ASYMPTOTIC BEHAVIOUR OF THE

CASTELNUOVO-MUMFORD REGULARITY

Dale Cutkovsky Deparment of Mathematics, University of Missouri Columbia, MO 65211, USA

Jürgen Herzog Fachbereich Mathematik, Universität-GHS Essen Postfach 103764, Germany

Ngô Viêt Trung Institute of Mathematics Box 631, Bo Ho, Hanoi, Vietnam

Dedicated to F. Hirzebruch on the occasion of his seventieth birthday

1. Introduction

Let I be any homogeneous ideal in a polynomial ring A = k[X_{1}, …, X_{r}], where k is an arbitrary field. Let

0 → … → F_{j} → … → F_{0} → I → 0

be the minimal free resolution of I. Write a_{j} for the maximum degree of the generators of F_{j}. The Castelnuovo-Mumford regularity reg(I) is defined to be the maximum of a_{j} − j, j ≥ 0. It is an important invariant which measures the complexity of the ideal I. For instances, reg(I) is the maximum degree n such that there is an index j ≥ 0 such that H_{𝔪}^{j}(I)_{n−j} ≠ 0, where H_{𝔪}^{j}(I) denotes the jth local cohomology module of I with respect to 𝔪, or Tor_{j}(k, M)_{n+j} ≠ 0, see e.g. Eisenbud and Goto [EG], Bayer and Mumford [BM].

Recently, Swanson [S] has proved that there is a number D such that for all n ≥ 1,

reg(I^{n}) ≤ nD.

This result follows from a linear bound on the growth of associated primes of ideals which is closely linked with a version of the uniform Artin-Rees lemma along the line of Huneke’s uniform bounds in noetherian rings [Hu]. However, Swanson could not provide a formula for the number D in general.

Typeset by AMS-TEX

A possible candidate for D is reg(I). In fact, if dim A/I = 1, Geramita, Gimigliano and Pittelloud [GGP] and Chandler [C] showed that reg(I^{n}) ≤ n reg(I) for all n ≥ 1. This result can be easily generalized to the case depth A/I^{n} ≥ dim A/I − 1 for all n. The same bound also holds for a Borel-fixed monomial ideal I by the Eliahou-Kervaire resolution [EK]. See [SS] and [HT] for explicit linear bounds for reg(I^{n}) when I is an arbitrary monomial ideal.

The problem of bounding reg(I^{n}) is also of interest in algebraic geometry. Given a smooth complex projective variety X ⊂ ℙ^{r}, and let I_{X} be the ideal sheaf of the embedding of X. The Castelnuovo-Mumford regularity of I_{X} is defined to be the the least integer t such that H^{i}(ℙ^{r}, I_{X}(t−i)) = 0 for all i ≥ 1. Let d_{X} denote the minimum of the degrees d such that X is a scheme-theoretic intersection of hypersurfaces of degree at most d. Bertram, Ein and Lazarsfeld [BEL] have shown that there is a number e such that

H^{i}(ℙ^{s}, I_{X}^{n}(a)) = 0

for all a ≥ nd_{X} + e, i ≥ 1. The proof used the Kodaira vanishing theorem. See [B] and [W] for related recent results.

In this paper we will use a simpler method to improve the above results of Swanson, Bertram, Ein and Lazarsfeld as follows.

Theorem 1.1. Let I be an arbitrary homogeneous ideal. Let d(I) denote the maximum degree of the homogeneous generators of I. Then

(i) There is a number e such that reg(I^{n}) ≤ nd(I) + e for all n ≥ 1.

(ii) reg(I^{n}) is a linear function for all n large enough.

Since we always have d(I) ≤ reg(I), it follows that

lim \frac{reg(I^{n})}{n} = lim \frac{d(I^{n})}{n}.

It is clear that the common limit is a positive number ≤ d(I). Therefore, the difference between reg(I^{n}) and n reg(I) can be arbitrarily large if d(I) < reg(I).

Part (i) of the above result implies that for an arbitrary projective variety X ⊂ ℙ^{r}, there is a number e such that

H^{i}(ℙ^{s}, I_{X}^{n}(a)) = 0

for all a ≥ nd_{X} + e, i ≥ 1.

However, part (ii) does not have a similar geometric version. In fact, it does not hold if we replace I^{n} by its saturation Ĩ^{n} though I^{n} and Ĩ^{n} define the same projective scheme. We will give examples of homogeneous ideals of ‘fat’ points for which reg(Ĩ^{n}) ist not a linear function for large n (Example 4.2) and, if the characteristic of k is positive, even not a periodic one (Example 4.3).

