On the Castelnuovo-Mumford regularity and the arithmetic degree of ...

Math. Z. 229, 519–537 (1998)

c Springer-Verlag 1998

On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals Lˆe Tuˆan Hoa, Ngˆo Viˆet Trung Institute of Mathematics, Box 631, B`o Hˆo, Hanoi, Vietnam Received April 28, 1997; in final form July 16, 1997

Dedicated to Professor Mario Fiorentini on the occasion of his 75-th birthday Abstract In the first part of this paper we show that the CastelnuovoMumford regularity of a monomial ideal is bounded above by its arithmetic degree. The second part gives upper bounds for the Castelnuovo-Mumford regularity and the arithmetic degree of a monomial ideal in terms of the degrees of its generators. These bounds can be formulated for an arbitrary homogeneous ideal in terms of any Gr¨obner basis. 1. Introduction Let I be a homogeneous ideal in a polynomial ring R := k[X1 , . . . , Xn ], k any field. Given any homogeneous prime ideal P in R, we define multI (P ) to be the length of the largest ideal of finite length in the ring RP /IRP . The arithmetic degree of I is defined as follows: X adeg(I) := multI (P ) deg(P ), where P runs over all homogeneous primes in R. This notion was introduced by Bayer and Mumford [B-M] in order to study the complexity of I. It is obvious that multI (P ) 6= 0 if and only if P is an associated prime of I and that multI (P ) = `(RP /IRP ) if P is a minimal associated prime Mathematics Subject Classification (1991): 13H15 The authors are partially supported by the National Basic Research Program. The first author is also supported by Massey University, New Zealand

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of I. Hence adeg(I) is a natural generalisation of the classical degree notion deg(I) which takes into account only those associated primes P of I with dim R/P = dim R/I. We refer the readers to the excellent survey of Vasconcelos [V] on various refined degree notions. The most important invariant in computational algebraic geometry is perhaps the Castelnuovo-Mumford regularity. There are several characterizations for the Castelnuovo-Mumford regularity of a graded R-module E [E-G]. But we shall use here the following definition: i reg(E) := min{t| HM (E)n−i = 0 for all n > t, i ≥ 0}, i (E) where HM n−i denotes the (n − i)-graded part of the i-th local cohomology module of E with respect to the maximal graded ideal M of R. It is well-known that reg(I) provides upper bounds for the degrees of the syzygies of I, for instances, D ≤ reg(I), where D denotes the maximal degree of the minimal generators of I. Due to a result of Bayer and Mumford [B-M, Proposition 3.6] we also know that adeg(I) ≤ reg(I)n . Note that

reg(I) = reg(R/I) + 1. The first part of this paper deals with the problem of bounding reg(I) in terms of adeg(I). We will prove the following result: Theorem 1.1. Let I be a monomial ideal in R. Then reg(I) ≤ adeg(I). Our proof is based on a Bezout-type theorem for the arithmetic degree of monomial ideals found in [S-T-V]. Moreover, we will also characterize the class of unmixed monomial ideals for which reg(I) = adeg(I) (Theorem 2.6). The above inequality is better than the bound D ≤ adeg(I) of [S-T-V, Theorem 3.1]. Unfortunately, it does not hold for an arbitrary homogeneous ideal I in R. An example taken from [S-V2] and [V] shows that adeg(I) can be arbitrarily less than reg(I). The second part of this paper gives upper bounds for reg(R/I) and adeg(I) in terms of the degrees of the minimal generators of I. In general, upper bounds for reg(R/I) and adeg(I) are doubly exponential in terms of D [B-M]. Let in(I) denote the initial ideal of I with respect to an arbitrary term order. Since reg(R/I) ≤ reg(R/in(I)) and adeg(I) ≤ adeg(in(I)) (see e.g. [B-M] and [S-T-V]), it makes sense to concentrate on the case of monomial ideals. In this case we get much better bounds for reg(I) and adeg(I) in terms of D.

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Theorem 1.2. Let I be a monomial ideal in R with the minimal generators {m1 , ..., ms }. Assume that deg(m1 ) ≥ · · · ≥ deg(ms ). Set u = min{s, n} and e = dim R/I + u − n. Then (i) reg(R/I) ≤ deg(m1 ) + · · · + deg(mu ) − u. (ii) adeg(I) ≤ deg(m1 ) · · · deg(mu ) − e. Equalities hold in (i) or (ii) if and only if I is a complete intersection. Note that e ≥ 0 and that there are examples which show that one can not replace u by ht(I). The bound for reg(R/I) is a consequence of the bound reg(R/I) ≤ deg(F ) − ht(I), where F is the least common multiple of m1 , . . . , ms . This bound is inspired of a similar bound proved by Bruns and Herzog [BH, Theorem 3.1]. As a consequence (Corollary 3.3) we will give a bound for reg(I q ) which is better than a bound recently found by Smith and Swanson [S-S]. The bound for adeg(I) is a significant improvement of the bound adeg(I) ≤ deg(m1 ) · · · deg(ms )−e of [S-T-V, Theorem 3.1], and the proof here is completely different. Our approach also leads to another bound for adeg(I) which uses to be better but involves non-conventional invariants of I (Theorem 4.3). Applying Theorem 1.2 to the initial ideal of an aribtrary homogeneous ideal I we immediately obtain the following consequences which give upper bounds for reg(R/I) and adeg(I) in terms of the degrees of the elements of a Gr¨obner basis of I. Corollary 1.3. Let I be any homogeneous ideal in R. Let {g1 , ..., gs } be a Gr¨obner basis of I with respect to an arbitrary term order. Assume that deg(g1 ) ≥ · · · ≥ deg(gs ). Set u = min{s, n} and e = dim(I) − u + n. Then (i) reg(R/I) ≤ deg(g1 ) + · · · + deg(gu ) − u. (ii) adeg(I) ≤ deg(g1 ) · · · deg(gu ) − e. It should be mentioned that for a generic choice of the variables of R, reg(R/I) is bounded above by deg(g1 ) − 1 [B-S]. However, it is difficult to know when a given choice of variables is generic. The above bound for adeg(I) is an option in getting a Bezout-type theorem for the arithmetic degree (cf. [S-T-V, Corollary 5.6]). 2. Relationship between reg(I) and adeg(I) In this section we will prove Theorem 1.1 and we will characterize the class of unmixed monomial ideals for which reg(R/I) = adeg(I) − 1. First we shall prepare some lemmas.

