+ S be a graded surjection satisfying (ker TT) ² =0.

Since there is an injection

-+ H2(S,A,A® kerll)

s

we deduce that A is liftable to R iff A is liftable as a graded algebra. We would like to generalize this result to arbritarysurjections of complete local rings. This seems

difficult. However if we assume

for v > 0 or v < 0 (called negative or positive grading respectively), then the statement above follo~s from 2.6 of chapter 2 when S is a field k • In fact, let .1_ be the category of artinian local V-algebras with residue fields k ,

Vfmv =

k, and let Def ⁰(A/k, - ) , resp Def(A/k, - ) , be the graded deformation functor, resp non-graded deformation functor on .1.. with hulls R⁰(A) and R(A) respectively.

Consider the local V-morphism

R(A) -+ R⁰(A)

Theorem 2.6 states that this morphism has a section whenever A has negative or positive grading. This follows from the existence of an isomorphisms.

Here degT

=

1 if we have negative grading.

In chapter 3 we enter into projective geometry assuming the graded algebras to be positively gzeded and generated by elements of degree 1. Y.le compare the groups

- 3 -

with the corresponding groups Ai(S,X,M{v)) in projective geometry, X= Proj(A). The groups Ai(s,x, -) were intro-

duced by Illusie in [I] and by Laudal [L1]. If X is S-smooth, then

i "' i rv

A (S,X,M)

=

H (X, ~ ®8M)

where 8X is the sheaf of S-derivati.oas on X. If the depth of M with respect to the ideal

ro

m =Jot ~

v~1

is sufficiently big, the groups VHi(S,A,M) and

coincide. For instance, if dept~A ~ 4 ,

\IH1(s,A,A):::: A1(S,X,Ox(v)) and

2 r __ 2

vH (S,A,A)~A {S,X,OX(v))

Thisimplies that the deformations of A and X correspond uniquely to each other. wnen dept~A ~ 3 a rigidity theorem of Schlessinger, see (2.2.6)in

[K,L]j

is generalized by the injection

Now these depth conditions are usually rather crude, and the exact sequences in which these groups fit are in many cases a better tool.

In chapter 3 we also relate the groups corresponding to embeddings. Let ~:B ~ A be a surjective morphism of graded S-algebras such that B₀

=

A₀

=

S and let

f : X = Proj(A) ~ X = Proj{B)

be the induced embedding. We would like to compare the groups and A ⁱ (S,f,M(v)). ^"' If f is locally a complete intersection, one knows that

where Nf is the normal bundle of X in Y. Again putting depth conditions on M we conclude that

and

coincide. If depthmA ~ 2 , then

. 1

vHl(B,A,A) ~ A (S,f,fx(v)) and

2 ~ ~..,.._ 2

vH (B,A,A)~A (S,f,OX(v))

From this follows that if B is S

=

k-£ree then

on 1 where D.ef⁰(cP,-) is the graded deformation functor of cp and where Hilbx(-) is the local Hilbert functor at X • From this and the isomorphism

we generalize a theorem of Pinkham [P] as follows. If A has

- 5 -

negative grading and depthmA ~ 1 and if X

=

Proj(A[T]) is the projective cone of X in ~~¹

=

Proj(B(T]) , then there is a smooth morphism of functors

Hilbx(-) ~ Def(A/k,-)

In chapter 4 we investigate the conditions of negative and positive grading. We shall assume A to be the minimal cone of a closed subsoheme·

bedding we prove that the minimal cone

By twisting the em- B of X

S

P ~ ^for large N very often has negative or positive grading. For instance, if X is S-smooth B will have negative grading.

If X is of pure dimention > 2 and locally Cohen-Maoauley.

then B will have positive grading. Combining these two re- sults we deduce a theorem of Schlessinger [S3]. See also

(M].

Using these results we find that the smooth unliftable projective variety of Serre [se] gives rise to a graded

k-algebra which is unliftable to characteristic zero. This is done in chapter 5o His example is of the form X = Y/G. Y is a complete intersection of dimention 3 and the order oi G divides the characteristic.

The possibility of using this example to get an unliftable k-algebra may be looked upon as the beginning of this paper.

The proof given here is due to O.A. Laudal and the ~uthor.

We end chapter 5 by proving that if ord(G) did not divide the characteristic and if Y was a complete intersection of dimention ~ 3 then X

=

Y/G would have been everywhere

liftable.J:This paper contains all the results of [K]. I would like to thank O.A.Laudal for reading the manuscript.

CHAPTER 1

pohomology groups of~aded algebr~~·

Rings will be commutative with unit. Let ~-aljg be the category of S-algebras and

the £ull subcategory of free S-algebras. Given an S-algebra A and an A-module M, we define

H

ies·

_.

'

A M)

, =

11·m(i)

--t-

Der₈ (-,M)

(SF/A)0

where Der₈(-,M) is the functor on (§~jA)⁰ with values in

.A~ defined by

qJ

Der₈(-,M)(F ~A)

=

Der₈(F,M) M being an F-module via ~.

If S ~ A is a graded S-algebra and if M is a graded A-module, we may cor..sider the category of graded S-algebras

.~al~ and the corresponding category

of free graded S-algebras. Let

be the functor defined by

~

kDer8(-,M,(F ^{~ A)}

=

kDer8^(F,M)

=

Then we put M_ini..t!.on ^1..:..1

!DE Der8^(F~M)\D is graded oft degree k

kHi(S,A,M)

=

lim(i)

~er 8 ^(-,M)

<--

~Ao

- 7 -

As mentioned in the introduction, the groups Hi{S,A,-) and

0Hi{S,A,-) classifies formal deformations. Recall that if

1T

R-+ S

is any surjection with nilpotent kernel, we say that an R-algebra A¹ is a lifting or deformation of A to R if there is given a cocartesian diagram

R >-A'

1T

~ ~

S >A such that

Two liftings A' and A" are considered equivalent if there is an R-algebra isomorphism A' ~ A" reducing to the identity on A. If ~ : A -+ B is a morphism of S-algebras and A' and

B' are liftings of A and B respectively, we say that a morphism

~· : A' -+ B'

is a lifting or deformation of ~ with respect to A' and

B' if ~® ido

=

^~. We define graded liftings of graded algebras 1l ^u

and graded liftings of graded morphisms in exactly the same way.