Our method is based on a natural bigrading of the Rees algebra R = ⊕_{n≥0}I^{n}t^{n} given by setting deg xt^{n} = (deg x, n) for all homogeneous element x of I^{n}. We will show that

H_{𝔪}^{j}(I^{n})_{a} ≃ H_{M}^{j}(R)_{(a,n)}
Tor_{j}^{A}(k, I^{n})_{a} ≃ Tor_{j}^{R}(R/M, R)_{(a,n)}

for all numbers a, n, where M is the ideal of R generated by the elements of 𝔪. Therefore, we only need to study the bigraded structure of H_{M}^{j}(R) and Tor_{j}^{R}(R/M, R) in order to estimate reg(I^{n}).

The proof of Theorem 1.1 (i) and (ii) will be found in Section 2 and Section 3, respectively. We would like to mention that (i) has been also obtained by Lavila-Vidal and Zarzuela by a different method (private communication) and that linear programming is used to prove (ii) and the same method can be applied to give linear bounds for reg(I_{1}^{n_{1}} … I_{m}^{n_{m}}), where I_{1}, …, I_{m} are arbitrary homogeneous ideals, and for reg(\overline{I^{n}}), where \overline{I^{n}} denotes the integral closure of I^{n}.

Acknowledgement. The third author would like to thank the Max-Planck-Institut für Mathematik, Bonn, for financial support and hospitality during the summer of 1997 when this paper was worked out. The authors are grateful to K. Chandler, K. Smith and I. Swanson for sending them the preprints [C] and [SS] and to G. Valla for stimulating discussions.

2. Linear bound for the regularity

We begin with some observation on the bigraded structure of local cohomology modules which we shall need in the proof of Theorem 1.1 (i).

Let R = ⊕_{a,n≥0}R_{(a,n)} be a noetherian bigraded ring and E = ⊕_{a,n∈ℤ}E_{(m,n)} be a bigraded R-module. We may consider R as an ℕ-graded ring with R_{n} = ⊕_{a≥0}R_{(a,n)} and E as a ℤ-graded module with E_{n} = ⊕_{a≥0}E_{(a,n)}. Let 𝔪 be any graded ideal of the graded ring R_{0}. Since E_{n} is an R_{0}-module, the local cohomology module H_{𝔪}^{i}(E_{n}) is well defined for all i ≥ 0.

Let M denote the ideal generated by the elements of 𝔪, i.e. M = ⊕_{n≥0}𝔪R_{n}. We shall see that H_{𝔪}^{i}(E_{n}) is a ℤ-graded component of the local cohomology module H_{M}^{i}(E).

Lemma 2.1. H_{𝔪}^{i}(E_{n})_{a} = H_{M}^{i}(E)_{(a,n)} for all numbers a, n.

Proof. We shall use the characterization of local cohomology modules by means of the Koszul complexes (see e.g. [BH], [H1]). Let x_{1}, …, x_{r} be a family of generating elements for 𝔪. Set x^{s} = x_{1}^{s}, …, x_{r}^{s} and denote by H^{i}(x^{s}, ·) the ith cohomology of the Koszul complex functor associated with x^{t}. Then

H_{𝔪}^{i}(E_{n}) = lim_{⟶} H^{i}(x^{t}, E_{n}),
H_{M}^{i}(E) = lim_{⟶} H^{i}(x^{t}, E).

Since the elements x_{1}, …, x_{r} have degree zero in the ℕ-graded ring R, we have

H^{i}(x^{t}, E_{n}) = H^{i}(x^{t}, E)_{n}.

¿From this it follows that

H_{𝔪}^{i}(E_{n}) = lim_{⟶} H^{i}(x^{t}, E)_{n} = H_{M}^{i}(E)_{n}.

It is clear that the equation H_{𝔪}^{i}(E_{n}) ≃ H_{M}^{i}(E)_{n} also reflects the bigraded structure in the sense that H_{𝔪}^{i}(E_{n})_{a} = H_{M}^{i}(E)_{(a,n)}. □

¿From now on let R = ⊕_{n≥0}I^{n}t^{n} be the Rees algebra of a homogeneous ideal I in a polynomial ring A = k[X_{1}, …, X_{r}]. As I is homogeneous, we may view R as a bigraded ring with

R_{(a,n)} = (I^{n})_{a}t^{n}.

Let 𝔪 = (X_{1}, …, X_{r}) be the maximal graded ideal of A. By Lemma 2.1 we have

H_{𝔪}^{i}(I^{n})_{a} = H_{M}^{i}(R)_{(a,n)}

for all numbers a, n. Therefore, we may get information on the graded structure of H_{𝔪}^{i}(I^{n}) by the bigraded structure of H_{M}^{i}(R).

Assume that I is generated by s homogeneous polynomials. Then R may be represented as a factor ring of the bigraded polynomial ring S = k[X_{1}, …, X_{r}, Y_{1}, …, Y_{s}]. Let N denote the ideal of S generated by X_{1}, …, X_{r}. It is clear that

H_{M}^{i}(R)_{(a,n)} ≃ H_{N}^{i}(E)_{(a.n)}

for all numbers a, n. We will use a bigraded minimal free resolution of R over S to study the the bigraded structure of H_{N}^{i}(R).

