COMMUTATIVE ALGEBRA Contents - Columbia University

COMMUTATIVE ALGEBRA

00AO

Contents

1. Introduction

4

2. Conventions

4

3. Basic notions

5

4. Snake lemma

7

5. Finite modules and finitely presented modules

7

6. Ring maps of finite type and of finite presentation

9

7. Finite ring maps

10

8. Colimits

11

9. Localization

14

10. Internal Hom

19

11. Characterizing finite and finitely presented modules

20

12. Tensor products

21

13. Tensor algebra

26

14. Base change

28

15. Miscellany

29

16. Cayley-Hamilton

31

17. The spectrum of a ring

33

18. Local rings

37

19. The Jacobson radical of a ring

38

20. Nakayama's lemma

39

21. Open and closed subsets of spectra

40

22. Connected components of spectra

42

23. Glueing properties

43

24. Glueing functions

46

25. Zerodivisors and total rings of fractions

48

26. Irreducible components of spectra

49

27. Examples of spectra of rings

50

28. A meta-observation about prime ideals

53

29. Images of ring maps of finite presentation

56

30. More on images

59

31. Noetherian rings

62

32. Locally nilpotent ideals

64

33. Curiosity

66

34. Hilbert Nullstellensatz

66

35. Jacobson rings

68

36. Finite and integral ring extensions

75

37. Normal rings

80

38. Going down for integral over normal

83

This is a chapter of the Stacks Project, version 74af77a7, compiled on Jun 27, 2023. 1

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39. Flat modules and flat ring maps

85

40. Supports and annihilators

92

41. Going up and going down

94

42. Separable extensions

97

43. Geometrically reduced algebras

99

44. Separable extensions, continued

101

45. Perfect fields

103

46. Universal homeomorphisms

104

47. Geometrically irreducible algebras

109

48. Geometrically connected algebras

113

49. Geometrically integral algebras

114

50. Valuation rings

115

51. More Noetherian rings

119

52. Length

121

53. Artinian rings

123

54. Homomorphisms essentially of finite type

125

55. K-groups

126

56. Graded rings

128

57. Proj of a graded ring

130

58. Noetherian graded rings

134

59. Noetherian local rings

136

60. Dimension

139

61. Applications of dimension theory

143

62. Support and dimension of modules

144

63. Associated primes

146

64. Symbolic powers

149

65. Relative assassin

150

66. Weakly associated primes

153

67. Embedded primes

157

68. Regular sequences

158

69. Quasi-regular sequences

161

70. Blow up algebras

164

71. Ext groups

166

72. Depth

170

73. Functorialities for Ext

173

74. An application of Ext groups

173

75. Tor groups and flatness

174

76. Functorialities for Tor

179

77. Projective modules

179

78. Finite projective modules

181

79. Open loci defined by module maps

185

80. Faithfully flat descent for projectivity of modules

187

81. Characterizing flatness

187

82. Universally injective module maps

189

83. Descent for finite projective modules

195

84. Transfinite d?vissage of modules

196

85. Projective modules over a local ring

199

86. Mittag-Leffler systems

200

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87. Inverse systems

201

88. Mittag-Leffler modules

202

89. Interchanging direct products with tensor

206

90. Coherent rings

211

91. Examples and non-examples of Mittag-Leffler modules

213

92. Countably generated Mittag-Leffler modules

215

93. Characterizing projective modules

216

94. Ascending properties of modules

218

95. Descending properties of modules

218

96. Completion

220

97. Completion for Noetherian rings

224

98. Taking limits of modules

227

99. Criteria for flatness

229

100. Base change and flatness

235

101. Flatness criteria over Artinian rings

236

102. What makes a complex exact?

239

103. Cohen-Macaulay modules

242

104. Cohen-Macaulay rings

246

105. Catenary rings

247

106. Regular local rings

249

107. Epimorphisms of rings

250

108. Pure ideals

253

109. Rings of finite global dimension

256

110. Regular rings and global dimension

260

111. Auslander-Buchsbaum

263

112. Homomorphisms and dimension

264

113. The dimension formula

266

114. Dimension of finite type algebras over fields

267

115. Noether normalization

269

116. Dimension of finite type algebras over fields, reprise

272

117. Dimension of graded algebras over a field

274

118. Generic flatness

274

119. Around Krull-Akizuki

279

120. Factorization

283

121. Orders of vanishing

287

122. Quasi-finite maps

291

123. Zariski's Main Theorem

294

124. Applications of Zariski's Main Theorem

298

125. Dimension of fibres

300

126. Algebras and modules of finite presentation

302

127. Colimits and maps of finite presentation

305

128. More flatness criteria

314

129. Openness of the flat locus

319

130. Openness of Cohen-Macaulay loci

322

131. Differentials

324

132. The de Rham complex

329

133. Finite order differential operators

332

134. The naive cotangent complex

336

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135. Local complete intersections

