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課程名稱 |
近世代數二
Modern Algebra (Ⅱ) |
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開課學期 |
107-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
林惠雯 |
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課號 |
MATH5002 |
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課程識別碼 |
221 U6590 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
必修 |
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上課時間 |
星期三6,7(13:20~15:10)星期五6,7(13:20~15:10) |
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上課地點 |
天數201天數201 |
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備註 |
總人數上限：30人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1072MATH5002_Algebra |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
1. Homological Algebra:
Tensor product; Free, projective, injective and flat modules, Ext and Tor functors, Cohomology of groups, Crossed homomorphisms and H^1(G, A), Group extensions and H^2(G, A), Introduction to Derived category, Spectral sequence.
2. Commutative Algebra:
Noetherian rings and Artinian rings, Associated primes and Primary decomposition, Going-up and Going-down theorems, Dedekind domains, Completion.
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課程目標 |
For students who are interested in various related fields, we hope to equip students with a solid foundation in algebra. |
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課程要求 |
Undergraduate Algebra and Modern Algebra I |
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預期每週課後學習時數 |
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Office Hours |
備註： 星期一 1:20~2:10 |
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指定閱讀 |
Dummit-Foote: Abstract Algebra |
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參考書目 |
1. Joseph J. Rotman: An introduction to homological algebra
2. S.I.Gelfand and Yu.I.Manin: Methods of homological algebra
3. Atiyah, MacDonald, Introduction to commutative algebra, Westview Press (1994)
4. N. Jacobson, Basic Algebra II |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
作業 |
30% |
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2. |
期中考 |
35% |
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3. |
期末考 |
35% |
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週次 |
日期 |
單元主題 |
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第1週 |
2/20,2/22 |
Basic properties of modules; Free modules / (Sec. 10.1: 13, 19; Sec. 10.2: 4, 13) ; (Sec. 10.3: 18, 22) |
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第2週 |
2/27,3/01 |
Tensor product of modules; 連假 / (Sec. 10.4: 5, 7, 9, 14, 15) |
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第3週 |
3/06,3/08 |
Projective, Injective and Flat modules / (課堂作業 1, 2 + Sec. 10.5: 2, 14) ; (Sec. 10.5: 21, 25) |
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第4週 |
3/13,3/15 |
Homology functors; Ext and Tor (I) / (Sec. 17.1: 2, 3, 4, 6) ; (Sec. 17.1: 8, 9) |
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第5週 |
3/20,3/22 |
Ext and Tor (II); Group cohomology / (Sec. 17.1: 10, 17; Sec. 17.2: 1, 2) ; (Sec. 17.2: 3, 9) |
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第6週 |
3/27,3/29 |
Extensions of groups; 1st and 2nd group cohomology/ (作業課堂指定) ; (Sec. 17.4: 1, 2) |
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第7週 |
4/03,4/05 |
春假 |
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第8週 |
4/10,4/12 |
Derived category / (作業課堂指定) |
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第9週 |
4/17,4/19 |
Derived functors; Spectral sequence(I) / (作業課堂指定) |
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第10週 |
4/24,4/26 |
Spectral sequence(II) / (作業課堂指定); 期中考 |
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第11週 |
5/01,5/03 |
Primary decomposition (Sec. 15.2: 2, 33, 35, 43) ; Integral extensions (Sec. 15.3: 2, 5) |
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第12週 |
5/08,5/10 |
Hilbert's Nullstellensatz & Correspondence between geometry and algebra / (Sec. 15.2: 6, 11, 12, 21) ; (Sec. 15.1: 24, 26) |
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第13週 |
5/15,5/17 |
The prime spectrum of a ring (Sec. 15.5: 1, 2, 15, 19, 20); Local property (Sec. 15.4: 2, 11) |
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第14週 |
5/22,5/24 |
Going up and going down (作業課堂指定) ; Noether's normalization (Sec. 15.3: 15, 16 & Sec. 15.4: 29) |
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第15週 |
5/29,5/31 |
Artin rings (Sec. 16.1:6, 9, 12, 13); DVR (Sec. 16.2: 4, 8) |
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第16週 |
6/05,6/07 |
Dedekind domains ;端午節 (Sec. 16.3: 7, 11, 12, 18) |
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第17週 |
6/12,6/14 |
Completion |
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第18週 |
6/21 |
期末考 |
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