First we have the following description of H_{N}^{i}(S).

Lemma 2.2.

H_{N}^{i}(S) = 0, i ≠ r,
H_{N}^{r}(S) = k[X_{1}^{α_{1}} … X_{r}^{α_{r}}Y_{1}^{β_{1}} … Y_{s}^{β_{s}}| α_{1}, …, α_{r} < 0; β_{1}, …, β_{s} ∈ ℕ].

Proof. Since S is a direct product of copies of A = k[X_{1}, …, X_{r}], we have

H_{N}^{i}(S) = H_{𝔪}^{i}(A) ⊗_{A} S.

It is well-known [H1] that

H_{𝔫}^{i}(A) = 0, i ≠ r,
H_{𝔫}^{r}(A) = k[X_{1}^{α_{1}} … X_{r}^{α_{r}}| α_{1}, …, α_{r} < 0].

Hence the conclusion is immediate. □

Let d_{1}, …, d_{s} be the degree of the homogeneous generators of I. Then the bigrading of the polynomial ring S is given by

bideg X_{i} = (1, 0), i = 1, …, r,
bideg Y_{j} = (d_{j}, 1), j = 1, …, s.

This can be used to obtain information on the bigraded vanishing of H_{N}^{r}(S).

Corollary 2.3. H_{N}^{r}(S)_{(a,n)} = 0 for all a ≥ nd(I) − r + 1.

Proof. Note that d(I) = max{d_{1}, …, d_{s}}. Since

bideg X_{1}^{α_{1}} … X_{r}^{α_{r}}Y_{1}^{β_{1}} … Y_{s}^{β_{s}} = (α_{1} + ⋯ + α_{r} + β_{1}d_{1} + ⋯ + β_{s}d_{s}, β_{1} + ⋯ + β_{s}),

using Lemma 2.2 we get

H_{N}^{r}(S)_{(a,n)} = k[X_{1}^{α_{1}} … X_{r}^{α_{r}}Y_{1}^{β_{1}} … Y_{s}^{β_{s}}| α_{1}, …, α_{r} < 0;
α_{1} + ⋯ + α_{r} + β_{1}d_{1} + ⋯ + β_{s}d_{s} = a, β_{1} + ⋯ + β_{s} = n].

If a ≥ nd(I) − r + 1, then

α_{1} + ⋯ + α_{r} = a − (β_{1}d_{1} + ⋯ β_{s}d_{s})
≥ a − (β_{1} + ⋯ β_{s})d(I) = a − nd(I) ≥ 1 − r.

Hence at least one of the numbers α_{1}, …, α_{r} must be non-negative. From this it follows that H_{N}(S)_{(a,n)} = 0. □

The following result gives Theorem 1.1 (i).

Theorem 2.4. With the above notations let

0 → ⋯ → ⊕_{t}S(−a_{tj}, −b_{tj}) → ⋯ → ⊕_{t}S(−a_{t1}, −b_{t1}) → S → R → 0

be a bigraded minimal free resolution of R over S. Put c_{0} = 0, c_{j} = max_{t}{a_{tj} − b_{tj}d}, j ≥ 1, and e = max{c_{j} − j| j = 1, …, r − 1}. For all n ≥ 1 we have

reg(I^{n}) ≤ nd(I) + e.

Proof. First we will study the graded vanishing of H_{N}^{i}(R), i = 1, …, r. Rewrite the above resolution of R as follows:

0 → ⋯ → F_{j} → ⋯ → F_{1} → F_{0} = S → R → 0

Let a_{j} denote the image of the map F_{j} → F_{j−1} for j ≥ 1. Then there are the exact sequences

0 → I_{j} → F_{j−1} → I_{j−1} → 0

where I_{0} = R. Consider the derived exact sequence of local cohomology modules of these exact sequences. For i < r, we use Lemma 2.2 to deduce that

H_{N}^{i}(R) ≃ H_{N}^{i+1}(I_{1}) ≃ … ≃ H_{N}^{r−1}(I_{r−i−1})

and that there is an injective map H_{N}^{r−1}(I_{r−i−1}) → H_{N}^{r}(I_{r−i}) and a surjective map H_{N}^{r}(F_{r−i}) → H_{N}^{r}(I_{r−i}). For i = r we also have a surjective map H_{N}^{r}(F_{0}) → H_{N}^{r}(R). Therefore, for all i ≥ 0, H_{N}^{i}(R)_{(m,n)} = 0 if H_{N}(F^{r−i})_{(m,n)} = 0. By Corollary 3.2,

H_{N}^{r}(S(−a_{tj}, −b_{tj}))_{(m,n)} = 0

for m − a_{tj} ≥ (n − b_{tj})d − r + 1, where d = d(I). Therefore, H_{N}^{r}(F^{r−i})_{(m,n)} = 0 if m ≥ (n − b_{tr−i})d + a_{tr−i} − r + 1 for all t. The latter condition is satisfied if m ≥ nd + c_{r−i} − r + 1. Hence

H_{N}^{i}(R)_{(m,n)} = 0

for all m ≥ nd + c_{r−i} − r + 1.

