ON MINIMAL HORSE-SHOE LEMMA

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1 ON MINIMAL HORSE-SHOE LEMMA GUO-JUN WANG FANG LI Abstract. The main aim of this paper is to give some conditions under which the Minimal Horse-shoe Lemma holds and apply it to investigate the category K p (A) of p-koszul modules. Let A = i A i be a nice algebra. It is shown that the graded A-projective cover in the category BGr(A) always exists. We give an explicit construction of the graded A-projective cover of the graded A-module M = i M i in BGr(A) by using the minimal generating spaces of M. Moreover, we also investigate some properties of the graded A-projective cover similar to the nongraded case. Mainly, some sufficient conditions are given under which the minimal Horse-shoe Lemma holds. Finally, as an application of the minimal Horse-shoe Lemma, we show that the category K p (A) of p-koszul modules is closed under direct sums, direct summands, extensions and cokernels. 1. Introduction Let R be an arbitrary ring with identity 1 R, it is well known that, for an R-module M, the projective cover of M does not necessarily exist. Let R be above, then every finitely generated R-module M has a projective cover if and only if R is semiperfect, which is also well known. In this paper, we prove that if A = i A i is a nice algebra, 2 Mathematics Subject Classification. 18G5; 16S37; 16W5. Key words and phrases. nice algebra, graded A-projective cover, minimal graded projective resolution. This work is supported by the Program for New Century Excellent Talents in University (No.4-522), the National Natural Science Foundation of China (No ) and the Natural Science Foundation of Zhejiang Province of China (No.1228). 1

2 2 GUO-JUN WANG FANG LI the graded A-module M = i M i BGr(A) must have a graded A- projective cover, which has some similar properties as the nongraded case. We all know that the Horse-shoe Lemma plays an important role in Homological algebra. But it is only true for projective resolution. Obviously, a minimal projective resolution is better than a common projective resolution when computing the homology group. Thus there is a natural question: Does the Horse-shoe Lemma hold for the minimal case? Since for an arbitrary category C, the projective cover in C does not always exist. Thus, in order to answer the above question, we must find a category in which the projective cover always exists. In this paper, we define the category BGr(A), where A = i A i is a nice algebra, whose objects are the graded A-modules which are bounded below. That is, for each M BGr(A), M has the form: M = i j M i and M i = for all i < j. Since M[j] = i M i, where [ ] is the shift functor. Thus for each M BGr(A), we can always assume that M = i M i. Our main answer is that for an exact sequence K M N in NBGr(A) with K JM = JK, the Minimal Horse-shoe Lemma holds (see Theorem 3.9). Finally, as an application of the minimal Horse-Shoe Lemma, we investigate the category of p-koszul modules. The concept of Koszul objects was introduced by Priddy in [8]. Recently many people are interested in the Koszul objects([4], [5], [6], [7], etc.). Koszul objects have perfect homological properties. We study the category of p-koszul modules, denoted by K p (A), and show that K p (A) is closed under direct sums, direct summands, extensions and cokernels. The paper is organized as follows. In section 2, we give some basic definitions and notations. In section 3, firstly we show that the graded A-projective covers in the category BGr(A) always exist. Moreover, we give an explicit construction of the graded projective cover for the graded A-module M = i M i by using the minimal generating spaces {S, S 1,...} of M. Some sufficient conditions under which the minimal Horse-shoe Lemma holds are giving as well, which are the main results. In the last section, as an application of the minimal Horse-shoe Lemma,

