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Ideals with Stable Betti Numbers
Advances in Mathematics 152, 7277 (2000) doi:10.1006aima.1998.1888, available online at http:www.idealibrary.com on
Ideals with Stable Betti Numbers Annetta Aramova and Jurgen Herzog FB6 Mathematik und Informatik, Universitat GHS Essen, Postfach 103764, 45117 Essen, Germany E-mail: [A.Aramova,juergen.herzog]uni-essen.de
and Takayuki Hibi Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan E-mail: hibimath.sci.osaka-u.ac.jp Received November 17, 1998; accepted December 4, 1998
Componentwise linear ideals were introduced earlier to generalize the result that the StanleyReisner ideal I 2 of a simplicial complex 2 has a linear resolution if and only if its Alexander dual 2* is CohenMacaulay. It turns out that I 2 is componentwise linear if and only if 2* is sequentially CohenMacaulay. In this paper we discuss Betti number properties of componentwise linear ideals. Let I be a graded ideal of a polynomial ring S and Gin(I ) the generic initial ideal of I with respect to the reverse lexicographic term order on S. Our main result is that I and Gin(I ) admit the same graded Betti numbers if and only if I is componentwise linear. For the proof of this fact, we describe some properties of the Betti diagram of a generic initial ideal. Combinatorial implications for shifted complexes will also be discussed. 2000 Academic Press
1. GINS AND COMPONENTWISE LINEAR IDEALS Let K be a field of characteristic 0 and S=K[x 1 , ..., x n ] the polynomial ring over K with each deg x i =1. We work with the reverse lexicographic term order on S induced by x 1 > } } } >x n . For a graded ideal I of S, let Gin(I) denote the generic initial ideal of I with respect to this term order, see, e.g., [8, 10]. If I is a graded ideal of S, then we write I ( j ) for the ideal generated by all homogeneous polynomials of degree j belonging to I. Moreover, we write I d for the ideal generated by all homogeneous polynomials of I whose degree is greater than or equal to d. We say that a graded ideal I/S 72 0001-870800 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
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is componentwise linear if I ( j ) has a linear resolution for all j. We refer the reader to [11, 13] for detailed information on componentwise linear ideals. Let [ ; i, i+ j (I )] i, j denote the graded Betti numbers of a graded ideal I. It is known that ; i, i+ j (I ) ; i, i+ j (Gin(I)) for all i and j [10]. The main result of the present paper is the following: Theorem 1.1. A graded ideal I of S satisfies ; i, i+ j (I )=; i, i+ j (Gin(I )) for all i and j if and only if I is componentwise linear. In order to prove Theorem 1.1, the following result concerning the homological data of generic initial ideals will be required. Theorem 1.2.
Let I be a graded ideal generated in degree d. Then
(a)
If ; i, i+ j (Gin(I )){0, then ; i $, i $+ j (Gin(I )){0 for all i $
(b)
If ; 0, j (Gin(I)){0, then ; 0, j $(Gin(I )){0 for all d j $< j.
Proof. Since the generic initial ideal is strongly stable, statement (a) follows from the EliahouKervaire resolution [9] of stable ideals. Let g 1 , ..., g m be the generators of I of degree d. Suppose that ; 0, j&1(Gin(I)) =0. Then consider the ideal I j&2 . Since Gin(I j&2 )=Gin(I) j&2 , we may assume that ; 0, d+1(Gin(I))=0. We have to show that Gin(I) is generated in degree d. It follows from ; 0, d+1(Gin(I))=0 that all S-polynomials of degree d+1 reduce to zero with respect to [ g 1 , ..., g m ]. By virtue of Remark 1.3 below, this implies that [ g 1 , ..., g m ] is a Grobner basis of I, equivalently, Gin(I) is generated in degree d. In fact, since (in(g 1 ), ..., in(g m )) is a strongly stable ideal, its first syzygy module is generated in degree d+1. Remark 1.3. Recall the following fact from Grobner basis theory which is a stronger version of the well known Buchberger criterion [8, Sect. 15.4]: A set of generators G=[ g 1 , ..., g m ] of I is a Grobner basis of I if and only if each S-polynomial which comes from a minimal set of homogeneous generators of the first syzygy module of the ideal (in( g 1 ), ..., in( g m )) reduces to zero with respect to G. Before starting our proof of Theorem 1.1, we state the following: Lemma 1.4. Let I and J be graded ideals of S generated in degree d with the same graded Betti numbers. Then I d+1 and J d+1 have the same graded Betti numbers. Proof.
