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

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Chapter 4: Graded minimal free resolution of ideals 67 Example 4.3.6. Let S = K[x1, . . . , x4] and J2 = (x1x3, x2x4) be an ideal of S. Then I2 = (x1x3, x2x4, x5x6x7) is an ideal of R = K[x1, . . . , x7] and similar to Example 4.3.5 we have β R 0 (R/I2) = 1, β R 1 (R/I2) = 3, β R 2 (R/I2) = 3, and β R 3 (R/I2) = 1. Furthermore, using [16, Excersice 4.4.16 (b)] I2 =(x1x3, x2x4, x5x6x7) = (x1, x2x4, x5x6x7) ∩ (x3, x2x4, x5x6x7) =(x1, x2, x5x6x7) ∩ (x1, x4, x5x6x7) ∩ (x3, x2, x5x6x7) ∩ (x3, x4, x5x6x7) =(x1, x2, x5) ∩ (x1, x2, x6) ∩ (x1, x2, x7) ∩ (x1, x4, x5) ∩ (x1, x4, x6) ∩ (x1, x4, x7)∩ (x3, x2, x5) ∩ (x3, x2, x6) ∩ (x3, x2, x7) ∩ (x3, x4, x5) ∩ (x3, x4, x6) ∩ (x3, x4, x7). Hence I2 is the Stanley-Reisner ideal of a pure simplicial complex ∆2 which consists of 12 facets all of dimension 3. One can easily see that dim K[∆2] = dim R/I2 = dim ∆2 + 1 = 3 + 1 = 4. ✷
Chapter 5 Bass numbers of Local Cohomology modules In this chapter we study the Bass numbers, the dual notion of the Betti numbers, of local cohomology modules. In fact it is an important problem in local cohomology to determine when the set of associated primes of such modules is finite, and so when the Bass numbers of them is finite. Here we are interested in the following contexts: • Finiteness of the support (and associated primes) of local cohomology modules • Artinian local cohomology modules Let R be a commutative Noetherian ring, a an ideal of R and M a finitely generated R-module. Let t be a non-negative integer. We proved that • If H i a(M) has finite support for all i < t, then Ass (H t a(M)) is finite. • If H i a(M) is Artinian for all i < t, then HomR(R/a, H t a(M)) need not be Artinian but it has a finitely generated submodule N such that HomR(R/a, H t a(M))/N is Artinian. For the second issue note that it is already known that if the local cohomology 68
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Betti numbers of modules over Noeth
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Abstract Betti numbers are topologi
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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
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Chapter 1: Preliminaries 14 a minim
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Page 97 and 98: Bibliography 86 [58] M. E. Rossi an
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