3 ON MINIMAL HORSE-SHOE LEMMA 3 we investigate the category K p (A) of p-koszul modules. We show that K p (A) is closed under direct sums, direct summands, extensions and cokernels. 2. Preliminaries Firstly let us give some easy definitions which will be used later. Definition 2.1. Let A = i A i be a graded algebra, if (1) A is semisimple (2)A i A j = A i+j. Then we call that A is a nice algebra. Throughout this paper, A = i A i is a nice algebra over a field K and J = i 1 A i is the graded Jacobson radical of A. Let Gr(A) denote the category of graded A-modules and degree zero morphisms. Let BGr(A) denote the full subcategory of Gr(A) consisting of graded A- modules which are bounded below and degree zero morphisms. gr(a) is the full subcategory of Gr(A) consisting of finitely generated graded modules. Obviously, gr(a) BGr(A). We also denote the pure module category by gr s (A) (that is, if M gr s (A), then M is generated in degree s), which is a full subcategory of gr(a). Definition 2.2. Let M BGr(A) and P is a graded A-projective module, if there exists a degree zero epimorphism: P f M with kerf JP, then we will call that P is the graded A-projective cover of M. It is easy to see that the definition of graded projective cover is similar to that in the nongraded case. Moreover, it has some properties which are similar to that in the nongraded case (for example, see Proposition 3.3). Definition 2.3. Let A = i A i be a graded algebra. If for every graded A-module M = i M i, M has a graded projective cover. Then A is called a graded semiperfect algebra. Definition 2.4. Let f n f n 1 f 1 f P n Pn 1 P1 P M be a minimal graded projective resolution of M in BGr(A) over a nice algebra A. If for each i, we have JKerf i = Kerf i J 2 P i, then we say M to be a nice module.

4 4 GUO-JUN WANG FANG LI We denote the category of nice modules by NBGr(A). If we replace the nice algebra A by Koszul algebra, then the above is the definition of Quasi-Koszul module. Definition 2.5. [5] Let A be a nice algebra and M gr(a). Suppose that Q is a minimal graded projective resolution of M: f n f n 1 f 1 f Q : Q n Qn 1 Q1 Q M. If for all n, Q n is generated in degree δ(n), where δ(n) = { np 2 + s if n is even, (n 1)p s if n is odd, 2 and p 2, s is an integer. Then we call that M is a p-koszul module. It is easy to see that the definition of p-koszul modules is slightly different from that in [5](where it is called a d-koszul module). We denote the category of p-koszul modules by K p (A), which is a full subcategory of gr(a). Let M be a p-koszul module. From the definition of p-koszul module above, it is evidently that M is generated in degree δ() = s. We all know that the concept of p-koszul algebra is introduced by Roland Berger as a generalization of Koszul algebras [3]. E.L.Green and coauthors generalized the kind of algebras to the non-connected case [5]. We denote the category of p-koszul module by K p (A), which is a full subcategory of gr(a). Obviously, Koszul algebras and p-koszul algebras are nice algebras, and quasi-koszul modules are nice modules. In particular, when p = 2, p-koszul algebras are just the Koszul algebras and p-koszul modules are just the Koszul modules. 3. Main results Now we will show that the graded A-projective covers in the category BGr(A) always exist and we give an explicit construction of the graded A-projective cover. Theorem 3.1. let M = i M i BGr(A). Then M has a graded A-projective cover.

5 ON MINIMAL HORSE-SHOE LEMMA 5 Proof. Let S, S 1,, S n, be the minimal generating spaces of M, and degs i = i. Since M = i M i is a graded A-module, it has the following form: M = S (S 1 + A 1 S ) (S 2 + A 2 S + A 1 S 1 ). Set S M = S S 1 S 2, which can be considered as an A -module. Let P = A A S M. We will show that P is the graded projective cover of M = i M i. Firstly, we claim that P = A A S M is a graded A-projective module. It is obvious that P is graded since P i = A A S i. Let K N be exact in Gr(A). Since Hom A (P, K) = Hom A (A A S M, K) = Hom A (S M, Hom A (A, K)) = Hom A (S M, K). A is semisimple since A is a nice algebra. Thus Hom A (S M, ) is an exact functor. It follows that we have the following commutative diagram: Hom A (S M, K) Hom A (S M, N) id id Hom A (P, K) Hom A (P, N) where id is an isomorphism. Therefore Hom A (P, K) Hom A (P, N) is exact and hence P is a graded A-projective module. Now we define a morphism f: f : P = A A S M M via f : a i s i a i s i. Obviously f is a A-map and f is an epimorphism. Finally, we claim that Kerf JP = J A S M since A i A j = A i+j for all i, j, where J = i 1 A i is the graded Jacobson radical of A and it is easy to check. Corollary 3.2. If A = i A i is a nice algebra, then A is a graded semiperfect algebra. Proof. It is immediate from Theorem 3.1. Remark 3.1. The above theorem shows that if A = i A i is a nice algebra, then the graded A-projective covers in BGr(A) always exist.