The exact sequence 0 Ä I d+1 Ä I Ä K(&d ) ;0, d Ä 0
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induces the long exact sequence } } } Ä Tor i+1(K, I d+1 ) (i+1)+ ( j&1) Ä Tor i+1(K, I ) (i+1)+( j&1) ;
0, d Ä Tor i+1(K, K ) (i+1)+ j&(d+1) Ä Tor i (K, I d+1 ) i+ j Ä Tor i (K, I ) i+ j
;
0, d Ä Tor i (K, K) i+ j&d Ä } } } .
It then follows that ; i, i+ j (I d+1 )=; i, i+ j (I ) for all i and for all j{d, d+1. Also, ; i, i+ j (I d+1 )=0 if j d. Now, if j=d+1, then the above long exact sequence becomes 0, d 0 Ä Tor i+1(K, I ) i+1+d Ä Tor i+1(K, K ) ;i+1
Ä Tor i (K, I d+1 ) i+d+1 Ä Tor i (K, I ) i+d+1 Ä 0. n ) ; 0, d (I)&; i+1, i+1+d (I). Hence, ; i, i+d+1(I d+1 )=; i, i+d+1(I )+( i+1 The same formulae are valid for ; i, i+ j (J ). This completes the proof.
K
We are now in the position to give a proof of Theorem 1.1. Proof of Theorem 1.1. First, suppose that I is componentwise linear. The following formula for the graded Betti numbers of a componentwise linear ideal I is known [11]: ; i, i+ j (I )=; i (I ( j ) )&; i (mI ( j&1) ). Here m is the irrelevant maximal ideal (x 1 , ..., x n ) of S. Since a strongly stable ideal is componentwise linear and since Gin(I ) is strongly stable, the same formula is valid for Gin(I ). Therefore, it suffices to prove that ; i (I ( j ) )=; i (Gin(I ) ( j ) ) and ; i (mI ( j&1) )=; i (m Gin(I ) ( j&1) ). Since I ( j ) has a linear resolution, it follows from the BayerStillman theorem [3], which guarantees that the regularity of an ideal and that of its generic initial ideal coincide, that Gin(I ( j ) )=Gin(I ) ( j ) . Since I ( j ) and Gin(I ( j ) ) have the same Hilbert function, and since the Betti numbers of a module with linear resolution are determined by its Hilbert function, the first equality follows. To prove the second one, we note that mI ( j&1) has again a linear resolution and that, by the same reason as before, m Gin(I) ( j&1) =Gin(mI ( j&1) ). Second, suppose that I and Gin(I ) have the same graded Betti numbers. Let max(I ) (resp. min(I)) denote the maximal (resp. minimal) degree of a homogeneous generator of I. To show that I is componentwise linear, we work with induction on r=max(I )&min(I ). Set d=min(I ). Let r=0. Since I and Gin(I ) have the same graded Betti numbers, it follows that Gin(I ) is generated in degree d. Since Gin(I ) is a strongly stable ideal, we have that Gin(I ) has a linear resolution, hence I has a linear resolution.
