$\textrm{Ext}$ -modules]]> An intersection multiplicity in terms of Ext -modules C-Y. Jean Chan The main aim of this paper is to discuss the relation between Serre's intersection multiplicity and the Euler form. The Euler form is defined to be an alternating sum of the length of $\textrm{Ext}$ -modules and is used by Mori and Smith to develop intersection theory over noncommutative rings. We show that they differ by a sign and that this relation is closely related to Serre's vanishing theorem.

]]> The main aim of this paper is to discuss the relation between Serre's intersection multiplicity and the Euler form. The Euler form is defined to be an alternating sum of the length of Ext -modules and is used by Mori and Smith to develop intersection theory over noncommutative rings. We show that they differ by a sign and that this relation is closely related to Serre's vanishing theorem. B H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. 27:3669 D1 S. P. Dutta Generalized Intersection Multiplicities of Modules, Trans. Amer. Math. Soc., 276 , no. 2, (1983), 657-669. 84i:13024 D2 S. P. Dutta Generalized Intersection Multiplicities of Modules II , Proc. Amer. Math. Soc., 93 , no. 2, (1985), 203-204. 86k:13026 DHM S. P. Dutta, M. Hochster and J.E. McLaughlin, Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985), 253-291. 86h:13023 F W. Fulton Intersection Theory, Second edition, Springer-Verlag, Berlin, 1998. 99d:14003 GS H. Gillet and C. Soule, K-theorie et nullite des multiplicites d'intersection, C. R. Acad. Sci. Paris, Ser. I, no. 3, t. 300 (1985), 71-74. 86k:13027 Mts H. Matsumura Commutative Ring Theory , Cambridge University Press, Cambridge, England, 1986. 88h:13001 Mori I. Mori, Intersection multiplicity over noncommutative algebras, to appear in J. Algebra. MSmith I. Mori and S. P. Smith , Bezout's theorem for noncommutative projective spaces, J. Pure Appl. Algebra 157 (2001), 279-299. R1 P. C. Roberts , Multiplicities and Chern Classes in Local Algebra, Cambridge Tracts in Mathematics 133, Cambridge University Press (1998). 99:13 R2 P. Roberts , The vanishing of intersection multiplicities of perfect complexes , Bull. Amer. Math. Soc. 13 (1985), 127-130. 87c:13030 R3 P. Roberts, Intersection Theorems, Commutative Algebra, Proc. MSRI Microprogram, Springer-Verlag, 417-436. 90j:13024 R4 P. Roberts, The MacRae Invariant and the First Local Chern Character, Trans. Amer. Math. Soc., 300 , no. 2, (1987), 583-591. 88c:14010 R5 P. Roberts, Local Chern Characters and Intersection Multiplicities, Proceedings of Symposia in Pure Mathematics, 46 , (1987), 389-400. 89a:14011 Rot J. Rotman , An Introduction to Homological Algebra Academic Press, New York, 1979. 80k:18001 S J.-P. Serre , Algebre Locale-multiplicites, Lecture Notes in Mathematics 11 , Springer-Verlag, New York, 1961. 34:1352

H. BASS, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. MR 27:3669

S. P. DUTTA Generalized Intersection Multiplicities of Modules, Trans. Amer. Math. Soc., 276, no. 2, (1983), 657-669. MR 84i:13024

S. P. DUTTA Generalized Intersection Multiplicities of Modules II, Proc. Amer. Math. Soc., 93, no. 2, (1985), 203-204. MR 86k:13026

S. P. DUTTA, M. HOCHSTER AND J.E. MCLAUGHLIN, Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985), 253-291. MR 86h:13023

W. FULTON Intersection Theory, Second edition, Springer-Verlag, Berlin, 1998. MR 99d:14003

H. GILLET AND C. SOUL´E, K-théorie et nullité des multiplicités d'intersection, C. R. Acad. Sci. Paris, Sér. I, no. 3, t. 300 (1985), 71-74. MR 86k:13027

H. MATSUMURA Commutative Ring Theory, Cambridge University Press, Cambridge, England, 1986. MR 88h:13001

I. MORI, Intersection multiplicity over noncommutative algebras, to appear in J. Algebra.

I. MORI AND S. P. SMITH, Bézout's theorem for noncommutative projective spaces, J. Pure Appl. Algebra 157 (2001), 279-299.

