Let (
pISSN 1859-1388 eISSN 2615-9678 Hue University Journal of Science: Natural Science Vol 129, No 1D, 67–70, 2020 ON THE HILBERT COEFFICIENTS AND REDUCTION NUMBERS Ton That Quoc Tan* Faculty of Natural Sciences, Duy Tan University, 254 Nguyen Van Linh St., Thanh Khe, Da Nang, Vietnam * Correspondence to Ton That Quoc Tan (Received: 02 May 2020; Accepted: 28 June 2020) Abstract Let (𝑅, 𝑚) be a noetherian local ring with 𝑑𝑖𝑚(𝑅) = 𝑑 ≥ and 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − Let 𝐼 be an 𝑚-primary ideal of 𝑅 In this paper, we study the non-positivity of the Hilbert coefficients 𝑒𝑖 (𝐼) under some conditions Keywords: Hilbert coefficients, reduction numbers, Castelnuovo-Mumford regularity, 𝑚-primary ideals, the depth of associated graded rings and Introduction 𝐼 is a parameter 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼)) ≥ 𝑑 − , then ideal such 𝑒𝑖 (𝐼) ≤ for that 𝑖= Let (𝑅, 𝑚) be a noetherian local ring of dimension 1, … , 𝑑 In [5], Puthenpurakal obtained remarkable 𝑑 ≥ and 𝐼 an 𝑚 -primary ideal of 𝑅 Let ℓ( ) results that if 𝐼 is an 𝑚-primary ideal of a ring 𝑅 denote the length of an 𝑅 -module The Hilbert- with dimension such that 𝑟(𝐼) ≤ , then Samuel function of 𝑅 with respect to 𝐼 is the 𝑒3 (𝐼) ≤ function 𝐻𝐼 : ℤ ⟶ ℕ0 given by ℓ(𝑅/𝐼 𝑛 ) 𝐻𝐼 (𝑛) = { The main result of this paper is to give an 𝑖𝑓 𝑛 ≥ 0; 𝑖𝑓 𝑛 < improvement of the result of Linh-Trung [4] There exists a unique polynomial 𝑃𝐼 (𝑥) ∈ ℚ[𝑥] (called the Hilbert- Samuel polynomial) of degree 𝑑 such that 𝐻𝐼 (𝑛) = 𝑃𝐼 (𝑛) for 𝑛 ≫ and it is written by Theorem 3.3 Let (𝑅, 𝑚) a noetherian local ring with 𝑑𝑖𝑚(𝑅) = 𝑑 ≥ and 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − Let 𝐼 be an 𝑚 -primary ideal of 𝑅 such that 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼)) ≥ 𝑑 − For 𝑖 = 1, … , 𝑑 , 𝑖𝑓 𝑟(𝐼) ≤ 𝑖 − 𝑡ℎ𝑒𝑛 𝑒𝑖 (𝐼) ≤ 𝑑 𝑃𝐼 (𝑛) = ∑ (−1)𝑖 ( 𝑖=0 𝑛+𝑑−𝑖−1 ) 𝑒𝑖 (𝐼) 𝑑−𝑖 Then, the integers 𝑒𝑖 (𝐼) is called Hilbert coefficients of 𝐼 The aim of this paper is to study the non-positivity of Hilbert cofficients In 2010, Mandal-Singh-Verma [1] showed that 𝑒1 (𝐼) ≤ for all parameter ideals 𝐼 of 𝑅 If 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − , McCune [2] showed that 𝑒2 (𝐼) ≤ and Saikia-Saloni [3] proved that 𝑒3 (𝐼) ≤ for every parameter ideal 𝐼 Recently, Linh-Trung [4] proved