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Table of contents: "A Singular Introduction to Commutative Algebra"

Preface (ps-file)

1 Rings, Ideals and Standard Bases
1.1 Rings, Polynomials and Ring Maps
1.2 Monomial Orderings
1.3 Ideals and Quotient Rings
1.4 Local Rings and Localization
1.5 Rings Associated to Monomial Orderings
1.6 Normal Forms and Standard Bases
1.7 The Standard Basis Algorithm
1.8 Operations on Ideals and their Computation
1.8.1 Ideal Membership
1.8.2 Intersection with Subrings
1.8.3 Zariski Closure of the Image
1.8.4 Solvability of Polynomial Equations
1.8.5 Solving Polynomial Equations
1.8.6 Radical Membership
1.8.7 Intersection of Ideals
1.8.8 Quotient of Ideals
1.8.9 Saturation
1.8.10 Kernel of a Ring Map
1.8.11 Algebraic Dependence and Subalgebra Membership
2. Modules
2.1 Modules, Submodules and Homomorphisms
2.2 Graded Rings and Modules
2.3 Standard Bases for Modules
2.4 Exact Sequences and free Resolutions
2.5 Computing Resolutions and the Syzygy Theorem
2.6 Modules over Principal Ideal Domains
2.7 Tensor Product
2.8 Operations on Modules and their Computation
2.8.1 Module Membership Problem
2.8.2 Intersection with Free Submodules
2.8.3 Intersection of Submodules
2.8.4 Quotients of Submodules
2.8.5 Radical and Zerodivisors of Modules
2.8.6 Annihilator and Support
2.8.7 Kernel of a Module Homomorphism
2.8.8 Solving Systems of Linear Equations
3. Noether Normalization and Applications
3.1 Finite and Integral Extensions
3.2 The Integral Closure
3.3 Dimension
3.4 Noether Normalization
3.5 Applications
3.6 An Algorithm to Compute the Normalization
3.7 Procedures
4. Primary Decomposition and Related Topics
4.1 The Theory of Primary Decomposition
4.2 Zero-dimensional Primary Decomposition
4.3 Higher Dimensional Primary Decomposition
4.4 The Equidimensional Part of an Ideal
4.5 The Radical
4.6 Procedures
5. Hilbert Function and Dimension
5.1 The Hilbert Function and the Hilbert Polynomial
5.2 Computation of the Hilbert-Poincare Series
5.3 Properties of the Hilbert Polynomial
5.4 Filtrations and the Lemma of Artin-Rees
5.5 The Hilbert-Samuel Function
5.6 Characterization of the Dimension of Local Rings
5.7 Singular Locus
6. Complete Local Rings
6.1 Formal Power Series Rings
6.2 Weierstrass Preparation Theorem
6.3 Completions
6.4 Standard bases
7. Homological Algebra
7.1 Tor and Exactness
7.2 Fitting Ideals
7.3 Flatness
7.4 Local Criteria for Flatness
7.5 Flatness and Standard Bases
7.6 Koszul Complex and Depth
7.7 Cohen-Macaulay Rings
7.8 Further Characterization of Cohen-Macaulayness
A. Geometric Background
A.1 Introduction by Pictures (ps-file)
A.2 Affine Algebraic Varieties
A.3 Spectrum and Affine Schemes
A.4 Projective Varieties
A.5 Projective Schemes and Varieties
A.6 Morphisms between Varieties
A.7 Projective Morphisms and Elimination
A.8 Local versus Global Properties
A.9 Singularities
B. SINGULAR - A Short Introduction (ps-file)
B.1 Downloading Instructions
B.2 Getting Started
B.3 Procedures and Libraries
B.4 Data Types
B.5 Functions
B.6 Control Structures
B.7 System Variables
B.8 Libraries
B.9 SINGULAR and Maple
B.10 SINGULAR and Mathematica
B.11 SINGULAR and MuPAD

References (ps-file)
Index (ps-file)
Algorithms

 

SINGULAR Examples

SINGULAR Examples

algebraic dependence, 87
annihilator, 186

Betti numbers, 135
- graded, 137

classification of singularities, 493
computation
- in fields, 5
- in polynomial rings, 7
- in quotient rings, 25
- of d(I,K[x]), 222
- of Hom, 106
- of the dimension, 211
- of Tor, 340
computing with radicals, 27
counting nodes, 489
creating ring maps, 8
cyclic decomposition, 159

deformation of singularities, 495
degree, 289
- of projection, 469
- of projective variety, 478
diagonal form, 154
dimension, 289
- embedding, 304
- of a module, 302

elimination
- and resultant, 431
- of module components, 180
- of variables, 71
- projective, 466
equidimensional
- decomposition, 263
- part, 261
estimating the determinacy, 491

finite maps, 196
finiteness test, 324
Fitting ideal, 186, 345
flat locus, 356
flatness test, 369
flattening stratification, 352

global versus local rings, 35
graded
- Betti numbers, 137
- rings and modules, 116

highest corner, 60
Hilbert
- function, 289
- polynomial, 299
Hilbert-Poincare series, 282
homogeneous resolution, 137

ideal membership, 68
image of module homomorphism, 99
independent set, 220
initial ideal, 299
injective, 420
integral
- closure of an ideal, 202
- elements, 195
intersection
- of ideals, 79
- of submodules, 102, 181
inverse of a power series, 316

Jacobian criterion, 304
Jordan normal form, 163

kernel
- of a ring map, 85
- of module homomorphism, 99, 187
Koszul complex, 378

leading data, 11
linear combination of ideal members, 68
local and global dimension, 472
lying over theorem, 225

maps induced by Hom, 96
matrix operations, 94
Milnor and Tjurina number, 488
minimal
- associated primes, 209
- presentations, 109
module
- annihilator, 186
- membership, 178
- presentation of, 104
- quotient, 102
- radical and zerodivisors, 184
monomial orderings, 16
morphisms of projective varieties, 455
multiplicity, 480

Noether normalization, 216
non-normal locus, 232
normal form, 51, 123
normalization, 230

Poincare series, 299
presentation of a module, 104
primary
- decomposition, 258
- test, 252
projective
- closure, 443
- elimination, 466
- Nullstellensatz, 437
- subschemes, 446
properties of ring maps, 20

quotient
- of ideals, 81
- of submodules, 102, 183

radical, 184, 265
- membership, 78
realization of rings, 42
reduction to zero-dimensional case, 257
regular
- sequences, 372
- system of parameters, 304
regularity test, 400
resolution, 135
- homogeneous, 137

saturation, 83, 446
Schreyer resolution, 150
singular locus, 309
solving equations, 76
- linear, 189
standard bases, 59, 124
subalgebra membership, 87
submodules, 104
- intersection of, 102, 181
- of An, 98
sum of submodules, 102
surface plot, 407, 413
surjective, 420
syzygies, 141

tangent cone, 480
tensor product
- of maps, 170
- of modules, 172
- of rings, 175
test
- for Cohen-Macaulayness, 386, 392, 394
- for flatness, 354
- for local freeness, 347

Weierstrass polynomial, 321

Zariski closure of the image, 74
zero-dimensional primary decomposition, 253
zerodivisors, 184
z-general power series, 320