Luận văn thạc sỹ về cấu trúc của vành nửa nhóm số định nghĩa bởi iđêan định thức

THAI NGUYEN UNIVERSITY UNIVERSITY OF SCIENCES ĐỖ VĂN KIÊN ON THE STRUCTURE OF NUMERICAL SEMIGROUP RINGS DEFINED BY DETERMINANTAL IDEALS A DISSERTATION PRESENTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS THAI NGUYEN − 2022 THAI NGUYEN UNIVERSITY UNIVERSITY OF SCIENCES ĐỖ VĂN KIÊN ON THE STRUCTURE OF NUMERICAL SEMIGROUP RINGS DEFINED BY DETERMINANTAL IDEALS Speciality: Algebra and Number theory Speciality code: 46 01 04 A DISSERTATION PRESENTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisors: Assoc Prof Dr Naoyuki Matsuoka Assoc Prof Dr Đoàn Trung Cường THAI NGUYEN − 2022 Abstract The computation of the minimal free resolution and/or the defining ideal of an algebra over a field k is one of the important problems in commutative algebra In this dissertation, we consider the case where the base ring is a numerical semigroup ring defined by a determinantal ideal Let R = k[H] be the semigroup ring associated to a numerical semigroup H of embedding dimension n It is well known by J Herzog (see [33]) that if n = and R is not Gorenstein, then the defining ideal of R is generated by the maximal minors of a × matrix and the minimal free resolution of R is explicitly determined Herzog’s result has inspired us to study the semigroup ring R in which its the defining ideal has the form a determinantal ideal, namely generated by maximal minors of a × n matrix Determinantal ideals are interesting from a commutative algebra perspective because they possess a large number of interesting syzygies (see [20]) In this dissertation, we look for characterizations that make the defining ideal of R a determinantal ideal We compute generators of the defining ideal and describe the minimal free resolution of R We first focus on the case pseudo-Frobenius numbers of H being multiple of a fixed integer Next, we consider the case the numerical semigroup ring having maximal embedding dimension Lastly, we provide some ring-theoretic properties of numerical semigroup rings defined by determinantal ideals The dissertation consists of four chapters In Chapter 1, we present i basic concepts related to the main results of the dissertation In Chapter 2, which is joint work with Shiro Goto, Naoyuki Matsuoka, and Hoang Le Truong, we investigate the relationship between the behavior of the pseudo-Frobenius numbers of H and generators of the defining ideal of R We give a precise description of a minimal set of generators of the defining ideal, provided the pseudo-Frobenius numbers of H are multiples of a fixed integer In this case, the minimal free resolution of R is explicitly determined and R is an almost Gorenstein ring In Chapter 3, which is joint work with Naoyuki Matsuoka, we consider the case where R has maximal embedding dimension We give characterizations so that the defining ideal of R is a determinantal ideal We find out explicitly a minimal set of generators of the determinantal defining ideal of R Moreover, the minimal free resolution of R is also explicitly described and we show that the symbolic Rees algebra of the defining ideal is Noetherian and Cohen-Macaulay Finally, in Chapter 4, which is joint work with Nguyen Thi Anh Hang and Hoang Le Truong, we introduce two notions of canonical stretched rings and sparse stretched rings We investigate the canonical stretched and sparse stretched properties of the numerical semigroup rings mentioned in the main results