Nonexistence of Negative Weight Derivations on Graded Artin Algebras: A Conjecture of Halperin

Journal of Algebra 216, 1᎐12 Ž1999. Article ID jabr.1998.7750, available online at http:rrwww.idealibrary.com on

Nonexistence of Negative Weight Derivations on Graded Artin Algebras: A Conjecture of Halperin Hao Chen* Department of Mathematics, Zhongshan Uni¨ ersity, Guangzhou, Guangdong 510275, People’s Republic of China Communicated by J. T. Stafford Received March 7, 1996

1. INTRODUCTION A classical result of A. Borel wBox states that the Serre spectral sequence for rational cohomology of the universal bundle GrH ª BH ª BG collapses if GrH is a homogeneous space of equal rank pairs Ž G, H . of compact Lie groups. In 1976 S. Halperin made the following conjecture wH, Mex. Halperin Conjecture. Suppose that F ª E ª B is a fibration with simply connected base B and that the Ž rational . cohomology algebra of the fibre is an Artin algebra of the form H *Ž F , Q . s Qw x 0 , x1 , . . . , x n xrŽ f 0 , f1 , . . . , fn . . Then the Serre spectral sequence for this fibration collapses. Here we note that x 0 , x 1 , . . . , x n are cohomology classes and we can take their dimensions to be their weights. In this weight type, f 0 , f 1 , . . . , f n are relations of these cohomology classes in some cohomology groups. Therefore they are weighted homogeneous polynomials. Actually the above conjecture is equivalent to the following conjecture about the nonexistence of negative weight derivations wH, Mex. * Research supported by NNSF ŽChina.. 1 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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Halperin Conjecture Ž Equi¨ alent Form.. Let x 0 , x 1 , . . . , x n be weighted ¨ ariables and f 0 , f 1 , . . . , f n be weighted homogeneous polynomials in P s C w x 0 , x 2 , . . . , x n x. Suppose that R is an Artin algebra of the form C w x 0 , x 1 , . . . , x n xrŽ f 0 , f 1 , . . . , f n .. Then there is no non-zero negati¨ e weight deri¨ ation on R.

Here we note that R and DerŽ R, R ., the R-module of derivations on R, are naturally graded by the Žweighted. degrees Žsee Definition 2.1. on C w x 0 , x 1 , . . . , x n x. The Žnon.existence of negative weight derivations appears in many other contexts which may also serve as a motivation for the present work. For example, their nonexistence has also been conjectured by J. Wahl for positive dimensional quasi-homogeneous isolated singularities wWa1x and by S. Yau for moduli algebras of isolated quasi-homogeneous hypersurface singularities wYa1, Ch-Xu-Yax. For a study of the Žnon.existence of negative weight derivations from the point of view of singularity theory, we refer the reader to wKa1, Ka2, Wa, Wa2, M-S, Ch-Xu-Ya, Ch1x. The Halperin conjecture is true when the fibre F is homogeneous wBo, Me, Hx. Also this conjecture has been proved to be true for the two-variable case wTh1, Th2x. The purpose of this note is to give a proof for the Halperin conjecture for the three-variable case. This is a generalization of the work of wCh-Xu-Yax. On the other hand we should mention that the Halperin conjecture has been proved in the case of ‘‘large’’ graded Artin algebras Ži.e., when the degrees of f 0 , f 1 , . . . , f n are sufficiently large. in our other paper wCh2x.

2. NEW WEIGHT TYPES DEFINITION 2.1. Let ␣ 0 , ␣ 1 , . . . , ␣ n be positive integers. A monomial x 0i 0 x 1i1 ⭈⭈⭈ x ni n in the polynomial ring P s C w x 0 , x 1 , . . . , x n x is said to be of Žweighted. degree i 0 ␣ 0 q i1 ␣ 1 q ⭈⭈⭈ qi n ␣ n . A polynomial in P is said to be weighted homogeneous of degree d if it is a C-linear combination of monomials of degree d. It is clear that P is graded. If I ; P is a homogeneous ideal, i.e., an ideal generated by weighted homogeneous polynomials, we know that R s PrI s [k R k is also graded. Let DerŽ R, R . be the R-module of derivations on R. There is a natural grading on DerŽ R, R . defined by DerŽ R, R . k s  D g DerŽ R, R .: DŽ R i . ; R iqk for any possible i4 . Derivations in DerŽ R, R . k for k - 0 are called negative weight derivations. The main problem considered in this paper is whether DerŽ R, R . k for k - 0 is zero for some k and some R. We denote the weight of a derivation D by wtŽ D ..

