Betti numbers of modules over Noetherian rings with ... - IPM

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Chapter 4: Graded minimal free resolution of ideals 65 [50]. Hence p1, . . . , pr are the minimal prime ideals of I by [16, Theorem 2.1.6]. Thus Ass (R/I) = {p1, . . . , pr}. Remark 4.3.2. The ideal I is generated by a regular sequence on R. Thus the Castelnuovo-Mumford regularity of R/I is k1 + · · · + kt − t. Let ∆ be a simplicial complex and ∆ ∗ denote the Alexander dual of ∆, i.e., the simplicial complex ∆ ∗ = {F ⊆ [n] : [n] − F /∈ ∆} Corollary 4.3.3. Consider the graded version of Theorem 4.3.1. Then the regularity of R/I∆∗ is reg(R/I∆∗) = proj.dimR/I − 1. Proof. Using the primary decomposition of I∆∗ we have I∆ ∗ = (x1,1, . . . , x1,t1) ∩ · · · ∩ (xs,1, . . . , xs,ts). By a known result of Eagon and Reiner, K[∆] is Cohen-Macaulay if and only if I∆∗ has a linear resolution. Furthermore, proj.dim(K[∆]) = reg(I∆∗) by a result of Terai. In view of Theorem 4.2.1, R/I is Cohen-Macaulay and proj.dim(K[∆]) = t. Therefore, reg(I∆∗) = t and so reg(R/I∆∗) = t − 1. In the following we have some examples. Example 4.3.4. Let S = K[x1, x2] and J = (x1x2) be an ideal of S. Obviously we have β S 0 (S/J) = 1, β S 1 (S/J) = 1. Now let I0 = (x1x2, x3x4) be an ideal of R = K[x1, . . . , x4]. Then (4.6) implies that ⎧ ⎪⎨ ⎪⎩ β R 0 (R/I0) = β S 0 (S/J) = 1, β R 1 (R/I0) = β S 1 (S/J) + β S 0 (S/J) = 1 + 1 = 2, β R 2 (R/I0) = β S 1 (S/J) = 1.
Chapter 4: Graded minimal free resolution of ideals 66 Furthermore, applying [16, Excersice 4.4.16 (b)], it is easy to see that I0 =(x1x2, x3x4) = (x1, x3x4) ∩ (x2, x3x4) =(x1, x3) ∩ (x1, x4) ∩ (x2, x3) ∩ (x2, x4), Hence I0 is the Stanley-Reisner ideal of a pure simplicial complex ∆0 consisting of 4 facets all of dimension 1. As a result dim K[∆0] = dim R/I0 = dim ∆0 + 1 = 1 + 1 = 2. ✷ Example 4.3.5. Let S = K[x1, . . . , x4] and J1 = (x1x3, x2x4) be an ideal of S. Then I1 = (x1x3, x2x4, x5x6) is an ideal of R = K[x1, . . . , x6] and using Example 4.3.4 we have ⎧ ⎪⎨ ⎪⎩ β R 0 (R/I1) = β S 0 (S/J1) = 1, β R 1 (R/I1) = β S 1 (S/J1) + β S 0 (S/J1) = 2 + 1 = 3, β R 2 (R/I1) = β S 2 (S/J1) + β S 1 (S/J1) = 1 + 2 = 3, β R 3 (R/I1) = β S 2 (S/J1) = 1. Furthermore, by the help of [16, Excersice 4.4.16 (b)] I1 =(x1x3, x2x4, x5x6) = (x1, x2x4, x5x6) ∩ (x3, x2x4, x5x6) =(x1, x2, x5x6) ∩ (x1, x4, x5x6) ∩ (x3, x2, x5x6) ∩ (x3, x4, x5x6) =(x1, x2, x5) ∩ (x1, x2, x6) ∩ (x1, x4, x5) ∩ (x1, x4, x6) ∩ (x3, x2, x5) ∩ (x3, x2, x6) ∩ (x3, x4, x5) ∩ (x3, x4, x6). Thus I1 is the Stanley-Reisner ideal of a pure simplicial complex ∆1 which consists of 8 facets all of dimension 2. One can easily see that dim K[∆1] = dim R/I1 = dim ∆1 + 1 = 2 + 1 = 3. ✷
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Betti numbers of modules over Noeth
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Abstract Betti numbers are topologi
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Abstract v to test and check the li
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Contents vii 4.3 Analysis of a spec
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List of Figures 1.1 The simplicial
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xi Dedicated to my father, my mothe
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Chapter 1: Preliminaries 2 ideal. I
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Chapter 1: Preliminaries 4 Let F
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Chapter 1: Preliminaries 6 i < 0, t
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Chapter 1: Preliminaries 8 Remark 1
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Chapter 1: Preliminaries 10 the rel
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Chapter 1: Preliminaries 12 1.4 Loc
Page 25 and 26: Chapter 1: Preliminaries 14 a minim
Page 27 and 28: Chapter 1: Preliminaries 16 Example
Page 29 and 30: Chapter 1: Preliminaries 18 Figure
Page 31 and 32: Chapter 2: Linear resolution of pow
Page 49 and 50: Chapter 3: Growth of the Betti sequ
Page 65 and 66: Chapter 4 Graded minimal free resol
Page 67 and 68: Chapter 4: Graded minimal free reso
Page 75: Chapter 4: Graded minimal free reso
Page 79 and 80: Chapter 5 Bass numbers of Local Coh
Page 81 and 82: Chapter 5: Bass numbers of Local Co
Page 93 and 94: Bibliography [1] T. D. Alwis. Free
Page 95 and 96: Bibliography 84 [28] D. Eisenbud an
Page 97 and 98: Bibliography 86 [58] M. E. Rossi an
Page 99 and 100: Appendix A: Algorithms for our crit
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