By Lemma 2.1 we get

H_{𝔪}^{i}(I^{n})_{m−i} = H_{N}(R)_{(m−i,n)} = 0

for m−i ≥ nd+c_{r−i} −r +1, i = 1, …, r. Since nd+e ≥ nd+c_{r−i} −r +i, this vanishing holds if m ≥ nd + e. Note that H_{𝔪}^{0}(I^{n}) = 0. Then we obtain reg(I^{n}) ≤ nd + e. □

Corollary 2.5. Let X ⊂ ℙ^{r} be an arbitrary projective variety. Let I_{X} be the ideal sheaf of the embedding and d_{X} the minimum of the degrees d such that X is a scheme-theoretic intersection of hypersurfaces of degree at most d. Then there is a number e such that

H^{i}(ℙ^{r}, I_{X}^{n}(a)) = 0

for all a ≥ nd_{X} + e, i ≥ 1.

Proof. The conclusion follows from a well-known theorem of Serre which says that H^{i}(ℙ^{r}, I_{X}^{n}(a)) ≃ H_{𝔪}^{i+1}(I^{n})_{a} for i ≥ 1, see e.g. [H1]. □

3. Asymptotic behaviour of regularity

Let I be a homogeneous ideal in A = k[X_{1}, …, X_{r}]. In this section we will show that reg(I^{n}) is not only bounded by a linear function, but, for n ≫ 0, is a linear function. The approach will be similar as in the previous section.

For any A-module E each we set

reg_{i}(E) = max{a| Tor_{i}(k, E)_{a} ≠ 0} − i.

Since reg(E) = max{reg_{i}(E)| i ≥ 0}, Theorem 1.1 (ii) follows from the following result.

Theorem 3.1. Let I be an arbitrary homogeneous ideal. Then for all i ≥ 0, the function reg_{i}(I^{n}) is linear for n ≫ 0.

Recall that for any homogeneous ideal J, d(J) denotes the maximal degree of the homogeneous generators of J. It is well-known that d(J) is nothing else than reg_{0}(J). The next result encodes the fact that the linear functions associated with reg_{0}(I^{n}) and reg(I^{n}) have the same slope.

Corollary 3.2. Let I be an arbitrary homogeneous ideal. Then

lim \frac{d(I^{n})}{n} = lim \frac{reg(I^{n})}{n},

and this common limit is a positive integer ≤ d(I).

Proof. Let reg(I^{n}) = an + b and reg_{0}(I^{n}) = cn + d for n ≫ 0. Since reg_{0}(I^{n}) ≤ reg(I^{n}) for all n, it follows that c ≤ a. On the other hand, we have reg(I^{mn}) ≤ reg_{0}(I^{n})m + e for large n and all m ≥ 0. This implies that an ≤ reg_{0}(I^{n}) = cn + d for all large n. Therefore, a ≤ c, and so a = c. It is clear that c is a positive integer ≤ d(I). □

In order to prove Theorem 3.1 we shall consider the Rees algebra R = ⊕_{n≥0}I^{n}t^{n} as a factor ring of the bigraded polynomial ring S = k[X_{1}, …, X_{r}, Y_{1}, …, Y_{s}] as in Section 3. Let 𝔪 = (X_{1}, …, X_{r}) be the maximal graded ideal of A and N = 𝔪S.

Lemma 3.3. Tor_{i}^{A}(k, I^{n})_{a} ≃ Tor_{i}^{S}(S/N, R)_{(a,n)} for all a, n, i ≥ 0.

Proof. Consider a minimal free S-resolution of the Rees algebra R:

𝔽 : 0 → ⋯ → F_{j} → ⋯ → F_{1} → F_{0} = S → R → 0

Taking the n-homogeneous component is an exact functor, so that the sequence

𝔽_{n} : 0 → ⋯ → (F_{j})_{n} → ⋯ → (F_{1})_{n} → (F_{0})_{n} → R_{n} → 0

is exact. Since the modules (F_{i})_{n} are free A-modules, 𝔽_{n} is a free A-resolution for R_{n} = I^{n}. We have Tor_{i}^{S}(S/N, R) = H_{i}(𝔽/𝔪𝔽) so that

Tor_{i}^{S}(S/N, R)_{n} ≃ H_{i}(𝔽_{n}/𝔪𝔽_{n})

which is isomorphic to Tor_{i}^{A}(k, I^{n}). Hence Tor_{i}^{A}(k, I^{n})_{a} ≃ Tor_{i}^{S}(S/N, R)_{(a,n)}. □

Remark. The above free resolution 𝔽_{n} of I^{n} is not minimal in general. For instance, let

I = (X_{1}^{2}, X_{1}X_{2}, X_{2}^{2}) ⊂ A = k[X_{1}, X_{2}].

