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

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Chapter 1 Preliminaries In this chapter we collect some basic facts which will be used throughout of this thesis. Our sources for most of this material are: for basic commutative algebra, [26, 51]; for the theory of Cohen-Macaulay rings and modules, [16]; for homological algebra, [69]; and for Rees rings, [67]. In this dissertation we will be concerned primarily with commutative rings R (with identity) which are of either of the following types • Noetherian local rings; or • homogeneous K-algebras, K a field. A homogeneous K-algebra (so called a standard graded algebra) is a positively graded ∞ ring R = Ri, where Ri denotes the ith graded component of R, such that R0 = K, i=0 each Ri is a finite dimensional vector space over K, and R is generated as a K- algebra in degree 1: R = R0[R1]. Equivalently, R is of the form R = S/I, where S = K[x1, · · · , xn] is a polynomial ring over K with deg xi = 1 and I is a homogeneous 1
Chapter 1: Preliminaries 2 ideal. If R is local, then we let m denote the maximal ideal and we let K denote the residue class field R/m. If R is homogeneous, then m will denote the irrelevant maximal ideal m = R1 ⊕ R2 ⊕ · · · . When R is a homogeneous K-algebra, automatically I is a homogeneous ideal of R. Furthermore M stands for a finitely generated R-module unless otherwise specified. 1.1 Betti numbers and graded Betti numbers The Betti numbers of a finitely generated module M over a commutative Noethe- rian local unit ring R are the minimal numbers bi for which there exists a long exact sequence bn fn 0 → R −→ R bn−1 b1 → . . . → R f1 b0 f0 −→ R −→ M → 0, which is called a minimal free resolution of M. The Betti numbers are uniquely determined by requiring that bi be the minimal number of generators of Kerfi−1 for all i ≥ 0. Graded modules arise naturally in homology. For example for every integer i, there exists an ith singular homology group of a space Hi(X), and usually the “total homology” of the space is considered to be the direct sum ⊕iHi(X). This makes the “total” homology of X a Z- module graded over the natural numbers N. There are several reasons to support studying Graded Betti numbers in detail. A minimal free resolution of a finitely generated graded module M over a commutative Noetherian N-graded ring R in which all maps are homogeneous module homomorphisms, i.e., the homomorphisms that map every homogeneous element to a homogeneous element
Page 1 and 2: Betti numbers of modules over Noeth
Page 3 and 4: Abstract Betti numbers are topologi
Page 5 and 6: Abstract v to test and check the li
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: xi Dedicated to my father, my mothe
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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