ON CASTELNUOVO-MUMFORD REGULARITY OF PROJECTIVE CURVES

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 5, Pages 1293–1299 S 0002-9939(99)05184-9 Article electronically published on August 5, 1999 ON CASTELNUOVO-MUMFORD REGULARITY OF PROJECTIVE CURVES ISABEL BERMEJO AND PHILIPPE GIMENEZ (Communicated by Wolmer V. Vasconcelos) Abstract. We give an effective method to compute the regularity of a saturated ideal I defining a projective curve that also determines in which step of a minimal graded free resolution of I the regularity is attained. Introduction Let S := K[x0 , . . . , xn ] be a polynomial ring over an algebraically closed field K, and let I be a homogeneous ideal of S defining a subscheme X of projective nspace PnK . The Castelnuovo-Mumford regularity (or simply regularity) of I, reg I, is defined as follows: if (0.1) 0→ βp M ϕp ϕ1 S(−epj ) −→ · · · −→ β0 M ϕ0 S(−e0j ) −→ I → 0 j=1 j=1 is a minimal graded free resolution of I, setting ei := max {eij ; 1 ≤ j ≤ βi }, then reg I := max {ei − i; 0 ≤ i ≤ p}. In other words, reg I is the smallest integer m for which I is m-regular, i.e. eij ≤ m + i for all i, j (see [2, Def. 3.2] for equivalent definitions). When I is saturated (i.e. when it is the largest ideal defining X), we call this the regularity of X (see [2, Sect. 1]). The regularity is a numerical invariant of the ideal I and is, as said in [6], “an important measure of how hard it will be to compute a free resolution”. In fact, knowing it beforehand avoids unnecessary computation in large degrees while obtaining the minimal graded free resolution of I through Buchberger’s syzygy algorithm (see [3]). In this paper, we shall essentially be concerned with the regularity of a saturated ideal I defining a subscheme X of PnK of dimension one. In Section 1, we show a general property of finitely generated graded S-modules asserting that the regularity of M is determined by the tail of the minimal graded free resolution (Proposition 1.1). As a consequence we obtain that, in our case, reg I is equal to either en−1 − n + 1 or en−2 − n + 2, i.e. the regularity is always attained at one of the last two steps of the resolution. Received by the editors June 23, 1998. 1991 Mathematics Subject Classification. Primary 13D45; Secondary 14Q05, 13D40. Key words and phrases. Regularity, projective curves, Hilbert functions. The first author was supported in part by D.G.U.I., Gobierno de Canarias. The second author was supported in part by D.G.I.C.Y.T., PB94-1111-C02-01. c 2000 American Mathematical Society 1293 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1294 ISABEL BERMEJO AND PHILIPPE GIMENEZ Assuming that K[xn−1 , xn ] is a Noether normalization of S/I, we give in Section 2 an effective method to compute the regularity of I that does not require the knowledge of a minimal graded free resolution of I (Theorem 2.7). The idea is to introduce an arithmetically Cohen-Macaulay curve whose regularity is closely related with that of X. For this reason, we first focus on the Cohen-Macaulay case (Theorem 2.4). These two theorems together with an effective criterion to determine whether X is arithmetically Cohen-Macaulay (Proposition 2.1), give an algorithm to compute the regularity of I. Using Section 1, this algorithm also determines in which step of a minimal graded free resolution of I, reg I is attained. 