TT

Assume that R -+ S satisfi-es (kern) 2 = 0 Then it is known that

There is an element

o{A) 6 H {S,A,A 2 ® ker rr)

s

whiCh is zero if and only if A can be lifted to R.

If

a(A)

=

0 , then the set of non-equivalent liftings is a prin- cipal homogeneous space over H 1 ( S, A, A ® ker TT)

s

There is an element

a(cp, A' ,B') E H1 (S9A,B ~ ker 1)

s

which is zero if and only if cp can be lifted to R with

respect to A' and B^{1 •} If a(cp; A',B')

=

0 then the set of liftings is a principal homogeneous space over

Der8(A,F ^® kern)

s

The elements a(A). and cr(cp; A',B') are called obstructions.

Then corresponding theorems in the graded case are

There is an element

cr (A) E H2(S,A,A ® ker TT)

0 0

s

which is zero if and only if A can be lifted to a graded R-algebra. If cr₀ (A) = 0 , then the set of non-equivalent

liftings is a principel homogeneous space over H1(s,A,A ® kern)

0

s

There is an element

cr ( cp • A ¹ B 1 ) E ₀H 1 ( S, A, B ® ker TT )

0 ' ,

s

which is zero if and only if cp can be lifted as a graded morphism to R with respect to A' and B' • Moreover, i f

cr₀(cp; A¹,B') = 0, then the set of graded liftings is a principal homogeneous space over ₀H⁰(S,A,B® kern)=

s

Ders (A, B ® ker n)

0

s

- 9 -

In [L1]we find proofs of 1.2 and 1.3 and these can easily be carried over to the graded case.

If we want to compare the graded and non-graded theories of deformation, we need to know the relations between the groups ₀Hi(s,A,M) and Hi(s,A,M) • This is given by the following theorem. A proof of this can also be found in [I].

Theorem

1 • 6

Let S ~ A be a graded ringhomomorphism an let M be

a graded A-module. If S is ~oetherian and S ~ A is finite~y

generated, then there is a canonical isomorphism

00 .

.U kH¹ (S,A,M) ~ Hi(S,A,M)

k=-^_a::J

for every i ~ 0.

R~~ In general, there is an injection

co .

ll kH¹ (S,A,M) ~ Hi(S,A,M)

k=- co

for every i > 0.

Pr_oqi Let

( SgF

I

A) :tg ~-

?J£!..1

^A

be the full subcategory defined by the objects ~ : F ~ A where F is a finitely generated 8-algebra.

Look at the diagram of categories (?!,/A) fg ) §]../A

i i

( SgF

I

A) f~ Sfi"f/ A

where all functors are forgetful. These induce morphisms

lim(i)Der8(-,M) ^{••tc-· -} lim(i)Der8(-,M) ⁼ Hi(s,A,M)

E-- <E--

{SF/A)

_, fgt --

§!/A ^~

t

lim(i)Der8(-,M) ~ lim(i)Der8^(-,M)

<:--- 4E:--

(?£/A)

^f g §£;!./A

I claim that these maps are all isomorphisms for i

?.

o.

This will prove 1.6 since there is a canonical isomorphism of functors

en

*) k~- c:JcDer₈(-,M) ~ Der_{8 (-,M)}

For i = 0 , the contention of

*

is easily proved. For

i > 0 let us prove that the ~ight hand vertical morphisms

are isomorphisms.

Let F ^~ A be a graded S-algebrasurjection and let F. = F x F x •••• xF

1 A A A

Consider the complex

lim{q) Der

8(-,M) ~

lim(q) Der

8(-,M) ~

.:E- ~

SF/F

~-~ 0 §!_/F 1

(i+1)-times

~ lim(q) Der

8(-,M)

^~ f -

SF/Fi

where the differensials are the alterating sum of group- morphisms

lim(q) Der8{-,M) ~ lim(q) Der8(-,M)

<:::- "'IE--

§[/Fi-1 pF/Fi

induced by the projections F_{1 -+} Fi_ 1• In this situation there

- 11 -

is a Leray spectral sequence given by the term Ep,q ⁼ HP(lim(q) Der8(-,M))

~

~/F.

converging to

lim{·) Der₈(-,M) = H(•)(S,A,M)

~

g/A

For a proof see (2.1.3) in [L1].

Similarely, there is a Leray spectral sequence with

converging to

'Ep,q-_{2 -} HP(lim(q)

+--

§$1/F.

Der

8(-,M))

lim(·)

D~r 8 ^{-,M)

~

§J!Z./A

To show that the morphisms lim{i)

~

.~F/A

Der

8(-,M)

^{~ lim(i)}

~

SgF/A

are isomorphisms, we use induction on i • If it is an isom- orphism for i ~ n and for every object A in Sg-al~, we conclude that the morphism

E~,q -+ 'E ,p,q

2 2

is an isomorphism for q ^~ n and every p.

