指導教授：朱樺臺灣大學：數學研究所李亮瑩Lee, Liang-YingLiang-YingLee2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264008In this note, we give self-contained proofs of the following three theorems: Theorem A (Auslander-Buchsbaum formula) The depth of a Noetherian local ring A is equal to the sum of the depth and the projective dimension of a nitely generated A-module M with nite projective dimension. Theorem B (Serre theorem) A Noetherian local ring is regular if and only if its global dimension is nite. Theorem C A regular local ring is a unique factorization domain.Contents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Abstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Abstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Introduction 5 2 Preliminary results 7 2.1 Prime Avoidance Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Nakayama Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Exact sequence and Localization . . . . . . . . . . . . . . . . . . . . . 8 2.4 Basic homology tool . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Noetherian and Artinian . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Dimensions, Degrees, and Principal ideal Theorem 12 3.1 Artin-Rees Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Hilbert Funcition of a Graded Module . . . . . . . . . . . . . . . . . 13 3.3 Hilbert-Samuel Function over a Local Ring . . . . . . . . . . . . . . . 14 3.4 Principal Ideal Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Some Homological Algebra 18 4.1 The Functor Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 The Functor Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Projective and Injective Dimension . . . . . . . . . . . . . . . . . . . 20 4.4 Projective Dimension over a Local Ring . . . . . . . . . . . . . . . . . 24 4.5 Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 Depth 30 5.1 Regular sequence and Depth . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Auslander-Buchsbaum formula . . . . . . . . . . . . . . . . . . . . . . 33 5.3 The Koszul complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6 Regular local rings 37 6.1 Regular local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.2 Serre Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.3 A regular local ring is UFD . . . . . . . . . . . . . . . . . . . . . . . 40448360 bytesapplication/pdf論文使用權限：不同意授權正則局部環唯一分解整環整體維數投影維數深度thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264008/1/ntu-103-R99221022-1.pdf