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Now, suppose that r>0. Since Gin(I d+1 )=Gin(I ) d+1 , our induction hypothesis and Lemma 1.4 imply that I d+1 is componentwise linear. Thus, it suffices to prove that I ( d ) has a linear resolution. Suppose this is not the case. Then, by the BayerStillman theorem, Gin(I ( d ) ) has regularity >d. Moreover, since Gin(I ( d ) ) is strongly stable, its regularity equals max(Gin(I ( d ) )). It follows from Theorem 1.2 that Gin(I ( d ) ) has a generator of degree d+1. Now, ; 0, d+1(I)=dim I d+1 &dim(mI ( d ) ) d+1 =dim I d+1 &dim(I ( d ) ) d+1 , and ; 0, d+1(Gin(I ))=dim Gin(I ) d+1 &dim(m Gin(I ) ( d ) ) d+1 =dim Gin(I ) d+1 &dim(m Gin(I ( d ) )) d+1 >dim Gin(I ) d+1 &dim Gin(I ( d ) ) d+1 , because (m Gin(I ( d ) )) d+1 is properly contained in Gin(I ( d ) ) d+1 . Hence ; 0, d+1(Gin(I ))>; 0, d+1(I ), a contradiction. This completes our proof.
K
The following result supports a conjecture of Buchsbaum and Eisenbud [5]. Corollary 1.5. Let I be a componentwise linear ideal of projective p ) for all i 0. dimension p. Then ; i (I ) ( i+1 Proof. In fact, the conjecture is known to be true for monomial ideals, see, e.g., [12]. Theorem 1.1 guarantees that the Betti numbers of a componentwise linear ideal are attained by a monomial ideal. K We conclude this section by noting that Theorem 1.1 and Theorem 1.2(b) are not valid in positive characteristic. Indeed, if characteristic p>0, then I=(x p, y p ) provides a counterexample.
3. SHIFTED COMPLEXES Let, as before, S=K[x 1 , ..., x n ] be the polynomial ring over a field K. Let 2 be a simplicial complex on the vertex set [n]=[1, ..., n] and I 2 the StanleyReisner ideal of 2. Let 2* be the Alexander dual 2*=[F/[n] : [n]"F Â 2]
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of 2, and let 2 s denote the algebraic shifting (cf. [4]) of 2 introduced by G. Kalai [14]. It is shown in [11] (and [13]) that I 2 is componentwise linear if and only if 2 is sequentially CohenMacaulay. We now present the exterior algebra version of Theorem 1.1 whose proof is completely analogue. We refer the reader to [2] for information on Grobner basis theory in exterior algebras. Theorem 2.1. Let J be a graded ideal of the exterior algebra. Then J and Gin(J) have the same graded Betti numbers if and only if J is componentwise linear. Let J 2 be the StanleyReisner ideal of 2 in the exterior algebra E over K. Then the following facts are known: (a) I 2 is componentwise linear if and only if J 2 is componentwise linear [11]; i+ j&1 (b) ; i, i+ j (EJ 2 )= ik=0 ( k+ j&1 ) ; k, k+ j (SI 2 ) for all i and all j [1], in particular, the graded Betti numbers of J 2 are determined by those of I 2 , and vice versa. Corollary 2.2. Let I 2 be the StanleyReisner ideal of a simplicial complex 2. Then I 2 is componentwise linear if and only if I 2 and I 2 s have the same graded Betti numbers. Proof. Recall that J 2 s =Gin(J 2 ). Hence, Theorem 2.1 together with the above observations (a) and (b) complete the proof. K For a componentwise linear ideal I 2 , the h-triangle of 2* and the graded Betti numbers of I 2 are related as follows [11, 13]: For all j, we have : ; i, i+ j (I 2 ) t i = : h i, n& j&1(2*)(t+1) i. i 0
i 0
Combining Corollary 2.2 with the above formula and noting that I 2* is componentwise linear if and only if 2 is sequentially CohenMacaulay over K, it follows that the h-triangles of 2 and 2 s coincide if 2 is sequentially CohenMacaulay over K (since (2*) s =(2 s )*). This is one direction of [6, Theorem 5.1]. The combinatorial version of Theorem 1.2 is the following: Theorem 2.3. Let 2 be a pure simplicial complex of dimension d. Then 2 is CohenMacaulay if and only if 2 s has no facet of dimension d&1. Proof. Note that 2 is pure of dimension d if and only if J 2* in the exterior algebra is generated in degree n&d&1. Now, the analogue of Theorem 1.2 for the exterior algebra yields that Gin(J 2* ) is generated in
IDEALS WITH STABLE BETTI NUMBERS
77
degree n&d&1 if and only if Gin(J 2* ) has no generator of degree n&d. Since Gin(J 2* )=J (2*) s and since (2*) s =(2 s )*, we have the desired conclusion by using the fact [14] that a simplicial complex is CohenMacaulay if and only if its shifted complex is pure. K