NEGATIVE WEIGHT DERIVATIONS

3

It is well known that derivations on R can be thought of as derivations on P preserving the ideal I. On the other hand any derivation on P is of the form D s p 0 ⭸r⭸ x 0 q p1 ⭸r⭸ x 1 q ⭈⭈⭈ qpn ⭸r⭸ x n ,

Ž 2.1.

where p 0 , p1 , . . . , pn are polynomials in P. Thus we only consider the negative weight derivations on P which preserve the ideal I in the following part of this note. From now on we always assume ␣ 0 G ␣ 1 G ⭈⭈⭈ G ␣ n , which we may do without loss of generality. If a derivation D as in Ž2.1. on P is of negative weight, it is clear that each pi is a weighted homogeneous polynomial of degree less than ␣ i . Thus we know that pi is a polynomial of variables x iq1 , x iq2 , . . . , x n for i s 0, 1, . . . , n, i.e., the negative weight derivation D on P has to be the following form for some 0 F k F n, D s p 0 Ž x 1 , . . . , x n . ⭸r⭸ x 0 q p1 Ž x 2 , . . . , x n . ⭸r⭸ x 1 q ⭈⭈⭈ qpk Ž x kq1 , . . . , x n . ⭸r⭸ x k ,

Ž 2.2.

where pk is a non-zero polynomial. The purpose of this section is to introduce some new weight types for any given negative weight derivation D on the weighted polynomial ring P. These new weight types are the natural generalization of the original weight type Ž ␣ 0 , ␣ 1 , . . . , ␣ n . and will play a crucial role in this paper and our other paper wCh2x. DEFINITION 2.2. Let D be a negative weight derivation on the weighted polynomial ring P as in Ž2.2.. The following weight types Ž l 0 , l 1 , . . . , l n . controlled by parameters Ž ⑀ 0 , ⑀ 1 , . . . , ⑀ k , ␮ kq1 , . . . , ␮ n . are called new weight types associated with D, l n s ␣ n ␮n l ny 1 s ␣ ny1 ␮ ny1 ⭈⭈⭈ l kq 1 s ␣ kq1 ␮ kq1 ,

Ž 2.3.

where ␮ kq 1 , . . . , ␮ n are real parameters. If l qq1 , l qq2 , . . . , l n are defined, l q is defined as follows Ži. If the coefficient pq of ⭸r⭸ x q is zero l q s ␣ q ␮q ,

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where ␮ q is a real parameter; Žii. If the coefficient pq of ⭸r⭸ x q is a non-zero polynomial i qq 1 l q s ⑀ q q min  Q s degrees of monomials x qq1 ⭈⭈⭈ x ni n appearing

in the expansion of pq Ž x qq1 , . . . , x n . 4 , Ž 2.4. where ⑀ q is a real parameter and the Q-degrees are defined with respect to the weight type Ž l qq 1 , . . . , l n .. As in the original weight type Ž ␣ 0 , ␣ 1 , . . . , ␣ n . we can define Q-degrees of monomials with respect to this new weight type Ž l 0 , l 1 , . . . , l n . and Q-weighted homogeneous polynomials. EXAMPLE 2.1. For any negative weight derivation D on the weighted polynomial ring P, the new weight type associated with D is the original weight type Ž ␣ 0 , ␣ 1 , . . . , ␣ n . if we take the ␮’s parameters to be 1 and the ⑀ ’s parameters to be ywtŽ D .. EXAMPLE 2.2. Let P s C w x 0 , x 1 , x 2 x be the weighted polynomial ring of three variables with the weight type Ž ␣ 0 , ␣ 1 , ␣ 2 . and D be a negative weight derivation on P of the form D s p 0 Ž x 1 , x 2 . ⭸r⭸ x 0 q cx 2m⭸r⭸ x 1 ,

Ž 2.5.

where p 0 is a weighted homogeneous polynomial of degree ␣ 0 q wt D, c is a non-zero constant and m is the positive integer ␣ 1 q wt D. We consider the new weight type associated with D controlled by the parameters

⑀ 0 s ym ␣ 2 ⑀ 1 s ym ␣ 2

Ž 2.6.

␮ 2 s 1. Then we have l2 s ␣ 2 l1 s 0 l 0 s ym ␣ 2 q min  i 2 ␣ 2 : x 1i1 x 2i 2 ’s appearing in the expansion of p 0 Ž x 1 , x 2 . 4 . Ž 2.7. This new weight type will be used in our proof for the main result, Theorem 3.1. In our proof for the main result, we can introduce a coordinate change; by this coordinate change we only need to prove the

