Luận án Các hệ số hilbert trong vành cohen-macaulay và một số lớp mở rộng

→ 0 82 and S is Cohen-Macaulay, we get the following commutative diagram 0 // u/qu ι //R/q ϕdq,R  ϵ // S/qS //  _ ϕdq,S  0 Hdm(R) ∼= //Hdm(S) with the exact first row. Let x ∈ 0 :S/qS m. Then, since ϕdq,R is surjective on the socles, we get an element y ∈ 0 :S/qS m such that ϕdq,S(x) = ϕdq,R(y). Thus ϵ(y) = x, because the canonical map ϕdq,S is injective. Therefore [u+ q] : m = q : m+ u. Thanks to Lemma 2.1.8, it can be proven that [qn + u] :R m = qn :R m+ u (4.1) for all n ≥ 1. Since S is Cohen-Macaulay, we have the following exact sequence 0→ u/qn+1u→ R/qn+1R→ S/qn+1S → 0. It follows from (4.1) and applying HomR(k, ∗) to the above sequence that we obtain the exact sequence 0→ HomR(k, u/qn+1u)→ HomR(k,R/qn+1)→ HomR(k, S/qn+1S)→ 0. Thus ℓR([qn+1 :R m]/qn+1) = ℓR([qn+1u :u m]/qn+1u)+ℓR([(qn+1+u) :R m]/qn+1+u). Since q is m-primary, by [13, Lemma 4.2], the degree of the polynomials ℓR([qn+1u :u m]/qn+1u) is dim u − 1. Since dim u < dimR and the degree of the polynomials ℓR([qn+1 :R m]/qn+1), ℓR([(qn+1 + u) :R m]/(qn+1R+ u)) are dimR − 1, the conclusion follows by comparing the coefficients of the polynomials in the above inequality, as required. By [53, Lemma 2.1], we get that f0(q, R) = f0(q, S) = rd(S) = rd(R). 83 On the other hand, by [53, Theorem 1.1], we have e1(qS : mS, S)− e1(qS, S) = rd(S) = rd(R). Since (q : m)S ⊆ qS : mS, we have ℓR(S/(q : m)n+1S) ≥ ℓR(S/(qS : mS)n+1), for all n ≥ 0. Since S is a regular local ring and q ⊆ m2, by [5, Theorem 2.1], we have qS : mS is integral over qS and so e0(qS : mS, S) = e0((q : m)S, S) = e0(qS, S). Therefore, we have e1((q : m)S, S) ≤ e1(qS : mS, S). Hence e1((q : m)S, S)− e1(qS, S) ≤ rd(R). On the other hand, by Lemma 1.1.8, we have e1(q, R) =  e1(q, S) if dim u ≤ d− 2, e1(q, S)− e0(q, u) if dim u = d− 1, e1(q : m, R) =  e1(q : m, S) if dim u ≤ d− 2, e1(q : m, S)− e0(q : m, u) if dim u = d− 1. If dim u < d − 1, then we have e1(q : m, R) − e1(q, R) = e1(q : m, S) − e1(q, S) ≤ rd(R) = f0(q, R), as required. Now, we can assume that dim u = d− 1. Then we have e1(q : m, R)− e1(q, R) ≤ rd(R)− e0(q : m, u) + e0(q, u). By the Prime Avoidance Theorem, we can choose a parameter element x ∈ q of R such that (x) ∪ u = 0. Put R = R/(x), and S = S/(x). Then we have the following exact sequence 0→ u→ R→ S → 0. 84 Consequently, e0(m, R) = e0(m, S) + e0(m, u) ≥ 2, so that by [24, Proposition 2.3], qR : m = (q : m)R is integral over qR. Thus, e0(q : m, R) = e0(q, R). Since x is regular in S, e0(q : m, S) = e0(q, S). Hence, we have e1(q : m)− e1(q) ≤ rd(R)− e0(q : m, u) + e0(q, u) = rd(R)− (e0(q : m, R)− e0(q : m, S)) + (e0(q, R)− e0(q, S)) = rd(R) = f0(q, R). Now we consider the case (a). Since dimR ≥ 2, by [24, Proposition 2.3], we have mIn = mqn for all n. Therefore In ⊆ qn : m for all n. Consequently, we obtain ℓR(R/qn+1)− ℓR(R/In+1) = ℓR(In+1/qn+1) ≤ ℓR((qn+1 : m/qn+1). Hence we have e1(q : m, R)− e1(q, R) ≤ f0(q, R), as required. In [52] and [53], H.L. Truong provided the characterizations of Cohen- Macaulay rings in terms of their Chern numbers, irreducible multiplicities, index of reducibility of a parameter of R, and the Cohen-Macaulay type provided R is unmixed. Notice that the necessary and sufficient conditions of these characterizations need to hold true for all parameter ideals of R. Recently, N.T.T. Tam and H.L. Truong [48] gave the characterizations of Cohen-Macaulay rings in terms of their Chern numbers, irreducible mul- tiplicities, and the type, which was introduced by S. Goto and N. Suzuki ([25, Definition (01)]). Let us now state the main results of this section and its corollaries. 85 Theorem 4.1.4. Assume that R is unmixed of dimension d ≥ 2. Then the following statements are equivalent. i) R is Cohen-Macaulay. ii) For all C-parameter ideals q ⊆ m2, we have N (R) = f0(q) = e1(q : m)− e1(q) = rd(R). iii) For some C-parameter ideal q ⊆ m2 of R, we have N (R) ≤ e1(q : m)− e1(q). iv) For some C-parameter ideal q ⊆ m2 of R , we have N (R) ≤ f0(q, R). v) For some C-parameter ideal q ⊆ mg(R)+1 of R, we have f0(q) ≤ rd(R). vi) For some C-parameter ideal q ⊆ mg(R)+1 of R, we have e1(q : m)− e1(q) ≤ rd(R). Proof. i)⇒ ii), i)⇒ iii) and i)⇒ v) follow from Lemma 3.1.7. iii)⇒ iv) and v)⇒ vi) are trivial. iv)⇒ i) and vi)⇒ i) are immediate from Lemma 4.1.1 and 2.2.5. The following result establishs the relationship among the stable value N (R/u), Chern numbers e1(q : m), e1(q), irreducible multiplicities f0(q), and the Cohen-Macaulay type rd(R), and also shows that the differ- ence e1(q : m)− e1(q) ranges among only finitely many value, where q runs over all C-parameter ideals of R. 86 Proposition 4.1.5. Assume that dimR ≥ 2 and q is a C-parameter ideal of R. Then we have N (R/u) ≥ f0(q) ≥ e1(q : m)− e1(q) ≥ rd(R), if either R is non-regular or q ⊆ mg(R)+1. Proof. Put S = R/u, IR = q :R m and IS = qS :S mS. Then S is unmixed with dimS = d. If R is non-regular or q ⊆ mg(R)+1, then by Lemma 4.1.1 and 3.1.7, we obtain N (S) ≥ f0(q, S). Thus by [48, Lemma 2.1], Lemma 4.1.2 and Corollary 2.3.5, we have N (S) ≥ f0(q, S) ≥ f0(q, R) ≥ e1(q :R m, R)− e1(q, R) ≥ rd(R), and this completes the proof. For every parameter ideal q of R, by [32, Theorem 3.6], we have e1(q) ≤ 0. Thus we can establish the relationship among the above invari- ants as follows. Remark 4.1.6. Assume that dimR ≥ 2 and q is a C-parameter ideal of R. Then we have N (R/u) ≥ f0(q) ≥ e1(q : m)− e1(q) ≥ e1(q : m), if either R is non-regular or q ⊆ mg(R)+1. Let Ξi(R) = {ei(q : m) | q is a C-parameter ideal of R}, for all i = 1, . . . , d. Now, we can give an upper bound for the set Ξ1(R). Corollary 4.1.7. Assume that R is non-regular and dimR ≥ 2. Then Ξ1(R) ⊆ (−∞,N (R/u)]. 