10.

P. C. ROBERTS, Multiplicities and Chern Classes in Local Algebra, Cambridge Tracts in Mathematics 133, Cambridge University Press (1998). CMP 99:13

11.

P. ROBERTS, The vanishing of intersection multiplicities of perfect complexes, Bull. Amer. Math. Soc. 13 (1985), 127-130. MR 87c:13030

12.

P. ROBERTS, Intersection Theorems, Commutative Algebra, Proc. MSRI Microprogram, Springer-Verlag, 417-436. MR 90j:13024

13.

P. ROBERTS, The MacRae Invariant and the First Local Chern Character, Trans. Amer. Math. Soc., 300, no. 2, (1987), 583-591. MR 88c:14010

14.

P. ROBERTS, Local Chern Characters and Intersection Multiplicities, Proceedings of Symposia in Pure Mathematics, 46, (1987), 389-400. MR 89a:14011

15.

J. ROTMAN, An Introduction to Homological Algebra Academic Press, New York, 1979. MR 80k:18001

16.

J.-P. SERRE, Algèbre Locale-multiplicités, Lecture Notes in Mathematics 11, Springer-Verlag, New York, 1961. MR 34:1352 ]]> 2001 American Mathematical Society Proceedings of the American Mathematical Society Proc. Amer. Math. Soc. proc 130 02 20010525 October 11, 1999 June 15, 2000 May 25, 2001 327 327-336 10 S0002-9939-01-06022-1 10.1090/S0002-9939-01-06022-1 0002-9939 1088-6826 English 13D22 13H15 14C17 13D07 2000 http://www.ams.org/jourcgi/jour-getitem?pii=S0002-9939-01-06022-1 Introduction Assume A is a regular local ring. Let M and N be finitely generated A -modules such that M N has finite length. In the 1950s, Serre defined the intersection multiplicity of M and N as ( M, N) (-1) i( i (M, N) ). To develop intersection theory over noncommutative rings, Mori and Smith attempted to generalize intersection multiplicity using the Euler form (M,N) (-1) i ( i (M, N)) for two arbitrary finitely generated Noetherian right modules. The intersection multiplicity is defined as (-1) N (M,N), where the dimension of modules over a noncommutative ring is defined by using the Euler form ( cf. MSmith ). It should be pointed out that the tensor product of two right modules is not well-defined. Therefore the -modules cannot be applied in noncommutative cases. Mori and Smith established Bezout's Theorem for noncommutative spaces for this intersection multiplicity. We refer the readers to Mori and Smith's paper MSmith for complete details. Since the idea is to generalize results in commutative algebraic geometry, there should be relations between these two intersection multiplicities. In Mori , Mori conjectured (M, N) (-1) M (M,N). This paper presents the study of the relation between (M,N) and (M, N) over commutative Noetherian rings. More generally, we will see that the formula relating (M,N) and (M,N) is closely related to Serre's vanishing theorem. In section 2, we prove that the above formula holds over a regular local ring. Our proof relies on Serre's vanishing theorem. theorem Vanishing, Roberts R2 , Gillet and Soule GS vanishing Assume M and N are finitely generated modules over a complete intersection A such that M and N both have finite projective dimension and M N has finite length. If M N A , then (M, N) 0. theorem However, as we try to generalize the result over a complete intersection, the original proof cannot be applied in this generality. The properties of complete intersections and the formula itself suggest that we apply an argument similar to the one used in Roberts' proof of the vanishing theorem. This will be discussed in the third section. We further prove the formula for a Gorenstein ring of dimension 5 . This is also inspired by the vanishing theorem under the same condition R3 . From the discussion in the earlier sections, it seems likely that a proof of the vanishing theorem will also give a proof of the formula that we are interested in: equation eq:1st ( M, N) (-1) M (M,N). equation In the last section, we investigate the relation between the vanishing theorem and this formula. We will see that they are equivalent over a regular local ring. The equivalence is not known in general. It is known that all finitely generated modules over a regular local ring have finite projective dimension, so the sum in the definitions of both multiplicities have finitely many terms involved. Over a more general ring, even if the two modules satisfy the same conditions as in Serre's definition, it is not necessary that the expressions for (M,N) and (M,N) are only finite sums. There are many ways to fix this. We mainly discuss the case with the condition that both modules have finite