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5

conclusion in the case that ␣ 2 s 1 Žsee Section 3.. Therefore let us consider the situation of Example 2.2 when ␣ 2 s 1. EXAMPLE 2.2 Žcontinued.. Let P s C w x 0 , x 1 , x 2 x be the weighted polynomial ring with the weight type Ž ␣ 0 , ␣ 1 , 1. and D be a negative weight derivation on P of the form as in Ž2.5.. Suppose that x 1t x 2u is the term in the expansion of p 0 Ž x 1 , x 2 . with the smallest possible exponent u of x 2 . We can assume u - m without loss of generality Žsee Section 3.. Hence we have l 0 s ym q u l1 s 0

Ž 2.8.

l 2 s 1. We note l 0 - 0. Let A k be the set of weighted homogeneous polynomials of degrees k in P, Alk be the set of polynomials in A k with Q-degrees not less than l, and Ckl be the set of weighted homogeneous polynomials with respect to both weight types Ž ␣ 0 , ␣ 1 , 1. and Ž l 0 , l 1 , l 2 . of degrees k and Q-degrees l. PROPOSITION 2.1. Ži. ; DŽ A ky wt D .;

Suppose that l G ␣ 1 q wtŽ D .. Then we ha¨ e Alk

Žii. Suppose that l - ␣ 1 q wt D. Then we can take elements of the form x 0u x 1¨ x 2w , with u ␣ 0 q ¨ ␣ 1 q w s k, ul 0 q w s l, and w - m, as the l i 0 i 1 i 2yl 0 Ž Ž .. base of Ckl rCkl l Ž A) for i 2 y l 0 k q D A kywtŽ D . . Moreo¨ er x 0 x 1 x 2 i 0y1 i 1qt q1 i 2 l l l . Ž . Ž .. G m s i 0ri1 q 1 ex 0 x 1 x 2 in CkrCk l Ž A) k q D A kywt D , Ž . where e is a non-zero constant only depending on p 0 x 1 , x 2 and c. Proof. Ži. Let x 2k g Alk with l G ␣ 1 q wt D. Note k s l G ␣ 1 q wt D s m, we have x 2k s DŽ e⬘ x 1 x 2ky m . for a non-zero constant e⬘. Hence x 2k g DŽ A kywt D .. For any x 1u x ¨2 g Alk with l G ␣ 1 q wt D s m, we have l s ¨ G m and x 1uq 1 x ¨2 ym g A kywt D . Thus x 1u x ¨2 s DŽ e⬙ x 1uq 1 x ¨2 ym . for a non-zero constant e⬙. Let x 0 x 1u x ¨2 g Alk with l s ¨ q l 0 G ␣ 1 q wt D s m. We have DŽ x 0 x 1uq 1 x ¨2 ym . s p 0 Ž x 1 , x 2 . x 1uq 1 Kx ¨2 ym q e⵮ x 0 x 1u x ¨2 for a nonzero constant e⵮. For any term x 1i1 x 2i 2 in the expansion of p 0 Ž x 1 , x 2 ., we have ¨ y m q i 2 s ¨ y Ž m y i 2 . G ¨ q l 0 Žby the definition of l 0 . G m. Hence e⵮ x 0 x 1u x ¨2 s DŽ x 0 x 1uq 1 x ¨2 ym . q terms x 1t 1 x 2t 2 with t 2 G m. Therefore we have x 0 x 1u x ¨2 g DŽ A kywt D .. Assume that we have proved that x 0h x 1u x ¨2 ’s, with arbitrary u, ¨ , fixed h g Alk , and l G ␣ 1 q wt D s m, are in DŽ A ky wt D .. Now we want to prove that x 0hq 1 x 1u⬘ x ¨2 ⬘ Žin Alk . with l G ␣ 1 q wt D is in DŽ A ky wt D .. We have D Ž x 0hq1 x 1u⬘q1 x ¨2 ⬘ym . s Ž h q 1. p 0 Ž x 1 , x 2 . x 0h x 1u⬘q1 x ¨2 ⬘ym q e⬙⬙ x 0hq 1 x 1u⬘ x ¨2 ⬘ for a non-zero constant e⬙⬙. Argued as above we find that every monomial in the first term is in DŽ A ky wt D .. Hence the second term is in DŽ A ky wt D .. We get the conclusion.

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Žii. For any monomials x 0hq 1 x 1u⬘ x ¨2 ⬘ with ¨ ⬘ G m, we note that s linear combination of terms x 0h x 1u⬘q1 x 2G ¨ ⬘ql 0 mod DŽ A kywt D . by the above argument. Moreover we have that monomials of the form x 1u x ¨2 with ¨ G m are in DŽ A kywt D .. Therefore the only possible monomil Ž .. are of the form indicated als in the base of Ckl rCkl l Ž A) k q D A kywt D in Žii.. The other conclusion of Žii. follows directly from the above argument. Q.E.D. x 0hq 1 x 1u⬘ x ¨2 ⬘

The following proposition gives a clear description about the set Ckl . PROPOSITION 2.2. Ži. ␣ 1.