of Chapters and ii Declaration This dissertation was written on the basic of my research works carried out at University of Sciences, Thai Nguyen University under the guidance of Assoc Prof Dr Naoyuki Matsuoka and Assoc Prof Dr Doan Trung Cuong This work has not been submitted for any other degree or professional qualification Thai Nguyen, May 2022 The author Do Van Kien iii Acknowledgements First and foremost I would like to thank my academic advisors Associate Professor Naoyuki Matsuoka and Associate Professor Doan Trung Cuong I would like to thank them for their time, ideas, thoughtful comments, suggestions and patient guidance throughout my studies and writing of this dissertation In addition to accompanying me in research, I and two of my supervisors have had many trips together during my PhD studies They gave me a lot of insight into mathematics in life I am very grateful to them for everything I would like to thank Prof Nguyen Tu Cuong, Prof Shiro Goto, Prof Le Tuan Hoa, Prof Le Thi Thanh Nhan, and Prof Ngo Viet Trung for their efforts to organize the Japan-Vietnam Joint Seminar (JVS) on Commutative Algebra over the years The JVSs are very meaningful to Vietnamese PhD students like me The JVSs have promoted research cooperation between Japanese and Vietnamese mathematicians We heard many interesting lectures as well as thoroughly understood the challenges faced by both Vietnamese and Japanese researchers Actually, my thesis could not have been completed without the JVSs I would like to show my appreciation to Prof Shiro Goto for his valuable advice and suggestions during my stay in Japan He is not only a passionate teacher but also the one who is always holding up high the flag of “by and for young mathematicians” He was willing to spend hours iv explaining to me a problem I didn’t understand It is a motivation for me to complete this dissertation I would like to thank Tran Do Minh Chau, Naoki Endo, Shinya Kumashiro, Ryotaro Isobe, Tran Thi Phuong and Hoang Le Truong for their valuable comments and suggestions on my research results during my stay in Meiji University We worked very hard on different topics I also would like to thank my colleagues, PhD students in Thai Nguyen University as well as in Hanoi Institute of Mathematics They have always accompanied me, encouraged, and shared with me in research I sincerely thank Board of Rectors, Board of Deans of Mathematics and Information Department, Thai Nguyen University of Sciences for giving me the best study environment so that I can complete this dissertation I sincerely thank Board of Rectors, Board of Deans of Mathematics Department, Hanoi Pedagogical University for giving me the most favorable conditions to both complete my studies and ensure the teaching at my host university I am sincerely grateful to all the staff members of Hanoi Institute of Mathematics Although I am not a PhD student at the Institute, they gave me the best conditions to study as a PhD student of the Institute Finally, I would especially like to thank my parents, my wife and my two children for their love, encouragement, and especially for their patience during the time I was working intensively to complete my PhD studies The author Do Van Kien v Contents Notations Introduction Chapter Preliminaries 13 1.1 Numerical semigroups and numerical semigroup rings 13 1.1.1 Notable elements 14 1.1.2 Symmetry of numerical semigroups 19 1.2 Numerical semigroup rings and symmetry of numerical semigroups 22 1.2.1 