Yanqi Lake Lectures on Algebra

Introduction......Page 6
1.1 Review on ring theory......Page 8
1.2 Localization of rings and modules......Page 10
1.3 Radicals and Nakayama's lemma......Page 14
1.4 Noetherian and Artinian rings......Page 16
1.5 What is commutative algebra?......Page 17
2.1 The support of a module......Page 20
2.2 Associated primes......Page 21
2.3 Primary and coprimary modules......Page 23
2.4 Primary decomposition: the main theorem......Page 24
2.5 Examples and remarks......Page 26
3.1 Integral extensions......Page 30
3.2 Nullstellensatz......Page 33
3.3 Flatness: the first glance......Page 35
3.4 Structure of flat modules......Page 38
3.5 Faithful flatness and surjectivity......Page 41
4.1 Going-up and going-down......Page 44
4.2 Subsets in the spectrum......Page 47
4.3 Graded rings and modules......Page 49
4.4 Filtrations......Page 51
4.5 Theorems of Artin–Rees and Krull......Page 52
5.1 Completions......Page 56
5.2 Further properties of completion......Page 59
5.3 Hilbert–Samuel polynomials......Page 61
5.4 Definition of Krull dimension......Page 63
5.5 Krull's theorems and regularity......Page 67
6.1 Dimensions in fibers......Page 70
6.2 Calculation for polynomial algebras......Page 72
6.3 Noether normalization and its consequences......Page 73
7.1 Review of discrete valuation rings......Page 78
7.2 Auxiliary results on the total fraction ring......Page 79
7.3 On normality......Page 80
7.4 Serre's criterion......Page 82
7.5 Introduction to depth......Page 83
8.1 Preparations in homological algebra......Page 88
8.2 Auxiliary results on depth......Page 89
8.3 Koszul complexes and depth......Page 91
Index......Page 96