87 Theorem 4.1.8. Assume that R is a non-regular unmixed local ring of dimension d ≥ 2. Then Ξ1(R) ⊆ (−∞,N (R)]. Moreover, N (R) ∈ Ξ1(R) if and only if R is Cohen-Macaulay. Proof. R is a non-regular unmixed local ring, by Remark 2.1.8, we have Ξ1(R) ⊆ (−∞,N (R)]. Now, we assume that R is Cohen-Macaulay. Since R is non-regular, by Lemma 3.1.7, we get e1(q : m) = N (R). Conversely, assume that e1(q :R m, R) = N (R). By Prosition 4.1.5, we have N (R) = e1(q : m) ≤ e1(q : m)− e1(q) ≤ f0(q) ≤ N (R). Therefore N (R) = f0(q). Thus R is Cohen-Macaulay, because of Lemma 4.1.1. This completes the proof. The following consequence provides a characterization of Gorenstein rings. Corollary 4.1.9. Assume that R is unmixed of dimension d ≥ 2. Then the following statements are equivalent. i) R is Gorenstein. ii) We have N (R) = 1. iii) For some C-parameter ideal q ⊆ mg(R)+1 of R, we have f0(q) = 1. iv) For some C-parameter ideal q ⊆ mg(R) + 1, we have e1(q : m)− e1(q) = 1. Proof. Since R is Gorenstein, we obtain N (R) = r(R) = 1, that is i)⇒ ii) is true. On the other hand, by Proposition 4.1.5, for all C-parameter ideal q ⊆ mg(R)+1 of R, we get that 1 = N (R) ≥ f0(q, R) ≥ e1(q :R m)− e1(q) ≥ rd(R) = 1. 88 It means that , ii) ⇒ iii) and iii) ⇒ iv) hold. Now, we prove iv) ⇒ i). Let q be a C-parameter ideal of R such that e1(q :R m) − e1(q) = 1 and q ⊆ mg(R)+1. Then we have e1(q : m)− e1(q) ≤ rd(R). By Theorem 4.1.4, R is Cohen-Macaulay. Therefore, we have 1 = e1(q :R m)− e1(q) = rd(R). Hence, R is Gorenstein, as required. 4.2 Finiteness of sets of Chern numbers In this section, we analyze finiteness of sets Ξi(R), for each i = 1, . . . , d. And then we deduced that the local cohomology modules {Him(R)}i<d are of finite length. For each n, i ≥ 1, we put Ωni (R) = {ei(q :R m, R) | q is a C-parameter ideal of R contained in mn}. Note that Ωti(R) ̸= ∅ and Ωti(R) ⊆ Ωt′i (R) ⊆ Ξi(R), for all t ≥ t′ ≥ 1. Moreover, we get by Lemma 3.1.7 that if R is a non-regular generalized Cohen-Macaulay ring, then Ξi(R) and Ωni (R) are finite for all i ≥ 1 and n ≥ 1. Let x = x1, x2, . . . , xd be a C-system of parameters of R. Put q[n] = (xn1 , xn2 , . . . , xnd) and I [n] = q[n] :R m, for all n ≥ 1. For each t ≥ 1, let Ωtx,1(R) = {e1(I [n], R) | n ≥ t}. Note that Ωtx,1(R) ̸= ∅ and Ωtx,1(R) ⊆ Ωt′x,1(R) for all t ≥ t′ ≥ 1. Lemma 4.2.1. Assume that R is unmixed with dimR = d ≥ 2. Suppose that there exist a C-system of parameters x = x1, . . . , xd of R and an integer t ≥ 2 such that the set Ωtx,1(R) is finite. Let h = max{|X| : X ∈ Ωtx,1(R)}+N (R). 