1. Where is the regularity attained? Let M be a finitely generated graded S-module and consider a minimal graded free resolution of M : ϕp with Fi = ϕ1 ϕ0 0 → Fp −→ · · · −→ F0 −→ M → 0 , (1.1) βi M S(−eij ). We denote by ei := max {eij ; 1 ≤ j ≤ βi }. j=1 Using spectral sequences, Schenzel proved that the regularity of M is determined by the tail of (1.1) ([10, Thm. 3.11]). We propose here a different proof of this issue based on an observation of Herzog relating the vanishing of a row in some matrix in (1.1) and the regularity of M when M is Cohen-Macaulay ([11, Cor. B.4.1]). Our treatment is both elementary and carries some additional information. Proposition 1.1. Let M be a finitely generated graded S-module and let (1.1) be a minimal graded free resolution of M . Denoting c := n + 1 − dim M , one has: e 0 < e1 < · · · < ec . Proof. Assume the claim is false. Then for some i, 1 ≤ i ≤ c, the matrix Mi describing ϕi : Fi → Fi−1 has a zero row. Consider now the head of the minimal graded free resolution (1.1) of M : ϕc ϕc−1 ϕ1 ϕ0 Fc −→ Fc−1 −→ · · · −→ F0 −→ M → 0 and apply HomS (., S) to this complex. Setting N := Coker ϕ⋆c , one gets (1.2) ϕ⋆ ϕ⋆ ϕ⋆ 1 2 c F1⋆ −→ · · · −→ Fc⋆ −→ N → 0 F0⋆ −→ which is a complex whose homology is ExtiS (M, S) = 0 for i < c. Thus, (1.2) is the head of a minimal graded free resolution of N , contradicting the fact that the matrix describing ϕ⋆i , the transpose of Mi , has a zero column. Consider a homogeneous ideal I of S and a minimal graded free resolution (0.1) of I. The following is a direct consequence of the above proposition. Corollary 1.2. reg I = max {ei − i; n − dim S/I ≤ i ≤ p}. 2. How to compute the regularity? Let I be a homogeneous ideal of S defining a not necessarily reduced projective curve C in PnK . Assume that K[xn−1 , xn ] is a Noether normalization of S/I (i.e. K[xn−1 , xn ] ֒→ K[x0 , . . . , xn ]/I is an integral ring extension). Monomials in S will License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CASTELNUOVO-MUMFORD REGULARITY OF PROJECTIVE CURVES 1295 αn n+1 0 be denoted by xα := xα . Let in (I) denote 0 · · · xn , with α = (α0 , . . . , αn ) ∈ N the initial ideal of I with respect to the reverse lexicographic order. Consider the evaluation morphism θ (resp. χ): K[x0 , . . . , xn ] → K[x0 , . . . , xn−2 ] defined by xn 7→ 0 (resp. xn 7→ 1), xn−1 7→ 0 (resp. xn−1 7→ 1) and xi 7→ xi for i ∈ / {n − 1, n}. Let I˜ be the ideal of S generated by χ(in (I)). I˜ is a primary monomial ideal such that in (I) ⊆ I˜ and I˜ defines a projective curve C̃ ⊆ PnK of degree deg C̃ = deg C (see [5, Lemme 1]). Denote by I0 the ideal I0 := θ(I)S ⊂ S. As in (I0 ) = θ(in (I))S, then in (I0 ) ⊆ in (I) and so the degree of the curve C0 ⊆ PnK defined by I0 satisfies deg C0 ≥ deg C. Define F := {α = (α0 , . . . , αn−2 ) ∈ Nn−1 | x(α,0,0) ∈ I˜ − in (I0 )} ⊂ Nn−1 . As K[xn−1 , xn ] is a Noether normalization of S/I, F is finite (possibly empty). The following is a criterion to determine, in terms of F , whether S/I is Cohen-Macaulay (i.e. whether C is an arithmetically Cohen-Macaulay projective curve). It implies that S/I is Cohen-Macaulay