Recall that

E^o, 2 -q c l1'm(q) D ers -, ( M)

=

Hq(S F M)

, '

+--

§~jF

Since F E obSgF , we get

~ l~(q)

^{Ders(-,M) = 0}

<

SgF/F

for q

?:

1

as well. Since £or r > 2 the differensials of the spectral sequence are of bidegree (r,1-r) , and since for p and q given, EP~? = EP~ q for some

r ,

we easily deduce isomorphisms

Ep, q _{ro ---,} ' 1EP' q _ro

for every p and q with p+q < n+1 • Hence there is an isomorphism

lim(n+1 )

~

§]jA

Ders(-,M)~lim(n+

¹

)

^Ders(-,M)

~

I~.fiE./ A

Let R ~S be a graded surjection such that {ker rr)²

=

⁰

It is easy to see that the injection

OH2(s A₁A ® ker rr)

~H

²

(s

^{.. A,A ®kern)}

I S S

Q.E.D.

maps the obstruction cr₀(A) onto cr(A) • For definitions of the obstructions see [L1]. This proves

Co;olla:cy 1.7

Let R ~S be a graded surjection such that (kerrr) ² = 0.

If A is a graded S-algebra, then A can be lifted to R iff A can be lifted to R as a graded algebra

Remark

Let FA be the set of non-equivalent liftings of A to

- 13 -

R and the corresponding set of graded liftings. If A' is a graded lifting of A to R , then ~e are isomorp- hisms and obvions vertical injections fitting into the diagram

Hence there is a projection

Now 1.7 can be generalized as follows. Let

be a graded ~bxahotlomorphism Assume there are liftingS. A' B'·

,

not necessarily graded, of A and B such that cp is liftable to R with respect to

A

and B' • Then ^cp admits a graded lifting to R with respect p{A•) and p{B' )

.

We omit the proof.

Similar results for graded S-modules and for graded module morphisms are valid.

and

CHAPTER 2

.P~~~at:Lon. f_¥E_c_t<?£S and formal moduli.

For the rest of this paper we shall deform only finitely generated algebras.

Let IT R ~ R• be a surjective ringhomomorpPism If (ker TI) 2 = 0 then 1.7 say that A is liftable to R iff A is liftable to

drop the condition

R as a graded algebra •. We would like to

.· 2

(ker TI)

=

0 in 1.7. To do this we shall introduce defamation functors.

Let V be a noetherian local ring with maximal ideal mv and residue field k = V /mv • Let 1 be the category

whose objects are artinian local V-algebras with residue fields k and whose morphisms are local V-homomorphisms. Let S be a finitely generated k-algebra and assume that we can find graded liftings SR of S to R for any R..E obl such that for any morphism TI : R ~ R 1 of

1

there is a morphism SR ~ SR• with

For each R , fix one SR with this property and let

be a finitely generated graded 8-algebra. Relative to the choice of liftings we define

l

A¹ is a graded lifting of A to

- 15 -

It is easy to see that Def⁰(A/S,-) is a covariant functor on 1 with values in ^S~ • This is the ^~ded deformation functor or A/S. Correspondingly, we denote by Def(A/S,-) the non-graded deformation functor of A/S •

Recall that a morphism of covariant functors

F ), G

on 1 is smooth iff the map

F(R) 4F(R' ) x G(R) G(R' )

is surjective whenever R ~R' is surjective. The tangent space tF of F is defined to be

tF

⁼

F(k(e])

when k[e] E ob

1

is the dual ring of numbers.

Definition 2.1

A pro-1 object R(A/S) , or just R(A) is called a hull for Def(A/S,-) if there is a smooth morphism of functors

Honii (R(A) ,-)

~

Def(A/S,-)

-

on

1

which induces an isomorphism on their tangent spaces.

R⁰(A) is similarely defined as the hull of Def⁰(A/S,-) • By 1.2 and 1.4 we see that

Def(A/S,k[t]) = H¹(S,A,A) Def⁰(A/S,k(t])

=

0H¹(S,A,A) Look at the canonical morphism of functors

Def⁰(A/S,-) ~Def(A/S,-) and the corresponding V-.ntorphism

R(A) -4-R⁰ (A)

If this morphism splits we have solved the problem mentioned at the beginning of this paragraph.

In [11] we find a very general theorem decribing these hulls. Following

[L1]

we notice that since

generated S-algebra, the group Hi(S,A,A)

A is a finitely for a given i is finite as an A-module. We pick a countabel basis { v.}

J

for Hi(S,A,A) as a k-vectorspace and define a topology on Hi in which a oasis for the neighbourhoods of zero are those subspaces containing all but a finite number of these v . •

J Let

for i = 1, 2 and let

be the completion of Symv(Hi*) in the topology induced by

"*

the topology on H^{1 ,} i.e. the topology in which a basis for the neighoourhoods of zero are those ideals containing some power of the maximal ideal and intersecting in an open subspace. If Hi is a finite k-vectorspace then Ti is a converg~nt power series algebra on

t .

The result we need is the following. See

(4.2.4.)

in

[L1].

Theorem 2.2

There is a morphism of complete local rings

such that

" 1\

R(A) ';;; T1 ®2 V T

- 17 ....

Short remark on the pr~~~·

To simplify ideas, assume V = k and H1(&,A,A) finite as a k-vectorspace. Let lh c 1

of 1 consisting of objects R

be the full subcategory statisfying mR n

=

^{0 •} Put

and _R2

=

_{T2 •} 1 ^{If R E} ob1₂ , then by 1.2

Def(A/S~R) = H 1 (S,A,A) ®mR = Hom~ (H 1

*

,mR) ⁼ Hom1(T2,R) c 1

k

-

Hence R2 represents the functor Def(A/S,-) on

1

_{2 •} ~t

A2 be the universal lifting of A to cr2

T~

^--f.k

~T~

is the composition,

then R2 ⁼ T1 2 = T1 ®2 2 T2 By induction we shall assume that

cr. l

are constructed such that

R: =

T!

^®2 k

1 1 T.

l

SR _.12

k •

and such that A2 is liftable to SR1 . Consider the following diagram

cr n-1

.