Then R = S/(f_{1}, f_{2}, f_{3}) with f_{1} = X_{2}Y_{1} −X_{1}Y_{2}, f_{2} = X_{2}Y_{2} −X_{1}Y_{3} and f_{3} = Y_{2}^{2} −Y_{1}Y_{3}. One sees easily that (f_{1}, f_{2}, f_{3}) is a height 2 perfect ideal, and hence the Rees algebra R has the S-resolution

0 → S(−5, −2)^{2} → S(−3, −1)^{2} ⊕ S(−4, −2) → S → R → 0.

Thus, if we want to compute a resolution of I^{2}, we have to take the second component of the above resolution, and get

0 → A(−5)^{2} → A(−5)^{6} ⊕ A(−4) → A(−4)^{6} → I^{2} → 0,

which, of course, is not minimal.

By Lemma 3.3 we have

reg_{i}(I^{n}) = max{a| Tor_{i}^{S}(S/N, R)_{(a,n)} ≠ 0} − i.

Notice that each Tor_{i}(S/N, R) is a finitely generated bigraded module over the bigraded polynomial ring S/N = k[Y_{1}, …, Y_{s}] with bideg Y_{i} = (d_{i}, 1), i = 1, …, s. Then Theorem 3.1 follows from the following property of such modules.

Theorem 3.4. Let E be any finitely generated bigraded module over k[Y_{1}, …, Y_{s}]. The function

ρ_{E}(n) := max{a| E_{(a,n)} ≠ 0}

is linear for n ≫ 0.

Proof. Put T = k[Y_{1}, …, Y_{s}]. It is clear that for a given exact sequence of bigraded T -modules

0 → E” → E → E′ → 0

we have ρ_{E}(n) = max{ρ_{E”}(n), ρ_{E′}(n)} for all n ∈ ℕ. Therefore, since there exists a sequence of bigraded submodules

0 = E_{0} ⊂ E_{1} ⊂ … ⊂ E_{i−1} ⊂ E_{i} = M

of E such that E_{j}/E_{j−1} is cyclic for j = 1, …, i, we may assume that E is cyclic.

We represent E as a quotient T/J. Let < be any term order, and denote by in(J) the initial ideal of J with respect to this term order. It is clear that T/J has a k-basis consisting of the residues classes of all the monomials which do not belong to in(J), and it is well-known that the residue classes of the same monomials modulo J form a (bigraded) k-basis of T/J. Therefore ρ_{E}(n) = ρ_{T/ in(J)}(n) for all n ≥ 0, and we may assume that J itself is a monomial ideal.

Let J be generated by the monomials Y_{1}^{c_{i1}} … Y_{s}^{c_{is}} for i = 1, …, p. For any a = (a_{1}, …, a_{s}) ∈ ℕ^{s} let y^{a} denote the residue class of Y_{1}^{a_{1}} …, Y_{s}^{a_{s}} in S/J. Let B_{n} denote the minimal basis of (S/J)_{n}. Then

ρ_{E}(n) = max{v(a)| y^{a} ∈ B_{n}}

with v(a) = ∑_{i} a_{i}d_{i}. Note that y^{a} ∈ B_{n} if and only if ∑_{j} a_{j} = n, and for all i = 1, …, p there exists an integer 1 ≤ j ≤ s with a_{j} < c_{ij}.

Let L denote the set of maps {1, …, p} → {1, …, s}, and consider for each f ∈ L the subset

B_{n,f} = {y^{a}| ∑_{j} a_{j} = n, a_{f(i)} < c_{if(i)} for i = 1, …, s}.

It is clear that B_{n} = ∪_{f∈L}B_{n,f}. Define ρ_{f}(a) = max{v(a)| y^{a} ∈ B_{n,f}}. Then

ρ_{E}(n) = max{ρ_{f}(n)| f ∈ L}.

Thus it suffices to show that the functions ρ_{f}(n) are linear for all f ∈ F and all n ≫ 0.

Let {j_{1}, …, j_{k}} be the image of f, and suppose that j_{1} < j_{1} … < j_{k}. We set c_{j_{t}} = min{c_{ij(i)}| j(i) = j_{t}} − 1 for t = 1, …, k. Then

B_{n,f} = {y^{a}| ∑_{j} a_{j} = n and a_{j_{t}} ≤ c_{j_{1}}, for t = 1, …, k},

and ρ_{f}(n) is given by the maximum of the linear functional v(a) on the convex bounded set

C_{n} = {a| ∑_{j} a_{j} = n, and a_{j_{t}} ≤ c_{j_{1}} for t = 1, …, k}.

This is a rather trivial example of linear programming. The solution is the following.

