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

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Abstract iv order) of the defining equations of the Rees ring of I. More specifically, we provide an upper bound for the Castelnuovo-Mumford regularity of powers of I and give a simple criterion in terms of the Rees algebra of I to check regularity of its powers. This gives some information on the nonlinear invariants of I. Another goal of thesis is to study the Betti numbers of the canonical module of a Cohen-Macaulay ring. In other words, for the growth of the Betti sequence of the canonical module, we provide some generalizations of the known case where the radical cube of the maximal ideal is zero. This leads to some new results on even more general contexts. We note that this study is worthwhile since canonical modules play a central role in the local duality theorem relating local cohomology with certain Ext functors. The study of the graded minimal free resolution of some ideals over polynomial rings and the finiteness of the Bass numbers, the dual notion of the Betti numbers, of local cohomology modules are the other goals of this thesis. The organization of this thesis is as follows. In Chapter 1 we recall some basic definitions and known facts on Betti numbers and graded Betti numbers, Hilbert func- tions, local cohomology modules, Castelnuovo-Mumford regularity, Koszul complexes, Taylor resolutions, and Stanley-Reisner rings. The Castelnuovo-Mumford regularity provides links between local cohomology theory and the syzygies of finitely generated graded modules over polynomial rings over a field. In Chapter 2 as we mentioned above, we provide a careful overview of the Castelnuovo-Mumford regularity and its asymptotic behavior. We also recall the Rees ring of a homogenous ideal with special attention to its bigraded structure. This leads us to derive a criterion and algorithms
Abstract v to test and check the linear resolution of equigenerated ideals. Chapter 3 is devoted to the problem of growth of the Betti sequence of the canonical module where we provide some generalization of the case of radical cube zero. In Chapter 5 we study the Bass numbers of local cohomology modules in detail and give some answers to the problem of finiteness of this number which is equivalent with the finiteness of the set of associated primes of Artinian and minimax local cohomology modules. The graded minimal free resolution of some ideals are studied in Chapter 4. We also provide a mechanism to construct pure Cohen-Macaulay simplicial complexes. The results in Chapter 2, 3, 4 and 5 have been published in [9, 11, 12] or are submitted to [10, 13].
Page 1 and 2: Betti numbers of modules over Noeth
Page 3: Abstract Betti numbers are topologi
Page 7 and 8: Contents vii 4.3 Analysis of a spec
Page 9 and 10: List of Figures 1.1 The simplicial
Page 11 and 12: xi Dedicated to my father, my mothe
Page 13 and 14: Chapter 1: Preliminaries 2 ideal. I
Page 15 and 16: Chapter 1: Preliminaries 4 Let F
Page 17 and 18: Chapter 1: Preliminaries 6 i < 0, t
Page 19 and 20: Chapter 1: Preliminaries 8 Remark 1
Page 21 and 22: Chapter 1: Preliminaries 10 the rel
Page 23 and 24: 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
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Chapter 3: Growth of the Betti sequ
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Chapter 4 Graded minimal free resol
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Chapter 4: Graded minimal free reso
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Chapter 5 Bass numbers of Local Coh
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Chapter 5: Bass numbers of Local Co
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Bibliography [1] T. D. Alwis. Free
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Bibliography 84 [28] D. Eisenbud an
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Bibliography 86 [58] M. E. Rossi an
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Appendix A: Algorithms for our crit
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Betti numbers of modules over Noetherian rings with ... - IPM

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