^If

We shall try to constrQct ^fJ _n T2 n ~ · n T 1 such that the diagram

above commutes. In fact it is enough to define as a k-linear map. Let

Then the diagram

T1 n ^{I t )} R' n

Til l

^n'

T1 n-1~ Rn-1

is commutativ~ and kerrr • is a k-module via

cr n on H ^2*

Let A

n-1 be any lifting of _~ to SR • The obstruction for iifting _{An_ 1} to SRj

n

n-1 is given by

cr (An-1) E H2 (SRn-1'\1-1 'An-1 ^® ker TT') ~ H2 (s,A,A) ®kerTT•

k

Hom (H2*, ker IT')

Let cr _n be any k-linear map fitting into the commutative diagram H2*

))

I I

R n .. ., :::> ker IT IT

and put

Thus killing the obstruction of lifting, we conclude that

A is liftable to

-n-1 ~·

R(A)

=

lim

Rn

t!:--

Pu.t

and cr = lim cr •

+--

ⁿ

- 19 -

Laudal proves that this R(A) is a hull for Def(A/S,-) • If V

4=k ,

just as in the general step, we let v 2 be the largest quotient of V/mv2 to which S ~ A is liftable.

Any lifting S ~ A may serve as a zero point for the iso- v2 2

morphism

Def(A/S,R) ~ H1(S,A,A) ® mR k

where RE ob_:L,2 For the rest we may proceed as before.

Corollary 2.2.a.

Let V be a regular local ring such that S ^~ A is lift- able to V/mv2 • Then R(A) is regular iff the composition

H2~ T2 ~T1 is zero.

Proof.

It follows from the fact that the image of the composition

is in mT 1 2 Q.E.D~

Similar results are true for R⁰ (A) • If

0Hi* = Hom~(

0

Hi(S,A,A),k) i=1,2 and

for i=1,2 is the completion of Symv(₀H¹ .* ⁾ in the corresponding topo- logy, then there is a morphism of complete local rings

such that

The canonical injections

induces surjections

tor i = 1J2

These surjections can be assumed to fit nicely into a commutative diagram

in such a way tllat the induced morphism R(A)~R⁰(A) rnakes the diagram

Def⁰(A/S,-) ~Def(A/S,-)

t ¹

Hom(R⁰{A),-) ,~ Hom(R(A),-) commutative.

We shall only sketch a proof of this commutativity since we will not use it much. We need an easy lemma, see (4.2.3)

in [L1].

Consider the commutative diagram

- 21 -

whose objects and morphisms are in

1.

Assume rr 1 and n2 surjective and (kerrr_{1 )}²

=

(kerrr_{2 )}²

=

^{0 • If} A1 is

a lifting o£ A to sR_{1 ,} and ~ = A_{2 ®} R_{2 ,} then

R1

H2(SR ,A₁ ,A₁ ® ker rr _{1 )}

--->

^H2(SR ,A_{2 ,A}2 ^® kerT? 2 )

1 R · 2 R2

maps the obstruction a(A_{1 )} onto a(A_{2 ) •}

x~oof __ of_!he.commulativity.

As in the nproof" of 2.2, let us assume V • k and

H1(S,A,A) finite as a k-vectorspace. We constructed Rn and ^C1 _n in such a way that

R(A)

=

lim R

~ n cr _, = lim a _n

~

In the graded case we shall use the notations

cr = lim (cr )

o ^~ on

Now and represents this deformation functors on is the universal lifting of A to SR we

2

A2 0 °R₂ is the graded universal lifting

R2

easilY see that to

Let n > 3 and let A

n-1 be a lifting of A2 . to By induction we may assume the commutativity of

and that

a n-1

A ® ⁰R

n-1 R n-1 n-1

>

By 2.3 a commutative diagram

is found,

Since

hence T1

(J n

n

then Rn

TT n

Rn-1

ker rr _n is a

)

) o

2·*

H-

oT1 n commutative

't ^oR _n

OTT n

) OR

n-1

•)

diagram

is a surjective map of k-vectorspaces, v!e deduce from the surjecti"'"T-ity ot

H 1 (S,A,A) ® ker 1T

k n

---J'

^{H 1} (S,A,A) ^{0 ker0rr}

7 k n

(using 1.2 and 1.7) that there is a lifting An and A _n-1 to such that is a graded lifting of

A _n-1 ^{® 0}

Fn-1 .

R n-1 The case

v *

^{1t makes ll.O tr_puble. Q.E.D.}

From this we get

Proposition 2.4

Let v be a regular. local ring such that S ^-+ A is liftable to V/mv2· Then

i) If R(A) is a regular local ring, so is R⁰ (A) _• ii) If R⁰ (A) is regular, then the morphism

R(A) ) ⁰R(A) splits

Proof

,....,._,~"'"""'.-""''--~~ ..

i) follows from the cammutativity of the diagram 2* ) H2:l!"

H

t

⁰

^~

T2

>

^{0 T2}

cr

1

^°a

'V

T1 ₎

0

T1 using 2.2.a

If R⁰(A) is regular, then °T1= R⁰(A) • The obvious surjection

induces an injection

which defines a one-sided inverse of R(A) --7 R⁰(A) •

The surjections Q.E.D.

fori

=

1,2 induce. morphisms

If the corresponding diagram

oom1.ttes ,

then R(A) ---,>R⁰ (A) splits. In general there seem to be no reasons for this diagram to commute. However imposing some rather natural conditions on the graded algebra A , the commutativity can be proved.

Defj,uition ~.5

We say that s·--:;)A has negative grading (resp. positive grading) if

(resp.

\IH1(S,A,A) = 0

\IH (S,A,A) 1 = 0

for ^\1 > 0 for ^\1 < 0 If A has positive or negative grading, then the diagram above commutes, proving

Theorem 2,6

If S ^~ A has negative or positive grading, then R(A)

--?