Suppose that ℓ is the smallest integer such that j_{t} = t for t < ℓ and j_{ℓ} > ℓ. In other words, we have a_{1} < c_{1}, …, a_{ℓ−1} < c_{ℓ−1} and no bound on a_{ℓ} (except that ∑_{j} a_{j} = n).

If ℓ = s + 1, then ∑_{j} a_{j} can be at most ∑_{j} c_{j}, so that for n ≫ 0, B_{n,f} = 0 and hence ρ_{f}(n) = 0.

If ℓ ≤ s, let n ≥ c_{1} + c_{2} + ⋯ + c_{ℓ−1}. We claim that v has its maximal value for a = (c_{1}, …, c_{ℓ−1}, n − ∑_{j=1}^{ℓ−1} c_{j}, 0, …, 0). Then

v(a) = ∑_{j=1}^{ℓ−1} d_{j}c_{j} + d_{ℓ}(n − ∑_{j}^{ℓ−1} c_{j})

which is a linear function on n, as we wanted to show.

Indeed, if a = (a_{1}, …, a_{s}) ∈ C_{n}, and if for some 1 ≤ i < j ≤ s we have a_{i} < c_{i} and a_{j} > 0, then a′ = (a_{1}, …, a_{i} +1, …, a_{j} −1, …, a_{s}) also belongs to C_{n} and v(a′) ≥ v(a) since d_{i} ≥ d_{j}, by assumption. This argument shows that if we fill up the first ‘boxes’ as much as possible, we must reach the maximal value of v. The resulting a with maximal value is exactly the one described above. □

Remark. With the same method as above one can prove the prove the following modifications of Theorem 3.1:

(i) Let I_{1}, …, I_{m} be graded ideals in the polynomial ring A. Then there exist integers a_{1}, …, a_{m} with a_{j} ≤ d(I_{j}) for j = 1, …, m, and an integer b such that

reg(I_{1}^{n_{1}} … I_{m}^{n_{m}}) = a_{1}n_{1} + … + a_{m}n_{m} + b

for all n_{1}, …, n_{m} ≫ 0. For the proof one considers the multi-Rees ring A[I_{1}t_{1}, …, I_{m}t_{m}].

(ii) Let \overline{J} denote the integral closure of an ideal J. Then reg(\overline{I^{n}}) is a linear function for n ≫ 0. This follows from the fact that ⊕_{n≥0}\overline{I^{n}}t^{n} is a finitely generated bigraded module over the Rees-algebra of I.

4. Regularity of saturations of ideals

In this section we will study the regularity of the saturation Ĩ^{n} of I^{n}.

Proposition 4.1. Let I be an arbitrary homogeneous ideal. There is a number e such that

reg(Ĩ^{n}) ≤ nd(I) + e

for all n ≥ 1.

Proof. We have

H_{𝔪}^{i}(Ĩ^{n}) ≃ \{ \begin{matrix}0, & i = 0, 1, \\ H_{𝔪}^{i}(I^{n}), & i ≥ 2. \end{matrix}

Hence the conclusion follows from Theorem 2.4. □

Now we will present examples which show that reg(Ĩ^{n}) is not a linear function for n ≫ 0. The ideal I will be the ideal of certain ‘fat’ points.

Example 4.2. Let p_{1}, …, p_{s} be distinct points on a rational normal curve in ℙ^{r}, s ≥ 2. Let ℘_{1}, …, ℘_{s} denote their defining prime ideals in A = k[X_{0}, …, X_{r}], where k is an arbitrary algebraically closed field, and I = ℘_{1} ∩ … ∩ ℘_{s}. Then

Ĩ^{n} = ℘_{1}^{n} ∩ … ∩ ℘_{s}^{n}.

By [CTV, Proposition 7] we know that

reg(A/Ĩ^{n}) = max \{2n − 1, [\frac{ns + r − 2}{r}]\} .

Note that reg(Ĩ^{n}) = reg(A/Ĩ^{n}) + 1. If s ≥ 2r, then

reg(Ĩ^{n}) = [\frac{ns + 2r − 2}{r}] .

In this case, if s is not divided by r, reg(Ĩ^{n}) is not a linear function but a periodic one whose values depend on the residue of s modulo r.

The following example show that in general reg(Ĩ^{n}) is even not a periodic function.

Example 4.3. For any p > 0 such that p is congruent to 2 mod 3, there exists a field k of characteristic p and an ideal I ⊂ k[x, y, z] such that the regularity of the saturated

powers Ĩ^{n} is not (eventually) periodic. In fact, reg({ \atop I^{5(n+1)}}) ≤ 29n + 7 if n is not a

power of p and reg({ \atop I^{5(n+1)}}) = 29n + 8 if n is a power of p.