89 Then mhHim(R) = 0 for all i ̸= d. Proof. We use induction on the dimension d of R. Suppose that d = 2. Since R is unmixed, R is generalized Cohen-Macaulay, and so N (R) = 2r1(R) + r2(R). We chosse n ≥ t. Put q = (xn1 , xn2) and I = q :R m. Then, since x1, x2 is a C-system of parameters of R, by Lemma 3.1.7 we have ℓ(H1m(R)) = −e1(I, R) + r1(R) + r2(R) ≤ |e1(I, R)|+N (R) ≤ h. Hence, mhH1m(R) = 0. Suppose that d ≥ 3 and our assertion holds true for d − 1. We put A = R/(x1). Then dimA = d − 1 and by the fact 3.1.4, we have AssA ⊂ AsshA ∪ {m}. It follows that UA0 = H0m(A). Let B = A/H0m(A). Then B is unmixed. Now, by Lemma 3.1.4, x′ = x2, x3, . . . , xd is a C-system of parameters of A. Thus, x′ is also a C-system of parameters of B. On the other hand, it follows from Lemma 3.1.4 that [(xm2 , xm3 , . . . , xmd ) + H0m(A)] :A mA = [(xm2 , xm3 , . . . , xmd ) :A mA] + H0m(A), for all m ≥ 1. Therefore, we have the following. Claim 4.2.2. Ωtx′,1(B) ⊆ Ωtx,1(R). Proof. For each m ≥ t, we put y2 = xm2 , . . . , yd = xmd , q = (y1, y2 . . . yd), I = q :R m, q′ = (y2, . . . , yd). Since x is a C-system of parameters of R, x1 is a superficial element of R with respect to I and q. Therefore, by Lemma 1.1.7, we have e1(I, R) = e1(I ′, A), where I ′ = q′A :A mA = (q :R m)A. It follows from Lemma 1.1.8, d ≥ 3 and [q′ +H0m(A)] :A mA = [q′ :A mA] + H0m(A) that we have e1(I ′, A) = e1(q′B :B mB,B). 90 Consequently, e1(q′B :B mB,B) = e1(I, R) ∈ Ωtx,1(R). Hence Ωtx′,1(B) ⊆ Ωtx,1(R), as required. It follows from the above claim that the set Ωtx′,1(B) is finite. Thus by the hypothesis inductive on d, we have mh′Him(B) = 0, for all i ̸= d− 1, where h′ = max{|X| : X ∈ Ωtx′,1(B)} + N (B). On the other hand, by Lemma 3.1.4, we have N (R) = N (A) = N (B) + r0(A). Therefore h′ ≤ h, and so mhHim(B) = 0, for all i ̸= d−1. Hence mhHim(A) = 0, for all 1 ≤ i ≤ d− 2. Now since R is unmixed, by [33, Lemma 3.4], x1 is R-regular and x1H1m(R) = 0. Therefore it follows from the exact sequence 0 //R x1 //R //A // 0 that we have the surjective maps Him(A) // // 0 :Hi+1m (R) x1 for all i ≤ d− 2 and the injective map H1m(R)   //H1m(A), since x1H1m(R) = 0. Thus we have mh[0 :Hi+1m (R) x1] = 0 and m hH1m(R) = 0 for all 1 ≤ i ≤ d − 2. It follows from Him(R) = ⋃ n≥1 [0 :Him(R) m n] that we have mhHim(R) = 0 for all i ̸= d, as required. Lemma 4.2.3. Assume that d ≥ 2. Suppose that R/u is generalized Cohen-Macaulay. Then q :R m = q :R/u m for all parameter ideals q of R/u contained in m2. Proof. Put S = R/u. Let q is a parameter of S, IR = q :R m and IS = q :S mS. Then q is also a parameter of R. Since S is generalized Cohen- Macaulay and q ⊆ m2, by Lemma 3.1.6, we have (IS)2 = qIS. Thus by [23, 91 Theorem 1.1], we obtain e1(IS, S) = ℓR(S/IS)− e0(q, S)+ e1(q, S). By [23, Theorem 3.1], we have ℓR(S/IS)− e0(q, S) = e1(IS, S)− e1(q, S) ≥ e1(IR, S)− e1(q, S) ≥ ℓR(S/IRS)− e0(q, S). Then we have IS = IR since IR ⊆ IS. Theorem 4.2.4. Assume that d ≥ 2. Then the following statements are equivalent. i) R/u is