R⁰ (A)

splits as a local V-homomorphism.

In the same direction We have the following more general result.

Theorem 2.7

Assume S ~ A has negative {resp. positive) grading and put B

=

A[T] with degT

=

1 (resp degT

=

-1). Then there is

- 25 -

a V-isomorphism

R⁰ (B) ::: R(A) life shall need some preparations.

Let A and B be graded S-algebras and 'l : B ~A

an S-algebra homomorphism, not necessarily graded. For every

i ~ 0 ' ' induces maps

Let ~i/0 be the composed map

OHi ( S, B, B) --~) Hi ( S , B, B) ) Hi ( S, B, A) 'l'i

Lemma 2o8

If 'l'i and 'l'i/0 are isomorphisms for i = 1 and in- jections for i

=

^{2 ,} then there is a local V-isomorphism

B-_~mark 2 ⁰ 8 ~

Let ^{TT :} R~ R' be a surjection in 1 such that ker TT is a k-module via R ~k • Look at

k--:>

s

h A ' and

w"

w ere ^.1 are liftings to SR• ^{and B I lS .} a gra e ^{d d}

lifting to SRr • Consider the diagram

H2(S,A,A) 0 kern ) H²(S,B,A) _{0 ker n}

~ ~ 2 ¹⁰

^{0 kern}

k 1J:'2 0 ker TI k

By [L1], the obstructions for deforming A¹ and B¹ respec- tively map on the same element in H2 (S,B,A) 0 kern.

]?,FOOf of ? • ~

We shall use the notation

for the completion O.f

The morphisms

'fi H i (S,A,A) -~) ^Hi(S,B,A)

~:i/o ; Jli(S,B,B) --~) Hi(S,B,A) induce morphisms

i = 1, 2

which by the ."proof" of 2. 2 and by 2 .8.a fit into a comnutative diagram

oT~(< ^m2~T2 _{J:~ A}

(] ⁰

l l ^ta

oT1~

B T1~T1

~ A

- 27 -

The horizontal maps are surjections and isomorphisms by the assumptions of 2.8.

Q.E.D Remark 2.8 b

If ~i is an isomorphism for i = 1 and an injection for i

=

² the morphism

induces a morphism

R(A)

>

R(B) Now we turn to the proofs of 2.6 and 2.7 Proof of

2.7

Let

be the composition

B ⁼ A(T] ---~~) A( T] /( T-1 ) ^f'.J A and let j be the canonical injection

j A~ B

Let M be any B-module.

j induces maps

Using the exact sequence .

. l

. . JM . . 1

~ Hl(A B M) Hl(S B M) ----"'- Hl(S,A,M) ~ Hl+ (A.B.M)

/ 9 ' --;Jo ' ' ~ ---;il" ' '

and the fact that

for i > 1

)

we deduce that j~ are isomorphisms for i > 1

¥i are the inverse maps of ji for i

=

^1,2

Hence by 2.8 it is enough to prove that

However

is an isomorphism for i = 1 and an injection for i = 2.

Look at the diagram

0Hi(S, A, B) - - - j Hi(S, A, B) ----7 Hi( S,A, A)

U II

Hi ( S, A, A) ® k [ T] ~ Hi ( S, A, A) k

where the lower horizontal map is induced by sending T to 1. If deg T = 1 and if i > 1 , (1i)- 1¥i/o is given by the composition

. 0 . 0 .

0H¹(S,B,B) ~ ~ v H¹ (S,A,A)T-v ~a v H¹ (S,A,A)

V=- co V=- co

which is an injection for all i

?:

^{1 •} If A has negative grading, then by definition ¥ 1/o is an isomorphism. The

case degT ^{= -1} is similar. Q.E.D.

Proof of 2.6 Let

~ : B ~ B/(T)

=

^A

be the canonical surjection. Then

- 29 -

are isomorphisms for i > 1 • By 2.8 b there is a morphism

deduced form the commutative diagram

The horizontal maps are induced by {~i)-^{1 o} ~i • Moreover

by 2. 7 and its proof, the isomorphism.

R⁰(B) ~ R(A)

is deduced from the commutative diagram

The horisontal maps are induced by

However if degT

=

1 , this morphism is given by

. 0 . .

OH¹(S,B,B) ~ l v Hl(S,A,A) ---).H¹(S,A,A)

V=- o:J

which splits. Using a one-sided inverse; i.e. a projection for i = 2 , a commutative diagram

is found, inducing the isomorphism R⁰(B) :: R(A) We claiJll that the composed map

R⁰ (A) ~ R⁰ (B) :::: R(A)

is a one-sided inverse of R(A) ~ R⁰ (A) This is trivial if we look at the diagram

oT~--) oT2 ---} ^T2

.1:1. B A

"o(A)~

⁰

l

⁰

1

^{cr (A)}

oT1 _A~ oT1 .-..,)) T1

B A

The composition of. the horizontal maps are induced by the obvious projections

0Hi(S,A1A) ~Hi(S,AtA) since (~i)-¹o ~i are given by

0Hi(S,B,B)

~ l

v Hi(S,A,A)T-v __ _.) ₀Hi(S,A,A)

V=- CXJ

sending T to 0 • The case degT

=

^-1 is similarily treated.

Q.E.D ..

Theorem 2.7 can be generalized in the following way. Let

be the localization of B in the multiplicative system {1,T,T2 , ••• } and put degT

=

1 • Then for IJ:_Y£!. finitely generated S-algebra A , then is an isomorphism

We omit details of a proof.

- 31 ...