if and only if S/ in (I) is Cohen-Macaulay, and that S/I0 and S/I˜ are Cohen-Macaulay. Proposition 2.1. S/I is Cohen-Macaulay if and only if F = ∅. Proof. Observe that F = ∅ is equivalent to in (I0 ) = in (I). As S/I is CohenMacaulay if and only if {xn−1 , xn } is a regular sequence on S/I ([9, Ch. 3, Prop. 4.4]), we shall prove that in (I0 ) = in (I) if and only if {xn−1 , xn } is a regular sequence on S/I. Assume that in (I0 ) = in (I). Let f ∈ (I : xn ). Then f ∈ I because otherwise the remainder r of the division of f by a Gröbner basis of I w.r.t. the reverse lexicographic order is nonzero and in (r) ∈ / in (I). As xn in (r) ∈ in (I) and in (I) = in (I0 ), this is impossible. Similarly, let f ∈ ((I, xn ) : xn−1 ). For the same reason as above, f ∈ (I, xn ) because in (I, xn ) = (in (I), xn ) and in (I) = in (I0 ). Conversly, if {xn−1 , xn } is a regular sequence on S/I, then the monomials in a minimal set of generators of in (I) are not divisible by either xn−1 or xn . Thus, in (I0 ) = in (I). As already stated, C0 is arithmetically Cohen-Macaulay by Proposition 2.1 and deg C0 ≥ deg C. The difference between deg C0 and deg C is indeed a measure of how far C is from being arithmetically Cohen-Macaulay. Corollary 2.2. C is arithmetically Cohen-Macaulay if and only if deg C = deg C0 . Proof. The difference deg C0 − deg C is equal to #F . In fact, deg C0 is equal to / in (I0 )} because the Hilbert polynomial of S/I0 is PI0 (T ) = #{α ∈ Nn−1 | x(α,0,0) ∈ P n−1 | x(α,0,0) ∈ in (I0 )}. By a similar α∈E / 0 (T + 1 − |α|) where E0 = {α ∈ N n−1 (α,0,0) ˜ |x ∈ / I}. argument deg C̃ = #{α ∈ N Assume that S/I is Cohen-Macaulay. We will give an effective method to compute reg I that does not require the knowledge of a minimal graded free resolution of I. Set E := {(α0 , . . . , αn−2 ) ∈ Nn−1 | x(α,0,0) ∈ in (I)}. As K[xn−1 , xn ] is a Noether normalization of S/I, for s ≫ 0 and α ∈ Nn−1 one has that |α| ≥ s implies α ∈ E. Define the regularity of E, H(E), as the smallest integer s satisfying this property. Denote by H(I) the regularity of the Hilbert function HI of S/I, i.e. the smallest integer s0 such that for s ≥ s0 , HI (s) = PI (s) (PI (T ) is the Hilbert polynomial of S/I). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1296 ISABEL BERMEJO AND PHILIPPE GIMENEZ Lemma 2.3. H(E) = H(I) + 2. Proof. As the value at s of HI is HI (s) = #{(α0 , . . . , αn ) ∈ Nn+1 |α0 + · · · + αn = s and (α0 , . . . , αn−2 ) ∈ / E} , P / E satisfies |α| ≤ H(E) − 1 so then PI (T ) = α∈E / (T + 1 − |α|). Any element α ∈ H(I) ≤ H(E) − 1. It is now easy to check that HI (s0 ) = PI (s0 ) for s0 = H(E) − 2 and that HI (s0 ) > PI (s0 ) for s0 = H(E) − 3. Theorem 2.4. Let I ⊂ S be the homogeneous defining ideal of an arithmetically Cohen-Macaulay projective curve C ⊂ PnK . Then reg I = H(E). Proof. By the previous lemma, one has to prove that reg I = H(I) + 2. From [6, Prop. 20.20], one gets that reg I = reg (I, xn−1 , xn ). As dim S/(I, xn−1 , xn ) = 0, then reg (I, xn−1 , xn ) coincides with the regularity H(I, xn−1 , xn ) of the Hilbert function of S/(I, xn−1 , xn ) ([3, Lemma 1.7]). The result now follows from the equality H(I, xn−1 , xn ) = H(I) + 2. Example 2.5. Consider the ideal I ⊂ K[x, y, z, t] generated by f1 = x17 y 14 − y 31 , f2 = x20 y 13 , f3 = x60 − y 36 z 24 − x20 z 20 t20 . The reduced Gröbner basis of I w.r.t. the reverse lexicographic order is {f1 , f2 , f3 , y 48 , x3 y 31 }, so in (I) = (x17 y 14 , x20 y 13 , x60 , y 48 , x3 y 31 ). Then K[x, y, z, t]/I is Cohen-Macaulay (Proposition 2.1) and reg I = 72 (Theorem 2.4). (0,48) • • (3,31) (17,14)• • (20,13) ❅ ❅ ❅❅ • reg I = 72 (60,0) As already observed, S/I is Cohen-Macaulay if and only if S/ in (I) is CohenMacaulay. Thus, we get the following consequence of Theorem 2.4 which can also be obtained from [3, Thm. 2.4 (b)]. Corollary 2.6. If I satisfies the conditions of Theorem 2.4, then reg I = reg in (I). Let’s assume now that I is a saturated ideal defining a nonarithmetically CohenMacaulay projective curve C ⊂ PnK . We shall give a relation between reg I and reg I0 to obtain, as in Theorem 2.4, an effective method to compute reg I that does not require the knowledge of a minimal graded free resolution of I. In this case F 6= ∅ (Proposition 2.1) and one has the partition introduced in [5]: {(α0 , . . . , αn ) ∈ Nn+1 αn 0 | xα / in (I)} = 0 · · · xn ∈ αn−2 0 ˜ ∪ R, {(α0 , . . . , αn ) ∈ Nn+1 | xα / I} 0 · · · xn−2 ∈ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CASTELNUOVO-MUMFORD REGULARITY OF PROJECTIVE CURVES 1297 S where R = α∈F {α×[N2 −Eα ]} for Eα = {(αn−1 , αn ) ∈ N2 | x(α,αn−1 ,αn ) ∈ in (I)}. Therefore, the value at s ∈ N of the Hilbert function HI of S/I is HI (s) = HI˜(s) + #{β ∈ R ||β| = s} , where #{β ∈ R ||β| = s} is constant for s ≫ 0. Denote by H(R) (resp. H(Eα )) the smallest integer s0 such that for s ≥ s0 , #{β ∈ R ||β| = s} (resp. #{(αn−1 , αn ) ∈ N2 − Eα | αn−1 + αn = s}) is constant. It is clear that H(R) ≤ max α∈F {|α| + H(Eα )}. Theorem 2.7. Let I ⊂ S be a saturated ideal defining a nonarithmetically CohenMacaulay projective curve C ⊂ PnK . Then reg I = max {reg I0 , H(R) + 1}. Proof. Since the field K is infinite and K[xn−1 , xn ] is a Noether normalization of S/I and I is a saturated ideal, then there exists κ ∈ K − {0} such that xn − κxn−1 is a nonzero divisor on S/I. If we denote by Iκ the ideal (I, xn − κxn−1 ) of S, then reg I = reg Iκ by [6, Prop. 20.20]. sat is the saturation of Iκ , one deduces from [3, LemOn the other hand, if (Iκ ) mas 1.6, 1.7, 1.8] that reg Iκ = max {s0 , H(Iκ , h)} where h is a linear form which sat is a nonzero divisor on S/(Iκ ) , and s0 is the smallest integer such that, for any s ≥ s0 , (Iκ : h)s = (Iκ )s . sat Since S/(Iκ ) is a finite K[xn ]-module of dimension 1, then K[xn ] is a Noether sat by [9, Ch. 2, Rem. 6.5.0]. Thus, xn is a nonzero divisor normalization of S/(Iκ ) sat and reg Iκ = max {s0 , H(Iκ , xn )}, s0 being the smallest integer such on S/(Iκ ) that, for any s ≥ s0 , (Iκ : xn )s = (Iκ )s . Let us prove now that reg Iκ = max {H(I) + 1, H(Iκ , xn )}. Indeed, as for any s, .x ϕ n Ss /(Iκ )s −→ Ss /(Iκ , xn )s → 0 0 → Ss−1 /(Iκ : xn )s−1 −→ is an exact sequence, where ϕ is the canonical morphism, and as H(Iκ ) = H(I) + 1, one has max {s0 , H(Iκ , xn )} = max {H(I) + 1, H(Iκ , xn )}. On the other hand, H(Iκ , xn ) = reg I0 because (Iκ , xn ) = (I0 , xn−1 , xn ) and I0 defines an arithmetically Cohen-Macaulay curve (see proof of Theorem 2.4). ˜ Finally, max {H(I)+ 