REFERENCES 1. A. Aramova and J. Herzog, Almost regular sequences and Betti numbers, Amer. J. Math., to appear. 2. A. Aramova, J. Herzog, and T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra 191 (1997), 174211. 3. D. Bayer and M. Stillman, A criterion for detecting m-regularity, Invent. Math. 87 (1987), 111. 4. A. Bjorner and G. Kalai, An extended Euler Poincare theorem, Acta Math. 161 (1988), 279303. 5. D. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3, Amer. J. Math. 3 (1977), 447485. 6. A. M. Duval, Algebraic shifting and sequentially CohenMacaulay simplicial complexes, Electron. J. Combin. 3 (1996), R21. 7. J. A. Eagon and V. Reiner, Resolutions of StanleyReisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), 265275. 8. D. Eisenbud, ``Commutative Algebra with a View Towards Algebraic Geometry,'' Graduate Texts in Math., Vol. 150, Springer-Verlag, New YorkBerlin, 1995. 9. S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 125. 10. M. Green, Generic initial ideals, in ``Six Lectures on Commutative Algebra'' (J. Elias, J. M. Giral, R. M. Mir'o-Roig, and S. Zarzuela, Eds.), Progress in Math., Vol. 166, pp. 119185, Birkhauser, Basel, 1998. 11. J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J., in press. 12. J. Herzog and M. Kuhl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), 16271646. 13. J. Herzog, V. Reiner, and V. Welker, Componentwise linear ideals and Golod rings, Mich. Math. J. 46 (1999), 211223. 14. G. Kalai, Characterization of f-vectors of families of convex sets in R d. Part I. Necessity of Eckhoff's conditions, Israel J. Math. 48 (1984), 175195.
Ideals with Stable Betti Numbers Annetta Aramova and Jurgen Herzog FB6 Mathematik und Informatik, Universitat GHS Essen, Postfach 103764, 45117 Essen, Germany E-mail: [A.Aramova,juergen.herzog]uni-essen.de
and Takayuki Hibi Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan E-mail: hibimath.sci.osaka-u.ac.jp Received November 17, 1998; accepted December 4, 1998
Componentwise linear ideals were introduced earlier to generalize the result that the StanleyReisner ideal I 2 of a simplicial complex 2 has a linear resolution if and only if its Alexander dual 2* is CohenMacaulay. It turns out that I 2 is componentwise linear if and only if 2* is sequentially CohenMacaulay. In this paper we discuss Betti number properties of componentwise linear ideals. Let I be a graded ideal of a polynomial ring S and Gin(I ) the generic initial ideal of I with respect to the reverse lexicographic term order on S. Our main result is that I and Gin(I ) admit the same graded Betti numbers if and only if I is componentwise linear. For the proof of this fact, we describe some properties of the Betti diagram of a generic initial ideal. Combinatorial implications for shifted complexes will also be discussed. 2000 Academic Press
1. GINS AND COMPONENTWISE LINEAR IDEALS Let K be a field of characteristic 0 and S=K[x 1 , ..., x n ] the polynomial ring over K with each deg x i =1. We work with the reverse lexicographic term order on S induced by x 1 > } } } >x n . For a graded ideal I of S, let Gin(I) denote the generic initial ideal of I with respect to this term order, see, e.g., [8, 10]. If I is a graded ideal of S, then we write I ( j ) for the ideal generated by all homogeneous polynomials of degree j belonging to I. Moreover, we write I d for the ideal generated by all homogeneous polynomials of I whose degree is greater than or equal to d. We say that a graded ideal I/S 72 0001-870800 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
IDEALS WITH STABLE BETTI NUMBERS
73
is componentwise linear if I ( j ) has a linear resolution for all j. We refer the reader to [11, 13] for detailed information on componentwise linear ideals. Let [ ; i, i+ j (I )] i, j denote the graded Betti numbers of a graded ideal I. It is known that ; i, i+ j (I ) ; i, i+ j (Gin(I)) for all i and j [10]. The main result of the present paper is the following: Theorem 1.1. A graded ideal I of S satisfies ; i, i+ j (I )=; i, i+ j (Gin(I )) for all i and j if and only if I is componentwise linear. In order to prove Theorem 1.1, the following result concerning the homological data of generic initial ideals will be required. Theorem 1.2.