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Lemma 2.1. Let I be an arbitrary homogeneous ideal and x an arbitrary linear form in R. Then reg(R/I) ≤ max{reg(R/I : x) + 1, reg(R/(I, x))}.

Proof. From the exact sequence x

0 → R/I : x →→ R/I → R/(I, x) → 0 we derive, for i ≥ 0, the exact sequence of local cohomology modules: i i i HM (R/I : x)q−1 → HM (R/I)q → HM (R/(I, x))q .

Hence the conclusion is immediate from the definition of the CastelnuovoMumford regularity. Lemma 2.2. Let I be an arbitrary homogeneous ideal and x an arbitrary homogeneous element in R. Then (i) adeg(I : x) ≤ adeg(I). (ii) adeg(I : x) ≤ adeg(I) − 1 if x belongs to a minimal associated prime of I. Proof. Given any homogeneous prime P in R, the multiplication by x induces an injective map RP /(I : x)RP → RP /IP . Thus, by the definition of multI (P ), multI:x (P ) ≤ multI (P ) Summing up over all primes of R we then obtain (i). If x belongs to a minimal associated prime P of I, then RP /IRP is of finite length and (I, x)RP /IRP , the image of RP /(I : x)RP in RP /IRP , is a proper ideal. Therefore multI:x (P ) = `((I, x)RP /IRP ) < `(RP /IRP ) = multI (P ). From this (ii) immediately follows. The following result is the aforementioned Bezout-type theorem for the arithmetic degree of monomial ideals. It follows from [S-T-V, Lemma 3.3 and Lemma 3.4]. Lemma 2.3. Let I be a monomial ideal and m a monomial of R. Then adeg(I, m) ≤ adeg(I) deg(m).

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Proof of Theorem 1.1 . Let I be a monomial ideal in R. If adeg(I) = 1, I is a prime ideal. In this case, reg(I) = adeg(I) = 1. If n = 1, I is a principal ideal (X1t ). In this case, reg(I) = adeg(I) = t. Therefore, we may assume that adeg(I) > 1 and n > 1. Choose a variable X which belongs to a minimal associated prime of I. By Lemma 2.1 we have (1)

reg(I) ≤ max{reg(I : X) + 1, reg(I, X)}.

Note that I : X is also a monomial ideal. By induction on the value of the arithmetic degree we may assume that reg(I : X) ≤ adeg(I : X). Using Lemma 2.2 (ii) we obtain (2)

reg(I : X) + 1 ≤ adeg(I : X) + 1 ≤ adeg(I).

Put S = R/(X) and J = (I, X)/(X). Then S is a polynomial ring over k in n − 1 variables and J is a monomial ideal in S. By induction on n we may assume that reg(J) ≤ adeg(J). We have reg(I, X) = reg(J) and adeg(I, X) = adeg(J). Therefore, (3)

reg(I, X) ≤ adeg(I, X) ≤ adeg(I),

where the last inequality follows from Lemma 2.3. Combining (1), (2) and (3) we obtain reg(I) ≤ adeg(I). The proof of Theorem 1.1 is now complete. The following example ([S-V2, Example 2] and [V, Example 4.2 (b)]) shows that Theorem 1.1 is false for an arbitrary homogeneous ideal I which is not generated by monomials. Example. Let I = (X12 , X1 X2 , X22 , X1 X4t − X2 X3t ), t ≥ 1. Then I is a primary ideal with adeg(I) = 2 and reg(I) = t + 1. Hence adeg(I) can be arbitrary less than reg(I). Next we will look for conditions under which the inequalities of Lemma 2.2 (ii) and Lemma 2.3 become equalities. Lemma 2.4. Let I be a monomial ideal which has no embedded associated primes. Let X be a variable of R which belongs to an associated prime P of I. Then adeg(I : X) = adeg(I) − 1 if and only if the P -primary component of I has the form (Xi1 , . . . , Xir , X t ), t ≥ 1, and P is the only associated prime of I containing X.