In [CS, Section 6] the first author and Srinivas construct a counterexample to Zariski’s Rieman-Roch problem in char p > 0. There one can find a non singular projective curve C of genus 2 over a field k of characteristic p ≠ 0 as above with points η, q ∈ C such that

h^{1}(O_{C}(n(η − q) + q)) = \{ \begin{matrix}0 & if n is not a power of p \\ 1 & if n is a power of p. \end{matrix}

We will use this curve to construct our example.

Set D = 6q − η. Then D is a divisor on C such that deg(D) = 5 ≥ 2g + 1, where g = 2 is the genus of C. Thus D is very ample [H2, Corollary IV.3.2] and h^{0}(O_{C}(D)) = deg(D) = 1 − g = 4 [H2, Example IV.1.3.4 and Theorem IV.1.3]. Hence H^{0}(O_{C}(D)) gives an embedding of C as a curve of degree 5 in ℙ^{3}. We can project C onto a degree 5 plane curve γ with only nodes as singularities from a point in P^{3} not on C [H2, Theorem IV.3.10]. The arithmetic genus of γ is p_{a}(γ) = 2+n where n is the number of nodes of γ [H2, Exercise IV.1.8]. Since d = deg(γ) = 5, p_{a}(γ) = \frac{1}{2}(d − 1)(d − 2) = 6 [H2, Exercise I.7.2]. Thus γ has n = 4 nodes.

Let these singular points be q_{1}, …, q_{4}. Let π_{1} : S_{1} → ℙ^{2} be the blow up of these 4 points. Let F_{i} be the exceptional curves that map respectively to q_{i}. Let γ_{1} be the strict transform of γ. Then γ_{1} ≅ C since it is nonsingular. Let H_{1} = π^{−1}(H′) where H′ is a hyperplane on ℙ^{2}. Since the singular points are nodes,

π^{−1}(γ) = γ_{1} + 2F_{1} + ⋯ + 2F_{4}

and

F_{i} · γ_{1} = q_{i1} + q_{i2}

for (distinct) points q_{ij} on γ_{1}, 1 ≤ i ≤ 4, j = 1, 2. The divisor

5H_{1} · γ_{1} − 2q_{11} − ⋯ − 2q_{42} − η + 5q

has degree 13 since (H_{1} ·γ_{1}) = (H′ ·γ) = 5. Thus it is very ample [H2, Corollary IV.3.2], and there are points p_{1}, …, p_{13} ∈ γ_{1} such that

5H_{1} · γ_{1} − 2q_{11} − ⋯ − 2q_{42} − η + 5q ∼ p_{1} + ⋯ + p_{13}

where ∼ denotes linear equivalence.

Let π_{2} : S_{2} → S_{1} be the blowup of the points p_{1}, …, p_{13}, with respective exceptional curves E_{i} mapping to p_{i}. Let \overline{γ} ≅ C be the strict transform of γ_{1}, \overline{F}_{i} be the strict transform of F_{i} for 1 ≤ i ≤ 4. Let π : S_{2} → ℙ^{2} be the composed map. Let \overline{H} = π^{−1}(H′). Then

5\overline{H} ∼ π^{−1}(γ) = \overline{γ} + E_{1} + … + E_{13} + 2F_{1} + ⋯ + 2F_{4}.
\overline{γ} · \overline{γ} ∼ (5\overline{H} − E_{1} − ⋯ − E_{13} − 2F_{1} − ⋯ − 2F_{4}) · \overline{γ} ∼ η − 5q

By our construction, \overline{H} · \overline{γ} ∼ D = 6q − η. Thus

(5\overline{γ} + 4\overline{H}) · \overline{γ} ∼ η − q,
(\overline{γ} + \overline{H}) · \overline{γ} ∼ q.

Set A = 5\overline{γ} + 4\overline{H}, B = \overline{γ} + \overline{H}. Observe that (\overline{γ}^{2}) = −4 and (\overline{γ} · \overline{H}) = 5. Then H^{1}(O(m\overline{H})) = 0 for all m ≥ 0 and H^{1}(O\overline{γ}(m\overline{H} + n\overline{γ})) = 0 if 5m − 4n ≥ 3 since (m\overline{H} + n\overline{γ}) · \overline{γ}) = 5m − 4n and a divisor on a curve of genus g is nonspecial if deg > 2g − 2 [H2, Example IV.1.3.4]. Consideration of the cohomology of

0 → O(m\overline{H} + (n − 1)\overline{γ}) → O(m\overline{H} + n\overline{γ}) → O\overline{γ}(m\overline{H} + n\overline{γ}) → 0

and induction imply H^{1}(O(m\overline{H} +n\overline{γ})) = 0 if 5m−4n ≥ 3. The relations H^{2}(O(m\overline{H})) = 0 for all m ≥ 0 and H^{2}(O\overline{γ}(m\overline{H} +n\overline{γ})) = 0 for all m, n > 0 imply H^{2}(O(m\overline{H} +n\overline{γ})) = 0 for all m, n > 0.