Suppose Ckl / 0. We ha¨ e k y l can be di¨ ided by

Žii. Suppose that Ckl / 0. We ha¨ e that Ža. Ž l 0r␣ 0 . k F l F k, if k can be di¨ ided by ␣ 0 ; Žb. p⵮ q pl 0 F l F k, if k cannot be di¨ ided by ␣ 1. Here k s p␣ 0 q p⬘, where p s w kr␣ 0 x is the integral part of kr␣ 0 and 0 - p⬘ - ␣ 0 , p⬘ s p⬙ ␣ 1 q p⵮ where p⬙ s w p⬘r␣ 1 x and 0 F p⵮ - ␣ 1; Žiii. For any pair of positi¨ e integers k and l such that k y l can be di¨ ided by ␣ 1 , we ha¨ e that Ža. if k can be di¨ ided by ␣ 0 and Ž l 0r␣ 0 . k q Ž h q 1. ␣ 1 F l F 0 or 0 - l F k y hŽ t q 1. ␣ 1 , then dim Ckl G h q 1; Žb. if k cannot be di¨ ided by ␣ 0 and p⵮ q pl 0 q Ž h q 1. ␣ 1 F l F 0 or 0 - l F k y hŽ t q 1. ␣ 1 , then dim Ckl G h q 1. Proof. Ži. We note ␣ 0 y l 0 s t ␣ 1 q u y wt D q m y u s t ␣ 1 q u y wt D q ␣ 1 q wt D y u s Ž t q 1. ␣ 1. The conclusion follows directly. Žii. For Ža., we note l 0 - 0, l 1 s 0, and l 2 s 1. Thus Q y deg x 0Ž k r ␣ 0 . s min Q-degrees of monomials in Ckl 4 . For Žb. the conclusion follows similarly. Žiii. For fixed positive integer k divided by ␣ 0 , i.e., k s p␣ 0 and l s pl 0 q r ␣ 1 Žnote the condition in Žiii.., we have 1 q h F r F p from the condition p␣ 0 q Ž h q 1. ␣ 1 F l F 0. Then x 0py rqi x 1r tyiŽ tq1. x 2r uyr wt Dyil 0 g Ckl for i s 0, 1, . . . , h. We have dim Ckl G h q 1. In the case that 0 - l F k y hŽ t q 1. ␣ 1 , let k y l s hŽ t q 1. ␣ 1 q w␣ 1 with a positive integer w Žnote the condition in Žiii... We have x 0i x 1Ž hyi.Ž tq1.qw x 2lyi l 0 g Ckl for i s 0, 1, . . . , h. Therefore dim Ckl G h q 1. The conclusion of Žb. can be proved similarly. Q.E.D. 3. THE HALPERIN CONJECTURE FOR THE THREE-VARIABLE CASE In this section we give the proof for the main resultᎏthe Halperin conjecture in the three-variable case. We first prove a weak result in

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Proposition 3.1. From this result we can see that it is natural to introduce the new weight types. PROPOSITION 3.1. Let f 0 , f 1 , . . . , f n g C w x 0 , x 1 , . . . , x n x be weighted homogeneous polynomials. Suppose that R s C w x 0 , x 1 , . . . , x n xrŽ f 0 , f 1 , . . . , f n . is an Artin algebra and let D s pi Ž x iq1 , . . . , x n . ⭸r⭸ x i be a negati¨ e weight deri¨ ation on R. Then D is a zero-deri¨ ation on R. Proof. By local duality wA-G-V, Theorem 2 in 5.11x we know dim R M s 1 and R ) M s 0, where M s deg f 0 q deg f 1 q ⭈⭈⭈ qdeg f n y ␣ 0 y ␣ 1 y ⭈⭈⭈ y␣ n . Moreover R i = R Myi ª R M s C is a non-degenerate bilinear form. We note that for every weighted homogeneous polynomial g of degree M y ␣ i y wtŽ D ., there exists a weighted homogeneous polynomial h of degree M y wtŽ D . such that ⭸ hr⭸ x i s g. Thus h g Ž f 0 , f 1 , . . . , f n . from the fact R ) M s 0 and pi g s Dh g Ž f 0 , f 1 , . . . , f n . since D is a derivation on R. Therefore pi s 0 in R␣ iqwt D since g is arbitrary in R My ␣ iywt D . The conclusion is proved. Q.E.D. COROLLARY 3.1 wTh1, Th2x. The Halperin conjecture is true for the two-¨ ariable case. Proof. The statement follows immediately from the analysis Ž2.2. of negative weight derivations and the above Proposition 3.1. THEOREM 3.1. The Halperin conjecture is true for the three-¨ ariable case. Proof. Let D s p 0 Ž x 1 , x 2 . ⭸r⭸ x 0 q p1 Ž x 2 . ⭸r⭸ x 1 q p 2 ⭸r⭸ x 2

Ž 3.1.

be a negative weight derivation on the weighted polynomial ring P which preserves the ideal Ž f 0 , f 1 , f 2 .. We want to prove that it is a zero-derivation on R s PrŽ f 0 , f 1 , f 2 .. There are three possibilities here Ž1. Ž2. Ž3.

p 2 / 0; p 2 s 0, p 0 and p1 are non-zero polynomials; p 2 s 0, one of p 0 and p1 is the zero-polynomial.