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Proof. By the assumption every associated prime of I is minimal over I. Therefore, from the proof of Lemma 2.2 (ii) we can see that adeg(I : X) = adeg(I) − 1 if and only if the following condittions are satisfied: (i) `((I, X)RP /IRP ) = `(RP /IRP ) − 1, (ii) `((I, X)RP 0 /IRP 0 ) = `(RP 0 /IRP 0 ) for any associated prime P 0 6= P of I. The first condition means (I, X)RP = P RP or, equivalently, (Q, X) = P , where Q denotes the P -primary component of I. Since Q is a monomial ideal, (Q, X) = P if and only if Q has the form (Xi1 , . . . , Xir , X t ), t ≥ 1. The second conditions means (I, X)RP 0 = RP 0 or, equivalently, X 6∈ P 0 for any associated prime P 0 6= P of I. Lemma 2.5. Let I be a monomial ideal and X an arbitrary variable of R. Let I = ∩Qi be a minimal primary decomposition of I. Then (I, X) = ∩(Qi , X). It is a minimal primary decomposition of (I, X) if adeg(I, X) = adeg(I). Proof. It is clear that (I, X) ⊆ ∩(Qi , X). Conversely, if m is an arbitrary monomial of ∩(Qi , X) not divisible by X, then m ∈ Qi for all i, hence m ∈ I. This proves the relation (I, X) = ∩(Qi , X). Let Pi be the associated prime of Qi . Then (Pi , X) is a prime ideal and (Qi , X) is a (Pi , X)-primary ideal. Hence the set of the associated primes of (I, X) is contained in the set of the primes (Pi , X). Put S = R/(X), J = (I, X)/(X), and Pi = (Pi , X)/(X). We may consider S as a subalgebra of R and J as a contraction of I. Therefore, there is an injective homomorphism from S/J to R/I which maps Pi to Pi . From this it follows that mult(I,X) (Pi , X) = multJ (Pi ) ≤ multI (Pi ). If adeg(I, X) = adeg(I), then X X mult(I,X) (Pi , X) = multI (Pi ). Thus, mult(I,X) (Pi , X) = multI (Pi ) for all i and all primes (Pi , X) are different. Since mult(I,X) (Pi , X) = multI (Pi ) > 0, (Pi , X) must be an associated prime of (I, X). Hence we can conclude that (I, X) = ∩(Qi , X) is a minimal primary decomposition of (I, X). Now we are able to prove the following result.

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Theorem 2.6. Let I be a monomial ideal in R which has no embedded associated primes. Then reg(I) = adeg(I) if and only if for every associated prime P of I, there is a variable X such that the P -primary component of I is of the form (Xi1 , . . . , Xir , X t ), t ≥ 1, and P is the only associated prime of I which contains the variable X. If that is the case, reg(I) agrees with the maximal degree of the minimal generators of I. Notice that there are examples which shows that the condition reg(I) = adeg(I) does not imply that I has no embedded primary component. Such an example will be given after the proof of Theorem 2.6. Proof. Assume that reg(I) = adeg(I). Let P be an arbitrary associated prime of I. Let X be a variable of R which belongs to P . By the proof of Theorem 1.1 we must have an equality in (1). That means reg(I) = max{reg(I : X) + 1, reg(I, X)}. If reg(I) = reg(I : X) + 1, using Theorem 1.1 and Lemma 2.2 (ii) we get reg(I : X) ≤ adeg(I : X) ≤ adeg(I) − 1 = reg(I) − 1 = reg(I : X). Hence adeg(I : X) = adeg(I) − 1. By Lemma 2.5, the P -primary component of I must be of the form (Xi1 , . . . , Xir , X t ), t ≥ 1, and P is the only associated prime of I which contains X. If reg(I) = reg(I, X), using Theorem 1.1 and Lemma 2.3 we get reg(I, X) ≤ adeg(I, X) ≤ adeg(I) = reg(I) = reg(I, X). Hence adeg(I, X) = adeg(I) and reg(I, X) = adeg(I, X). Let Q be the P -primary component of I. By Lemma 2.5, (P, X) is an associated prime of (I, X) and (Q, X) is the (P, X)-primary component of (I, X). Using induction on n we may assume that (Q, X) has the form (Xi1 , . . . , Xir , Y t ), t ≥ 1, and (P, X) is the only associated prime of (I, X) which contains Y . Therefore, Q has the form (Xi1 , . . . , Xir , Y t ) and, by Lemma 2.5, P is the only associated prime of I which contains Y . This proves the necessary part of Theorem 2.6. Conversely, assume that for every associated prime P of I, there is a variable X such that the P -primary component of I is of the form (Xi1 , . . . , Xir , X t ), t ≥ 1, and P is the only associated prime of I which contains the variable X. Then multI (P ) = t. Let m denote the product of all such terms X t . Clearly, m is a minimal generator of I which has the maximal degree.

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It is well-known that deg(m) ≤ reg(I). On the other hand, by Theorem 1.1 we have reg(I) ≤ adeg(I) = deg(m). Therefore, reg(I) = adeg(I) = deg(m). So we have proved Theorem 2.6.