6 6 GUO-JUN WANG FANG LI Note that indeed in Theorem 3.1 we have given the construction of the graded projective cover of M BGr(A). Next we will investigate some basic properties of graded projective covers. Proposition 3.3. Let M = i M i BGr(A) and P f M be a graded projective cover, and Q is another graded projective module in BGr(A) with an epimorphism: Q g M satisfying Kerg JQ. Then P = Q. Proof. Consider the following diagram: Q P f g M Since Q is a graded projective module, there exists α : Q P, such that fα = g. Since g is an superfluous epimorphism, α is an epimorphism. That is Q α P is exact. Since P is projective, there is a monomorphism β : P Q, such that αβ = 1. Thus we get f = f 1 = fαβ = gβ. Similarly β is an epimorphism since f is an epimorphism. Thus β is an isomorphism. Therefore P = Q. Proposition 3.3 shows the uniqueness of the graded projective cover, which is similiar to that of the nongraded case. Proposition 3.4. Let M = i M i BGr(A) and P f M be the graded projective cover of M. Then M is generated in degree s if and only if P is generated in degree s. Proof. If From the proof of Theorem 3.1, we know that A A S M is one of the graded projective covers of M and by Proposition 3.3, the projective covers of M are unique up to isomorphism. Hence P = A A S M = A A S A A S 1 and let P i = A A S i. If M is generated in degree s, then M = A M s = A S s. In this case P = A A S s, evidently P is generated in degree s. only if By Theorem 3.1 and Proposition 3.3, P = A A S M and P i = A A S i. By hypothesis, P is generated in degree s, i.e.,

7 ON MINIMAL HORSE-SHOE LEMMA 7 P j = A j s P s for all j s. So we have A A S j = A j s A A S s. That is, M is generated in degree s. Corollary 3.5. Let M = i M i BGr(A) and P be the graded projective cover of M. Then M is generated in degrees j 1,j 2,, j n, if and only if P is generated in degrees j 1, j 2,, j n. Proof. By the same way as in proposition 3.4. Lemma 3.6. Let K f M g N be an exact sequence in Gr(A). Then there is an exact sequence K JM JM JN in Gr(A). Proof. Let f = f K JM and g = g JM. Since f is an embedding, f is a monomorphism. Let x = a i n i JN with a i J and n i N. Since g is an epimorphism, there exists m i M such that g(m i ) = n i. Set y = a i m i and we have g(y) = g( a i m i ) = g(a i m i ) = ai g(m i ) = a i n i, thus g is an epimorphism. Let x = a i m i kerg with a i J and m i M. So g(x) = g( a i m i ) =, since g is an epimorphism, there exists z K JM such that f(z) = x. Therefore Ker(g) Imf. Conversely, let x = f(y) Imf with y K JM. Since g(x) = gf(y) =, x Kerg, hence Imf Kerg. It follows that we have the exact sequence: K JM JM JN. Now we will give some sufficient conditions for the minimal Horseshoe Lemma holding: Lemma 3.7. Let K M N be an exact sequence in Gr(A). Then the following statements are equivalent: (1) K JM = JK, (2) A/J A K A/J A M is a monomorphism.

8 8 GUO-JUN WANG FANG LI Proof. Suppose K JM = JK. By Lemma 3.6, we have the following commutative diagram with exact rows and columns: JK K K/JK JM M M/JM JN N N/JN Obviously A/J A K = K/JK and A/J A M = M/JM. Thus A/J A K A/J A M is a monomorphism since K/JK M/JM is exact. Therefore assertion (1) implies assertion (2). Conversely, suppose A/J A K A/J A M is a monomorphism. It is easy to see that K/JK M/JM is a monomorphism. Consider the following commutative diagram: K/JK M/JM N/JN f K/K JM M/JM N/JN By Five Lemma, we get JK = K JM. Therefore we complete the proof of Lemma 3.7. Lemma 3.8. Let Ω P f M be an exact sequence in Gr(A) with M NBGr(A) and P being the graded projective cover of M. Then A/J A Ω A/J A JP is a monomorphism. Proof. We first claim that we have the exact sequence: Ω JP f JM, where f = f JP, which can be proved easily since Ω JP. Since M is a nice module, JΩ = Ω J 2 P = Ω J(JP). By Lemma 3.7, we have that A/J A Ω A/J A JP is a monomorphism. Now we can state and prove the main theorem in this paper. Theorem 3.9. Let K M N be an exact sequence in NBGr(A) with K JM = JK. Then the minimal Horse-shoe Lemma id id