In case Ž1., we can find a coordinate change which preserves the original weight type as in wCh1x. In this new coordinate system D is of the form c⭸r⭸ x 2 Žwhere c is a non-zero constant.. Hence we get the conclusion in cases Ž1. and Ž3. from Proposition 3.1. The only remaining case is Ž2.. In case Ž2., D is a negative weight derivation as in Example 2.2, Ž2.5.. If p 0 Ž x 1 , x 2 . can be divided by x 2m , i.e., u G m, D s x 2m Ž pX0 ⭸r⭸ x 0 q c⭸r⭸ x 1 . Žwhere c is a non-zero constant.. Hence we can find a coordinate change as in wCh1x to put D into the form x 2m⭸r⭸ x 1. The conclusion follows from

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Proposition 3.1 immediately. If the weight ␣ 2 ) 1 in the case Ž2., we can introduce a new variable xX2 with weight 1 as D⬘ s p 0 Ž x 1 , Ž xX2 .

␣2

. ⭸r⭸ x 0 q c Ž xX2 . ␣ f 0X Ž x 0 , x 1 , xX2 . s f 0 Ž x 0 , x 1 , Ž xX2 . . ␣ f 1X Ž x 0 , x 1 , xX2 . s f 1 Ž x 0 , x 1 , Ž xX2 . . ␣ f 2X Ž x 0 , x 1 , xX2 . s f 2 Ž x 0 , x 1 , Ž xX2 . . .

m␣2

⭸r⭸ x 1

2

2

Ž 3.2.

2

Then D⬘ is a negative weight derivation on R s C w x 0 , x 1 , xX2 xrŽ f 0X , f 1X , f 2X .. We note that R is an Artin algebra from Proposition 2 in p. 198 of wA-G-Vx. Hence we can assume ␣ 2 s 1 without loss of generality. From now on we only consider the situation in Example 2.2 Žcontinued.. Let deg f i s d i and M s d 0 q d1 q d 2 y ␣ 0 y ␣ 1 y 1. We have the following conclusions from local duality Žsee wA-G-Vx. Ži. R s [k G 0 R k and R k s 0 for k ) M; Žii. dim R M s 1 and R M is generated over C by detŽ ⭸ f ir ⭸ x j . 0 F i, j F 3 ; Žiii. R i = R Myi ª R M , defined by the multiplication in the Artin algebra R, is a non-degenerate bilinear form. In the following part of the proof we want to prove A M ; DŽ A Mywt D . q Ž f 0 , f 1 , f 2 ., i.e., R M s 0 Žnote R Mywt D s 0 for wt D - 0., which is a contradiction. Let x 1t x 2u be the monomial in the expansion of p 0 Ž x 1 , x 2 . with the smallest possible exponent u of x 2 as in Example 2.2 Žcontinued.. It is clear that A u is generated over C by x 2u since u - m - ␣ 1 F ␣ 0 . Therefore we can reduce to the case that u s 0 by shifting yu in the exponents of the variable x 2 and local duality Žiii. Žnote x 2u / 0 in R u , otherwise, D s p 0 Ž x 1 , x 2 . ⭸r⭸ x 0 in R and the conclusion follows directly from Proposition 3.1.. We assume X u s 0X in the following part of the proof. In this X case, if x 0i 0 x 1i1 x 2i 2 and x 0i 0 x 1i1 x 2i 2 are two elements in Ckl we have Ž i 0 y iX0 . l 0 q Ž i 2 y iX2 . s l y l s 0 and i 2 y iX2 can be divided by m by l 0 s ym. l Ž ... s 1 from Proposition Thus we have dimŽ Ckl rCkl l Ž A) k q D A kywt D 2.1Žii.. Let h s max l: AlM rAlM l Ž DŽ A Mywt D . q Ž f 0 , f 1 , f 2 .. / 04 . By Proposition 2.1Ži. we know h - ␣ 1 q wt D. Here we want to prove CMh ; CMh l h ŽŽ A) Ž .. q Ž f 0 , f 1 , f 2 .., which is a contradiction and thus M q D A Mywt D l A M ; DŽ A Mywt D . q Ž f 0 , f 1 , f 2 .. We note that there is a term of the form x ¨0 in the expansion of one of f 0 , f 0 , f 2 , otherwise, we have Ž f 0 , f 1 , f 2 . ; Ž x 1 , x 2 ., which is a contradiction

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to the fact that PrŽ f 0 , f 1 , f 2 . is an Artin algebra. Therefore we can assume without loss of generality that f 0 contains a term x ¨0 in its expansion. Case Ži..

h F Ž l 0 r␣ 0 . d 0 q M y d 0 .