Examples. 1) I = (X1 , X2 ) ∩ (X1 , X3 ) ∩ (X4 ) = (X1 X4 , X2 X3 X4 ). Then reg(I) = adeg(I) = 3. This shows that for an ideal satisfying the conditions of Theorem 2.8, a variable can appear in two or more primary components. 2) I = (X1 , X2 ) ∩ (X3 , X4 ) ∩ (X2 , X3 ) = (X1 X3 , X2 X3 , X2 X4 ). This square-free monomial ideal does not satisfy the conditions of Theorem 2.8. In fact, reg(I) = 2 and adeg(I) = 3. 3) I = (X1t ) ∩ (X1t+1 , X2 ) = (X1t+1 , X1t X2 ), t ≥ 1. This ideal has an embedded primary component. However, reg(I) = adeg(I) = t + 1.

3. Upper bounds for reg(R/I) The following result is inspired by a recent result of Bruns and Herzog [BH, Theorem 3.1 (a)] which shows that the monomial entries of the matrices in a minimal free resolution of a monomial ideal divide the least common multiple of the monomial generators. Theorem 3.1. Let I be a monomial ideal in R with the minimal monomial generators {m1 , . . . , ms }. Let F denote the least common multiple of m1 , . . . , ms . Then reg(R/I) ≤ deg(F ) − ht(I).

Proof. Without loss of generality we may assume that each variable Xi appears in at least a monomial mj . Then we will use the technique of polarization (or lifting) to reduce to a square-free monomial (seeQ e.g. [Sa V1, Chap. 2] or [B-H, Sect. 2]). For j = 1, . . . , Q s, if mj = Xi ij , aij > 0, we will replace mj by the monomial m0j = Xi Yi1 . . . Yi(aij −1) , where Yi1 , . . . , Yi(aij −1) are new variables. Let ai = max{degXi (mj ); j = 1, ..., s}. Put S = k[X1 , Y12 , ..., Y1(a1 −1) , ..., Xn , Yn1 , ..., Yn(an −1) ]

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and J = (m01 , ..., m0s ). Then J is a square-free monomial ideal in S. Obviously, R/I ∼ = S/(J, X1 − Y11 , ..., X1 − Y1(a1 −1) , . . . , Xn − Yn1 , . . . , Xn − Yn(an −1) ), and X1 − Y11 , ..., X1 − Y1(a1 −1) , . . . , Xn − Yn1 , . . . , Xn − Yn(an −1) is a regular sequence on S/J. Therefore, dim S/J = a1 + · · · + an − n + dim R/I = deg(F ) − ht(I), reg(S/J) = reg(R/I). On the other hand, we may view S/J as the Stanley-Reisner ring of a simplicial complex. Then reg(S/J) ≤ dim S/J by [S-V1, Chap. 2, Lemma 2.5 (i)]. Hence we can conclude that reg(R/I) ≤ deg(F ) − ht(I). Remark. From the aforementioned result of Bruns and Herzog we can only deduce that reg(R/I) ≤ deg(F ) − 1. Recently, Smith and Swanson [S-S] have studied the Castelnuovo-Mumford regularity of the powers of a monomial ideal (see also [S]). As an immediate consequence of Theorem 3.1 we obtain the following improvement of their main result. Corollary 3.2. Let I be a monomial ideal in R with the minimal monomial generators {m1 , . . . , ms }. Let F denote the least common multiple of m1 , . . . , ms . Let J be any ideal generated by products of at most q monomials from m1 , . . . , ms such that ht(I) = ht(J), q > 0. Then reg(R/J) ≤ q deg(F ) − ht(I).

Remark. Smith and Swanson actually studied the sum of certain powers of a set of monomial ideals I1 , . . . , Im [S-S, Theorem 3.1]. Assume that every variable lies in the radical of I1 + · · · + Im . They proved that q ) ≤ q max{nl, L}, reg(I1q + · · · + Im

where l is the largest exponent of a variable occuring in any of the generating sets of Ij and L is a somewhat intricated invariant of I1 , . . . , Im . But this situation is only a special case of the setting of Corollary 3.2. In fact we q may put I = I1 + · · · + Im and J = I1q + · · · + Im . Then deg(F ) ≤ nl. Therefore, from Corollary 3.2 we obtain the better bound: q reg(I1q + · · · + Im ) ≤ qnl − ht(I) + 1.

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q Moreover, this bound still holds if we replace I1q + · · · + Im by any ideal of q1 qm the form I1 + · · · + Im with 1 ≤ qj ≤ q.

Now we will deduce Theorem 1.2 (i) from Theorem 3.1. For that we shall need the following estimate for the dimension of the quotient ring of a monomial ideal. Lemma 3.3. Let I be a monomial ideal in R with the minimal monomial generators {m1 , . . . , ms }. Assume that every variable Xi appears in at least a monomial mj . Then dim R/I ≤ deg(m1 ) + · · · + deg(ms ) − s. Proof. If n = 1, then I = (m1 ) and the conclusion is immediate. If n > 1, we may assume, without loss of generality, that Xn is contained in a highestdimensional associated prime of I. Then dim R/I = dim R/(I, Xn ). Without restriction we may assume that m1 , . . . , mt , t < s, are the monomials not divisible by Xn and that the variables appearing in these monomials are X1 , ..., Xp , p < n. Let S = k[X1 , ..., Xp ] and J the ideal in S generated by the monomials (m1 , ..., mp ). Then dim R/(I, Xn ) = dim S/J + n − p − 1. By induction on n we may assume that dim S/J ≤ deg(m1 ) + · · · + deg(mt ) − t. Since all mt+1 , ..., ms are divisible by Xn and each variable Xp+1 , ..., Xn−1 appears in at least a monomial mj , j > t, we have deg(mt+1 ) + · · · + deg(ms ) ≥ s − t + n − p − 1. Hence dim R/I = dim S/J + n − p − 1 ≤ deg(m1 ) + · · · + deg(mt ) − t + n − p − 1 ≤ deg(m1 ) + · · · + deg(ms ) − (s − t + n − p − 1) − t + n − p − 1 = deg(m1 ) + · · · + deg(ms ) − s. For convenience we recall Theorem 1.2 (i) below.

