CAnotes - commutative algebra Lectures delivered by Jacob...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: commutative algebra Lectures delivered by Jacob Lurie Notes by Akhil Mathew Fall 2010, Harvard Last updated 12/1/2010 Contents Lecture 1 9/1 1 Unique factorization 6 2 Basic definitions 6 3 Rings of holomorphic functions 7 Lecture 2 9/3 1 R-modules 9 2 Ideals 11 Lecture 3 9/8 1 Localization 13 Lecture 4 9/10 1 Spec R and the Zariski topology 17 Lecture 5 [Section] 9/12 1 The ideal class group 21 2 Dedekind domains 21 Lecture 6 9/13 1 A basis for the Zariski topology 24 2 Localization is exact 27 Lecture 7 9/15 1 Hom and the tensor product 28 2 Exactness 31 3 Projective modules 32 Lecture 8 9/17 1 Right-exactness of the tensor product 33 2 Flatness 34 Lecture 9 [Section] 9/19 1 Discrete valuation rings 36 1 Lecture 10 9/20 1 The adjoint property 41 2 Tensor products of algebras 42 3 Integrality 43 Lecture 11 9/22 1 Integrality, continued 44 2 Integral closure 46 Lecture 12 9/24 1 Valuation rings 48 2 General remarks 50 Lecture 13 [Section] 9/26 1 Nakayamas lemma 52 2 Complexes 52 3 Fitting ideals 54 4 Examples 55 Lecture 14 9/27 1 Valuation rings, continued 57 2 Some useful tools 57 3 Back to the goal 59 Lecture 15 9/29 1 Noetherian rings and modules 62 2 The basis theorem 64 Lecture 16 10/1 1 More on noetherian rings 66 2 Associated primes 67 3 The case of one associated prime 70 Lecture 17 10/4 1 A loose end 71 2 Primary modules 71 3 Primary decomposition 73 Lecture 18 10/6 1 Unique factorization 76 2 A ring-theoretic criterion 77 3 Locally factorial domains 78 4 The Picard group 78 Lecture 19 10/8 1 Cartier divisors 81 2 Weil divisors and Cartier divisors 82 Lecture 20 10/13 1 Recap and a loose end 85 2 Further remarks on Weil( R ) and Cart( R ) 86 3 Discrete valuation rings and Dedekind rings 86 Lecture 21 10/15 1 Artinian rings 89 2 Reducedness 91 Lecture 22 10/18 1 A loose end 94 2 Total rings of fractions 95 3 The image of M S- 1 M 96 4 Serres criterion 97 Lecture 23 10/20 1 The Hilbert Nullstellensatz 99 2 The normalization lemma 100 3 Back to the Nullstellensatz 102 4 Another version 103 Lecture 24 10/22 1 Motivation 104 2 Definition 104 3 Properties of completions 105 Lecture 25 10/25 1 Completions and flatness 109 2 The Krull intersection theorem 110 3 Hensels lemma 111 Lecture 26 10/27 1 Some definitions 113 2 Introduction to dimension theory 114 Lecture 27 10/29 1 Hilbert polynomials 117 2 Back to dimension theory 119 Lecture 28 11/1 1 Recap 121 2 The dimension of an affine ring 122 3 Dimension in general 123 4 A topological characterization 123 Lecture 29 11/3 1 Recap 125 2 Another notion of dimension 126 3 Yet another definition 127 Lecture 30 11/5 1 Consequences of the notion of dimension 129 2 Further remarks 130 3 Change of rings 130 Lecture 31 11/8 1 Regular local rings 133 2 A bunch of examples 133...
View Full Document

Page1 / 172

CAnotes - commutative algebra Lectures delivered by Jacob...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online