that if 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − DOI: 10.26459/hueuni-jns.v129i1D.5803 Preliminary Let (𝑅, 𝑚) be a noetherian local ring of dimension 𝑑 and 𝐼 be an 𝑚-primary ideal of 𝑅 A numerical function 𝐻𝐼 : ℤ ⟶ ℕ0 ℓ(𝑅/𝐼 𝑛 ) 𝑛 ⟼ 𝐻𝐼 (𝑛) = { 𝑖𝑓 𝑛 ≥ 0; 𝑖𝑓 𝑛 < is said to be a Hilbert-Samuel function of 𝑅 with respect to the ideal 𝐼 It is well known that there exists a polynomial 𝑃𝐼 ∈ ℚ[𝑥] of degree 𝑑 such 67 Ton That Quoc Tan that 𝐻𝐼 (𝑛) = 𝑃𝐼 (𝑛) for 𝑛 ≫ The polynomial 𝑃𝐼 reduction of 𝐼 and no other reduction of 𝐼 is is called the Hilbert-Samuel polynomial of 𝑅 with contained in 𝐽 , then 𝐽 is said to be a respect to the ideal 𝐼, and it is written in the form reduction of 𝐼 If 𝐽 is a minimal reduction of 𝐼, then ∑𝑑𝑖=0 minimal the reduction number of 𝐼 with respect to 𝐽, 𝑟𝐽 (𝐼), is 𝑃𝐼 (𝑛) = 𝑖 𝑛+𝑑−𝑖−1 (−1) ( ) 𝑒𝑖 (𝐼), 𝑑−𝑖 given by where 𝑒𝑖 (𝐼) for 𝑖 = 0, … , 𝑑 are integers, called Hilbert coefficients of 𝐼 In particular, 𝑒(𝐼) = 𝑒0 (𝐼) 𝑟𝐽 (𝐼): = min{ 𝑛 | 𝐼 𝑛+1 = 𝐽𝐼 𝑛 } The reduction number of 𝐼, denoted 𝑟(𝐼), is and 𝑒1 (𝐼) are called the multiplicity and Chern given by coefficient of 𝐼, respectively 𝑟(𝐼): = min{𝑟𝐽 (𝐼) | 𝐽 is a minimal reduction of 𝐼} An 𝑥 ∈ 𝐼\𝑚𝐼 element is said to be superficial for 𝐼 if there exists a number 𝑐 ∈ ℕ 𝑛 𝑐 such that (𝐼 : 𝑥) ∩ 𝐼 = 𝐼 𝑛−1 for 𝑛 > 𝑐 If 𝑅/𝑚 is A the reduction the following lemma Lemma 2.2 [7, Proposition 3.2] exists A sequence of elements 𝑥1 , … , 𝑥𝑟 ∈ 𝐼\𝑚𝐼 is superficial sequence for 𝐼 if 𝑥𝑖 is between number of 𝐼, 𝑎𝑑 (𝐺(𝐼)) and 𝑟𝑒𝑔(𝐺(𝐼)) is given by infinite, then a superficial element for 𝐼 always said to be a relationship 𝑎𝑑 (𝐺(𝐼)) + 𝑑 ≤ 𝑟(𝐼) ≤ 𝑟𝑒𝑔(𝐺(𝐼)) superficial for 𝐼/(𝑥1 , … , 𝑥𝑖−1 ) for 𝑖 = 1, … , 𝑟 Suppose that 𝑑𝑖𝑚(𝑅) = 𝑑 ≥ and 𝑥 ∈ 𝐼\ Main result 𝑚𝐼 is a superficial element for 𝐼, then ℓ(0:𝑅 𝑥) < Throughout this section, (𝑅, 𝑚) is a noetherian ∞ and 𝑑𝑖𝑚(𝑅/(𝑥)) = 𝑑𝑖𝑚(𝑅) − = 𝑑 − The local ring of dimension 𝑑 and 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − following lemma give us a relationship between Let 𝐼 be an 𝑚 -primary ideal of 𝑅 In [8], Elias 𝑒𝑖 (𝐼) and 𝑒𝑖 (𝐼1 ), where 𝐼1 = 𝐼(𝑅/(𝑥)) considered the numerical function Lemma 2.1 [6, Proposition 1.3.2] Let 𝑅 be a noetherian local ring of dimension 𝑑 ≥ and 𝐼 an 𝑚- 𝜎𝐼 : ℕ 𝑘 ⟶ℕ ⟼ 𝜎𝐼 (𝑘) = 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼 𝑘 )) primary ideal of 𝑅 Let 𝑥 ∈ 𝐼\𝑚𝐼 be a superficial element for 𝐼 and 𝐼1 = 𝐼(𝑅/(𝑥)) Then Elias [8] showed that 𝜎𝐼 is a non-decreasing function and 𝜎𝐼 (𝑘) is a constant for 𝑘 ≫ This (i) 𝑒𝑖 (𝐼) = 𝑒𝑖 (𝐼1 ) for 𝑖 = 0, … , 𝑑 − 2; constant is denoted by 𝜎(𝐼) (ii) 𝑒𝑑−1 (𝐼) = 𝑒𝑑−1 (𝐼1 ) + (−1)𝑑 ℓ(0: 𝑥) If denote by 𝐺(𝐼) =⊕𝑛≥0 𝐼 𝑛 /𝐼 𝑛+1 By [9, Lemma 2.4], the associated graded ring of 𝑅 with respect to 𝐼 and 𝑖 𝑎𝑖 (𝐺(𝐼)) = sup{𝑛 |𝐻𝐺(𝐼) (𝐺(𝐼))𝑛 ≠ 0}, + 𝑖 where 𝐻𝐺(𝐼) (𝐺(𝐼)) is the 𝑖 -th local cohomology + module of 𝐺(𝐼) with respect to 𝐺(𝐼)+ The Castelnuovo-Mumford regularity of 𝐺(𝐼), 𝑟𝑒𝑔(𝐺(𝐼)), 𝑎𝑖 (𝐺(𝐼 𝑘 )) ≤ [ 𝑎𝑖 (𝐺(𝐼)) ] for all 𝑖 ≤ 𝑑 and 𝑘 ≥ 1, 𝑘 where [𝑎] = 𝑚𝑎𝑥{𝑚 ∈ ℤ | 𝑚 ≤ 𝑎} Thus, for 𝑖 ≥ 0, we have 𝑎𝑖 (𝐺(𝐼 𝑘 )) ≤ for 𝑘 ≫ is defined by and 𝑟𝑒𝑔(𝐺(𝐼)) = 𝑚𝑎𝑥{𝑎𝑖 (𝐺(𝐼)) + 𝑖 | 𝑖 ≥ 0} 𝜎(𝐼) ≥ 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼)) Recall that an ideal 𝐽 ⊆ 𝐼 is called a reduction of 𝐼 if 𝐼 68 𝑛+1 = 𝐽𝐼 𝑛 for 𝑛 ≫ If 𝐽 is a The following theorem (1) (2) gives positivity for the last Hilbert coefficient a non- pISSN 1859-1388 eISSN 2615-9678 Hue University Journal of Science: Natural Science Vol 129, No 1D, 67–70, 2020 Theorem 3.1 [10, Theorem 2.4] Let (𝑅, 𝑚) be a noetherian local ring of dimension 𝑑 ≥ and 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − Let 𝐼 be an 𝑚-primary ideal such that 𝑟(𝐼) ≤ 𝑑 − and 𝜎(𝐼) ≥ 𝑑 − Then, 𝑒𝑑 (𝐼) ≤ Therefore, 𝑚𝑎𝑥{𝑎𝑑−1 (𝐺(𝐽)), 𝑎𝑑 (𝐺(𝐽))} ≤ −1 From the proof of Theorem 3.1, we have 𝑑−1 𝑒𝑑 (𝐼) = −ℓ(𝐻𝐺(𝐽) (𝐺(𝐽))0 ) = + For 𝑘 ≫ , let 𝐽 = 𝐼 𝑘 , 𝑅 = 𝑅[𝐽𝑡] =⊕𝑛≥0 𝐽𝑛 In [4], Linh-Trung proved that if 𝑄 is a denote the Rees algebra of 𝑅 with respect to 𝐽, 𝑅+ parameter ideal such that 𝑑𝑒𝑝𝑡ℎ(𝐺(𝑄)) ≥ 𝑑 − , =⊕𝑛>0 𝑅𝑛 By [11, Theorem 4.1] and [11, Theorem then 𝑒𝑖 (𝑄) ≤ for all 𝑖 = 1, … , 𝑑 In this case, 3.8], we have 𝑟(𝑄) = 𝑑 (−1) 𝑒𝑑 (𝐼) = (−1) 𝑒𝑑 (𝐽) = 𝑃𝐽 (0) − 𝐻𝐽 (0) = ∑𝑑𝑖=0 (−1)𝑖 ℓ(𝐻𝑅𝑖 + (𝑅)0 ) Theorem 3.3 𝑖 𝐻𝐺(𝐼) (𝐺(𝐼)) + , = for 𝑖 = 0, … , 𝑑 − By Lemma 2.7], ]𝑟(𝐼) + − 𝑠(𝐼)[ 𝑟(𝐽) ≤ + 𝑠(𝐼) − 𝑘 ]𝑟(𝐼) + − 𝑑[ = + 𝑑 − ≤ 𝑑 − 𝑘 Hence, 𝑎𝑑 (𝐺(𝐽)) < On the other hand, 𝑎𝑖 (𝐺(𝐽)) ≤ for all 𝑖 ≥ from (1) By applying Theorem 5.2], we 𝑎𝑑−2 (𝐺(𝐽)) < get 𝑎𝑑−1 (𝐺(𝐽)) ≤ It follows that implies 𝑑−1 −ℓ(𝐻𝐺(𝐽) (𝐺(𝐽))0 ) ≤ + that From the is an Let (𝑅, 𝑚) a noetherian local 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼)) ≥ 𝑑 − For 𝑖 = 1, … , 𝑑 , if 𝑟(𝐼) ≤ 𝑖 − 1, then 𝑒𝑖 (𝐼) ≤ It is clear that the theorem holds for 𝑑 = Now, we consider 𝑑 > By [4, Theorem 1], the theorem holds for the case 𝑖 = In the case 𝑖 = 𝑑, by assumption, we have 𝑟(𝐼) ≤ 𝑑 − and 𝜎(𝐼) ≥ 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼)) ≥ 𝑑 − By applying Theorem 3.1, we obtain 𝑒𝑑 (𝐼) ≤ So, we need to prove for 𝑖 = 2, … , 𝑑 − Without loss of generality, we assume that 𝑅/𝑚 is infinite and 𝑥1 , … , 𝑥𝑑−𝑖 is a superficial 𝑑−1 (−1)𝑑 𝑒𝑑 (𝐼) = (−1)𝑑−1 ℓ(𝐻𝐺(𝐽) 𝐺(𝐽)0 ) + This theorem Let 𝐼 be an 𝑚 -primary ideal of 𝑅 such that 𝜎(𝐼) = 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐽)) ≥ 𝑑 − 2.2, we have 𝑎𝑑 (𝐺(𝐽)) + 𝑑 ≤ 𝑟(𝐽) From [9, Lemma [12, following ring with 𝑑𝑖𝑚(𝑅) = 𝑑 ≥ and 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − 𝑖 = ∑𝑑𝑖=0 (−1)𝑖 ℓ(𝐻𝐺(𝐽) 𝐺(𝐽)0 ) + Since The improvement of Linh-Trung’s result 𝑑 sequence for 𝐼 Let 𝑅𝑖 = 𝑅/(𝑥1 , … , 𝑥𝑖 ) and 𝐼𝑖 = 𝑒𝑑 (𝐼) = proof of Theorem 3.1, we obtain the following corollary Corollary 3.2 Let (𝑅, 𝑚) be a noetherian local ring of dimension 𝑑 ≥ and 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − Let 𝐼 be an 𝑚-primary ideal such that 𝑟𝑒𝑔(𝐺(𝐼)) ≤ 𝑑 − and 𝜎(𝐼) ≥ 𝑑 − Then, 𝑒𝑑 (𝐼) = For 𝑘 ≫ , set 𝐽 = 𝐼 𝑘 Since 𝑟𝑒𝑔(𝐺(𝐼)) ≤ 𝑑 − 2, 𝑚𝑎𝑥{𝑎𝑑−1 (𝐺(𝐼)) + 𝑑 − 1, 𝑎𝑑 (𝐺(𝐼)) + 𝑑} ≤ 𝑑 − Thus, 𝑎𝑖 (𝐺(𝐼)) ≤ −1 for 𝑖 = 𝑑 − 1, 𝑑 By [9, Lemma 2.4], 𝑎𝑖 (𝐺(𝐼 𝑘 )) ≤ [𝑎𝑖 (𝐺(𝐼))/𝑘] 𝐼𝑅𝑖 Then, 𝑒𝑖 (𝐼) = 𝑒𝑖 (𝐼𝑑−𝑖 ) from Lemma 2.1 From this hypothesis, it follows that 𝑑𝑖𝑚(𝑅𝑑−𝑖 ) = 𝑖 ≥ 2, 𝑑𝑒𝑝𝑡ℎ(𝑅𝑑−𝑖 ) ≥ 𝑖 − and 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼𝑑−𝑖 )) ≥ 𝑖 − We have 𝑟(𝐼𝑑−𝑖 ) ≤ 𝑟(𝐼) ≤ 𝑖 − From (2), we get 𝜎(𝐼𝑑−𝑖 ) ≥ 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼𝑑−𝑖 )) ≥ 𝑖 − Appying Theorem 3.1, we obtain 𝑒𝑖 (𝐼) = 𝑒𝑖 (𝐼𝑑−𝑖 ) ≤ for 𝑖 = 2, … , 𝑑 − The proof is complete Combining Theorem 3.3 and Corollary 3.2, we get the following corollary Corollary 3.4 Let (𝑅, 𝑚) be a noetherian local ring with 𝑑𝑖𝑚(𝑅) = 𝑑 ≥ and 𝑑𝑒𝑝𝑡ℎ(𝑅) ≥ 𝑑 − DOI: 10.26459/hueuni-jns.v129i1D.5803 69 Ton That Quoc Tan Let 𝐼 be an 𝑚 -primary ideal of 𝑅 such that 𝑑𝑒𝑝𝑡ℎ(𝐺(𝐼)) ≥ 𝑑 − For 𝑖 = 1, … , 𝑑 , if 𝑟𝑒𝑔(𝐺(𝐼)) ≤ 𝑖 − 2, then 𝑒𝑖 (𝐼) = References Rossi ME, Valla G Hilbert Functions of Filtered Modules Vol Berlin (DE): Springer-Verlag Berlin Heidelberg; 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