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Theorem 3.4. Let I be a monomial ideal in R with the minimal monomial generators {m1 , ..., ms }. Assume that deg(m1 ) ≥ · · · ≥ deg(ms ). Set u = min{s, n}. Then reg(R/I) ≤ deg(m1 ) + · · · + deg(mu ) − u. Equality holds above if and only if I is a complete intersection. Proof. We need to modify the above inequality as follows: Claim: Let F denote the least common multiple of m1 , . . . , ms . Let g1 , . . . , gt be monomials in I such that F is a divisor of G = lcm{g1 , . . . , gt }. Then reg(R/I) ≤ deg(g1 ) + · · · + deg(gt ) − t. To prove this claim we put I 0 = (g1 , . . . , gt ). As in the proof of Theorem 3.1 we may associate with I 0 a square-free monomial ideal J 0 = (g10 , . . . , gt0 ) in a new polynomial ring S such that dim S/J 0 = deg(G) − ht(I 0 ) and deg(gi0 ) = deg(gi ), i = 1, . . . , t. Therefore, using Theorem 3.1 and Lemma 3.3 we get reg(R/I) ≤ deg(F ) − ht(I) ≤ deg(G) − ht(I 0 ) ≤ deg(g10 ) + · · · + deg(gt0 ) − t = deg(g1 ) + · · · + deg(gt ) − t. It is clear that we may always choose g1 , . . . , gt ∈ {m1 , . . . , ms } with t = u. Since deg(gi ) ≤ deg(mi ), i = 1, . . . , u, we obtain reg(R/I) ≤ deg(m1 ) + · · · + deg(mu ) − u. Now assume that reg(R/I) = deg(m1 ) + · · · + deg(mu ) − u. We will show that this relation will imply that I is a complete intersection. We may assume that every variable appears in at least a monomial mi , i = 1, . . . , s, and that no monomial mi is a variable. Let X be an arbitrary variable. By Lemma 2.1 we know that reg(R/I) ≤ max{reg(R/I : X) + 1, reg(R/(I, X))}. Let v = min{r, n − 1}, where r is the number of monomials mi not divisible by X. By the above claim we have reg(R/(I, X)) ≤ deg(m1 ) + · · · + deg(mv ) − v. Since v < u and deg(mu ) ≥ 2, deg(m1 )+· · ·+deg(mv )−v < deg(m1 )+· · ·+deg(mu )−u = reg(R/I).

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Therefore, reg(R/(I, X)) < reg(R/I). From this it follows that reg(R/I) ≤ reg(R/I : X) + 1. Choose g1 , . . . , gu ∈ {m1 , . . . , ms } such that F = lcm{g1 , . . . , gu }. Let p be the number of monomials gi not divisible by X, i = 1, . . . , u. Then 0 ≤ p < u. Without restriction we may assume that X appears only in gp+1 , ..., gu . Put hi = gi , i = 1, . . . , p, and hi = gi /X, i = p + 1, . . . , u. Then h1 , ..., hu ∈ I : X and F/X = lcm{h1 , . . . , hu }. Let H denote the least commom multiple of the minimal monomial generators of I : X. Then H is a divisor of F/X. Hence H is a divisor of lcm{h1 , . . . , hu }. By the above claim we get reg(R/I : X) ≤ deg(h1 ) + · · · + deg(hu ) − u = deg(g1 ) + · · · + deg(gu ) − (u − p) − u. From this it follows that deg(m1 )+· · ·+deg(mu )−u ≤ deg(g1 )+· · ·+deg(gu )−(u−p)−u+1. Since deg(mi ) ≥ deg(gi ) and since u − p ≥ 1, we must have u − p = 1. So we have proved that among g1 , . . . , gu there is only a monomial divisible by X. Since X is arbitrarily chosen, g1 , . . . , gu must be a regular sequence. Hence I is a complete intersection if u = s. If u = n, R/(g1 , . . . , gn ) is an artinian Gorenstein ring. In this case, reg(R/I) = max{i| (R/I)i 6= 0}. If I is not a complete intersection, we would get reg(R/I) ≥ reg(R/(g1 , . . . , gn+1 )) ≥ reg(R/(g1 , . . . , gn )) + 1 = deg(g1 ) + · · · + deg(gn ) − n + 1 = deg(m1 ) + · · · + deg(mn ) − n + 1, a contradiction. Conversely, assume that I is a complete intersection. By considering the short exact sequences m