The conditions of negative and positive grading on S~A are only reasonable if the graded ring S sits in degree zero However, if S 4A is any graded morphism and S is S₀ smooth, then

splits i£ 3₀4 A has negative or positive grading. In be- eing more precise we shall assume that the ¹¹choice¹¹ of the liftings of S₀ and S are compatible, i.e. for any R E obl there is a morphism (S0^)R~ SR such that if R~R' is in 1 , then ~here is a commutative diagram

Then the maps

Def⁰(A/S,-) ~ Def⁰(A/S_{0 , - )} Def (A/S,-) ~ Def (A/S_{0 , - )}

are well defined and they are easily seen to be smooth.

Therefore the morphisms

R⁰ (A/S) ~ R⁰(A/S_{0 )} R (A/S) ~ , .. R (A/S_{0 )}

are still smooth. These maps fit into a commutative diagram

The right hand vertical morphism splits because S0^{~A has} negative or positive grading. By definition of smoothness the left hand vertical morphism also splits.

- 33. -

CHAPTER 3

R.~l_ai!.sml?_ to projective __g__eomet~.

As we know the graded theory of algebras are closely related to projective geometry. In what follows we shall compare the groups ~Hi(s,A,M) with Ai(s,x, M(~)) when X = Proj(A) • Moreover if

cp: B---7 A

is a surjective graded morphism and

f : Proj(A) ~ Proj(B)

is the induced embedding, we shall relate the groups to A ⁱ (S,f,M(~)) ^rv •

be any S-scheme, ^M any quasicoherent

and let f : ^X~ Y be a morphism of S-schemes. Then there are groups

for every i > 0 • Using [L1] we shall summarize some pro- perties needed in the sequel.

i) (3.1.12) in [L1]states that Ai(S,X,M) is the abut- ment of a spectral sequence given by the term

If U= Spec(A) is an open affine subscheme of X , the Ox-Module Aq(S,~), or just !q(~)

,

is given by

'"'-'

If X is affine, say X= Spec(A), and M = M for some A-module M , we deduce

If X is S-smooth, we find

Ai(s,x,~) = Hi(x,ex ®N)

s

where ex =

A

^0(0x) is the sheaf of S-derivations.

ii) By (3.1.14) in [L1] Ai(S,f,~) is the abutment of the spectral sequence given by

If V= Spec(B) is any open affine subscheme of Y , then by definition

Therefore if f is affine, say f-¹(V)

=

Spec(A),

iii) Let Z c X be locally closed. By (3.1.16) there is an exact sequence

~ ·A~(S,X,!i) --7 An(S,X,M) ~ An(S,X-Z,!1) --j-An~¹ (S,X,M)---;

whe~e the gr~up~~ A~(s.x,~) ia the abu~ent of a spectral se- quence given by the term

Ep2 q = AP(s,X,Hi(J1))

If X

=

Spec(A) and Z

=

V(I) for a suitable ideal I c A we write

n n ^~

HI(S,A,M)

=

^Az(S,X,M)

iv) Let f : X-?Y be an affine morphism of 8-schemes.

By (3.2.3) there is a long exact sequence

- 35 -

Let S be noetherian and let A and B be finitely generated, positively graded S-algebras generated by its elements of degree 1. Assume A₀

=

B₀ = S • Let

be a surjective graded S-algebra morphism and let f : X = Proj(A) ~ Proj(B) = Y be the corresponding embedding. Put

co m = .ll.

1\V

\1=1

and X' = Spec(A) - V(m)

Let

IT : X' ~ X

be the obvious morphism. ^IT is an affine smooth surjection.

If M is a graded A-module, we shall denote by J'IIa the localization of M in

!--

,,a,a , •••• 2

l .

^Let !VI( a) be the homogeneous piece of _lYI·a of degree zero.

1

Let b E B such that a :=(,J(b) • Since ^l,

is flat, a theorem from [A] gives the isomorphism

However

: •'

Therefore

Hence

Put

Then by (ii)

proving

Using (i) we find

Therefore

This proves

With notations as above there is an iSomorphism

where

of degree "'·

Proof ._ ...

_

^..

i '' "' . i rv

A (S,f,M(v)) ~VA (B,~ ,M)

is the homogeneous piece of A (B, X' ⁱ ,M) ^rv

Going back to the definitions of Ai(S,f,M(v)) and

- 37 -

i ' ~) ^[

J

A (B,X ,M in L1 we deduce a morPhism

i ,..._, i f' "')

A (S,f,M(v)) ~VA (B,X 4~

The corresponding morphism of spectral sequences

is an isomorphism for every p and q

Q.E.D.

Jpeorem 3.2

If cp : B ~A is surjective and if

dept~M

?.

n n an integer then the morphisms

are isomorphisms for i < n and injections for i = n Proof

By iii) there is a long exact sequence

----+Hi (BAM) -4Hi(B AM) -4-Ai(B X' M) ~Hi+¹(B AM)~

m ' ' ' ' ' ' ~ m ' '

Since depthM

?.

n , we conclude that

for q < n-1

Moreover H⁰(B,A,-)

=

⁰ since ~ is surjective. By the spectral sequence of iii) we deduce

u!(B,A,M)

=

0 for i < n

Q.E.D.

If depth m A> 2 -c-

\)H1(B,A,A) ~ ^A1(S,f,OX(v))

are isomorphisms and injections respectively.

Let us apply this result to the case

s

^{= k, k a field.}

Vle denote by

Hilbf(-)

the local Hilbert functor relative to ^y at f

'

defined on the category 1 . (See the beginning of chapter 2 and use

by the functor Def⁰(A/B,-) defined in chapter 2 using tri- vial liftings of B •

.Qorollar4 3.

