Introduction to commutative algebra and algebraic geometry

✍ Scribed by Ernst Kunz

• Publisher: Birkhauser
• Year: 2013
• Tongue: English
• Leaves: 253
• Series: Modern Birkhäuser classics
• Category: Library

✦ Synopsis

Algebraic varieties -- Dimension -- Regular and rational functions on algebraic varieties Localization -- The local-global principle in commutative algebra -- On the number of equations needed to describe an algebraic variety -- Regular and singular points of algebraic varieties -- Projective resolutions

✦ Table of Contents

Cover......Page 1
Introduction to Commutative Algebra and Algebraic Geometry......Page 4
Contents......Page 8
Foreword......Page 10
Preface to the English Edition......Page 12
Terminology......Page 13
1. Affine algebraic varieties......Page 16
2. The Hilbert Basis Theorem. Decomposition of a variety into irreducible components......Page 25
3. Hilbert's Nullstellensatz......Page 31
4. The spectrum of a ring......Page 37
5. Projective varieties and the homogeneous spectrum......Page 44
Exercises......Page 52
References......Page 53
1. The Krull dimension of topological spaces and rings......Page 54
2. Prime ideal chains and integral ring extensions......Page 59
3. The dimension of affine algebras and affine algebraic varieties......Page 64
4. The dimension of projective varieties......Page 74
References......Page 76
1. Some properties of the Zariski topology......Page 78
2. The sheaf of regular functions on an algebraic variety......Page 81
3. Rings and modules of fractions. Examples......Page 88
4. Properties of rings and modules of fractions......Page 93
5. The fiber sum and fiber product of modules. Gluing modules......Page 103
Exercises......Page 106
References......Page 107
1. The passage from local to global......Page 108
2. The generation of modules and ideals......Page 119
3. Projective modules......Page 125
Exercises......Page 135
References......Page 137
1. Any variety in n-dimensional space is the intersection of n hypersurfaces......Page 138
2. Rings and modules of finite length......Page 142
3. Krull's Principal Ideal Theorem. Dimension of the intersection of two varieties......Page 145
4. Applications of the Principal Ideal Theorem in Noetherian rings......Page 156
5. The graded ring and the conormal module of an ideal......Page 164
Exercises......Page 174
References......Page 177
1. Regular points of algebraic varieties. Regular local rings......Page 178
2. The zero divisors of a ring or module. Primary decomposition......Page 191
3. Regular sequences. Cohen-Macaulay modules and rings......Page 198
4. A connectedness theorem for set-theoretic complete intersections in projective space......Page 207
References......Page 209
1. The projective dimension of modules......Page 211
2. Homological characterizations of regular rings and local complete intersections......Page 220
3. Modules of projective dimension < 1......Page 226
4. Algebraic curves in A3 that are locally complete intersections can be represented as the intersection of two algebraic surfaces......Page 234
References......Page 237
A. Textbooks......Page 239
B. Research Papers......Page 240
List of Symbols......Page 244
Index......Page 247