Gorenstein numerical semigroup rings and symmetric numerical semigroups 23 1.2.2 Almost Gorenstein numerical semigroup rings and almost symmetric of numerical semigroups 26 Chapter Defining ideals of numerical semigroup rings with pseudo-Frobenius numbers being multiples of a fixed integer 31 2.1 RF-matrices associated to pseudo-Frobenius numbers and EagonNorthcott complexes 2.2 Defining ideals versus pseudo-Frobenius numbers 32 36 Chapter Defining ideals of numerical semigroup rings of maximal embedding dimension 47 3.1 Characterizations of determinantal defining ideals 48 3.2 Noetherian and Cohen-Macaulay properties of symbolic Rees algebra of determinantal defining ideals Chapter Canonical stretched rings 55 61 4.1 Canonical stretched rings 61 4.1.1 Characterizations of canonical stretched rings 64 4.1.2 Canonical stretched property of certain numerical semigroup rings 69 4.2 Sparse stretched rings 71 4.2.1 Sparse stretched rings 71 4.2.2 Sparse stretched property of certain numerical semigroup rings 83 Conclusions 86 List of author’s related publications 87 References 89 Notations N the set of non-negative integers Z the set of integer numbers Q the set of rational numbers R the set of real numbers ∅ the empty set A⊆B the set A is a subset of B A⊊B the set A is a proper subset of B A\B set difference A minus B the cardinality of a set X |X|, ♯X  ∨ a1 , , , , a n  the numerical semigroup generated by {a1 , , ai−1 , ai+1 , , an } AssR (M ) the set of associated prime ideals of M dimR M the dimension of an R-module M depthR (M ) the depth of an R-module M det A the determinant of a matrix A edim(M ) the embedding dimension of a module M DVR discrete value ring e(I, M ) the multiplicity of M w.r.t I e(M ) the multiplicity of M w.r.t the maximal ideal m ℓR (M ) the length of an R-module M gcd the greatest common divisor The ideals Ri give a strictly decreasing sequence R = R0 ⊋ R1 = m ⊋ R2 ⊋ ⊋ Rn = C ⊋ Rn+1 ⊋ , which induces the chain of duals: R ⊊ (R : R1 ) ⊊ ⊊ (R : Rn ) = R ⊊ (R : Rn+1 ) = t−1 R ⊊ For each non-negative integer i, we define D(i) = {sj | sj ≤ si and si − sj ∈ Val(R)} The set D(i) is often called the set of divisors of si , and its cardinality is denoted νi = |D(i)| Let I be an ideal of R We define the Frobenius number of the ideal I is the largest valuation of an element not belongs to I denoted by F(I), that is, F(I) = max{Val(a) | a ̸∈ I and a ∈ K} With the above notations, we have the following results Lemma 4.2.2 Let I be an m-primary ideal of R Then, we have F(I) = c(I) − = max{Val(a) | a ̸∈ I and a ∈ I : m} Proof First, we show that F(I) = c(I) − Indeed, for all a ∈ K such that Val(a) > c(I) − then Val(a) ≥ snI It follows that a ∈ RnI = I : R, whence a ∈ I On the other hand, let b ∈ K such that Val(b) = c(I) − Then Val(b) < c(I) = snI It implies that b ∈ / RnI Hence, b ∈ / I Therefore, we have c(I) − = F(I) as required Now let F′ (I) = max{Val(a) | a ̸∈ I and a ∈ I : m} Then, it is clear that F′ (I) ≤ F(I) On the other hand, assume that there is an element a ∈ K \ I such that Val(a) = F(I) but a ∈ / I : m Then, there exists α ∈ m such that aα ∈ / I It implies that Val(aα) ≤ Val(a) Hence, Val(a) + Val(α) ≤ Val(a) We get Val(α) ≤ which is a contradiction to α ∈ m Thus, F(I) ≤ F′ (I) as required Lemma 4.2.3 Let I be an m-primary ideal of R Then the following assertions are true 74 1) ((I : m) ∩ ms(R/I) ) \ I = {a ̸∈ I | Val(a) = F(I)} 2) If I = I1 ∩ I2 then F(I) = max{g(I1 ), g(I2 )} Moreover, if I = Ts i=1 Ii , where Ii are irreducible then F(I) = max {g(Ii )} 1≤i≤s 3) νnI ≤ ℓR (R/I) + Proof 1) The proof of assertion 1) employs the following claim Claim 4.2.4 For all a, b ∈ (I : m) \ I, if deg b ≤ deg a then aR ⊆ bR Proof of Claim 4.2.4 Let a, b ∈ (I : m) \ I such that deg b ≤ deg a Suppose deg b < deg a but aR ⊈ bR Then, since R is a DVR, we have bR ⊆ aR It follows that b = as for some s ∈ R Since deg b < deg a, we get b ∈ / mdeg(a) which implies that as ∈ / mdeg(a) Hence, since a ∈ mdeg(a) , one has Val(a) ≥ deg(a) > Val(as) = Val(a) + Val(s) Therefore, Val(s) < This is impossible because s ∈ R Thus, if deg b < deg a then aS ⊆ bR Now if deg(a) = deg(b) then aR = bR Indeed, without loss of generality we may assume bR ⊆ aR Then b = au for some u ∈ R Since b ∈ mdeg(a) \ mdeg(a)+1 , we have deg(a) + > Val(a) + Val(u) ≥ deg(a) Therefore, Val(a) + Val(u) = deg(a) We get Val(u) = This implies that u is invertible and hence aR = bR We now will show that (I : m) ∩ ms(R/I) \ I ⊆ {a ∈ K \ I | Val(a) = F(I)} Indeed, for all a ∈ (I : m) ∩ ms(R/I) \ I and for all b ∈ (I : m) \ I, then Proposition 4.1.3 says that deg(b) ≤ deg(a) Hence, aR ⊆ bR so that Val(b) ≤ Val(a) Therefore, by Lemma 4.2.2 we get Val(a) = F(I) as 75 desired For the reverse inclusion, we take a ∈ K \ I such that Val(a) = F(I) Then, we have am ⊆ I Because if otherwise then there exists b ∈ m such that ab ∈ / I It implies that Val(a) + Val(b) = Val(ab) ≤ Val(a), whence Val(b) = This is impossible because b ∈ n Thus, a ∈ I : m On the other hand, we take an element b ∈ ((I : m) ∩ ms(R/I) ) \ I, then Val(b) = F(I) Therefore, Val(a) = Val(b) and we get a = bu for some a unit u in S Hence, a ∈ ((I : m) ∩ ms(R/I) ) \ I as desired, that is, (I : m) ∩ ms(R/I) \ I ⊇ {a ∈ K \ I | Val(a) = F(I)} 2) Because K \ I ⊇ (K \ I1 ) ∪ (K \ I2 ), we get F(I) ≥ g(Ii ) for all i = 1, We suppose that F(I) > g(Ii ) for all i = 1, Then, there exist ∈ Ii such that Val(ai ) = F(I) for all i = 1, Hence, since a2 ∈ / I1 we get F(I) = Val(a2 ) ≤ g(I1 ) which is a contradiction Thus, F(I) = max{g(I1 ), g(I2 )} Now we decompose each I1 , I2 into the intersection of two ideals and application the result above for I1 , I2 The process continues we get the latter assertion in 2) Here note that the process will stop when Ii ’s are irreducible 3) One has νnI = |{sj | sj ≤ snI and snI −1 − sj ∈ Val(R)}| ≤ |{sj | sj ≤ snI and sj − e ̸∈ I}| + = |{sj | sj ≤ snI and sj − e ̸∈ I} ∪ {snI − 1}| ≤ ℓS (S/I) + Theorem 4.2.5 Let I be an m-primary ideal of R Then, we have s(R/I) ≤ s(R/I)e(m) ≤ F(I) The equalities hold in (4.8) if and only if R is a DVR 76 (4.8) Proof Let Ass(R/I) = {Qi }1≤i≤n the set of associated prime ideals of R/I Because ms(R/I) ⊈ I, we have (ms(R/I) ∩ I + ms(R/I)+1 )/ms(R/I)+1 and (ms(R/I) ∩ Qi + ms(R/I)+1 )/ns(R/I)+1 are proper vector subspaces of ms(R/I) /ms(R/I)+1 Moreover, since |R/m| = ∞, we get n ms(R/I) ∩ Q + ms(R/I)+1 [ ms(R/I) ms(R/I) ∩ I + ms(R/I)+1 i ∪ ( ) = ̸ s(R/I)+1 s(R/I)+1 ns(R/I)+1 m m i=1 Hence, we can choose a ∈ ms(R/I) \ (I ∪ Sn i=1 Qi ) which is a superficial element of ms(R/I) Then, since the analytic spread of ms(R/I) is 1, we have aR is a reduction of ms(R/I) Therefore, e(aR) = e(ms(R/I) ) ≥ s(R/I)e(m) On the other hand, we have e(aR) = ℓR (R/aR) = ℓR (R/aR) + ℓR (R/R) − ℓR (aR/aR) = ℓR (R/aR) = Val(a) ≤ F(I) We therefore get the second inequality in 4.8 Let us recall the following lemma that gives an explicit formula for the number of the set of divisors of si Lemma 4.2.6 ([10, Lemma 