0 → R/(m1 , ..., mi−1 ) →i R/(m1 , ..., mi−1 ) → R/(m1 , ..., mi ) → 0, i = 1, ..., s, we can easily derive that reg(R/I) = deg(m1 ) + · · · + deg(ms ) − s. The proof of Theorem 3.4 is now complete. The following example shows that we can not replace u by ht(I) in Theorem 3.4. Example. Let t ≥ 2 and n > c ≥ 1. Set t−1 t−1 , . . . , X1 Xnt−1 , . . . , Xc Xc+1 , . . . , Xc Xnt−1 ). I = (X1t , . . . , Xct , X1 Xc+1

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Then ht(I) = c. Note that t−1 I : X1 = (X1t−1 , X2t , . . . , Xct , Xc+1 , . . . , Xnt−1 ) ,

t−1 ,... , (I, X1 ) = (X1 , X2t , . . . , Xct , X2 Xc+1

t−1 X2 Xnt−1 , . . . , Xc Xc+1 , . . . , Xc Xnt−1 ).

Using the exact sequence X

0 → R/I : X1 →1 R/I → R/(I, X1 ) → 0, we can show by induction on c that   n(t − 2) + c if i = 0, i max{j| HM (R/I)j 6= 0} = −∞ if i 6= 0, n − c,   1+c−n if i = n − c. Therefore, reg(R/I) = n(t − 2) + c, while deg(m1 ) + · · · + deg(mn ) − n = n(t − 1) deg(m1 ) + · · · + deg(mc ) − c = c(t − 1). It is clear that reg(R/I) > c(t − 1) if t > 2.

4. Upper bounds for adeg(I) The following observation is crucial for our approach. Lemma 4.1. Let I be any homogeneous ideal and x a homogeneous form in R. Then adeg(I) ≤ adeg(I : x) + adeg(I, x). Proof. Let P be any homogeneous prime in R. From the exact sequence x

0 → RP /(I : x)RP →→ RP /IRP → RP /(I, x)RP → 0 we can derive the exact sequence of local cohomology modules HP0 RP (RP /(I : x)RP ) → HP0 RP (RP /IRP ) → HP0 RP (RP /(I, x)RP ). Note that the largest ideal of finite length in RP /IRP can be identified with the local cohomology module HP0 RP (RP /IRP ). Then we get multI (P ) ≤ multI:x (P ) + mult(I,x) (P ).

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Summing up this relation over all homogeneous primes P of R we will get the conclusion. Now we will prove Theorem 1.2 (ii). For convenience we restate it below. Theorem 4.2. Let I be a monomial ideal in R with the minimal monomial generators {m1 , ..., ms }. Assume that deg(m1 ) ≥ . . . ≥ deg(ms ). Set u = min{s, n} and e = dim R/I + u − n. Then adeg(I) ≤ deg(m1 ) · · · deg(mu ) − e. Equality holds above if and only if I is a complete intersection. Proof. If n = 1, I is a principal ideal (X1l ). In this case, adeg(I) = deg(I) = l (e = 0). Let n ≥ 2. Set di = deg(mi ), i = 1, . . . , s. Given a monomial ideal J we write u(J) and e(J) for the corresponding invariants defined similarly as u and e for I. If ds = 1, ms is a variable, say Xn . Let J be the ideal in S = k[X1 , . . . , Xn−1 ] generated by m1 , . . . , ms . Then R/I ' S/J and u − 1 ≤ u(J). Hence e(J) = dim S/J + u(J) − (n − 1) ≥ dim R/I + u − n = e. Since u(J) ≤ u, using the induction hypothesis we get adeg(I) = adeg(J) ≤ d1 · · · du − e(J) ≤ d1 · · · du − e. Assume that di ≥ 2, i = 1, . . . , s. By a suitable order of the variables we may write m1 = X1l1 · · · Xilr , where l1 ≥ 1, . . . , lr ≥ 1. Moreover, we may assume that X1 belongs to a highest-dimensional associated prime of I. This means dim R/(I, X1 ) = dim R/I. Put I1 = (I, X1 ), I2 = (I : X1 , X1 ), . . . , Il1 = (I : X1l1 −1 , X1 ),

Il1 +1 = (I : X1l1 , X2 ), . . . , Il1 +l2 = (I : X1l1 X2l2 −1 , X2 ), . . . , l

r−1 , Xr ), . . . , Il1 +···+lr−1 +1 = (I : X1l1 · · · Xr−1

l

r−1 Xrlr −1 . Il1 +···+lr = I : X1l1 · · · Xr−1

Note that l1 + · · · + lr = d1 . A multiple application of Lemma 4.1 yields adeg(I) ≤ adeg(I1 ) + adeg(I2 ) + · · · + adeg(Id1 ). Now we will estimate adeg(Ii ), i = 1, . . . , d1 .