1, reg I0 } = max {H(R)+ 1, reg I0 }. Indeed, as in (I0 ) ⊆ I, ˜ + 2 = reg I˜ ≤ reg I0 by Lemma 2.3, Theorem 2.4 and Corollary 2.6. If then H(I) ˜ the result follows from the previous H(R) and H(I) are smaller or equal to H(I), inequality. Otherwise, it is easy to check that H(R) = H(I) and we are done. Remark 2.8. It is worth noting that knowledge of in (I) and some extra combinatorial work give the value of reg I. In fact, in (I0 ) is generated by the minimal generators of in (I) which are not divisible by either xn or xn−1 because in (I0 ) = θ(in (I))S. Taking E = {α ∈ Nn−1 | x(α,0,0) ∈ in (I0 )}, one gets reg I0 = H(E) by Theorem 2.4. On the other hand, H(R) is also obtained from in (I). Example 2.9. For any ℓ ≥ 1, consider the saturated ideal Iℓ = (f1 , f2 , f3 , hℓ ) ⊂ K[x, y, z, t] generated by f1 , f2 , f3 of the Example 2.5 and by hℓ = y 41 z ℓ − y 40 z ℓ+1 . One can check that {f1 , f2 , f3 , hℓ , y 48 , x3 y 31 , y 40 z ℓ+8 } is the reduced Gröbner basis of Iℓ w.r.t. the reverse lexicographic order. Then in (Iℓ ) = (x17 y 14 , x20 y 13 , x60 , y 41 z ℓ , y 48 , x3 y 31 , y 40 z ℓ+8 ). The set F is not empty and independent of ℓ. It is License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1298 ISABEL BERMEJO AND PHILIPPE GIMENEZ represented by the following diagram: • ← F = {(i, j); 0 ≤ i ≤ 2, 40 ≤ j ≤ 47} (0,40) • • •• • Then for ℓ ≥ 1, K[x, y, z, t]/Iℓ is not Cohen-Macaulay by Proposition 2.1. Observe that for any ℓ ≥ 1, in (Iℓ )0 coincides with in (I), where I is the ideal (f1 , f2 , f3 ) of the Example 2.5. The regularity of (Iℓ )0 is then reg (Iℓ )0 = 72. Now Eα = (ℓ+8, 0)+N2 for any α = (i, 40) ∈ F , and Eα = (ℓ, 0) + N2 for any α = (i, j) ∈ F with j ≥ 41. So H(R) + 1 = max α∈F {|α| + H(Eα )} + 1 = 50 + ℓ and reg Iℓ = max {72, 50 + ℓ} by Theorem 2.7. Remark 2.10. Observe that in the previous example, in (Iℓ ) is a saturated ideal for sat any ℓ ≥ 1, but it is not true in general that I = I sat implies that in (I) = in (I) . 2 2 For example, the ideal I ⊂ K[x, y, z, t] generated by x − 3xy + 5xt, xy − 3y + 5yt, xz − 3yz, 2xt − yt and y 2 − yz − 2yt is saturated since z − t is a nonzero divisor on K[x, y, z, t]/I and in (I) = (yzt, y 2 , xt, xz, xy, x2 ) is not saturated because z −κt is a zero divisor on K[x, y, z, t]/ in (I), for any κ ∈ K. In this example, reg I 6= reg in (I) as reg I = 2 by Theorem 2.7 (reg I0 = H(R) + 1 = 2) and one can check with [4] that reg in (I) = 3. Nevertheless, if in (I) is also saturated one gets directly from Theorem 2.7 that reg I = reg in (I) . In particular, if xn is a nonzero divisor on S/I, one has in (I) = in (I)sat and the above equality also comes from [3, Thm. 2.4 (b)]. The last result of this section says that the method obtained from Theorems 2.4 and 2.7 to compute the regularity of I also determines when the regularity is attained at the last step of a minimal graded free resolution of I. Corollary 2.11. Let I ⊂ S be a saturated ideal defining a projective curve C ⊂ PnK . Then reg I is attained at the last step of a minimal graded free resolution of I if and only if either S/I is Cohen-Macaulay or reg I = H(R) + 1. Proof. When S/I is Cohen-Macaulay, the result is a consequence of Corollary 1.2. Assume that S/I is not Cohen-Macaulay. As