51]) For each i ∈ N let δ(i) be the number of gaps in the interval from to si − and let G(i) be the number of pairs of gaps whose sum equals si Then, νi = i − δ(i) + G(i) + The following theorem is an extension of a result of M Bras-Amorós [11, Theorem 1] Theorem 4.2.7 Let I be an m-primary ideal of R Then F(I) + ≤ ℓR (R/I) + 2δ(R) Proof It is straightforward to see that the intersection of two ideals satisfying the result also satisfies the result Now, by Lemma 4.2.3 2), it suffices to show that the result holds for the irreducible ideal I Also, by Lemma 77 4.2.3 3), it is sufficient to show that νnI − + 2δ(R) ≥ c(I) Indeed since snI ≥ c, δ(nI ) = δ(R), snI = nI + δ(R) Then, by Lemma 4.2.6, we get νnI − + 2δ(R) = (nI − δ(R) + G(nI ) + 1) − + 2δ(R) = nI + δ(R) + G(nI ) = snI + G(nI ) ≥ snI Definition 4.2.8 An m-primary ideal I of R is called maximum sparse, if the equality in Theorem 4.2.7 holds Example 4.2.9 Let R = k[[t4 , t6 +t7 , t15 ]] ⊆ R = k[[t]], where char(k) ̸= It is not difficult to see that Val(R) = ⟨4, 6, 13, 15⟩ Let I = (t12 − t16 , t14 + t15 + t16 + t17 )R By a direct computation, we get ℓR (R/R) = 7, F(I) = 23 ¯ and ℓR (R/I) = 10 Therefore F(I) + = 2ℓR (R/R) + ℓR (R/I) Hence, I is a maximal sparse ideal Notice that R is not a numerical semigroup ring Remark 4.2.10 Let R be as in Setting 4.2.1 then R always contains a maximal sparse ideal Indeed, if we let Val(R) = {λ0 < λ1 < · · · } then since |N\Val(R)| < ∞, there exists i > such that λi less than twice of the Frobenius number of Val(R) Then G(i) = so that I := Val(R)\D(i) is a maximum sparse ideal of Val(R) ( [11, Theorem 2]) Hence, the preimage Val−1 (I) is a maximum sparse ideal of R Lemma 4.2.11 If I is a maximum sparse ideal of R then I is a canonical ideal Proof Suppose that I is a maximum sparse ideal of R Since I is an mprimary ideal and R is a one-dimensional integral domain, we get < depthR (I) ≤ dimR I ≤ dim R = Hence, I is a Cohen-Macaulay Rmodule of dimension one On the other hand, since F(I) + = ℓR (R/I) + 78 2ℓR (R/R), yields that F(Val(I)) + = | Val(R) \ Val(I)| + 2ℓR (R/R) Thus, Val(I) is a maximum sparse ideal of the value ring Val(R) Therefore, Val(I) = Val(R) \ D(i) for some i such that G(i) = (see [10, 12]) We get that Val(I) is irreducible (by [5, Proposition 1]) Hence, I also is irreducible (by [5, Theorem 3]) Therefore, ℓR (I : m/I) = Hence, by [55, Satz 3.3], I is a canonical ideal of R Now, let us give a characterization for the Gorensteinness of R Theorem 4.2.12 The following statements are equivalent 1) R is Gorenstein 2) There is a maximum sparse ideal I of R such that I is a principal ideal 3) Every canonical ideal of R is maximum sparse Proof 1) ⇒ 2) Suppose that R is Gorenstein By Remark 4.2.10 we take I to be a maximum sparse ideal Then, Lemma 2.2.6 says that I is a canonical ideal of R Since the residue field S/n is infinite, there exists Q = (a) a reduction of I Then, since R is Gorenstein, we have I = Q (by [27, Lemma 3.7]) Hence, I is principal 2) ⇒ 1) Suppose I = aS is a maximum sparse ideal of S Then I is irreducible Hence, since a is regular on R, we get rR (R) = rR (R/aR) = rR (R/I) = ℓR (I : m/I) = Therefore, R is Gorenstein 1) ⇒ 3) Suppose that R is Gorenstein and I is a canonical ideal of R Since I ∼ = R, we have I = aR for some a ∈ I Notice that a is unit in Q(R), consequently ℓR (I/(I : R)) = ℓR (aR/(aR : R)) = ℓR (aR/a(R : R)) = ℓR (R/C) 79 Moreover, by Theorem 4.2.7, ℓR (R/C) ≤ ℓR (R/R) But, by [14, Theorem 1], ℓR (R/R) ≤ ℓR (R/C) This implies that ℓR (R/R) = ℓR (R/C) Thus, ℓR (I/(I : R)) = ℓR (R/R) so that I is maximum sparse 3) ⇒ 1) Taking the canonical module ωR as in Lemma 1.2.2 If ωR = R then it is clear that R is Gorenstein If otherwise, by writing ωR = R + n a P i R b i i=1 for some n > and some , bi ∈ R, bi ̸= 0, ≤ ∀i ≤ n and let x = b1 b2 bn , then xωR is a canonical ideal of S Let xzR is a minimal reduction of xωR , where z ∈ ωR Then, by ([19, Lemma 1.11]), we have ℓR (zR/ωR ) = ℓR (R/C) Moreover since C is an ideal of both R and R, we have xωR C = xωR RC = xzRC = xzC Therefore, ℓR (R/R) = ℓR (xzR/xzR) = ℓR (xzR/xωR ) + ℓR (xωR /xzR) = ℓR (R/C) + ℓR (xωR /xzR) = ℓR (xωR /xzC) = ℓR (xωR /xωR C) ≤ ℓR (xωR /CxωR )( because xωR C ⊆ CxωR ) On the other hand, by the assumption, xωR is maximum sparse Therefore, we obtain the equality above Hence, xωR C = CxωR It follows that xzC = xωR : R = x(ωR : R) = xC We get zC = C Note that z is in R If z ∈ m then by Nakayama’s lemma we get C = which is a contradiction Therefore, z is unit, whence R = xω Consequently, R is Gorenstein The proof is completed Now we define sparse stretched rings which is a special class of canonical stretched rings Definition 4.2.13 We say that (R, m) is a sparse stretched ring, if there is a maximum sparse ideal I ⊆ m2 of R such that R/I is a stretched ring 80 Proposition 4.2.14 If R is a sparse stretched ring then R is a canonical stretched ring Proof The assertion is immediate from Lemma 4.2.11 and the definition of sparse stretched rings Theorem 4.2.15 Assume that the Hilbert function of R is non-decreasing Then the following two conditions are equivalent 1) R is a sparse stretched ring 2) There exists an m-primary ideal I ⊆ m2 such that µR (mvR (I) /I) = 1, I : m ⊆ mvR (I) and s(R/I)(Val(z) − 1) = 2δ(R) + edim(R) − for some z ∈ m \ mvR (I)+1 and z ̸∈ I When this is the case, vR (I) = Proof 1) ⇒ 2) Since R is a sparse stretched ring, there exists an irreducible ideal I ⊆ m2 of R such that R/I is a stretched ring By Corollary 4.1.5, we have µR (mvR (I) /I) = and I : m ⊆ mvR (I) Now we see that R also is a canonical stretched ring, Lemma 4.1.9 implies that there exists s(R/I) a basis z1 , , zedim(R) for m such that ms(R/I) + I = (z1 s(R/I) because ms(R/I) ⊈ I, z1 s(R/I) we have z1 s(R/I) ∈ / I Moreover, since z1 s(R/I) ⊆ I : m Therefore, z1 s(R/I) Lemma 4.2.3 1), we get F(I) = Val(z1 ) + I Then, m ⊆ ms(R/I)+1 ⊆ I, ∈ (I : m)∩ms(R/I) \I Hence, by ) = s(R/I) Val(z1 ) Since R/I is a stretched ring and I is a maximum sparse ideal, we have s(R/I)(Val(z1 ) − 1) = F(I) − s(R/I) = 2δ(R) + ℓR (R/I) − s(R/I) − = 2δ(R) + edim(R) − 81 2) ⇒ 1) Let A = R/I and n = m/I It follows from µA (nvR (I) ) = and Corollary 4.1.5 that the Hilbert function of A is given by (4.3) Therefore, by [52, Theorem 3.4], we have hR (vR (A) − 1) − hR (1) + ≤ dimk (0 :A n) Moreover, since :A n ⊆ mvR (A) and µ(nvR (I) ) = 1, dimk (0 :A n) = Then hR (vR (A) − 1) − hR (1) ≤ Since the Hilbert function hR of R is non-decreasing, vR (A) = Then, by Corollary 4.1.5, R/I is stretched As z ̸∈ mvR (I)+1 , z ̸∈ I and n2 is a principal ideal, it yields ns(A) = z¯s(A) A, where z¯ denote the image of z in A Hence, by the same argument as in the proof of 1) ⇒ 2), we have F(I) = s(A) Val(z) = 2δ(R) + edim(R) − + s(A) = 2δ(R) + ℓR (R/I) − Hence, I is a maximum sparse ideal of R Here let us show examples One of them is sparse stretched the other one is not Example 4.2.16 Let R = k[[t6 , t7 , , t11 ]] is a numerical semigroup