4

If dept~A ~ 2 , then there is an isomorphism of functor

on 1 . Proof

Both functors are prorepresentable. By (2.2) Def⁰{~,-) is prorepresented by

Sym(₀H1(B,A,A)*) A~ k

Sym{₀H2 (B,A,A)*) A

Using (5.1.1) in [L1], Hilbf(-) is prorepresented by the object

1 1\ A

Sym(A {f,Ox)*) 0 k

Sym{A2 (f,Ox)*) A

- 39 -

The natural morphism of functors

corresponds to a morphism between their prorepresenting objects.

This is nothing but the morphism induced by the natural maps in (3.2)

Q.E.D.

Assume B to be k-free and

f : X~.~

to be the induced embedding. In this case Hilbf(-) is also denoted by Hilbx(-) Both Hilbx(-) and Def⁰(~,-) are easily defined on 1 for V arbritary, and by the same arguments as before there is an isomorphism of functors

on 1 whenever depth A > 2. Even if depth A ~ 1 we de- duce this isomorphism in some cases. In fact, the sequence

is exact. The isomorphism therefore follows from

Recall that if I=ker ^{~ ~} B

H¹(B,A,H¹(A)) = HomA(I/I2 ,H¹(A))

o m o r t m

Furthermore X

S

~ = P and if n > 2

~(A) ~llH¹ (P,I(v))

If we define c by

and

s = min {deg fil !fi,-'frl is a minimal set of generators of I!

then

for c < s In

[E]

we find more or less a direct proof of

(3.4).

So far we have concentrated on deformations of embeddings.

One may ask for the relationship between the groups

and

This is given by our next theorem

There are canonical morphisms

i i ,....,

\IH (S,A,r.i)

---t

A (S,X,M(v))

for any i > o and any v. If n > 1 and if depth~ ~ n+2 , then the morphisms above are bijective for 1 < i < n and injective for i = n.

Proof

Consider the following two exact sequences

. i . "" i · "' . i . "' i+1 ""

-+A (S;Tr,!1) -j· A (S, X' ,r![) _,.A (S, X,"lf*M) ~A (S, lf,M) with

~: X' = Spec(A)-V(m)--) X= Proj(A)

- 41 -

as before. The spectral sequence given by

converges to

and it is easy to see that

Since depth H > n+2 then

for 1 < i < n Furthermore

H!(M)

⁼

0

^{for i < n+1}

implying that

H!_(s,A,M) = 0

for i < n+1 The theorem now follows from the two exact sequences stated at the beginning of this proof.

g~~-roJJ-~L~-§.

If depth A >3 m ^~·~ and

for every v , then

Q.E.D.

In the smooth case

and (3.6) reduces to a rigidity theorem of Schlessinger;

see

(2.2.6)

in [K~LJ

•

See also [Sv].

,Qor~.x 3.7

If depthmA ~ 4 and

A2{S,X,QX(v))

⁼

0

for every v, then

H (S,A,A) 2 = 0

If X has only a finite number of nonsmooth points, then

H1(x,A1{ox(v))

⁼

o

Moreover if the non-smooth points are complete intersections

H⁰(x,~²(ox(v)) ⁼

o

In this case we conclude

We will end this chapter by proving a geometic variant of (2.7) due to Pinkham [P]. We also need (3.4-).

Let R be k-free and cp : R--} 1\. be surjective, corresponding to X= Proj(A) ~~· Look at the diagram

R[T] ~ R

B = A(T] ~ A

where ^l'P =·'+' ® idk[T] and where the horizontal maps are induced

k

by sending T to 1. Put deg T

=

1.

- 43 -

Clearly

Def⁰ ('CP,-) ~ Def⁰ (B/k,-) is smooth. Hence

is smooth. If ^A has negative grading R(A) ~ R⁰ (B) The composition

whenever X ₌ Proj _{(B) •} This proves Theorem _3.d~

Let X be a closed subscheme of

~

^{and let A be its}

minimal cone. If

X

⁼ Proj(A[T])

is its projective c:o..'le i:p. r~¹ and if A has negative grading, then there is a smooth morphism of functors

Hilbx(-) --7 Def(A/k,-) on 1 (V arbritary)

CHAPTER 4

Positive ~d negative gradin~

In this paragraph we shall see that if

is closed and satisfies some weak conditions, then after a suitable twisting the minimal cone of the corresponding em- bedding will have positive of negative grading.

Suppose S noetherian and let

co

be a graded A₀

=

S algebra of finite type, generated by A1• Denote by ^111' the angmentation ideal of A; i.e.

00

In= .tt..~

\1:1 Assume moreover

Let M be any graded A-module and put

ro

M(d) = ll.Md\1

V~CD

In what follows we shall relate the groups

to the groups

~e~4.1

If X¹ = Spec(A)- V(m) and X¹(d) =Spec (A(d))- V(m(d)) then the groups

- 45 -

are isomorphic for every i fTOOf

The canonical morphism A(d)~A induces a morphism of schemes

thus a homomorphism

It suffices to prove that the corresponding morphism of spectral sequences

is an isomorphism for every p and q • Consider the commutative diagram

Then

This will prove J_heorem

4_.?

Let n be an integer and assume dept~M > n + 2 Then

are isomorphic for ⁱ < n and for every d > 1

Consider the exact sequences

---7 H!(s,A,M) (d) ....-rHi(S,A,M) (d)

~Ai(S,X' ,M)

(d)---}

Hi~

¹ (S,A,M) (d)

---1

~ ~(d)

(S,A(d) ,M(d))-+ Hi(s, A( d) ,M(d)H Ai(S,X' (d) ,M(d))

~H!~~~S,A(d),

Since depthM > n+2 is equivalent to the conditions

.llHi(X,M(v)) =

o

tor 1 < ⁱ < n

\)

we easily deduce

Hence

depthM(d) > n+2

~(S,A,~I)

⁼

~(d)

(S,A(d)'M(d)) = 0 for ⁱ < n+1 Q.E.D.