Let M s p␣ 0 q p⬘ and p⬘ s p⬙ ␣ 1 q p⵮ as in Proposition 2.2. We have pl 0 q p⵮ F h by Proposition 2.2. Note that d 0 s ¨ ␣ 0 . We have M y d 0 s Ž p y ¨ . ␣ 0 q p⬘ with p⬘ - ␣ 0 and p⬘ s p⬙ ␣ 1 q p⵮ with p⵮ - ␣ 1. Therefore Ž p y ¨ . l 0 q p⵮ F h y ¨ l 0 F M y d 0 from the condition of case Ži.. hy ¨ l 0 Whenever h y ¨ l 0 is positive or negative we have dim CMyd G 1 from 0 i 0 i1 i 2 hy ¨ l 0 Proposition 2.2. Thus we can find a monomial x 0 x 1 x 2 in CMyd . 0 Let f 0 Ž x 0 , x 1 , x 2 . s x ¨0 q f 0j1 q f 0j 2 q ⭈⭈⭈ be the expansion of f 0 according to A d 0 s [l G ¨ l 0 C dl 0 . It is clear that dim A¨dl00 s 1 and A¨dl00 is spanned hy ¨ l 0 by the monomial x ¨0 . Multiplying the x 0i 0 x 1i1 x 2i 2 in CMyd as above on both 0 sides we get x 0i 0 x 1i1 x 2i 2 f 0 s x ¨0 qi 0 x 1i1 x 2i 2 q x 0i 0 x 1i1 x 2i 2 f 0j1 q ⭈⭈⭈ .

Ž 3.3.

j iqh y¨ l 0 Note that the terms after the first term in Ž3.3. are in A M . On the other hand ji y ¨ l 0 G ␣ 1 from Proposition 2.2Ži.. Thus ji q h y ¨ l 0 G h q ␣ 1. We have all those terms are in DŽ A Mywt D . q Ž f 0 , f 1 , f 2 . by the definition of h. Therefore we have

x 0i 0 x 1i1 x 2i 2 f 0 s x ¨0 qi 0 x 1i1 x 2i 2 h in CMh r Ž CMh l A) M q D Ž A Mywt D . q Ž f 0 , f 1 , f 2 .

..

Ž 3.4.

On the other hand x ¨0 qi 0 x 1i1 x 2i 2 s ex 0r x 1q x 2w for a non-zero constant e and h Ž ... as in the base monomial x 0r x 1q x 2w of CMh rŽ CMh l Ž A) M q D A Mywt D Proposition 2.1. Here we should note that the dimension of the CMh rCMh l h Ž A) Ž .. is 1 as remarked as above. Thus by Ž3.4. we have M q D A Mywt D r q w h Ž .. x 0 x 1 x 2 g Ž f 0 , f 1 , f 2 . in CMh rCMh l Ž A) M q D A Mywt D . We get the conclusion. Case Žii..

h ) Ž l 0r␣ 0 . d 0 q M y d 0 .

Here we first assume that d 0 G 3 ␣ 0 and d1 G Ž ␣ 0 q ␣ 1 q ␣ 2 . q Ž ␣ 1 y 1. s ␣ 0 q 2 ␣ 1. The low-degree case will be treated later. In case Žii. we cannot use f 0 to get the conclusion as in case Ži.. Let us assume d 2 s deg f 2 s min d 0 , d1 , d 2 4 without loss of generality. Consider the expansion of f 2 according to A d 2 s [l G Ž l 0 r ␣ 0 . d 2 C dl 2 f 2 Ž x 0 , x 1 , x 2 . s f 2p q f 2p 1 q ⭈⭈⭈ ,

Ž 3.5.

where the terms after f 2p have Q-degrees G Q y deg f 2p q ␣ 1 Žsee Proposition 2.2Ži... This f 2p will be used to lead to a contradiction.