9 ON MINIMAL HORSE-SHOE LEMMA 9 holds. That is, there is a commutative diagram with exact rows and columns:... P 1 K P 1 M P 1 N P K P M P N where and K M N PK 1 P K K, PM 1 P M M, PN 1 PN N are the minimal graded projective resolution of K, M and N respectively; and PM i = P K i P N i, for i. Proof. By Lemma 3.6, we have the following exact sequence: K JM JM JN. Since JK = K JM. We have the exact sequence: JK JM JN. Hence we have the following commutative diagram with exact rows and columns: JK JM JN K M N K/JK M/JM N/JN We get the exact sequences: S K S M S N

10 1 GUO-JUN WANG FANG LI in Gr(A ) since K/JK = S K = S S 1, M/JM = S M and N/JN = S N as A -modules. Since A is semisimple and the functor A A is exact. Thus we have the following exact sequences: A A S K A A S M A A S N. We can assume that P K = A A S K, P M = A A S M and P N = A A S N since the uniqueness of the graded projective cover. We get the following commutative diagram: Ω(K) Ω(M) Ω(N) P K P M P N K M N The exact sequence P K P M P N splits since P N is graded projective. Thus P M = P K P N. From the proof of Lemma 3.8, obviously, we get the following commutative diagram: Ω(K) Ω(M) Ω(N) JP K JP M JP N JK JM JN

11 ON MINIMAL HORSE-SHOE LEMMA 11 Apply the functor A/J A to the above diagram: A/J A Ω(K) A/J A JP K A/J A JK f A/J A Ω(M) A/J A JP M A/J A JM A/J A Ω(N) A/J A JP N A/J A JN By Lemma 3.8, A/J A Ω(K) A/J A PK is a monomorphism. From the commutativity of the left upper square, we have f is a monomorphism. By Lemma 3.7, we have JΩ(K) = Ω(K) JΩ(M). Next we claim: If M NBGr(A) and P f M is the graded projective cover of M. Then Ω(M) = Kerf NBGr(A). In fact, we have a minimal graded projective resolution of M f n f n 1 f 1 f P n Pn 1 P1 P M such that JKerf i = Kerf i J 2 P i for all i since M NBGr(A). Hence we get a minimal graded projective resolution of Kerf : f n f n 1 f 1 P n Pn 1 P1 Kerf such that JKerf i = Kerf i J 2 P i for all i 1. Therefore Kerf NBGr(A). By the above claim, we have that Ω(K), Ω(M) and Ω(N) are contained in NBGr(A). Replace K, M and N by Ω(K), Ω(M) and Ω(N). By an easy induction, we have the minimal Horse-shoe Lemma holds. Proposition 3.1. Let K M N be an exact sequence in NBGr(A) with K, M and N are generated in the same degree. Then the minimal Horse-shoe Lemma holds. Proof. Since K,M and N are generated in the same degree, we can assume K = A K s, M = A M s and N = A N s. By Theorem 3.9, we only need to show K JM = JK. Since JK K and JK JM, JK K JM. Let x K JM be a homogeneous element, then degx s + 1. If x is not contained in JK, then x is a generator of K,

12 12 GUO-JUN WANG FANG LI but K is generated in degree s, which is a contradiction. Thus x JK, and JK K JM. So JK = K JM. By Theorem 3.9, we finish the proof of the proposition. Proposition Let K M N be a split exact sequence in BGr(A). Then the minimal Horse-shoe Lemma holds. Proof. Since K M N splits, we have M = K N. Let P K and P N are the graded projective covers of K and N respectively. That is, there exist two epimorphisms P K f K and P N g N such that kerf JP K and kerg JP N. Set P = P k P N and define ϕ as follows: ϕ : P M via ϕ(x y) = f(x) g(y) for x P K and y P N. Obviously, ϕ is an epimorphism. Let ϕ(x+y) =, clearly f(x) = = g(y). Thus x Kerf and y Kerg, Kerϕ = Kerf Kerg. Since JKerϕ = J(Kerf Kerg) = JKerf JKerg JP K JP N = J(P K P N ) = JP. Therefore P is the graded projective cover of M. Thus we have the split exact sequence: Ω(K) Ω(M) Ω(N). Replace K, M and N by Ω(K), Ω(M) and Ω(N), repeat the above argument and by an easy induction, we can show that the minimal Horse-shoe Lemma holds. Lemma Let K f M g N be an exact sequence in gr(a) and A be a graded algebra. Then