generalized Cohen-Macaulay and dim u ≤ d− 2. ii) There exists an integer t ≥ 2 such that the set Ωt1(R) is finite. Proof. i) ⇒ ii). Let q be a C-parameter ideals of R contained in mg(R)+1. We get by Proposition 4.1.5 that N (R/u) ≥ e1(q :R m, R)− e1(q, R) ≥ rd(R). On the other hand, the set {e1(q, R) | q is a parameter of R} is finite be- cause of [18, Theorem 4.5]. Thus Ωt1(R) is finite for all t ≥ g(R) + 1. ii) ⇒ i). We put S = R/u and u = dim u. Then there exists a C-system x = x1, . . . , xd of parameters of R such that 1) (x) ⊆ mt, 2) (xu+1, xu+2, . . . , xd)u = 0, 3) x is a C-system of parameters of S. Let q[n] = (xn1 , . . . , xnd)R, I [n] R = q[n] :R m and I [n] S = q[n]S :S mS, where n ≥ t. We have I [n]R ⊆ I [n]S , and so we get ℓR(S/(I [n]S )m+1) ≤ ℓR(S/(I [n]R )m+1S) 92 for all m ≥ 0. Thus e1(I [n]S , S) ≥ e1(I [n]R , S), because I [n]S is integral over q[n]S by [24, Proposition 2.3]. Therefore by Proposition 4.1.5, we have N (S) ≥ e1(I [n]S , S)− e1(q[n], S) ≥ e1(I [n]R , S)− e1(q[n], S) = e1(I [n]R , R)− e1(q[n], R) ≥ rd(R) = rd(S). (4.2) Since the set Ωt1(R) is finite, so is the set {e1(I [n]R , R) | n ≥ 0}. Thus the set {e1(q[n]R , R) | n ≥ 0} is finite. By Lemma 1.1.8, we have 0 ≥ e1(q[n]R , S) ≥ e1(q[n]R , R). Therefore, the set {e1(q[n]R , S) | n ≥ 0} is finite. By (4.2), the set Ωt′x,1(S) is finite for some t′. It follows from S is unmixed and Lemma 4.2.1 that S is generalized Cohen-Macaulay. Now suppose that u = d− 1. It follows from Lemma 3.1.7 that e1(I [n]S , S) = − d−1∑ j=1 d− 2 j − 1 ℓR(Hjm(S))− d∑ j=1 d− i j − 1 rj(S)  , which is independent of the choice of n. On the other hand, we have e0(I [n]R , u) = e0(q[n], u) = e0((xn1 , . . . , xnu), u) = nue0(q, u) ≥ nu. Thus by Lemma 1.1.8, we get that −e1(I [n]R , R) = −e1(I [n]R , S) + e0(I [n]R , u) = −e1(I [n]S , S) + nue0(q, u), which is in contradiction with the finiteness of Ωn1(R) and Ωn1(S). Hence u ≤ d− 2. Applying Theorem 4.2.4 we obtain the following result. Theorem 4.2.5. Assume that R is a non-regular unmixed local ring of dimension d = dimR ≥ 2. Then the following statements are equivalent. 93 i) R is generalized Cohen-Macaulay. ii) The set Ξ1(R) is finite. Proof. It is immediate from Theorem 4.2.4 and Lemma 3.1.7. Theorem 4.2.6. Assume that R is a non-regular with dimR = d ≥ 2. Then the following statements are equivalent. i) R/H0m(R) is Cohen-Macaulay. ii) The sets Ξi(R) are finite for all i ≥ 2 and there exists a C-parameter ideal q ⊆ mg(R)+1 of R such that e1(q : m)− e1(q) ≤ rd(R). Proof. i) ⇒ ii). Let S = R/u. Then since S is Cohen-Macaulay, R is generalized Cohen-Macaulay. Therefore since R is non-regular, by Lemma 3.1.7, we obtain Ξi(R) are finite for all i ≥ 2. Let q ⊆ mg(R)+1 be a C-parameter ideals of R. Then by [16, Lemma 2.4], we can assume that [u+ q] : m = u+ [q : m]. It follows