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Let J1 be the ideal in S1 = k[X2 , ..., Xn ] generated by the monomials mi not divisible by X1 . Then R/I1 = S1 /J1 . Let u1 = u(J1 ). Then u1 ≤ u−1 and e(J1 ) = dim S1 /J1 + u1 − (n − 1) = dim R/I + u1 − (n − 1) = e − (u − u1 − 1). Note that the monomial generators of J1 are contained in the set {m2 , ..., ms}. Using the induction hypothesis we get adeg(I1 ) = adeg(J1 ) ≤ d2 · · · du1 +1 − e + (u − u1 − 1) ≤ d2 · · · du1 +1 2u−u1 −1 − e ≤ d2 · · · du − e. For each i = 2, ..., d1 , the monomial ideal Ii contains a variable occuring in m1 . Similarly as above, we can associate with Ii a monomial ideal Ji in a polynomial ring Si in n − 1 variables such that R/Ii = Si /Ji . Since ht(Ji ) ≤ u(Ji ), e(Ji ) = dim Si /Ji + ht(JI ) − (n − 1) ≥ 0. Each ideal Ji has a generating set of less than s monomials which are divisors of the monomials m2 , ..., ms . Therefore, u(Ji ) ≤ u − 1, whence the induction hypothesis gives adeg(Ii ) = adeg(Ji ) ≤ d2 · · · du − e(Ji ) ≤ d2 · · · du ,

j = 2, . . . , d1 .

Summing up all the estimates for adeg(Ii ), i = 1, . . . , d1 , we then obtain adeg(I) ≤ (d2 · · · du − e) + (d1 − 1)d2 · · · du = d1 d2 · · · du − e Now assume that adeg(I) = d1 · · · du − e. By the above arguments we must have u = u(J1 ) + 1 and adeg(J1 ) = d2 · · · du − e(J1 ). Using induction on n we may assume that J1 is generated by a regular sequence of monomials of degree d2 , . . . , du , say m2 , . . . , mu . From this it follows that I is generated by the monomials divisible by X1 and m2 , . . . , mu . The above arguments also shows that adeg(Jd1 ) = d2 · · · du . Hence we may assume that the ideal Jd1 is generated by a regular sequence of monomials of degrees d2 , . . . , du . By the definition of Jd1 each of these monomials are either a divisor of m2 , . . . , mu or a proper divisor of mu+1 , . . . , ms . Since deg(m2 ) ≥ . . . ≥ deg(ms ), the generators of Jd1 must be exactly m2 , . . . , mu . Therefore l

r−1 I : X1l1 · · · Xr−1 Xrlr −1 = (Xr , m2 , . . . , mu ).

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From this we can deduce that s = u (otherwise mu+1 , ..., ms would be divisors of m1 ) and that the monomials m2 , . . . , mu are not divisible by any of the variables X1 , . . . , Xr . Hence I = (m1 , ..., ms ) is a complete intersection. Conversely, if I is a complete intersection, then adeg(I) = deg(I) = d1 · · · ds . Note that u = s and e = 0 in this case. The proof of Theorem 4.2 is now complete. Theorem 4.2 naturally leads to the question whether the number u can be replaced by ht(I). Note that dim R/I + ht(I) − n = 0. The following example will show that this question has a negative answer. Example. We consider again the example of Sect. 3: t−1 t−1 I = (X1t , . . . , Xct , X1 Xc+1 , . . . , X1 Xnt−1 , . . . , Xc Xc+1 , . . . , Xc Xnt−1 ) t−1 = (X1 , . . . , Xc ) ∩ (X1t , . . . , Xct , Xc+1 , . . . , Xnt−1 ), t ≥ 2, an > c ≥ 1.

Then ht(I) = n. It is easy to verify that


t−1 , . . . , Xnt−1 ) adeg(I) = 1 + ` (X1 , . . . , Xc )/(X1t , . . . , Xct , Xc+1 = (t − 1)n−c (tc − 1) + 1,

while d1 · · · du − e = tn − n + c and d1 · · · dc = tc . Remark. From Theorem 4.2 it follows, for an monomial ideal I, that adeg(I) ≤ Du , where D denotes, as usual, the largest degree of the generators of I. For an arbitrary homogeneous ideal I, Bayer and Mumford [B-M, Proposition 3.6] already gave the bound adeg(I) ≤ reg(I)n . If char(k) = 0, reg(I) ≤ n−1 [G]. Therefore, (2D)2 adeg(I) ≤ (2D)n2

n−1

.

It is a common belief that an upper bound for adeg(I) must be at least doubly exponential in terms of D as shown above. Since we could not find a proof for this fact in the literature, we include here the following argument. Let f, f1 , . . . , fs be polynomials in R such that f ∈ (f1 , . . . , fs ). Let a be the least number such that we can write f = f1 g1 + · · · + fs gs with deg(gi ) ≤ a. It is well known that a bound for a must be at least doubly exponential in terms of the largest degree D of f1 , . . . , fs [M-M]. Let