projective dimension added to the assumption, except the discussion in section 4. After the earlier version of this paper was done, the author found that S. P. Dutta has similar results. In D1 and D2 Dutta proved the vanishing theorem for Gorenstein rings of dimension 5 by proving that a formula holds true. This formula is equivalent to Formula ( eq:1st ) under certain conditions. We will discuss this in section 2 and section 4. The relation between (M,N) and (M,N) 1 over a regular local ring We first state the main theorem in this section. theorem thm:regular Let A be a regular local ring and let M and N be finitely generated A -modules such that M N has finite length. Let M denote A - M . Then, equation eq:2 (M,N) (-1) M (M, N). equation theorem rem Let A be a regular local ring and let M and N satisfy the hypotheses of Theorem thm:regular . Serre proved that M N A. When the ring is not regular, whether or not this inequality is true remains unknown if both modules have finite projective dimension. rem To relate Serre's intersection multiplicity and the Euler form, we write (M,N) as the Euler characteristic of the total complex of the tensor product of the finite resolutions of both modules. Then, we claim that (M, N) is equal to the Euler characteristic described above with the resolution of M replaced by its dual. By reducing M to the case of the ring modulo a prime ideal and computing the latter Euler characteristic, the theorem will be proven. Before proving the theorem, we explain the above idea in detail starting with some background on homological algebra. Recall that a perfect complex, G , is a bounded complex of finitely generated free modules. Let (G ) denote the support of the complex G , which is the set p (A) : (G ) p is not exact . We define the dimension of a complex to be the length of largest such that p 0, , p are distinct in (G ) and p 0 p . If G is a complex supported only at the maximal ideal, let (G ) i (-1) i ( H i(G ) ) , where H i(G ) is the i - th homology of G . We call (G ) the Euler characteristic of G . One can use this to interpret Serre's intersection multiplicity. Let M and N be assumed as above and let E and F be finite free resolutions of M and N respectively. Then equation eq:complex (M,N) ( E F ), equation where E F denotes the total complex of the tensor product of the two complexes. We recall two well-known facts from homological algebra. They imply an important property which allows us to consider the Euler form in terms of Euler characteristics in the later discussion. 0.5pc prop homology2 Let A be a commutative Noetherian local ring. Let N be a finitely generated module of finite projective dimension and let G be a bounded below complex of finitely generated A -free modules such that H i(G ) N has finite length for all i . Then, (G N) (-1) i (H i(G ), N). prop prop homology Let A be a commutative Noetherian local ring of dimension d with maximal ideal m . Let M and N be finitely generated A -modules with finite projective dimension such that M N has finite length. Let E and F be finite free resolutions of M and N respectively. Then, equation eq:dual (M,N) (E F ) i 0 s (-1) i( i (M, A),N) . equation prop Proposition homology2 is the consequence of the fact that spectral sequences preserve Euler characteristics. Let G in Proposition homology2 be (E ,A) in Proposition homology . Proposition homology follows from Proposition homology2 and the fact that (E i, A) N (E i, N) for all i . We will use Proposition homology in the proof of Theorem thm:regular and the following sections. We now begin to prove Theorem thm:regular . proof By taking a filtration of M , 0 M 0 M r M, such that M j M j-1 A p j for some prime ideal p j , we have (M,N) (A p j , N) and (M,N) (A p j , N). Moreover, for any finitely generated A -module P , the vanishing theorem implies that (P, N) A p P (P p ) (A p , N). We first assume M is of the form A q , where q is a prime ideal. Using Formula ( eq:dual ), we conclude array ccl (M, N) i 0 s (-1) i ( i (M,N)) i 0 s (-1) i ( i( M,A),N) i 0 s (-1) i ( i A (M, A) A q ) ( A q , N), array since ( i(E , A)) (M). It is known that, if M A q , then equation eq:lemma1 i A(M,A) A q array cc (A q ) q i A- M , 0 otherwise , array . equation which can be reduced by localization to showing that equation eq:gorenstein i A(k,A) array cc k i A , 0 otherwise . array . equation (This is a special case of more general results in Bass; moreover, Formula ( eq:gorenstein ) is often taken as a definition of Gorenstein rings.) Thus, (A q , N) (-1) A q (A q , N). This also shows the vanishing theorem for the Euler form when the first module is in the form of A q ; namely, (A q , N) 0 if A q N A . In general, (M,N) is a sum of (A p j , N) taken over all p j occurring in a filtration of M . Since M N A , by the vanishing theorem, those terms with prime ideals of dimension less than M must vanish; similarly for (M,N) . From the result of the special case proved above, we obtain (M,N) (-1) M (M,N). proof Let A be a Gorenstein ring of dimension d and let M be a finitely generated module of dimension m . We define d-m (M,A). We say that M is perfect if and only if M is Cohen-Macaulay and of finite projective dimension; in this case, proj.dim M M . If M is perfect, then is also perfect of dimension equal to M and M . There used to be a general version of the vanishing conjecture, which we call the generalized vanishing conjecture for simplicity, that states that if either M or N has finite projective dimension and (M N) , then M N A implies (M,N) 0 . The generalized vanishing conjecture is known to be false due to Dutta, Hochster, and McLaughlin's example DHM . We recall a theorem in Dutta D1 . theorem Dutta thm2.4 Let A be a Gorenstein ring. The generalized vanishing conjecture is true if and only if for any perfect module M and any Cohen-Macaulay module N with M N A , equation eq:length (M N) (M ) . equation theorem Since the vanishing theorem is true if either M or N is of the form A (x 1, , x k) , where x 1,, x k form a regular sequence (see Serre ), it is not hard to show that (M,N) (-1) M (M,N) in this case. Using finitely many short exact sequences with modules of this form, one can reduce Theorem thm:regular to the case in which M is perfect and N is Cohen-Macaulay. In this case, we have (M N) () and (M ) ( ) (N). It is not hard to see that (M,N) (-1) M (N) and (M,N) (MN) if M is perfect and N is Cohen-Macaulay with M N A . Thus, using Theorem thm2.4 , this approach gives another proof of Theorem thm:regular . For details we refer to Dutta D1 and D2 ; at the time these papers were written the vanishing theorem was still a conjecture. The relation between (M,N) and (M,N) 1 over a complete intersection The relation presented by Formulas ( eq:dual ) and ( eq:lemma1 ), and the vanishing theorem are key points in the proof of Theorem thm:regular . Over a complete intersection, these three statements are still true. However, not all modules, in particular the type of the ring modulo a prime ideal, have finite projective dimension. One may not apply the vanishing theorem as over a regular local ring. Following the proof of the vanishing theorem (see Roberts R2 ), we use the theory of local Chern characters to do further generalizations. Assume A is a Noetherian local ring. Let (A) Q k(A) Q be the Chow group on A tensored with rational number field. The local Chern character of a perfect complex G is (G ) 0( G ) 1(G ) where, for each i and k , i(G ) is a map i(G ): k(A) Q k-i ((G )) Q and A ((G )) is the Chow group on (G ) . The Todd class of G , (G ), is an element in ((G )) . In particular, if G is supported at the maximal ideal m , then (G ) (G ) A m . In this case, we will identify the image of the Todd class with a number in Q since ((G )) Q Q . The local Riemann-Roch formula (see Roberts Theorem 12.6.1 R1 ) implies that (G ) (G )((A)), which relates local Chern characters with Euler characteristics (G ) (G )((A)). For more details on the definitions and properties, we refer to Fulton and Roberts R1 . theorem thm:ci Let A be a complete intersection and let M and N be finitely generated A -modules with finite projective dimension such that M N has finite length and M N A . Then, equation eq:result1 (M,N) (-1) M (M, N). equation theorem proof Assume A d , M m and N n . Let E and F be finite free resolutions of M and N respectively. From Formulas ( eq:complex ) and ( eq:dual ), (M,N) (E F ). It thus suffices to show that equation eq:result2 ( E F ) (-1) M ( E F ). equation It is known that (A) A when A is a complete intersection. Since (E F ) contains the maximal ideal only, by the local Riemann-Roch formula, we have ( E F ) (E F )((A)) d(E F )( A ). From the multiplicativity of local Chern characters and the duality property, i(E ) (-1) i i(E ) (see Roberts Theorem 9.7.2 and Theorem 12.3.1 R1 ), we have array ccl d(E F )( A ) i j d i(E ) j(F )( A ) i j d (-1) i i(E ) j(F )( A ). array For any d(A) , j(F )() d-j ((F )) . Since F is a finite free resolution of N , F N n. Those j with j(F )() possibly nonzero are j d -n , which implies i n for i j d . On the other hand, the multiplicativity property, i(E ) j(F )() j(F ) i(E )() , implies i d-m by the same argument and we thus have d - m i n . By the assumption that m n d , we conclude that i(E ) j(F )() 0 except possibly when m n d , i d-m and j d-n . This argument was first pointed out in the proof of Serre's vanishing theorem in Roberts R2 . In Serre's vanishing theorem, m n d , so all terms vanish. Similarly, array ccl (E F ) i j d i(E ) j(F )( A ) d-m (E ) d-n (F )( A ). array Hence, ( E F ) (-1) d-m d-m (E ) d-n (F )( A ) (-1) d-m (E F ). This proves Formula ( eq:result2 ) and completes the proof of the theorem. proof The key point of the above proof, as in Roberts R2 , is to write both (M,N) and (M,N) in terms of local Chern characters. The condition that the ring is a complete intersection implies that most terms vanish and the remaining ones are easy to determine. In R4 Roberts proved that 1(G ) 0 if G is a perfect complex with support of codimension greater than or equal to 2 . Therefore, the vanishing conjecture holds for Gorenstein rings of dimension less than or equal to 5 (see Roberts R3 ); namely, (M,N) 0 if M N A and so is (M,N) . This case was also proved by Dutta D1 and D2 using homological methods. We will come back to Dutta's proof in the next section. From the proof of Theorem thm:ci , only one term possibly does not vanish if M N A and A is a complete intersection. Over a Gorenstein ring, more terms remain undetermined in both (M,N) and (M,N) . However, they agree with signs. We have the following theorem. theorem thm:dim5 Let A be a Gorenstein ring of dimension less than or equal to 5 . Let M and N be as in Theorem thm:ci . Then, (M,N) (-1) M (M,N). theorem proof Let E and F be finite free resolutions of M and N respectively and let d , m , and n be as in the previous proof. We prove the case when d A 5 . The proof for lower dimensional rings is similar. Since A is a Gorenstein ring, d-i (A) 0 if i is odd (see Proposition 12.4.4 R1 ). Therefore (M,N) (E F ) s 1,3,5 C(s), where C(s) denotes ch s(E F )( s(A)) and is equal to C(s) i j s (-1) i i(E ) j(F )( s(A)). If s 5 , 5(A) is the top term of the Todd class; then as in the proof for complete intersections, C(5) array ll (-1) 5-m ch 5-m (E )ch 5-n (F )( A ) if m n 5, 0 otherwise. array . If s 3 or 1 , each term has either zeroth Chern character or first Chern character. The zeroth Chern character can be identified with the alternating sum of the ranks of the free modules of the complex and so vanishes if the complex is not supported at a minimal prime ideal, namely, if the complex has codimension greater than zero. Also, the first Chern character vanishes if the complex has codimension 2 as was pointed out earlier R4 . As it was shown in Roberts R3 , the local Chern characters in C(s) all vanish if M N A . We summarize the possibly nonvanishing cases when the sum of the dimensions reaches A and leave the details to the reader: array lcl C(3) array cl 2(E ) 1(F )( 3(A)) if E 1 and F 4 , - 3(E ) 0(F )( 3(A)) if E 0 and F 5 , 0 otherwise , array . C(1) 0. array The same argument holds for (E F ) and there are exactly the same terms left with opposite signs as we wish to have. Therefore, we have (E F ) (-1) M (E F ) . This completes the proof of the theorem. proof A similar argument shows the vanishing theorem for the Euler form. One may also prove it as a corollary to Theorem vanishing (Serre's vanishing theorem) and Theorem thm:ci or Theorem thm:dim5 . corollary Let A , M , and N be as in Theorem thm:ci or Theorem thm:dim5 . If M N A , then (M,N) 0 . corollary Discussion In this section, we further investigate the relation between the vanishing theorem and the formula relating (M,N) to (M,N) . We remark that an example in Roberts Chap. 13, Section 3 R1 shows that Theorem thm:ci is false over a Cohen-Macaulay ring when both modules have finite projective dimension. So, the result of Theorem thm:ci remains unknown only over a Gorenstein ring of dimension greater than 5 . Over a higher dimensional Gorenstein ring, whether or not the vanishing theorem holds is still an open problem. Also, we do not know whether or not Theorem thm:ci is true over an arbitrary Gorenstein ring. However, in the following theorem, we