from the Lemma 3.1.7 and 1.1.8 that e1(q :R m, R)− e1(q, R) = e1(q :S m, S)− e1(q, S) = e1(qS :S mS, S)− e1(q, S) = rd(S) = rd(R), as required. ii)⇒ i). Since e1(q :R m, R)− e1(q, R) ≤ rd(R) for some C-parameter ideals q ⊆ mg(R)+1 of R, by Lemma 2.2.5, S is Cohen-Macaulay. Suppose that u = dim u ≥ 1. Then there exists a C-system x = x1, . . . , xd of parameters of R such that 94 1) (x) ⊆ m2, 2) (xu+1, xu+2, . . . , xd)u = 0, 3) x is a C-system of parameters of S. Let q[n] = (xn1 , . . . , xnd)R, I [n] R = q[n] :R m and I [n] S = q[n]S :S mS for all n ≥ 0. Since S is Cohen-Macaulay, we get by Lemma 4.2.3 that I [n]S = I [n]R Moreover, it follows Lemma 3.1.7 that ed−u(I [n]S , S) =  0 if u ≤ d− 2, rd(S) if u = d− 1. which is independent of the choice of n. On the other hand, we have e0(I [n]R , u) = e0(q[n], u) = e0((xn1 , . . . , xnu), u) = nue0(q, u) ≥ nu. Thus, by Lemma 1.1.8, we get that (−1)d−ued−u(I [n]R , R) = (−1)d−ued−u(I [n]R , S) + e0(I [n]R , u) = (−1)d−ued−u(I [n]S , S) + nue0(q, u) ≥ nu + rd(S), which is in contradiction with the finiteness of Ξd−u(R). Hence u = 0, and this complete the proof. Theorem 4.2.7. Assume that R is a non-regular with dimR = d ≥ 2. Then the following statements are equivalent. i) R is generalized Cohen-Macaulay. ii) The sets Ξi(R) are finite for all i = 1, 2, . . . , d. Proof. i)⇒ ii) follows from Lemma 3.1.7. ii) ⇒ i). We put S = R/u and u = dim u. Assume that u > 0. Since 95 Ξ1(R) is finite, it follows from Theorem 4.2.5 that S is generalized Cohen- Macaulay. Then there exists a C-system x = x1, . . . , xd of parameters of R such that 1) (x) ⊆ m2, 2) (xu+1, xu+2, . . . , xd)u = 0, 3) x is a C-system of parameters of S. Let q[n] = (xn1 , . . . , xnd)R, I [n] R = q[n] :R m and I [n] S = q[n]S :S mS for all n ≥ 0. Since S is generalized Cohen-Macaulay, we get by Lemma 4.2.3 that I [n]S = I [n] R . Moreover, it follows Lemma 3.1.7 that ed−u(I [n]S , S) = (−1)d−u  u∑ j=1 u− 1 j − 1 ℓR(Hjm(S))− u+1∑ j=1 d− i j − 1 rj(S)  , which is independent of the integer n. On the other hand, we have e0(I [n]R , u) = e0(q[n], u) = e0((xn1 , . . . , xnu), u) = nue0(q, u) ≥ nu. Thus, by Lemma 1.1.8, we get that (−1)d−ued−u(I [n]R , R) = (−1)d−ued−u(I [n]R , S) + e0(I [n]R , u) = (−1)d−ued−u(I [n]S , S) + nue0(q, u), which is in contradiction with the finiteness of Ξd−u(R) and Ξd−u(S). Hence u = 0, as required. Remark 4.2.8. i) If for all i = 1, . . . , d, we define Ξi(R) = {ei(q : m) | q is a C-parameter ideal of M contained in m2}, then Corollary 4.1.7, Theorem 4.1.8, 4.2.5, 4.2.6, and 4.2.7 remains valid even without the condition that R is non-regular. 