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Suppose that there are w q 1Ž w G 0. monomials in the expansion in the expansion of f 2p. By Proposition 2.2 and its proof we know that all monomials x 0i 0 x 1i1 x 2i 2 in C dp2 have their i 0 ’s distinct. Let g s max i 0 : x 0i 0 x 1i1 x 2i 2 in the expansion of f 2p 4 . Hence we have g G w and for the monomial x 0g x 1i1 x 2i 2 in the expansion of f 2p , we have i 2 G ywl 0 . Thus d 2 G g ␣ 0 y wl 0 and d 2 y w Ž t q 1. ␣ 1 G p G gl 0 y wl 0 . By Proposition 2.1Ži. and the definition of h we have h y p - ␣ 1 q wt D y gl 0 q wl 0 F ␣ 1 q wt D q Ž g y w . ␣ 0 F d 2 q ␣ 1 q wt D q w Ž l 0 y ␣ 0 . s d 2 q ␣ 1 q wt D y w Ž t q 1. ␣ 1. Since d 0 G 3 ␣ 0 is assumed we have M y d 2 G d 2 q ␣ 1 q wt D and h y p - M y d 2 y w Ž t q 1. ␣ 1. On the other hand p F d 2 y w Ž t q 1. ␣ 1 as argued above. We have h y p G ¨ l 0 q M y d 0 y d 2 q w Ž t q 1. ␣ 1 from the assumption of h in case Žii.. Therefore h y p G ¨ l 0 q d1 y Ž ␣ 0 q ␣ 1 q ␣ 2 . q w Ž t q 1. ␣ 1 G ¨ l 0 q ␣ 1 y 1 q w Ž t q 1. ␣ 1 by d i G ␣ 0 q 2 ␣ 1 is assumed. Let M y d 2 s wŽ M y d 2 .r␣ 0 x ␣ 0 q p⬘ with p⬘ - ␣ 0 and p⬘ s p⬙ ␣ 1 q p⵮ with p⵮ - ␣ 1 as in Proposition 2.2. We have wŽ M y d 2 .r␣ 0 x G ¨ since M y d 2 G d 0 from the condition d1 G ␣ 0 q 2 ␣ 1. Therefore h y p G wŽ M y d 2 .r␣ 0 x l 0 q ␣ 1 y 1 q w Ž t q 1. ␣ 1. From the above argument whenever h y p is positive or negative, it satisfies the condition in Proposition 2.2Žiii.. Hence we have dim hy p C My argument as in case Ži. gives us d 2 G0 w0 q0 1. A similar w w w h that x 0i 0 x 1i 1 x 2i 2 f 2p , . . . , x 0i 0 x 1i 1 x 2i 2 f 2p g C Mh l Ž A ) q D Ž A M y wt D . q M 0 0 0 w w w i i i i i i Ž f 0 , f 1 , f 2 .., where x 00 x 11 x 22 , . . . , x 00 x 11 x 22 are w q 1 distinct monomials hy p in CMy d 2 whose existences have been proved as above. As in Proposition 2.1 and the remark before case Ži., let x 0r ⬘ x 1r ⬙ x 2r ⬙ be the base monomial of h Ž .. the space CMh rCMh l Ž A) M q D A Mywt D . We have that x 0i 0 x 1i1 x 2i 2 f 2p s e0 x 0r ⬘ x 1r ⬙ x 2r ⵮ ⭈⭈⭈ i 0w i 1w i 2w p x 0 x 1 x 2 f 2 s e w x 0r ⬘ x 1r ⬙ x 2r ⵮ , 0

0

0

Ž 3.6.

where e0 , . . . , e w are constants. By Proposition 2.1Žii. we have e0 e1 ⭈⭈⭈ ew

a0 a1 sE , ⭈⭈⭈ aw

0 0

Ž 3.7.

where E is a non-singular matrix and Ž a0 , a1 , . . . , a w . t is a non-zero vector only depending on f 2 and the derivation D. Finally we get the conclusion h Ž Ž . Ž .. x 0r ⬘ x 1r ⬙ x 2r ⵮ g CMh l Ž A) M q D A Mywt D q f 0 , f 1 , f 2 . This is what we want to prove.