Let I be a graded ideal generated in degree d. Then
(a)
If ; i, i+ j (Gin(I )){0, then ; i $, i $+ j (Gin(I )){0 for all i $
(b)
If ; 0, j (Gin(I)){0, then ; 0, j $(Gin(I )){0 for all d j $< j.
Proof. Since the generic initial ideal is strongly stable, statement (a) follows from the EliahouKervaire resolution [9] of stable ideals. Let g 1 , ..., g m be the generators of I of degree d. Suppose that ; 0, j&1(Gin(I)) =0. Then consider the ideal I j&2 . Since Gin(I j&2 )=Gin(I) j&2 , we may assume that ; 0, d+1(Gin(I))=0. We have to show that Gin(I) is generated in degree d. It follows from ; 0, d+1(Gin(I))=0 that all S-polynomials of degree d+1 reduce to zero with respect to [ g 1 , ..., g m ]. By virtue of Remark 1.3 below, this implies that [ g 1 , ..., g m ] is a Grobner basis of I, equivalently, Gin(I) is generated in degree d. In fact, since (in(g 1 ), ..., in(g m )) is a strongly stable ideal, its first syzygy module is generated in degree d+1. Remark 1.3. Recall the following fact from Grobner basis theory which is a stronger version of the well known Buchberger criterion [8, Sect. 15.4]: A set of generators G=[ g 1 , ..., g m ] of I is a Grobner basis of I if and only if each S-polynomial which comes from a minimal set of homogeneous generators of the first syzygy module of the ideal (in( g 1 ), ..., in( g m )) reduces to zero with respect to G. Before starting our proof of Theorem 1.1, we state the following: Lemma 1.4. Let I and J be graded ideals of S generated in degree d with the same graded Betti numbers. Then I d+1 and J d+1 have the same graded Betti numbers. Proof.
The exact sequence 0 Ä I d+1 Ä I Ä K(&d ) ;0, d Ä 0
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induces the long exact sequence } } } Ä Tor i+1(K, I d+1 ) (i+1)+ ( j&1) Ä Tor i+1(K, I ) (i+1)+( j&1) ;
0, d Ä Tor i+1(K, K ) (i+1)+ j&(d+1) Ä Tor i (K, I d+1 ) i+ j Ä Tor i (K, I ) i+ j
;
0, d Ä Tor i (K, K) i+ j&d Ä } } } .
It then follows that ; i, i+ j (I d+1 )=; i, i+ j (I ) for all i and for all j{d, d+1. Also, ; i, i+ j (I d+1 )=0 if j d. Now, if j=d+1, then the above long exact sequence becomes 0, d 0 Ä Tor i+1(K, I ) i+1+d Ä Tor i+1(K, K ) ;i+1
Ä Tor i (K, I d+1 ) i+d+1 Ä Tor i (K, I ) i+d+1 Ä 0. n ) ; 0, d (I)&; i+1, i+1+d (I). Hence, ; i, i+d+1(I d+1 )=; i, i+d+1(I )+( i+1 The same formulae are valid for ; i, i+ j (J ). This completes the proof.
