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課程名稱 |
代數K-理論導論
Introduction to Algebraic K-Theory |
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開課學期 |
103-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
于 靖 |
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課號 |
MATH5176 |
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課程識別碼 |
221 U6480 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期二1,2(8:10~10:00)星期四@(~) |
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上課地點 |
天數304天數305 |
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備註 |
總人數上限：15人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1032MATH5176_KTheory |
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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 |
Introducing K_0, K_1, K_2 of rings, K_2 of fields, norm residue symbols. K-theory over number fields, global fields. Quillen's plus construction. The classifying spaces. Matsumoto's theorem. Higher K-groups of finite fields, local fields. |
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課程目標 |
The aim of this course is to introduce advanced undergraduate students and graduate students to algebraic K-theory. This is one of the most advanced area of researches in algebra with profound applications to algebraic number theory, algebraic topology and algebraic geometry. This theory involves works of Bass, Milnor, Quillen, Suslin, Bloch, Kato, and Voevodsky. We will approach this theory from the side of algebraic number theory. We want to tell students the far-reaching developments of this chapter of 20-th century mathematics. |
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課程要求 |
Algebra, basic algebraic number theory, basic algebraic topology. |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
S. Lang, Algebra, Chap. 1, GTM, Springer-Verlag.
N. Jacobson, Basic Algebra II.
M. Artin, Algebra. |
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參考書目 |
1. J. Milnor, Introduction to algebraic K-theory, Annals math studies, Princeton U. Press, 1971.
2. Srinivas, Algebraic K-theory, Birkhauser, 2007.
3. Rosenberg, Algebraic K-theory and applications, GTM, Springer,1994. |
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評量方式
(僅供參考) |
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週次 |
日期 |
單元主題 |
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第1週 |
2/24,2/26 |
Projective modules, Free modules, Exact sequences, Direct summands, Lifting problem, Commutative rings, Spe(R), Fields, Division rings, Rank, Basis, Tensor products, Grothendieck groups, Projective class groups, K_0(R), Matrix rings. |
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第2週 |
3/03,3/05 |
Semirings, H_0(R), Picard groups, Local rings, Dedekind domains, Localizations, Exterior powers, Determinants, Determinant of a homomorphism, K_1(R), Elementary matrices, Sternberg groups, |
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第3週 |
3/10,3/12 |
Semisimple algebra, K_2(R), [R, R], Trace map on End_R(P), GL(R), Commutators, SL(R) |
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