96 ii) In [29], A. Koura and N. Taniguchi proved that the Chern numbers of m-primary ideals in R range among only finitely many values if and only if dimR = 1 and R/H0m(R) is analytically unramified. Now we close this chapter with the following example of non-generated Cohen-Macaulay local rings R with |Ξ1(R)| = 1. Moreover, this example also shows that the statements (1), (4), and (5) in Theorem 4.1.4 will not be equivalent if the unmixed condition is removed from the hypothesis. Example 4.2.9. Let S = k[[X, Y, Z,W ]] be the formal power series ring over a field k. Put R = S/[(X, Y, Z) ∩ (W )]. Then (1) R is not unmixed with dimR = 3. Moreover, R is not generalized Cohen-Macaulay. (2) |Ξ1(R)| = 1. (3) We have f(q, R) = e1(q :R m)− e1(q, R) = rd(R), for all parameter ideals q in R. Proof. Using similar arguments as in the Example 2.3.8, we have R is not unmixed with dimR = 3 and R is not generalized Cohen-Macaulay. However, for all parameter ideals q of R, one has ℓR(R/qn+1) = ℓR(A/qn+1) + ℓR(B/qn+1) = ℓR(A/qA) n+ 3 3 + ℓR(B/qB) n+ 1 1  and ℓR([qn+1 :R m]/qn+1) = ℓR([qn+1 :A m]/qn+1) + ℓR([qn+1 :B m]/qn+1) = n+ 1 2 + 1, 97 for all integers n ⩾ 0. Since ℓR(R/In+1) = ℓR(R/qn+1) − ℓR([qn+1 :R m]/qn+1), we have e1(q :R m) = 0 + 1 = 1. Thus |Ξ1(R)| = 1. Moreover, we have f(q, R) = e1(q :R m)− e1(q, R) = rd(R) = 1, as required. 98 Conclusion of Chapter 4 The main results of this chapter include: • An upper bound for sets of Chern numbers e1(q : m, R) of local rings R, where q is a C-parameter ideal of R (Corollary 4.1.7). • Some characterizations of Cohen-Macaulayness of unmixed local rings R in terms of Hilbert coefficients e1(q : m, R), e1(q, R), irreducible multiplicity f0(q, R), the stable value N (R) and the Cohen-Macaulay type r(R), where q is a C-parameter ideal of R (Theorem 4.1.4). • Some characterizations of generalized Cohen-Macaulayness of local rings R in terms of the behavior of Hilbert coefficients ei(q : m, R) (i = 1, . . . , d), where q is a C-parameter ideal of R. (Theorem 4.2.5 and 4.2.7). 99 Conclusions The main results of this dissertation include: • A characterization of sequential Cohen-Macaulayness of local rings R in terms of Hilbert coefficients ei(q : m, R), ei(q, R) (i = 1, . . . , d), where and q is a distinguished parameter ideal of R (Theorem 2.3.1). • Some characterizations of Cohen-Macaulay local rings R in terms of sectional genera sg(q : m, R), sg(q, R), where q is a C-parameter ideal of R (Theorem 3.2.5, 3.2.4, 3.3.2 and 3.3.4) • An upper bound for sets of Chern numbers e1(q : m, R) of local rings R, where q is a C-parameter ideal of R (Corollary 4.1.7). • Some characterizations of Cohen-Macaulayness, and generalized Cohen-Macaulayness of local rings R in terms of the behavior of Hilbert coefficients ei(q : m, R) (i = 1, . . . , d), and q is a C-parameter ideal of R. (Theorem 4.1.8, Theorem 4.2.5, and Theorem 4.2.7). 100 List of author’s related publications [1] K. Ozeki, H.L. Truong and H.N. 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