K
We are now in the position to give a proof of Theorem 1.1. Proof of Theorem 1.1. First, suppose that I is componentwise linear. The following formula for the graded Betti numbers of a componentwise linear ideal I is known [11]: ; i, i+ j (I )=; i (I ( j ) )&; i (mI ( j&1) ). Here m is the irrelevant maximal ideal (x 1 , ..., x n ) of S. Since a strongly stable ideal is componentwise linear and since Gin(I ) is strongly stable, the same formula is valid for Gin(I ). Therefore, it suffices to prove that ; i (I ( j ) )=; i (Gin(I ) ( j ) ) and ; i (mI ( j&1) )=; i (m Gin(I ) ( j&1) ). Since I ( j ) has a linear resolution, it follows from the BayerStillman theorem [3], which guarantees that the regularity of an ideal and that of its generic initial ideal coincide, that Gin(I ( j ) )=Gin(I ) ( j ) . Since I ( j ) and Gin(I ( j ) ) have the same Hilbert function, and since the Betti numbers of a module with linear resolution are determined by its Hilbert function, the first equality follows. To prove the second one, we note that mI ( j&1) has again a linear resolution and that, by the same reason as before, m Gin(I) ( j&1) =Gin(mI ( j&1) ). Second, suppose that I and Gin(I ) have the same graded Betti numbers. Let max(I ) (resp. min(I)) denote the maximal (resp. minimal) degree of a homogeneous generator of I. To show that I is componentwise linear, we work with induction on r=max(I )&min(I ). Set d=min(I ). Let r=0. Since I and Gin(I ) have the same graded Betti numbers, it follows that Gin(I ) is generated in degree d. Since Gin(I ) is a strongly stable ideal, we have that Gin(I ) has a linear resolution, hence I has a linear resolution.
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Now, suppose that r>0. Since Gin(I d+1 )=Gin(I ) d+1 , our induction hypothesis and Lemma 1.4 imply that I d+1 is componentwise linear. Thus, it suffices to prove that I ( d ) has a linear resolution. Suppose this is not the case. Then, by the BayerStillman theorem, Gin(I ( d ) ) has regularity >d. Moreover, since Gin(I ( d ) ) is strongly stable, its regularity equals max(Gin(I ( d ) )). It follows from Theorem 1.2 that Gin(I ( d ) ) has a generator of degree d+1. Now, ; 0, d+1(I)=dim I d+1 &dim(mI ( d ) ) d+1 =dim I d+1 &dim(I ( d ) ) d+1 , and ; 0, d+1(Gin(I ))=dim Gin(I ) d+1 &dim(m Gin(I ) ( d ) ) d+1 =dim Gin(I ) d+1 &dim(m Gin(I ( d ) )) d+1 >dim Gin(I ) d+1 &dim Gin(I ( d ) ) d+1 , because (m Gin(I ( d ) )) d+1 is properly contained in Gin(I ( d ) ) d+1 . Hence ; 0, d+1(Gin(I ))>; 0, d+1(I ), a contradiction. This completes our proof.
K
The following result supports a conjecture of Buchsbaum and Eisenbud [5]. Corollary 1.5. Let I be a componentwise linear ideal of projective p ) for all i 0. dimension p. Then ; i (I ) ( i+1 Proof. In fact, the conjecture is known to be true for monomial ideals, see, e.g., [12]. Theorem 1.1 guarantees that the Betti numbers of a componentwise linear ideal are attained by a monomial ideal. K We conclude this section by noting that Theorem 1.1 and Theorem 1.2(b) are not valid in positive characteristic. Indeed, if characteristic p>0, then I=(x p, y p ) provides a counterexample.