We are specially interested in (4.2) for the case n = 1 and M=A. Let R be a graded S-free algebra, generated

by R1, such that

. .

A= R/I

Put ^p ₌ ~ ^=; Proj {R)

If ^N > 2

- 47 -

Futhermore by assumption depth A > 1 , thus

Recall also

.?r_o.:gosi t-iQ.n 4 • ..:1_

H⁰ _m (A)

=

⁰

Let d > 1 and assume

for ⁱ > 1

for all t > d • Then there is a natural isomorphism

for v > 1 Proof

Consider the long exact sequences

~·

H 1m(S,A,A)

(d)~

H 1 (S,A,A)

(d)~ K·(s'.X'~,.)(4) ~H

²

m(S,A,A)

^(d)

---7

--->

H1m(d) (S,A(d)'A(d)~ H1(S,A(d)'A(d)~ A1(S,X'(d)'OX'(d)) )

H m(d)(S,A(d)'A(d)) 2 ~ By assumption we have

since ^'V > 1 •

Futhermore

d\llf(S,A,H2m(A))

= _dvDer 8 ^(A,~H

1 (OX(t))) = 0 vH⁰(S,A(d)'H

2 {A))=

vDer

8

^(A(d)'HH

1

_(ox(dt)))

=

o

~d} t

Since

is surjective and since

we find

Similarely we prove that

Hence

dV~m(S,A,A)

= 0

\IH m(d)(S,A(d)'A(d)) 2

=

0

for ^v > 1

for v > 1.

for v > 1 for ^\1 > 1 The exact sequences above together with (4.1) prove the pro~

position

Q.E.D •

.92~

If ^{dept~A ~} 2 and if v is an integer such that H1(x,ox(dv+1))

= o

then

d\IH1(S,A,A) • 0 implies

- 49 -

Proof

··----·--...

By assumption

Moreover

Thus

Using the long exact sequences of the proof of (4.3) we find a diagram

which proves 4.4

Q.E.D.

(Negative grading of A(d))

Assume dept~A ~ 1 and suppose there is a d > 1 such that

Then

Proof

~----z•

H1(X,OX(t+1))

=

⁰

tH (S,A,A) 1

=

⁰

Use 4.3 for ^\1 = 1,2, ••••

for t > d

for \1 > 1

Q.E.D.

92rol~a~.4.6 (Positive grading of A(d))

Assume depth A ~ 2. Suppose there is a d > 1 such that for t > d then

for v < 0 Proof

Use 4.4 for v = -1,-2, •••

Q.E.D.

Let us put 4.5 and

4.6

together in the following theorem

"~~_2J'em 3~_7.

Let X

=

^Proj(A)

a) If X is S-smooth, then there is a graded S-algebra B having negative grading such that

X :. Proj (B)

b) If depth A ~ 2 and if there is an integer n such that H 1 {X,Ox(t)) = 0 for t < n then there is an S·algebra B having positive grading such that

X";! Proj(B)

c) If X satisfies the conditions of a) and b) then

X~ Proj{B)

for an S-algebra B which has both positive and negative grading

Proof

If X is S-smooth, then

-.51 -

for large ^~. In fact the sequence

is exact and

~H

⁰

(S,A,H

¹

m(A))

⁼

~Der 8 ^(A,tH

¹

^{(I(t))) =}

⁰

~A

¹

(S,X',Ox,)

⁼ 'JH1 (X',6x,) =

~H

¹

(X,rr*eX,~ =

0 for large ^~

.

^Thus

^(4.5)

proves a) •

(4.6)

proves since

VH1(S,A,A)

=

⁰

for small ^{\1 •} This follows from the surjection H (R,A,A) 1 ~ H (S,A,A) ¹

and from the fact that

b)

for small ~. Q.E.D.

For similar results, see [S3] and [M].

... ··

CHAPTER 5

The exi~?_tence of a k-algebra which is unliftabl_e_ to chalf!+8:=

,;teristj,c

zero.

In [se] Serre gives an example of a k-smooth projective variety X in characteristic p which caru1ot be lifted to characteristic zero. This means that for any complete local ring A of characteristic zero such that A/mA. =k, it is impossible to lift X to

A .

His vari tey is of the form

X ⁼ Y/G

w.hen Y is a complete intersection of dimention 3 and G is a finite group operating on Y without fixpoints. Futhermore the order of G divides p •

By (4. 7 a) there exists a graded k-algebra B with negative grading such that

X

=

^Proj(B)

Hence (2.6) proves that B cannot be lifted to any~oetheri­

an) _complete local ring A of characteristic cero. In fact the example of Serre satisties even (4.7 c), thus proving the existence of a graded k-algebra C satisfying vH (k,C,C) 1 = 0 for v

t

0 , such that X = Proj(.ll.) • (2.6) reduces to the almost trivial result

Clearly C is unliftable to any complete local ring 1\ of characteristic zero.

The reason why Serre's example works is obviously that p, the characteristic of k, divides the order of G • To see

- 53 -

this, let us prove

.~heorem 5.

L

Let B--7 A be an S-algebrahomomorphism having a B-linear retraction. Let I c A be an ideal such that the comosed morphism

U = Spec(A) - V(I)C...:,. Spec(A) ~ Spec(B)

is etale. If depth1A

?.:

n+2 , then there is an injection Hi(s, B,B)

4

Hi(s, A, A)

for i ~ n.

fT...QFC?.f..

By 4taleness Ai (B, U , 0 U) = 0 for all i , and the depth condition implies

for i < n+1 Using the exact sequence

we conclude

Hi(B,A,A)

=

^{0 for i < n+1}

However, there is an exact sequence

Hence

i < n Since the injection B---; A has a B-linear retraction

is injective for ^~y.i

Q.E.D.