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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 朱樺(Huah Chu) | |
| dc.contributor.author | Liang-Ying Lee | en |
| dc.contributor.author | 李亮瑩 | zh_TW |
| dc.date.accessioned | 2021-06-07T23:44:17Z | - |
| dc.date.copyright | 2014-07-29 | |
| dc.date.issued | 2014 | |
| dc.date.submitted | 2014-07-10 | |
| dc.identifier.citation | Bibliography
[1] Auslander, M. and Buchsbaum, D. Homological Dimension in Noetherian Rings. Trans. Amer. Math. Soc. 85, (1957), 390-405. [2] Auslander, M. and Buchsbaum, D. Codimension and Multiplicity, Ann. Math., 68, (1958), 625-657. [3] Auslander, M. and Buchsbaum, D. Unique Factorization in Regular Local Rings. Proc. Natl. Acad. Sci. U.S.A. 45, (1959), 733-734. [4] Auslander, M. and Goldman, O. Maximal orders. Trans. Amer. Math. Soc. 97 (1960). 1-24. [5] Cartan, H. and Eilenberg, S. Homological algebra, Princeton University Press, 1956. [6] Eisenbud, D. Commutative Algebra with a view toward Algebraic Geometry. Springer-Verlag, New York, 1970. [7] Kaplansky, I. Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72, (1952) 327-340. [8] Kaplansky, I. Commutative Rings, Allyn and Bacon, (1970). [9] Krull, W., Dimensiontheorie in Stellenringen, J. Crelle vol. 179 (1938). [10] MacRae, R. On the Homological Dimension of Certain Ideals. Proc. Oklahoma Conf, Dekker, (1974) 163-171 [11] MacRae, R. On an application of the Fitting Invariants. J. of Alg. 2 (1965) 153-169. [12] Matsumura, H., Commutative Ring Theory, Cambridge University Press, Cambridge, 1986. [13] Nagata, M. A remark on the unique factorization theorem. J. Math. Soc. Japan 9 (1957), 143-145. [14] Nagata, M. A General Theory of Algebraic Geometry over Dedekind Rings II. Am. J. Math., 80, (1958), 382-420. [15] Northcott, D. G., Ideal theory, Cambridge University Press, 1953. [16] Samuel,P., Anneaux Factoriels, Sao Paulo, (1955). [17] Serre, J-P., Sur la dimension homologique des anneaux et des modules noeth eriens. (French) Proceedings of the international symposium on algebraic number theory, Tokyo Nikko, (1955), pp. 175{189. Science Council of Japan, Tokyo, (1956). [18] Singh, B., Basic Commutative Algebra, World scienti c, (2011). | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/16716 | - |
| dc.description.abstract | In 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. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-07T23:44:17Z (GMT). No. of bitstreams: 1 ntu-103-R99221022-1.pdf: 448360 bytes, checksum: 4c110215ee794644b2f1cdb170b0e1dc (MD5) Previous issue date: 2014 | en |
| dc.description.tableofcontents | 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 . . . . . . . . . . . . . . . . . . . . . . . 40 | |
| dc.language.iso | en | |
| dc.subject | 整體維數 | zh_TW |
| dc.subject | 正則局部環 | zh_TW |
| dc.subject | 唯一分解整環 | zh_TW |
| dc.subject | 投影維數 | zh_TW |
| dc.subject | 深度 | zh_TW |
| dc.subject | Auslander | en |
| dc.subject | UFD | en |
| dc.subject | Buchsbaum | en |
| dc.subject | Serre | en |
| dc.subject | projective dimension | en |
| dc.subject | global dimension | en |
| dc.subject | depth | en |
| dc.subject | regular local ring | en |
| dc.title | 關於 Serre-Auslander-Buchsbaum 定理 | zh_TW |
| dc.title | On Serre-Auslander-Buchsbaum Theorem | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 102-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 陳榮凱(Jung-Kai Chen),黃一樵(I-Chiau Huang) | |
| dc.subject.keyword | 正則局部環,唯一分解整環,整體維數,投影維數,深度, | zh_TW |
| dc.subject.keyword | Serre,Auslander,Buchsbaum,UFD,regular local ring,depth,global dimension,projective dimension, | en |
| dc.relation.page | 45 | |
| dc.rights.note | 未授權 | |
| dc.date.accepted | 2014-07-10 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 數學研究所 | zh_TW |
| 顯示於系所單位： | 數學系 | |
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