3. SHIFTED COMPLEXES Let, as before, S=K[x 1 , ..., x n ] be the polynomial ring over a field K. Let 2 be a simplicial complex on the vertex set [n]=[1, ..., n] and I 2 the StanleyReisner ideal of 2. Let 2* be the Alexander dual 2*=[F/[n] : [n]"F Â 2]
76
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of 2, and let 2 s denote the algebraic shifting (cf. [4]) of 2 introduced by G. Kalai [14]. It is shown in [11] (and [13]) that I 2 is componentwise linear if and only if 2 is sequentially CohenMacaulay. We now present the exterior algebra version of Theorem 1.1 whose proof is completely analogue. We refer the reader to [2] for information on Grobner basis theory in exterior algebras. Theorem 2.1. Let J be a graded ideal of the exterior algebra. Then J and Gin(J) have the same graded Betti numbers if and only if J is componentwise linear. Let J 2 be the StanleyReisner ideal of 2 in the exterior algebra E over K. Then the following facts are known: (a) I 2 is componentwise linear if and only if J 2 is componentwise linear [11]; i+ j&1 (b) ; i, i+ j (EJ 2 )= ik=0 ( k+ j&1 ) ; k, k+ j (SI 2 ) for all i and all j [1], in particular, the graded Betti numbers of J 2 are determined by those of I 2 , and vice versa. Corollary 2.2. Let I 2 be the StanleyReisner ideal of a simplicial complex 2. Then I 2 is componentwise linear if and only if I 2 and I 2 s have the same graded Betti numbers. Proof. Recall that J 2 s =Gin(J 2 ). Hence, Theorem 2.1 together with the above observations (a) and (b) complete the proof. K For a componentwise linear ideal I 2 , the h-triangle of 2* and the graded Betti numbers of I 2 are related as follows [11, 13]: For all j, we have : ; i, i+ j (I 2 ) t i = : h i, n& j&1(2*)(t+1) i. i 0
i 0
Combining Corollary 2.2 with the above formula and noting that I 2* is componentwise linear if and only if 2 is sequentially CohenMacaulay over K, it follows that the h-triangles of 2 and 2 s coincide if 2 is sequentially CohenMacaulay over K (since (2*) s =(2 s )*). This is one direction of [6, Theorem 5.1]. The combinatorial version of Theorem 1.2 is the following: Theorem 2.3. Let 2 be a pure simplicial complex of dimension d. Then 2 is CohenMacaulay if and only if 2 s has no facet of dimension d&1. Proof. Note that 2 is pure of dimension d if and only if J 2* in the exterior algebra is generated in degree n&d&1. Now, the analogue of Theorem 1.2 for the exterior algebra yields that Gin(J 2* ) is generated in
IDEALS WITH STABLE BETTI NUMBERS
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degree n&d&1 if and only if Gin(J 2* ) has no generator of degree n&d. Since Gin(J 2* )=J (2*) s and since (2*) s =(2 s )*, we have the desired conclusion by using the fact [14] that a simplicial complex is CohenMacaulay if and only if its shifted complex is pure. K
REFERENCES 1. A. Aramova and J. Herzog, Almost regular sequences and Betti numbers, Amer. J. Math., to appear. 2. A. Aramova, J. Herzog, and T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra 191 (1997), 174211. 3. D. Bayer and M. Stillman, A criterion for detecting m-regularity, Invent. Math. 87 (1987), 111. 4. A. Bjorner and G. Kalai, An extended Euler Poincare theorem, Acta Math. 161 (1988), 279303. 5. D. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3, Amer. J. Math. 3 (1977), 447485. 6. A. M. Duval, Algebraic shifting and sequentially CohenMacaulay simplicial complexes, Electron. J. Combin. 3 (1996), R21. 7. J. A. Eagon and V. Reiner, Resolutions of StanleyReisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), 265275. 8. D. Eisenbud, ``Commutative Algebra with a View Towards Algebraic Geometry,'' Graduate Texts in Math., Vol. 150, Springer-Verlag, New YorkBerlin, 1995. 9. S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 125. 10. M. Green, Generic initial ideals, in ``Six Lectures on Commutative Algebra'' (J. Elias, J. M. Giral, R. M. Mir'o-Roig, and S. Zarzuela, Eds.), Progress in Math., Vol. 166, pp. 119185, Birkhauser, Basel, 1998. 11. J. Herzog and T. Hibi, Componentwise linear ideals, Nagoya Math. J., in press. 12. J. Herzog and M. Kuhl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), 16271646. 13. J. Herzog, V. Reiner, and V. Welker, Componentwise linear ideals and Golod rings, Mich. Math. J. 46 (1999), 211223. 14. G. Kalai, Characterization of f-vectors of families of convex sets in R d. Part I. Necessity of Eckhoff's conditions, Israel J. Math. 48 (1984), 175195.