ring and put I = (t12 , t13 , t14 , t15 , t16 ) Then F(I) = 17, ℓR (R/I) = and δ(R) = Hence, F(I) = ℓR (R/I)+2δ(R)−1 which implies I is a maximum sparse ideal of R Moreover, the Hilbert function of R/I is 0 Hence, R is a sparse stretched ring Example 4.2.17 Let R = k[[t15 , t19 , t21 , t22 ]] is a numerical semigroup ring over a field k with the maximal ideal m = (t15 , t19 , t21 , t22 ) We have δ(R) = 82 36, edim R = which implies 2δ(R) + edim(R) − = 75 Observe that R is not a spares stretched ring Indeed, suppose R is a spares stretched ring Then, by Theorem 4.2.15, there exist I ⊆ m2 and z ∈ m\mvR (I)+1 such that z2 ∈ / I and s(R/I)(Val(z) − 1) = 75 Hence, Val(z) ∈ {2, 4, 6, 16, 26, 76} But it is clear that {2, 4, 6, 16, 26}∩Val(R) = ∅ Therefore, Val(z) = 76 and s(R/I) = It implies that I = m2 so that z ∈ I which is a contradiction Remark 4.2.18 In Example 4.2.17, it is easy to see that the defining ideal of R is P = I2  x22 x23 x24 x31 x1 x2 x3 x4  so that R is a canonical stretched ring (by Theorem 4.1.11) Example 4.2.17 also implies that the rings which satisfy Theorem 4.1.11 are canonical stretched but may not spares stretched 4.2.2 Sparse stretched property of certain numerical semigroup rings We maintain the assumption that H be a numerical semigroup minimally generated by a1 , a2 , , an with a1 < a2 < < an and R = k[[H]] be the numerical semigroup local ring of H over a field k Then R = k[[t]] is a DVR We denote Val the usual valuation associated to R In this subsection, we consider the sparse stretched property for the numerical semigroup ring k[[H]] satisfying the first condition in Theorem 3.1.1 of Chapter We assume that H satisfies the condition (4) in Theorem 3.1.1 The main result of the present subsection is stated as follows Theorem 4.2.19 Suppose H = ⟨n, n + h + α, , n + h + (n − 1)α⟩ for some h ≥ and α > Then R = k[[H]] is a sparse stretched ring Hence, R also is a canonical stretched ring Proof Let m = (tn , tn+h+α , · · · , tn+h+(n−1)α ) is the maximal ideal of R Then m2 = (t2n , t2n+h+α + · · · + t2n+h+(n−1)α , t2n+2h+2α , · · · , t2n+2h+(n−1)α ) 83 We consider the ideal I = (t2n+h+α , t2n+h+2α + · · · + t2n+h+(n−1)α ) ⊆ R It immediately obtains that I ⊆ m2 Moreover, for all ≤ i ≤ n − 1, by Lemma 3.1.5, one has 2n + 2h + iα − (2n + h + iα) = h ∈ H Hence, t2n+2h+iα ∈ I for all ≤ i ≤ n − In other words, m2 /I is generated by only one element t2n + I Therefore, it is sufficient to show that I is a maximum sparse ideal Indeed, because R has maximal embedding dimension, Proposition 1.1.14 implies that X n−1 n−1 (n + h + iα) − n i=1 (n − 1)(n + nα + 2h) = 2n ℓR (R/R) = δ(H) = On the one hand, we observe that for all i ≥ then 2n + 2h + αn + i − (2n + h + α) = h + (n − 1)α + i > h + (n − 1)α = F(H) Hence, 2n+2h+αn+i−(2n+h+α) ∈ H which yields that t2n+2h+αn+i ∈ I for all i ≥ On the other hand, for all ≤ i ≤ n − 1, then 2n + 2h + αn − (2n + h + iα) = h + (n − i)α ∈ / H Hence, t2n+2h+αn ∈ / I Therefore, the Frobenius number of I is F(I) = 2n + 2h + αn Now, by Lemma 3.1.5, h is divisible by n We write h = nq for some q ∈ N Because F(I) = (2 + α + 2q)n, we easily get {0, n, 2n, , (2 + α + 2q)n} ⊆ H \ Val(I) (4.9) Furthermore, it is clear that {2n + h + iα − n |i = 1, 2, , n} ⊆ H \ Val(I) 84 (4.10) It is not difficult to see that the elements are indicated in 4.9 and 4.10 are all elements of H \ Val(I) In other words, ℓR (R/I) = |H \ Val(I)| = 2q + n + α + Therefore, we obtain F(I) = 2ℓR (R/R) + ℓR (R/I) − which yields that