343

136. Syntomic morphisms

349

137. Smooth ring maps

355

138. Formally smooth maps

362

139. Smoothness and differentials

368

140. Smooth algebras over fields

369

141. Smooth ring maps in the Noetherian case

373

142. Overview of results on smooth ring maps

375

143. ?tale ring maps

376

144. Local structure of ?tale ring maps

382

145. ?tale local structure of quasi-finite ring maps

386

146. Local homomorphisms

389

147. Integral closure and smooth base change

390

148. Formally unramified maps

392

149. Conormal modules and universal thickenings

393

150. Formally ?tale maps

396

151. Unramified ring maps

397

152. Local structure of unramified ring maps

400

153. Henselian local rings

403

154. Filtered colimits of ?tale ring maps

410

155. Henselization and strict henselization

412

156. Henselization and quasi-finite ring maps

418

157. Serre's criterion for normality

421

158. Formal smoothness of fields

423

159. Constructing flat ring maps

427

160. The Cohen structure theorem

430

161. Japanese rings

433

162. Nagata rings

438

163. Ascending properties

445

164. Descending properties

448

165. Geometrically normal algebras

450

166. Geometrically regular algebras

452

167. Geometrically Cohen-Macaulay algebras

454

168. Colimits and maps of finite presentation, II

454

169. Other chapters

460

References

461

1. Introduction 00AP Basic commutative algebra will be explained in this document. A reference is

[Mat70].

2. Conventions 00AQ A ring is commutative with 1. The zero ring is a ring. In fact it is the only ring

that does not have a prime ideal. The Kronecker symbol ij will be used. If R S is a ring map and q a prime of S, then we use the notation "p = R q" to indicate

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the prime which is the inverse image of q under R S even if R is not a subring of S and even if R S is not injective.

3. Basic notions

00AR

00AS 00AT 00AU 00AV 00AW 00AX 00AY 00AZ

00B0

00B1 00B2 00B3 00B4 00B5 00B6 00B7 00B8 00B9 00BA 00BB 00BC 00BD 00BE 00BF

00BG 00BH 00BI 00BJ 0543 00BK 00BL

00BM 00BN

The following is a list of basic notions in commutative algebra. Some of these notions are discussed in more detail in the text that follows and some are defined in the list, but others are considered basic and will not be defined. If you are not familiar with most of the italicized concepts, then we suggest looking at an introductory text on algebra before continuing.

(1) R is a ring,

(2) x R is nilpotent,

(3) x R is a zerodivisor,

(4) x R is a unit,

(5) e R is an idempotent,

(6) an idempotent e R is called trivial if e = 1 or e = 0,

(7) : R1 R2 is a ring homomorphism, (8) : R1 R2 is of finite presentation, or R2 is a finitely presented R1-

algebra, see Definition 6.1, (9) : R1 R2 is of finite type, or R2 is a finite type R1-algebra, see Definition

6.1, (10) : R1 R2 is finite, or R2 is a finite R1-algebra, (11) R is a (integral) domain,

(12) R is reduced,

(13) R is Noetherian,

(14) R is a principal ideal domain or a PID,

(15) R is a Euclidean domain,

(16) R is a unique factorization domain or a UFD,

(17) R is a discrete valuation ring or a dvr,

(18) K is a field,

(19) L/K is a field extension,

(20) L/K is an algebraic field extension,

(21) {ti}iI is a transcendence basis for L over K, (22) the transcendence degree trdeg(L/K) of L over K,

(23) the field k is algebraically closed,

(24) if L/K is algebraic, and /K an extension with algebraically closed, then there exists a ring map L extending the map on K,

(25) I R is an ideal,

(26) I R is radical,

(27) if I is an ideal then we have its radical I,

(28) I R is nilpotent means that In = 0 for some n N,

(29) I R is locally nilpotent means that every element of I is nilpotent,

(30) p R is a prime ideal,

(31) if p R is prime and if I, J R are ideal, and if IJ p, then I p or J p.

(32) m R is a maximal ideal,

(33) any nonzero ring has a maximal ideal,

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