13 ON MINIMAL HORSE-SHOE LEMMA 13 (1) If K and N are generated in degree s, then M is generated in degree s, (2) If M is generated in degree s, then N is generated in degree s and K is generated in degree t with t s, (3) If K and N are generated in degree s and t, respectively, then M is generated in degrees s and t. Proof. Consider the exact sequence: K n f Mn g Nn with n > s, where N = i s N i, K = i s K i and M = i s M i and f = f Kn, g = g Mn. Let x M n be a homogeneous element, then g(x) N n, since N is generated in degree s, hence there exists n = a i n i N n with a i A n s and n i N s, such that g(x) = a i n i. Since we have the exact sequence, K s f Ms g Ns and g is epic, so there exists m i M s, such that g(m i ) = n i. Let y = ai m i be a homogeneous element of M n, since g(y) = g( a i m i ) = ai g(m i ) = a i n i = g(x), so we have y x Kerg = Imf and thus x y = f( b i k i ) with b i A n s and k i K s, therefore x = ai m i + b i f(k i ) where m i, f(k i ) M s, it follows easily that M is generated in degree s, thus assertion (1) holds. Suppose M is generated in degree s, for each integer n > s, consider the exact sequence, K n f Mn g Nn. Let x N n be a homogeneous element and since g is epic, there exists y M n such that g(y) = x and y = a i m i with m i M s and a i A n s. Hence x = g(y) = g( a i m i ) = a i g(m i ) and g(m i ) N s and it follows that N is generated in degree s. And the last conclusion of assertion (2) is trivial and assertion (3) can be proved similarly. Proposition Let K M N be an exact sequence in gr(a) such that K and N are generated in the same degree, where A is a nice algebra. If K and N are p-koszul modules, then the minimal Horse-shoe Lemma holds.

14 14 GUO-JUN WANG FANG LI Proof. Since K and N are generated in the same degree, we can assume that K = A K s and N = A N s. By Lemma 3.12, we have that M = A M s. Thus we have the following commutative diagram with exact rows and columns Ω 1 (K) Ω 1 (M) Ω 1 (N) PK PM PN K M N where PK, P M and P N are the graded projective covers of K, M and N respectively; Ω 1 (K), Ω 1 (M) and Ω 1 (N) are the first syzygies of K, M and N respectively. Since the exact sequence PK P M P N splits. We have that P M = P K P N. It is obviously that P K and P N are generated in degree δ() since K and N are p-koszul modules. By Lemma 3.12, we have P M is generated in degree δ(). Since P 1 K and P 1 N are the graded projective covers of Ω1 (K) and Ω 1 (N) respectively. We have that Ω 1 (K) and Ω 1 (N) are generated in degree δ(1). By Lemma 3.12, Ω 1 (M) is generated in degree δ(1). Replace K, M and N by Ω 1 (K), Ω 1 (M) and Ω 1 (N), by induction, we can complete the proof. Remark 3.2. Though we have found some sufficient conditions for the minimal Horse-shoe Lemma holding in this note, we do not know the necessary conditions for the minimal Horse-shoe Lemma holding, which is more difficult and interesting. 4. The applications of the minimal Horse-shoe lemma In this section, we study the category K p (A) of p-koszul modules in detail. As an application of the minimal Horse-shoe lemma, we proved that the category K p (A) is closed under direct sums, direct summands, extensions and cokernels.