For all n ≥ 0 we have

0 → O(nA + \overline{H}) → O(nA + B) → O\overline{γ}(nA + B) → 0.

By above, H^{1}(nA + \overline{H})) = H^{2}(O(nA + \overline{H})) = 0 for all n ≥ 0. Thus

h^{1}(O(nA + B)) = h^{1}(O_{C}(n(η − q) + q)) = \{ \begin{matrix}0 & if n is not a power of p \\ 1 & if n is a power of p. \end{matrix}

The formulas ((nA + B + m\overline{H}) · E_{i}) > 0 and ((nA + B + m\overline{H}) · F_{i}) > 0 for all m, n ≥ 0 implies that R^{i}π_{∗}O(nA + B + m\overline{H}) = 0 for m, n ≥ 0, and H^{1}(O(nA + B + m\overline{H})) = H^{1}(π_{∗}O(nA + B + m\overline{H})). The relation

nA + B + m\overline{H} = (29n + 6 + m)\overline{H} − 5(n + 1)(E_{1} + ⋯ + E_{13} + 2F_{1} + ⋯ + 2F_{4}) implies

π_{∗}O(nA+B+m\overline{H}) = (I_{1}^{5(n+1)}∩⋯∩I_{13}^{5(n+1)}∩J_{1}^{10(n+1)}∩⋯∩J_{4}^{10(n+1)})⊗O(29n+6+m)

where I_{i} are the ideal sheaves of the points p_{i} and J_{j} are the ideal sheaves of the points q_{j} in ℙ^{2}.

Let ℘_{1}, …, ℘_{13} and ℘_{14}, …, ℘_{17} be the homogeneous primes in k[x, y, z] which sheafify to I_{1}, …, I_{13} and J_{1}, …, J_{4}, respectively. Set

I = ℘_{1} ∩ ⋯ ∩ ℘_{13} ∩ ℘_{14}^{2} ∩ ⋯ ∩ ℘_{17}^{2}.

Let 𝔪 = (x, y, z). Then

H_{𝔪}^{0}(Ĩ^{n}) = H_{𝔪}^{1}(Ĩ^{n}) = 0,
H_{𝔪}^{2}(Ĩ^{n}) = ⊕_{a∈ℤ}H^{1}(ℙ^{2}, I^{n}(a)),
H_{𝔪}^{3}(Ĩ^{n}) = ⊕_{a∈ℤ}H^{2}(ℙ^{2}, I^{n}(a)),

where I is the ideal sheaf of I. Putting everything together, we obtain

dim_{k}H_{𝔪}^{2}({ \atop I^{5(n+1)}})_{(s−1)} = \{ \begin{matrix}0 & if s > 29n + 7 \\ 0 & if s = 29n + 7 and n is not a power of p \\ 1 & if s = 29n + 7 and n is a power of p. \end{matrix}

H_{𝔪}^{3}({ \atop I^{5(n+1)}})_{(s−2)} = 0 if s ≥ 29n + 7.

References

[BM] D. Bayer and D. Mumford, What can be computed in algebraic geometry? In: D. Eisenbud and L. Robbiano (eds.), Computational Algebraic Geometry and Commutative Algebra, Proceedings, Cortona 1991, Cambridge University Press, 1993, 1-48.

[B] A. Bertram, An application of a log version of the Kodaira vanishing theorem to embedded projective varieties, Prepint.

[BEL] A. Bertram, L. Ein, and R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), 587-602.

[BH] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge, 1993

[CTV] M. Catalisano, N.V. Trung, and G. Valla, A sharp bound for the regularity index of fat points in general position, Proc. Amer. Math. Soc. 118 (1993), 717-724.

[CS] S.D. Cutkosky and V. Srinivas, On a problem of Zariski on dimensions of linear systems, Ann. of Math. 137 (1993), 531-559.

[C] K. Chandler, Regularity of the powers of an ideal, Preprint.

[EG] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicities, J. Algebra 88 (1984), 107-184.

[EK] S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1-25.

[GGP] A. Geramita, A. Gimigliano, and Y. Pitteloud, Graded Betti numbers of some embedded rational n-folds, Math. Ann. 301 (1995), 363-380.

[H1] R. Hartshorne, Local cohomology, Lect. Notes in Math. 49, Springer, 1967.

[H2] R. Hartshorne, Algebraic Geometry, Springer, 1977.

[HT] L. T. Hoa and N. V. Trung, On the Castelnuovo regularity and the arithmetic degree of monomial ideals, Math. Z., to appear.

[Hu] C. Huneke, Uniform bounds in noetherian rings, Invent. Math. 107 (1992), 203-223.

[S] I. Swanson, Powers of ideals, primary decompositions, Artin-Rees lemma and regularity, Math. Ann. 307 (1997), 299-313.

[SS] K. Smith and I. Swanson, Linear bounds on growth of associated primes, Preprint.