a consequence of the proof of Theorem 2.7, one has that reg I = H(R) + 1 if and only if reg I = H(I) + 1. Let M βn−1 0→ j=1 S(−en−1,j ) −→ · · · −→ β0 M S(−e0j ) −→ I → 0 j=1 be a minimal graded free resolution of I. The Hilbert series of S/I is Q(t) (1−t)n+1 Q(t) = 1 − (te01 + · · · + te0β0 ) + · · · + (−1)n (ten−1,1 + · · · + ten−1,βn−1 ) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use with CASTELNUOVO-MUMFORD REGULARITY OF PROJECTIVE CURVES 1299 and deg (Q(t)) = H(I) + n. Since deg (Q(t)) ≤ reg I + n − 1, and equality holds if and only if reg I + n − 1 = en−1 , and the result follows. In summary, avoiding the construction of a minimal graded free resolution of Iℓ , in Example 2.9, one can assert now that for any ℓ, 1 ≤ ℓ ≤ 21, the regularity of Iℓ is attained at the second step of a minimal graded free resolution of Iℓ but not at the third step. For ℓ ≥ 22, the regularity of Iℓ is attained at the third step of a minimal graded free resolution of I but can also occur at the second step. Acknowledgements We would like to thank Monique Lejeune-Jalabert and Wolmer V. Vasconcelos for helpful conversations, and Aron Simis for the corrections he suggested in a previous version of this paper. References 1. D. Bayer, The division algorithm and the Hilbert scheme, Thesis, Harvard University, Cambridge, MA, 1982. 2. D. Bayer and D. Mumford, What can be computed in Algebraic Geometry? In: Computational Algebraic Geometry and Commutative Algebra, Proceedings Cortona 1991 (D. Eisenbud and L. Robbiano, Eds.), Cambridge University Press, 1993, 1–48. MR 95d:13032 3. D. Bayer and M. Stillman, A criterion for detecting m-regularity, Invent. Math. 87 (1987) 1-11. MR 87k:13019 4. D. Bayer and M. Stillman, Macaulay, a system for computation in Algebraic Geometry and Commutative Algebra, 1992, available via anonymous ftp from math.harvard.edu. 5. I. Bermejo and M. Lejeune-Jalabert, Sur la compléxité du calcul des projections d’une courbe projective, to appear in Comm. in Algebra. 6. D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer, Berlin, Heidelberg, New York, 1995. 7. D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicities, J. Algebra 88 (1984) 89-133. MR 85f:13023 8. G.M. Greuel, G. Pfister and H. Schoenemann, Singular, a system for computation in Algebraic Geometry and Singularity Theory, 1995, available via anonymous ftp from helios.mathematik.uni-kl.de. 9. M. Lejeune-Jalabert, Effectivité de calculs polynomiaux, Cours de D.E.A., Institut Fourier, Grenoble, 1984-85. 10. P. Schenzel, On the use of Local Cohomology in Algebra and Geometry, In: Six Lectures on Commutative Algebra (J. Elias, J.M. Giral, R.M. Miró-Roig and S. Zarzuela, Eds.), Progress in Mathematics 166, Birkhauser, Boston, 1998. 11. W.V. Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Algorithms and Computation in Mathematics 2, Springer, Berlin, Heidelberg, New York, 1998. MR 99c:13048 Departamento de Matematica Fundamental, Facultad de Matematicas, Universidad de La Laguna, 38271-La Laguna, Tenerife, Spain E-mail address: [email protected] Departamento de Algebra, Geometria y Topologia, Facultad de Ciencias, Universidad de Valladolid, 47005-Valladolid, Spain E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use