15 ON MINIMAL HORSE-SHOE LEMMA 15 Proposition 4.1. Let M = M 1 M 2 gr(a). Then M is a p-koszul module if and only if M 1 and M 2 are p-koszul modules. Proof. We have the following split exact sequence: M 1 M M 2. By Proposition 3.11, we have the following commutative diagram with exact rows and columns:... P 2 M 1 P 2 M 1 P 2 M 2 P 2 M 2 P 1 M 1 P 1 M 1 P 1 M 2 P 1 M 2 P M 1 P M 1 P M 2 P M 2 M 1 M M 2 where P 2 M 1 P 1 M 1 P M 1 M 1, P 2 M 1 P 2 M 2 P 1 M 1 P 1 M 2 P M 1 P M 2 M, and P 2 M 2 P 1 M 2 P M 2 M 2, are the minimal graded projective resolution of M 1, M and M 2 respectively. If M is a p-koszul module, by definition, PM i 1 PM i 2 is generated in degree δ(i) for all i. Obviously, PM i 1 and PM i 2 are generated in degree δ(i). Hence M 1 and M 2 are p-koszul modules. Conversely, if M 1 and M 2 are p-koszul modules, then PM i 1 and PM i 2 are generated in degree δ(i) for all i. Evidently, PM i 1 PM i 2 is generated in degree δ(i) for all i. Therefore M is a p-koszul module. By an easy induction, we can prove the following corollary by taking the same method as in Proposition 4.1. Corollary 4.2. Let M = M 1 M 2 M n and M gr(a). Then M is a p-koszul module if and only if all M i are p-koszul modules.

16 16 GUO-JUN WANG FANG LI Now we can investigate the extension closure of the category K p (A) of p-koszul modules. Proposition 4.3. Let K M N be an exact sequence in gr(a) such that K and N are in gr s (A), where A is a nice algebra. If K and N are in K P (A), then M K P (A). Proof. Consider the exact sequence: K M N. Since K and N are in degree gr s (A) and K and M are p-koszul modules, By proposition 3.13, we have the following commutative diagram with exact rows and exact columns: where... PK 2 PK 2 P N 2 PN 2 PK 1 PK 1 P N 1 PN 1 PK PK P N PN K M N P 2 K P 1 K P K K, PK 2 PN 2 PK 1 PN 1 PK PN M, and PN 2 P N 1 P N N, are the minimal graded projective resolution of K, M and N respectively; Since K and N are p-koszul modules, we have both PK i and P N i are generated in degree δ(i) for all i. Thus PK i P N i are generated in degree δ(i) for all i. Therefore M is a p-koszul module. That is K P (A) is closed under extensins. The next corollary shows that the category K P (A) preserves cokernels.

17 ON MINIMAL HORSE-SHOE LEMMA 17 Corollary 4.4. Let K M N be an exact sequence in gr s (A). If K and M are p-koszul modules, then M/K = N is a p-koszul module. Proof. Similarly to Proposition 4.2, we have the following commutative diagram with exact rows and exact columns:... PK 2 PK 2 P N 2 PN 2 PK 1 PK 1 P N 1 PN 1 PK PK P N PN K M N where PK 2 P K 1 P K K, PK 2 P N 2 P K 1 P N 1 P K P N M, and PN 2 P N 1 P N N, are the minimal graded projective resolution of K, M and N respectively; Since PK i and P K i P N i are generated in degree δ(i) for all i.by Lemma 3.12, we have PN i is also generated in degree δ(i) for all i. Hence M/K = N is a p-koszul module. References 1. F W Anderson,K R Fuller, Rings and categories of Modules [M], springer-verlag M Auslander and I Reiten, Syzygy Modules for Noetherian Rings[J]. J Algebra. 1996, 183, R. Berger, Koszulity for nonquadratic algebras, J. Algebra, Vol. 239 (21)

18 18 GUO-JUN WANG FANG LI 4. A. Beilinson, V. Ginszburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc., Vol. 9 (1996) E. L. Green, E. N.Marcos, R. Martinez-Villa, Pu Zhang, D-Koszul algebras, J. pure and Appl. Algebra 193 (24) E. L. Green, R. Martinez-Villa, I. Reiten, φ. Solberg, D. Zacharia, On modules with linear presentations, J. Algebra 25(2) (1998) R. Martinez-Villa and D. Zacharia Approxiamtions with modules having linear resolution Journal of Algebra 266(23) S. Priddy, Koszul resolutions, Trans. Amer. Math. Soc., Vol. 152 (197) M.Scott Osborne, Basic Homological Algebra, GTM C. A. Weible, An Introduction To Homological Algebra, Cambridge Studies in Avanced Mathematics, Vol. 38, Cambridge Univ. Press, Guo-Jun Wang: Department of Mathematics, Zhejiang University, Hangzhou 3127, China address: wgj@zju.edu.cn Fang Li: Department of Mathematics, Zhejiang University, Hangzhou 3127, China address: fangli@cms.zju.edu.cn