I is a maximum sparse ideal Hence, R/I is a stretched ring This implies that R is a sparse stretched ring In this chapter, we introduce canonical stretched rings and sparse stretched rings We give the characterizations for these rings (Theorems 4.1.10 and 4.2.15) Besides, on the one hand, we have shown that the numerical semigroup ring k[[H]] satisfies the second condition in Theorem 2.2.1 is a canonical stretched ring On the other hand, we have also shown the numerical semigroup ring satisfies the first condition in Theorem 3.1.1 is a sparse stretched ring and hence also is canonical stretched ring 85 Conclusions 1) When the pseudo-Frobenius numbers of a numerical semigroup H are multiples of a fixed integer, we show a minimal system of generators of the defining ideal of the semigroup ring k[H] Furthermore, the minimal graded free resolution of k[H] is clearly described (Theorem 2.2.1); 2) When a numerical semigroup H has maximal embedding dimension, we give the characterization of the determinatal defining ideal of the semigroup ring k[H] in terms of pseudo-Frobenius numbers of H As an application, we give a minimal system of generators of the defining ideal and explicitly describe the minimal graded free resolution of k[H] (Theorem 3.1.1); 3) We show that the symbolic Rees algebra RS (P ) of the determinantal defining ideal of a numerical semigroup ring is Noetherian and Cohen-Macaulay Moreover, RS (P ) is Gorenstein if and only if the semigroup ring has embedding dimension (Theorem 3.2.2); 4) We introduce two notions of canonical stretched rings and sparse stretched rings We show that if P = I2  ℓ ℓ ℓ x22 x33 ··· xℓnn x11 x1 x2 ··· xn−1 xn  is the defining ideal of k[H] as in Theorem 2.2.1 with ℓi > for all i then k[H] is a canonical stretched ring (Theorem 4.1.11) We also show the numerical semigroup ring defined by determinantal ideal as in Theorem 3.1.1 always is a sparse stretched ring (Theorem 4.2.19) These results are based on the papers [26], [32], and [37] 86 List of author’s related publications [1] S Goto, D V Kien, N Matsuoka, and H L Truong, Pseudo-Frobenius numbers versus defining ideals in numerical semigroups rings, J Algebra, 508 (2018), 1–15 [2] D V Kien and N Matsuoka, Numerical semigroup rings of maximal embedding dimension with determinantal defining ideals, Springer INdAM Series, 40 (2020), 185–196 [3] N T A Hang, D V Kien, and H L Truong, Canonical stretched rings, Acta Math Vietnam., 47 (2022), 161–179 87 The results of this dissertation have been presented at • The weekly seminar of the Department of Mathematics and Informatics, Thai Nguyen University of Sciences • The weekly joint seminar of the Department of Algebra and the Department of Number Theory, Institute of Mathematics, Vietnam Academy of Science and Technology • “International Workshop on Commutative Algebra”, January 4–7, 2017, Thai Nguyen University of Sciences, Thai Nguyen city, Vietnam • “Meeting of the Catalan, Spanish and Swedish Math Societies”, June 12–15, 2017, Umeăa Universitet, Sweden ã International Workshop on Commutative Algebra, September 5–8, 2017, Ton Duc Thang University, Ho Chi Minh city, Vietnam • “Taiwan-Vietnam Workshop on Mathematics”, May 9–11, 2018, National Sun Yat-sen University, Kaohsiung, Taiwan • “The 9th Vietnam Mathematical Congress”, Autumn 14–18, 2018, Telecommunications University, Nha Trang, Vietnam • “The Vietnam-USA Joint Mathematical Meeting”, June 10–13, 2019, ICISE, Quy Nhon city, Binh Dinh, Vietnam 88