NEGATIVE WEIGHT DERIVATIONS

11

In the following part we treat the low-degree case d 0 - 3 ␣ 0 or d1 - ␣ 0 q 2 ␣ 1. The main point here is that we can analyze the Q-degrees of monomials in the expansion of f 0 and f 1 if their ordinary degrees are small. In the case d1 - ␣ 0 q 2 ␣ 1 , let us assume that the smallest Q-degree component in f 1 is of the form x 0 x 1 x 2q with an integer 0 F q F ␣ 1. hy p As in the previous arguments what we want to prove here is CMy d1 / 0 q where p s Q y degŽ x 0 x 1 x 2 . s q q l 0 . Note that Df 1 cannot be zero here Žotherwise Ž c0 x 1t ⭸r⭸ x 0 q c1 x 2m⭸r⭸ x 1 .Ž x 0 x 1 x 2q ., which is the smallest Q-degree component of Df 1 which has to be zero.. Moreover the smallest Q-degree component of f 2 has to be the factor of Ž c 0 x 1t ⭸r⭸ x 0 q 0 . Žsince Df s gf c1 x 2t ⭸r⭸ x 1 .Ž x 0 x 1 x 2q . s x 2q Ž cX0 x 1tq1 q cX1 x 0 xyl 1 1 2 from the condition DŽ f 0 , f 1 , f 2 . ; Ž f 0 , f 1 , f 2 . and the assumption d 0 G d1 G d 2 .. Let us suppose that the smallest Q-degree component of f 2 is cX0 x 1tq1 q 0 cX1 x 0 xyl 2 . Hence d 2 s ␣ 0 y l 0 ␣ 2 s ␣ 0 y l 0 . Therefore M y d1 s d 0 q d 2 y Ž ␣ 0 q ␣ 1 q 1. s Ž ¨ y 1. ␣ 0 q d 2 y ␣ 1 y 1 Žhere ¨ s d 0r␣ 0 by the fact x ¨0 is in the expansion of f 0 .. What we want to prove here is that x ¨0 x 2qy 1 Žnote that it has degree M y d1 . has its Q-degree F h y p. We have q y 1 q ¨ l 0 F ¨ l 0 q M y d 0 y q y l 0 F h y p by the assumption of h in case Žii.. Thus we have h y p s Q y degŽ x ¨0 x 2qy 1 . q r ␣ 1 with r F ¨ and hy p x ¨0 yr x 1r t x 2qy 1yr wt D g CMyd . We get the conclusion. The other possibilities 1 can be proved similarly. Q.E.D. In the arbitrary n q 1 variable case we cannot have a clear description of Ckl Ži.e., Propositions 2.1 and 2.2.. Thus only a partial result which gives a positive answer to the Halperin conjecture for ‘‘large’’ graded Artin algebras can be obtained wCh2x.

ACKNOWLEDGMENTS The author thanks Professor S. Halperin, J. M. Wahl, and Stephen S.-T. Yau for their help during this work.

REFERENCES wA-G-Vx

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, ‘‘Singularities of Differential Maps,’’ Vol. 1, Birkhauser, Basel, 1985. ¨ wBox A. Borel, Sur la cohomologie des espaces fibres principaux des espaces homogenes de groups de Lie compacts, Ann. Math. 57 Ž1953., 115᎐207. wCh-Xu-Yaux H. Chen, Y.-J. Xu, and S. Yau, Nonexistence of negative weight derivations of moduli algebras of weighted homogeneous singularities, J. Algebra 172 Ž1995., 243᎐254. wCh1x H. Chen, On negative weight derivations of moduli algebras of weighted homogeneous hypersurface singularities, Math. Ann. 303 Ž1995., 95᎐107.

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HAO CHEN

H. Chen, Nonexistence of negative weight derivations on large quasi-homogeneous singularities, preprint. S. Halperin, private communication, July 5, 1994. J.-M. Kantor, Derivation sur les singularities quasi-homogenes: Cas des courbes, C. R. Acad. Sci. Paris 287 Ž1978., 117᎐119. J.-M. Kantor, Derivation sur les singularities quasi-homogenes: Cas des hypersurface, C. R. Acad. Sci. Paris 288 Ž1979., 33᎐34. W. Meier, Rational universal fibration and flag manifolds, Math. Ann. 258 Ž1982., 329᎐340. S. Mori and H. Sumihiro, On Hartshorne’s conjecture, J. Math. Kyoto Uni¨ . 18 Ž1978., 523᎐533. J. C. Thoams, Sur les fibration de Serre pures, C. R. Acad. Sci. Paris Ser. A 290 Ž1980., 811᎐813. J. C. Thoams, Rational homotopy of Serre fibration, Ann. Inst. Fourier Ž Grenoble. 31 Ž1981., 71᎐90. J. M. Wahl, Derivations of negative weight and non-smoothability of certain singularities, Math. Ann. 358 Ž1982., 383᎐398. J. M. Wahl, Derivations, automorphisms and deformations of quasi-homogeneous singularities, in Proc. Sympos. Pure Math., Vol. 40, Part 2, pp. 613᎐624, Amer. Math. Soc., Providence, 1983. J. M. Wahl, A cohomological characterization of P n , In¨ ent. Math. 72 Ž1983., 315᎐322. S. Yau, Solvability of Lie algebras arising from isolated singularities and non-isolatedness of singularities defined by slŽ2, C . invariant polynomials, Amer. J. Math. 113 Ž1991., 773᎐778.