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h, h1 , . . . , hs be the homogenized forms of f, f1 , . . . , fs in the polynomial ring S = k[X0 , X1 , . . . , Xn ]. Put I = (h1 , . . . , hs ). Then a is also the least number such that X0a h ∈ I. By [S-T-V, Theorem 2.1] we know that a ≤ adeg(I : h). Since adeg(I : h) ≤ adeg(I), an upper bound for adeg(I) in terms of D must be at least doubly exponential in terms of D. The above approach also leads us to the following upper bound for adeg(I) which is usually better than the bound of Theorem 4.2 (as one can immediately realize). However, we need to introduce some non-conventional notations. Theorem 4.3. Let I be a monomial ideal in R with the minimal monomial generators m1 , . . . , ms . Let δ1 be the largest degree of the monomials divisible by X1 among m1 , ..., ms . Let δi , 2 ≤ i ≤ n, be the largest degree of the monomials which are obtained from m1 , ..., ms by deleting all variables X1 , ..., Xi−1 and which are divisible by Xi , where δi = 1 if no such a monomial exists. Then adeg(I) ≤ min{δ1 , a1 + 1} · · · min{δn , an + 1}. Proof. The case n = 1 is immediate. Let n ≥ 2. We may assume that every variable occurs in some monomial mi . For any ideal J we will denote the corresponding invariants of ai and δi by ai (J) and δi (J). If δn = 1, then Xn ∈ I. Put ms = Xn . Let J be the ideal in S = k[X1 , . . . , Xn−1 ] generated by m1 , . . . , ms−1 . Then ai (J) = ai and δi (J) = δi , i = 1, . . . , n − 1. By induction on n we get adeg(I) = adeg(J) ≤ min{δ1 , a1 + 1} · · · min{δn−1 , an−1 + 1} = min{δ1 , a1 + 1} · · · min{δn , an + 1}. Now we may assume that δn ≥ 2. By Lemma 4.1 we have adeg(I) ≤ adeg(I : Xn ) + adeg(I, Xn ). It is obvious that ai (I : Xn ) = ai and δi (I : Xn ) = δi , i = 1, . . . , n − 1, while an (I : Xn ) = an − 1 and δn (I : Xn ) = δn − 1. By induction on δ1 we get adeg(I : X1 ) ≤ min{δ1 , a1 + 1} · · · min{δn−1 , an−1 + 1}[min{δn , an + 1} − 1]. Now let J be the ideal in S = k[X1 , . . . , Xn−1 ] generated by the monomials mi not divisible by Xn . Then S/J = R/(I, Xn ), ai (J) ≤ ai and δi (J) ≤ δi , i = 1, . . . , n − 1. By induction on n we get adeg(I, Xn ) = adeg(J) ≤ min{δ1 , a1 + 1} · · · min{δn−1 , an−1 + 1}.

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Summing the above estimates for adeg(I : Xn ) and adeg(I, Xn ) we obtain adeg(I) ≤ min{δ1 , a1 + 1} · · · min{δn , an + 1}. The following examples show that in many cases the bound for adeg(I) in Theorem 4.3 is rather good. For instance, it can be attained by a noncomplete intersection. Examples. 1) Consider again the ideal t−1 t−1 I = (X1t , . . . , Xct , X1 Xc+1 , . . . , X1 Xnt−1 , . . . , Xc Xc+1 , . . . , Xc Xnt−1 ), t ≥ 2, n > c ≥ 1.

We already know that adeg(I) = (t−1)n−c (tc −1)+1, while d1 · · · du −e = tn − n + c. However, min{δ1 , a1 + 1} · · · min{δn , an + 1} = tc (t − 1)n−c . If t = 2, this bound coincides with adeg(I). Of course, I is a non-complete intersection in this case. 2) If I is a square-free monomial ideal in n variables, then it follows from Theorem 4.3 that adeg(I) ≤ 2n . This bound can be also derived from the fact that there are at most 2n monomial prime ideals in R.

Acknowledgement. The first named author would like to thank Massey University for financial support and hospitality during the preparation of this paper. The authors are grateful to J. Herzog for many inspiring conversations on this topics which led to the proof of Theorem 1.1. After this paper was submitted, the authors have learned in a private communication that recently J. St¨uckrad also obtained Theorem 1.1.

References [B-M] Bayer, D., Mumford, D.: What can be computed in algebraic geometry? In: D. Eisenbud and L. Robbiano (eds.): Computational Algebraic Geometry and Commutative Algebra, Proceedings, Cortona 1991, pp. 1–48, Cambridge University Press 1993 [B-S] Bayer, D., Stillman, M.: A criterion for detecting m-regularity. Invent. Math. 87, 1–11 (1987) [B-H] Bruns, W., Herzog, J.: On multigraded resolutions. Math. Proc. Camb. Phil. Soc. 118, 245–257(1995) [E-G] Eisenbud, D., Goto, S.: Linear free resolutions and minimal multiplicities. J. Algebra 88, 89–133(1984) [G] Galligo, A.: Theoreme de division et stabilite en geometrie analytique locale. Ann. Inst. Fourier 29, 107–184 (1979) [M-M] Mayr, E., Meyer, A. R.: The complexity of the word problem for commutative semigroups and polynomial ideals. Adv. in Math. 46, 305–329(1982)

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[S-S] Smith, K. E., Swanson, I.: Linear bounds on growth of associated primes. Preprint 1996. [S-T-V] Sturmfels, B., Trung, N. V., Vogel, W.: Bounds on degrees of projective schemes. Math. Ann. 302, 417–432(1995) [S-V1] St¨uckrad, J., Vogel, W.: Buchsbaum rings and applications. Berlin: Springer 1986 [S-V2] St¨uckrad, J., Vogel, W.: Castelnuovo bounds for locally Cohen-Macaulay schemes. Math. Nachr. 136, 307–320 (1988) [S] Swanson, I.: Powers of ideals, Primary decompositions, Artin-Rees lemma and regularity. Math. Ann. 307, 299–313(1997) [V] Vasconcelos, W.: The degrees of graded modules. In: Proceedings of Summer School on Commutative Algebra, Bellaterra 1996, Vol. II, pp. 141–196 CRM Publication 1996