are able to show that the fact that the formula holds implies the vanishing theorem. Let A be a Gorenstein ring and let M and N be finitely generated A -modules. We say M and N have Property A if they satisfy the following conditions: enumerate M N A . M N has finite length. Either M or N has finite projective dimension. enumerate We note that condition 2 implies that i(M,N) and i(M,N) have finite length and condition 3 ensures that both expressions for (M, N) and (M, N) have finite sums. In particular, if A is Gorenstein, a finitely generated module has finite projective dimension if and only if it has finite injective dimension. Therefore, with condition 2, it suffices to have one of the two modules having finite projective dimension in order to have both (M,N) and (M,N) well-defined. The idea of the proof of the following theorem is due to I. Mori. theorem relation Let A be a Gorenstein ring of dimension d . If the formula (M, N) (-1) M (M,N) holds for any two modules M and N which have Property A , then the vanishing theorem holds; namely, if any two modules P and Q have Property A , then P Q A implies (P,Q) 0. theorem proof Let P and Q be two modules which have Property A and P Q A . Suppose there exists an A -module P 1 such that P 1 and Q have Property A and P 1 P 1 . Therefore, PP 1 and Q also have Property A and the formula holds by the assumption. Thus, we may apply ( , Q) and (, Q) on P P 1 , which has dimension equal to P 1 , and get array lcl (P,Q) (P 1, Q) (P P 1, Q) (-1) P 1 (PP 1, Q) -(-1) P (P, Q) (-1) P 1 (P 1, Q) - (P, Q) (P 1, Q) . array This shows that (P, Q) 0 . We now prove the existence of the module P 1 . Assume P -1 . There exists a regular sequence on A with length d- , x 1, , x d - , such that the module Q (x 1, , x d- ) has finite length. Following a standard commutative algebra argument, such a regular sequence may be constructed by avoiding finitely many prime ideals at each step. Therefore, P 1 A (x 1, , x d- ) has dimension as required. Moreover, the Koszul complex on x 1, , x d- is a resolution of P 1 so P 1 has finite projective dimension. All three conditions in Property A hold. proof In section 3, we have seen that the vanishing theorem and Formula ( eq:result1 ) over a complete intersection or a small Gorenstein ring share essentially the same proof. It is worth pointing out that Theorem thm:regular together with Theorem relation shows the equivalence of the vanishing theorem and the formula over a regular local ring as Theorem thm2.4 also proves. It is not clear that this equivalence is true in general. Roberts in R5 further proved the vanishing theorem when the singular locus of A has dimension at most one. In this generality, there will be a lot more terms that remain undetermined than those in the proof of Theorem thm:dim5 and it is not obvious that the formula will hold. However, we do know that the formula holds if A is a Gorenstein ring with singular locus of dimension at most one. It is unknown whether or not the vanishing conjecture and the formula are equivalent over arbitrary Gorenstein rings. Although the generalized vanishing conjecture has a counterexample, Theorem thm2.4 can be revised so that the vanishing conjecture over a Gorenstein ring is a sufficient condition for Formula ( eq:length ) (see D2 ). Dutta proved the vanishing theorem for Gorenstein rings of dimension 5 by proving Formula ( eq:length ) is true. As the referee pointed out, Theorem relation can be derived from either Theorem thm2.4 or its revised form since the conditions of M and N in Theorem thm2.4 all satisfy Property A and the formulas in these two theorems are equivalent under the conditions in which M is perfect and N is Cohen-Macaulay. Dutta, Hochster and McLaughlin constructed an example over a complete intersection which shows that the generalized vanishing theorem is not true DHM ; that is, one of the modules has infinite projective dimension. Over the same ring, there exist two modules which have Property A , but the formula is not satisfied. These modules may be constructed by going through Theorem relation based on Dutta, Hochster and McLaughlin's example. Acknowledgments Many of ideas in this paper were developed in the discussions with Professor Paul C. Roberts. I would like to express my gratitude to Professor Roberts for sharing his broad knowledge on this subject. I also would like to thank the referee for the valuable suggestions and comments.